umd department of computer science | - lecture3jkatz/crypto/f02/lectures/crypto... ·...

128

Upload: others

Post on 10-Jul-2020

5 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

����������� ������������������������� � � �"!$#�%'&)( �$�*�����+�,-� ���.�/�*� � ��0��*��1����032$�45 ���6�7� �*��98��.����� 2�����:;���*<

=?>A@CBEDGFE> H

I J?KMLONQPMLSR�J?TVUWLYX[ZE\

]_^`] acb.dedgfhdgfMiSj3kml5b$npo7qrfsdgbto�uvjxwzyvj|{-}~n`��j�a��_����7,�����_� 2��h�m�������� ���.�~����0�� �6�7,-���*�7,+mV���`���6����*�7,-��7,-�m�"�"�h2����������*�E� ���A���

���$���������6�����?� �¢¡�3£¤¦¥V§�¨ª©¬«��[­-£-®m¡-¯c¡~«�°�¡¤A¯±¡-­�­²7³´¡±­p§m².£V¡�µ ©�­c0�� �6�7,-� ���t�*�7,+� ��© ¶�·g¶«�¤c²�¸6¸¹ ©º­-¨ª¤V©p»-¼m¨ª©¬«��m­�«�°�¡-¤Cµ½·¾¶«�¤c²�¸6¸�¿ÁÀµ½· ²�� ¹ ¶«�¤c²�¸6¸v£©Ã§�®|¡¤¦¨`¡�Ä.¨6­;Å�ÆÇ¡A®�²�°�¡'4vÉÈ ¿�Ê Å�ËvÌÍ45ÉÈ ¿/Ë�ÎÏV�Ы�¨6®m¡-¤CÆv«�¤ ¹ ­�·Ñ¨6®|¡M��0��.�*� �������§3¤s«´» ²$»©�¸Ò©6¨ª¥~¨6®�²�¨Ñ²±¯c¡-­�­-²É³´¡�¿ÓÆv²Ô­9­¡��¨6·�³�©6°�¡�t¨6®�².¨vÆÇ¡'«$»¦­7¡-¤¦°�¡£©Ã§�®|¡-¤-¨`¡ Ä$¨eÅÉ·Ç©º­'¡ Ä´².£7¨ª¸Ò¥z¡ Õ¼�²�¸�¨p«�¨6®m¡A�M0���������§3¤*«$»�²$»-©6¸Ö©ª¨ª¥�¨6®�²�¨Ç¯c¡¦­�­²7³´¡9¿×©�­C­¡-�3¨pÎ

Ø��A�m�É�Ù1.�����A���6+��º�_0� �����5� 2���� � 2m���.�m�-Úp� ���A��0����t�7��,- ��0�� ���.� � ,�2m�7�±�M�º���*�7,+� �/�ª�º���*��� �º�A��Ç�;�.������1.�É���'�A0� �´���S�6���� 2m�;+�����������Û���º�*�*���Ü�+�� ���.�c����� µW�VÝÞ'ß �´�3à��|á �ãâ¾®|¡M«���¡-äs¨ª©�¯c¡�§|² ¹ ©�­h²;§m¡-¤6¶¡�£7¨ª¸Ò¥�ä`­7¡ £¼|¤ ¡±¡-��£¤¦¥�§�¨ª©ª«��Э£-®|¡¯c¡7Î

å;à��_�3æ ç�� ��+��A�;µ Ì�èÉé|ê7ëÔì7í�Ývîm�� ���$�~¿WÀ�ï ���������´��Å9�Ç�h2��7�����

45�È ¿�Ê ÅVË½Ì 4v�È ¿ñð/ÅVË45ÉÈ ÅVË

Ì 4v�È Å.Ê ¿/Ë�ò745ÔÈ ¿/Ë4vÉÈ ÅVË ê �sëÉ�

+�� ���m1 �`�"�Â��0�03����,��� ���.������� � 2m���m�������� ���.�Â���9,-�.������� ���.�����"0� ��Ü���Ü3����������Ý � �.������� ���.�����m1t�É���c���º��A�7� � ��1��7��1.�����7��4v�È ÅVË�Ìôó�õ�öÉ÷Y45ÉÈ Å.Ê ¿/Ë�òÉ4vÉÈ ¿/ËpÝvøÑ+m�ù��6������´�~¿�ê Å9�"�h2��É�����

45ÉÈ Å.Ê ¿/Ë½Ì 45ÉÈûúcÌü�ªÅvý¢¿t�`ËÌ þ´ÿ í

�*��� 2����;45ÔÈ Å�ËvÌñþ ÿ íAò�ó�õ�öÉ÷Y45ÉÈ ¿/ËvÌñþ ÿ í�ÝM4v��+m1�1.�º�m1~�º�$�*�Â�sëÉ�9��2m�É���9� 2����;45�È ¿�Ê ÅVËvÌ 4vÉÈ ¿/Ë�������"�h�� �h�m�.����Ý

]_^�� w��5o.n`�����`npo7q�d¾f;o.u5j w�y5j|{}�n��xj[aM�_�� 2m�A�.�m�-Úp� ���±�A0���� ��� ��û�9�z��� �t1����|���7��,- ��0�� ���.��� ,V2m�7�A��Ýcî���;�.�m�±� 2��º�m1mùg���;,�����m���hÜ��A+�� �7��*�t�*�7���Â�A�� �±� 2����Â�.�m�c�A�7��� ��1���Ý�î�+m � 2m�V�A�� ��ù¾���.+��m��7� �*� � 2��� �� Ü���� �C�*��� �7���Â�����ÚpÜ3����A�7� � ��1��� mÜ�+��"���_���.+/,���/�A���E���~�*�7,- ��E�����~��1� ��9�.��tÜ���� �ù$��2´�c�m����� +��*�Ç�±���Ñ�º���*�7,-���Ñ�����2����������������.+m �A�7� � ��1����vçÙ����� +�������´+��7�*� ���.�z���Ñ��2m�� 2m���"�h,�������AÜ���*�*�ÉÝ( �z�6��,-�ù��"�;,�����m���Ý � 2m�C�m���|�E� 2m��� �7� ��2m�É���;�6��.+m1.2������v� 2������*�A0�� �6�7,-� ���c�7��,-��´0���ÐÜ���� �ù

���.+��m��7���*��� 2��� �;��� ���7���*���Ü���� �7Ý

ë

Page 2: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Þ'ß �´�3à��|á��QÏ�¶~���Mê��vê��A�~©�­/²~§m¡-¤6¶¡�£7¨ª¸Ò¥ ­¡ £7¼|¤*¡t¡-�3£7¤V¥V§�¨ª©¬«�� ­£-®|¡¯c¡z«�°�¡-¤~¯±¡-­�­²7³´¡�­`§|².£¦¡±µ½·¨6®|¡-�xÆÇ¡/¯M¼�­¦¨9®�²�°�¡�Ê �cÊ��WÊ µÓÊ! «�¤�·'¤s«�¼.³�®$¸Ö¥Ð­`§m¡ ²#"�©6�´³�·h© ¶Mµ ÌWèÉé|ê7ëÔì7íô¨6®|¡� ÆÇ¡/¯M¼�­¦¨9®|²�°�¡Ê �cÊ$� þ í ²�� ¹ ¨6®m¡'¸�¡�´³.¨6®�«s¶'²���¥C§|²�¤¦¨ª©¬£¼|¸�²�¤%".¡-¥�©º­�?»©6¨6­'&�Î

å;à��_�3æ �|�É�z�"�'��Ü�� � ���A,��0�2m� �*���|��ÅÔÝ Ø��A,���t03���7�z� 2��'0���������5� 2m�� �7,-�7�����C�����Ð�m�7,-��´0�� Å+�� ���m1~���� � 0��.� � ��Ü����)(���Ðú À*�MÝ � 2�����1.�����7�;+��9���;�A�.�*�cÊ �cÊ����,+�� �7�´�C�A�7� � ��1��7�9��2���,�2Ð,-�.+��º�0��.��� ��Ü�����2��É���/ �7� +����*�7�¢���[,��0�2�� �*�����MÅÐ�ª�m���*���c� 2��º�����1.+��A�7�´�M2��.�����;����7� ����A�t������m�.�±��<�7��7��,- ��0�� ���.�Ð� ,�2��7�A��ùg���9���.�m1/���9�Ç�A��� � +��A��,-��� �7,-� �m�7� �9���Ç� 2m�����7,- �´0�� ���.�Ð����1������� 2��~�VÝ9øÑ+m�9���Ê �cÊ.- Ê µÓÊ_� 2m�7�Â� 2�� �����h�������7��� �h�.�m�~�A�7� � ��1��~¿ À µ �6��h��2���,�2�45ÉÈ ¿�Ê ÅVËEÌãé|Ý � 2�+��ù¾� 2��� ,�2m�7�±� �������mÜ����º���*�7,+m � ����� 2m�A² §3¤¦©ª«�¤V©50� ��Ü���Ü����º���`�����g¿O���Ç���.�|Úp<� ���6��2���,V2c�"� ,���/��� ��+m ��Ü��,�2m���.� ���m1c� 2m�;�����*�*V��Ü�+m� ���.���É��� µ ��0�0� ��0�������*�7�����VÝ

]_^�/ i�o�b.d¾y10gj�b32�o�o��_k5476ãd��5j8�:9Ø��z�A�7�$� ���.�m�7�¢�7��������M� 2����M� 2m���.���-Úp� ���A�/0����¢���M�����*�7,+� �~����+��*�7�[�`����,-�Â�6�Ç�7�º�¬ù�«$»-°�©ª«�¼´­¦¸Ö¥�Ý�Ý�ÝÖ�VÝØ��~,��� �0�2m���� �A� 2����;���h���.�����É���Ý ( �c��1.���m�±���Â���Ü����*��� � �7��,- ��0�� ���.�Â� ,�2m�7�±�c� 2����h�º�;+��*�7�Â�*��7��,- ��0������Ç���A�7��� ��1��7�9����.� ç ����,-�'�*�~ø"��Ü �ª,�����S� 2m�7�*���`�"���A�7� � ��1��7�9¿3;9������¿�<7�VÝ � � � ����������ù���"�c��1.2$�Ñ2���0�0��7��� 2����E���~�7�7���7� �m���0�0���(|�m�����"��2����Ñ¿ ; ���9�6������E���7���*�ùm2����Ñ�*�.�A�C����������c��� ���.���Ü��.+���¿3;V�V�E�6�� ���m���A0�����ù3¿3;9�±��1.2´��Ü��'���Ðç � : �A�7� � ��1���ù_��9�±��1.2´� Ü������>=Ç�m1.������2_ù���C�±��1.2´� �0� �7�*�7�´�E�M���7�@?Ô�m�±����� �Ç�7Ý5çr0� ��0�� ���±�Ç�;�c��1.2$�Ñ�m�7� �� ���6 �.�Û�.+mÑ�7��,-��´0�� ���.�~��,�2m�7�A�;�º�Ç� 2����¡-°�¡�G© ¶¢� 2m�~���m����� ����A(����É���'¿3;'�������*��7�MÅB;7ùe� 2m���7��,- ��0�� ���.�Â����¿�<±� 2m�.+����� �7�±���º�Â�*�7,+m���ª�¬Ýû��Ý�ù_��Ü��*� �|���m1~Å�<'� 2��.+����z1.�����M���~����������c��� ���.�t��Ü��.+m��¿�<É�VÝDC����*�A� 2����ùg����� 2m�.+m1.2t �7���*�.����Ü�����ù� 2������º���m����� 2m�',���*���6���� 2m�'�.�m�-Úp� ���A��0����_Ý ( �S� 2��'���m�������� �E(����É����¿3;�������� 2m�7�t� ��7��ÅF;7ù�� 2�����m����� �� �Ð,�������±�±�7�������*�7����,-�.�±0�+m�*�A� 2m�G(���Ð���hútÌü¿3;�ýxÅF;-ÝGC��É�Í���´���ª+m� +m �A,��032m� �*���|� �� 2������� �;��Ü3�*� ���7�zÜ��~� 2m�;���m����� ����/,����Ü��9���7,- �´0��*�7�z���±�A�7�������*�7���H�( ��������c��������ù�� 2m�7�_ù��Ç�M,�����m������'�*�7,+mV���`����1.���º���*�I"��3«�Æ��z§3¸�²�©��3¨`¡ Ä.¨ ²�¨¬¨p².£J"É­±��� ���.�����É���M�6�Ç�

��������1.�����h�±������c���_�m�������� ���.�~�º���A����r�Ç��J(|�V�V�

���$���������6���K���Q­£-®|¡¯±¡C©º­;�*�7,+� � ��1.���º���*�1(|�m�����c03�������´�*�����v���*� ��,L(��'© ¶�©ª¨e©º­�­7¡ £7¼�¤ ¡h¡°�¡�zÆe®|¡-�¨6®|¡±² ¹ °�¡-¤V­-²�¤V¥z©�­ ³.©�°�¡-�²~­7¡ Õ¼�¡�3£V¡c«s¶E§|²�©�¤V­c�6¿3;-ê ÅF;�ÌM�ON´�6¿3;¦�*�VêJPJPJPÉêÉ�6¿�Q7ê ÅLQ�ÌM�ON´�6¿�Q¦�*��·ÑÆe®|¡-¤ ¡¿3;7êJPJPJP7ê*¿�Q9²�¤ ¡�¤s²�� ¹ «�¯M¸Ò¥A£®�«Ô­7¡-�3ÎR TS'«�¨`¡�¨6®�²�¨e¨6®|¡�­²�¯c¡U"�¡¥;©º­E¼�­¡ ¹ ¨6®$¤s«�¼�³�®�«�¼m¨6·5²�� ¹ ¨6®$©�­E­-²�¯c¡"�¡-¥�©º­C¼�­¡ ¹ ¶-«�¤�¨6®|¡M£©Ã§�®|¡¤¦¨`¡�Ä.¨Ñ«$»¦­7¡-¤¦°�¡ ¹ »-¥c¨6®|¡M² ¹ °�¡¤�­²�¤V¥�¨6®�².¨Ç©ª¨5©�­h¨ª¤¦¥Ô©6�´³/¨p« ¹ ¡�£¤¦¥�§�¨pÎV&

� 2m��Ü���� ��,�������7�S���Ç� �7,+m���������,�2�������7�ÐÜ´�z� 2����.���-Úp� ���A��0����Ð�������6�*�7�Ð ��6� �7� �*������­¡ £7¼|¤V©ª¨ª¥²7³$²�©6�m­-¨v£©Ã§�®|¡¤¦¨`¡�Ä.¨v«���¸Ö¥c²�¨¬¨p²$£�"É­¦Ý ( Ýû��Ý�ù�� 2������m�������� �±1��� �Ñ�m��0��������´�*�����@?Ô,��0�2m� �*���|�v0�������5Ü���6�� �Ü��7�º�m1A����(��7�z�*�*W*���7,- �´0��YX±�A0��� � �º,+������,��0�2�� �*�����ÝØ���,���G���±��1.���m����������7�G�±�� �~�º��� �������.+��9�`��0����������*� ��,L(�� 2������ 2��~��Ü��É������2m�É�ã��Ü��.+m�M�

£-®�«Ô­7¡-�/§�¸ ²�©6��¨`¡�Ä.¨�²�¨¬¨p².£J"���2m� �C� 2m�����m����� ����~1��� � �*�~,�2m���.�*�h��2���,�2z0��������´�*����� ���� �h�7��,- �´0��*�7�Ü´�/ç��º��,-��Ý�Ø�2������±��1.2´�Ñ� 2����E,-�� �7� 0��.���/�*�±���/ �7���_�����6�FZ/Ø��7�º�¬ùm� 2m�;���m�������� �~�c��1.2$��,-�.�´�* �.�g�����0�0�����,��� ���.�|Ú`������7�g0� ���*��,-�.��� 2����9�����6��7������1c����� ���*�~ç ����,-�'�*�cÜ��h�7��,-��´0��*�7�gÝ\[ 7ù3� 2m�'���m���V� �� ��±��1.2´��Ü��h��Ü����;�*�~���A0����*�.�����*�;øÇ��ÜÐ������� 2�� �Ü´���6���,-���6�� ��� 2m� �����*�h,��+�� ���Eç ����,-�;�*���7��,- ��0��,-� � ����� � 2����m1.�7Ý ç�1.�����_ù_��� �Ç�.+�����Ü�������,-�M���5��� �7��,- ��0�� ���.� � ,�2m�7�±�M,-�.+�����Ü��M�*�7,+m ���6��"¶¦¼m¨ª¼|¤*¡�A�7� � ��1��7������7��+����m�E� 2����E�*����E���e���*� ��,L(3Ý ( �m����V�±�������t�6�"�;1.�����'�±�A�� �;�6����±���_���������� ���.�/�����*��.�����/� 2m�;,-�.+���*���V�

þ

Page 3: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���$���������6���^]Q� ­-£®|¡-¯c¡�©�­[�*�7,+� ����1.�������*� ,V2m�.�*�7��0��������´�*���|�t���*� ��,Y(|�G© ¶ ©ª¨c©�­Ð­¡ £7¼|¤*¡ ¡°�¡�Æe®|¡-�Q¨6®|¡ ² ¹ °�¡-¤V­-²�¤V¥x£®�«7«Ô­7¡¦­Ð¯c¡¦­�­²7³´¡-­~¿3;êJPJPJP�ê*¿�Q ²�� ¹ ¨6®|¡�x³´¡7¨6­ ¨p«[­7¡�¡ £7© §3®|¡-¤-¨`¡ Ä$¨6­/ÅF;ÂÌ�_N|�ªÅF;¦�VêJPJPJPÉê ÅLQ�Î` ª�v³´²�©���·Ñ¨6®|¡�­-²�¯c¡R".¡-¥~©º­;¼�­7¡ ¹ ¨6®$¤*«�¼.³�®|«�¼m¨6·�²�� ¹ ¨6®$©�­h­²�¯c¡I"�¡¥�Æ�©6¸�¸�»�¡�¼�­7¡ ¹ ¶«�¤¨6®|¡ba�£®�²�¸6¸�¡�´³´¡Oc'£©Ã§�®|¡-¤-¨`¡ Ä$¨E«$»¦­7¡-¤¦°�¡ ¹ »¥�¨6®|¡M² ¹ °�¡-¤V­-²�¤V¥~¸ ².¨`¡-¤�«��3ÎV&

( �c��1.���m� ����Ñ�'�±�.�A�7�$�Ñ2m�������.+~�"�.+�����,-�.���*�*�+�,-�Ñ��� ,�2m�7�±� �*�7,+� � ��1.���º���*�Ñ�'� �º�m1.���d(|�m�É���0��������´�*�����/���*� ��,Y( �ª�¬Ýû��Ý�ù���2m� � � 2m�Ð���m����� ���� 1��� �/�*�G�*����¢� ���m1.���t0�������6¿3;7ê ÅF;�Ìe�ON|�6¿3;��*�VÝçY�����*� ���A� 2��.+m1.2$�M� 2m�����'� 2����'0�� �6�7,-�'� �7,+m������Ð�º� � 2����'�*��*� ���m1 ���h���,fc,+����h�*�t��,�2�������� Ç���Â�6��,-�ù�Ç��,���gWs0� �����hX��t� 2���� �7� ����� 2m�/���.�º���É���º�m1z�6���� �6����,-�.+m��*��ùv�Ç�/,�����û�M �7��������0� �É����� +�,V2¢�� 2m��� �7�Í+��$� ���|�"��1.�������C���1��� �.+��e����� ���.�A���_�m��3����� ���.�Aþ_ ��6+m�� 2m���A�� ��ù´+����m�5�*�.�±�E �7���*�.����Ü�����m�������� ���.���Ñ�*�7,+m������~��1.���º���*���±�����m1.���!(|�m�É���z0��������´�*���������*� ��,L(�©º­;0��.� ����Ü������V�Þ'ß �´�3à��|ái]gS'«E ¬­-¨p²�¨`¡-¸�¡¦­V­:&�¡-�3£7¤V¥V§�¨ª©¬«���­-£-®m¡-¯c¡h£V²��лV¡Ç§m¡¤�¶7¡ £É¨ª¸Ö¥±­¡ £7¼|¤*¡h²7³$²�©6�m­-¨."���«�Æ��A§3¸�²�©6��ä¨`¡ Ä.¨Ñ²�¨¬¨p².£J"É­Îcâ¾®$©�­;©º­h¨ª¤¦¼�¡�¡°�¡�t© ¶h¨6®|¡A¡�3£¤¦¥V§�¨ª©ª«��Э-£-®m¡-¯c¡'©º­C¤*²�� ¹ «�¯�©kjÉ¡ ¹ Îl5�.+��� �h���Y(��7�z�*�±0���É���h�A���V� ���.�����S� 2����E� 2���� �7�ã�.�z� 2��;2m�.�A��"��Y(3Ý

m n KMLpo_LY\�qsr�LutUqsvGo�q U wãLpo_Lyx

���±�ª����Ç�'2��7���h� ��7�z�`�"�c�m�1.��� �����h �7� +���� �ùm���m�6����±�������c� +��±�c�����<�7��2m� ���

z � �A0�� ���7,-� �����7��,-��´0����ÚpÜ3���E�A�7� � ��1��7�ù3�Ç�'�m��7���*�c� 2��� �D�ÚpÜ����y(���|�Ý

z Ø��~,�����m�������,�2��������0�� �6�7,-�h�*�7,+mV���`����1.������� �)(����É���Â0��������´�*�����'���*� ��,Y(|�/�6��ÉùS�6��h� 2�����±���*�*�Éù|,V2m�.�*�7��0��������´�*���|�Ç���*� ��,Y(|�V�"+������m1h�'�*� ���*�7���7� �"�7��,- ��0�� ���.�~� ,V2m�7�A��Ý5ç ,-�.���*�h��+m�7��,-����Ñ� 2����'���C� 2������"� £�²����3«�¨;².£®$©p¡-°�¡;§�¡-¤6¶¡�£7¨�­¡�£¤ ¡ £¥ Æe®|¡�x¡-�3£7¤V¥V§�¨ª©6�´³Ð¯M¼|¸Ò¨ª©Ã§3¸�¡/¯c¡¦­V­-²7³�¡¦­+������7� �"�Ç�'+��*�;�±�*� ���*��ª+��_�7��,- �´0�� ���.��� ,�2��7�A��Ý

( �������{� +��*� �A�7�´� ���.� � 2����ù��º��1��7�m�V���¬ù3�"���"���´���*���É���.���Ð�*� ���*��6+��¾�7��,-��´0�� ���.�t� �º��,-�;� 2���� ,��+��*�7�0� ��Ü����7�c�����Ð,���*�M� 2m�A�*�7���m�C����� �7,-�7�����C1����Ws�.+m�9���Ç� ����,JX|Ý;ç ���*�mùg��� ���9�~1��7�m�����S�+����'� 2����� 2m�h���7� ���*� ���*�;� 2m�;Ü���*�*�'�ª�É���.�������`�|���m1c+m0z�±�7�A�� ���VÝ���mù��m�z�"�D� +��*�C1.�����~+m0HZ ( �;� 2����C� 2m���7��������,- ��0��*��1����0�2´� ���;�"�`(|�m��� ���@Z5�~Ø��7���¬ùe���Ç� 2��

�m�������� ���.�~�º�E�*�´��2�����/�*�c��,V2��������ù3����J� �|� +��*�Ñ �7���}�/� 2m�h�m��3����� ���.�_ݦÝ7Ýø"����� �h�Ç�C �7�º�}�~� 2m�C�m��3����� ���.�_ù|����J� �����´��(~�����*�.�A�;�����*�V�����*�C�"�É�������¾0���7�*�7�$� �º�m1�~���3�$� ���.�

ë�Ý � 2m�C�6�.�����������m1A��������,-�9Ü��7,��+��*�;���e��� ��� ���±0�����,�����3�

���$���������6�����Ù� � ¡-�3£7¤V¥V§�¨ª©¬«��[­-£®|¡-¯c¡~«�°�¡¤±¯c¡¦­V­-²7³�¡c­p§|².£¦¡�µ ©�­�§�¡-¤6¶¡�£7¨ª¸Ò¥�­7¡ £¼|¤ ¡c© ¶V·¾¶«�¤c¨6®|¡¼���© ¶«�¤¦¯ ¹ ©�­¦¨ª¤¦©p»-¼m¨ª©ª«��«�°�¡¤'²���¥c­¡¨vè7¿3;ê*¿�<ÔìI� µ «s¶h¨ªÆv«~¯c¡¦­V­-²7³�¡¦­'²�� ¹ ¶«�¤'²�¸�¸�£7© §3®|¡-¤-¨`¡ Ä$¨6­�ÅÆÇ¡9®�²�°�¡h�Ñ4vÉÈ ¿3;ÔÊ ÅVË�Ì?45ÉÈ ¿�<$Ê ÅVË�Î�Ï-Ϊ¡7Î�·S© ¶Ñ¿3;'²�� ¹ ¿�<±²�¤ ¡;¨6®|¡'«���¸Ò¥9§|«�­�­V©p»-¸�¡h¯c¡¦­�­²7³´¡-­�·Ç²�� ¹ ¨6®|¡-­¡²�¤*¡c¡ Õ¼|©Ã§3¤*«$» ²$»¸�¡±²C§�¤V©¬«�¤¦©º·Ñ¨6®|¡-��¨6®|¡¥c¤ ¡-¯±²�©6��¡ Õ¼|©Ã§3¤*«$» ²$»¸�¡c¡-°�¡�в*¶¦¨`¡¤�«$»-­¡-¤¦°É©6�´³�Å�Î

�|����,-�C�"�;+��*�C� 2��;�*��� §m¡¤�¶7¡ £É¨5­7¡ £7¼�¤¦©ª¨ª¥Ôù��Ç�h2�����Ü���*�*��2m��0��C� 2����d~ �������� ���.��� ! ������ë;�����h�´+����Ô�����7�´�Ý ( ���6��,-�ù3� 2m�C�6�.�����É�����m1M0� �����S��2m�É����� 2������ 2m��/�� ���å;à��_�3æ [9�m�;������7,-� ���.�/���Ñ�7���*��Ý � ���7�������ùm���S�A� ,�2m�7�±�h� ��� ���*���7�d~ �������� ���.�ÐëC� 2��7��,�2m���.� ���m1A� 2�������*�*���Ü�+m� ���.���É���Eµ ���Ç� 2m��+���������V�ü�����*�*���Ü�+m� ���.���É���Ñ�*�.�A���`�"��Ú`�A�7� � ��1��C� +�Ü��*��v���c�A�7�������*�7���1.�����7�U~ �������� ���.� ! Ý

Page 4: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

C����;ù3����� +��A�9�"�h2��7���'�A� ,�2m�7�±�h� ��� ���*�6�|���m1�~ �������� ���.� ! ÝÇîS�,��¿ ; ê*¿ < ê ÅÔÝ"�|����,-�C4v�È ¿ ; Ê Å�Ë$�45�È ¿�<.Ê ÅVËMÌÁëz�"�t2��7����45ÉÈ ¿3;ÔÊ ÅVË�Ì 45ÉÈ ¿�<$Ê ÅVË�ÌÁëB��þ?�6��2m���z� 2m�7�*� 0� ��Ü���Ü3������� ���7�M�� � ��� � 2������0������±�7�$����0�0� ��0�������*�9�*�`~ �������� ���.� ! �VÝ�C��É�hù

45ÔÈ ¿3;ÉÊ ÅVË½Ì 45ÉÈ Å.Ê ¿3;*Ë�ò74v�È ¿3;sË45ÉÈ ÅVË

Ì ëB��þ9ò�45ÉÈ Å.Ê ¿ ; ËëB��þ9ò´�ª4v�È Å.Ê ¿3;*Ë_�¢45�È Å.Ê ¿�<¦Ë6�

Ì 45ÔÈ Å�Ê ¿3;*Ë45ÉÈ Å.Ê ¿3;*ËO�¢4vÉÈ Å.Ê ¿�<¦Ë P

�|����,-��� 2���� �h��+���� ���.��2m�.�������*V+m�'�6��;�� Ü����*��� �z¿3;ê*¿�<��Ç�A��+��*�C2��É���c45ÉÈ Å.Ê ¿3;*Ë�Ì 45ÉÈ Å.Ê ¿�<¦Ë�����²���¥~¿ ; ê*¿ < Ý

C����;ù�� �#(������z�� Ü����*��� �������*�*���Ü3+m� ���.���É���9µ �����z�� Ü����*V�� �c¿�ê ÅÔÝ5Ø���2��7�����

4v�È ¿�Ê ÅVË½Ì 4v�È Å.Ê ¿/Ë�ò745�È ¿/Ë4vÉÈ Å�Ë

Ì 45ÉÈ Å.Ê ¿/Ë�ò745ÉÈ ¿/Ëó�õ��ºö�÷ 4v�È Å.Ê ¿E�ÒËÉ45�È ¿E�ÖË

Ì 4v�È ¿/Ëpê

��2m� ���"��+��*�E� 2m���ª��,-�v� 2����Ç45ÉÈ Å.Ê ¿ � Ë�Ì?4vÉÈ Å.Ê ¿/Ë3�6��Ç�����m¿ � ù´�����±� 2m���ª��,-�v� 2���� ó�õ � ö�÷ 4v�È ¿ � Ë_Ìë�ÝvøÑ+m��� 2����E�º�E���|��,-� ���/���� �h�´+��� �7�/Ü���~ �������� ���.��ë�ÝØ��'���´�* �|��+�,-�9�.�m�h�±�� �;�h��+����������7�$�E�m��3����� ���.�~Ü��������C�.+m��±�������±��������,��� ���.�_�

���$���������6���K��� �[¡�3£¤¦¥V§�¨ª©ª«�� ­-£®|¡-¯c¡�«�°�¡¤'¯c¡-­�­-²É³´¡A­p§|².£¦¡'µ ©�­�§�¡-¤6¶¡�£7¨ª¸Ò¥�­¡�£¼|¤*¡A© ¶V·_¶-«�¤A²���¥¨ªÆv«~¯±¡-­�­²7³´¡-­�¿3;-ê*¿�<CÀ�µ ²�� ¹ ¶«�¤±���´��²�¸û³$«�¤¦©ª¨6®$¯��ãÆÇ¡h®�²�°�¡h�

45ÉÈ �M���h�5Ì�¿3;�ÊV��Ì^�_N|�6¿3;V�`Ë�Ì?4v�È �����'�5Ì�¿3;�ÊV�rÌK�_N��6¿�<É�`Ë�P

çÍ�"����Ð��Ü��.+m�9� 2����9�m�������� ���.�_ÝCØ?2����C�Ç�A2��É���c���9�������m�������� �[�ª����1���V��� 2������ ��2m����� �* ���º�m1�*�3Ws1.+��7� �YXM��2��º,�2����g�`�"�A0��.� � ��Ü������A�7� � ��1��7���º�vÜ��7���m1��*�7�´�Ý � 2m�9�m��� ��� ���.��W��M���h��X� ��6���"�*��� 2���.+m�*0�+m� ���5� 2m�A����1������� 2�� ��2m�7�t��� �����+��t�.�Ð���m0�+��\�MÝ � 2�+��ù��Ç�M �h��+�����h� 2���� � 2m�M0� ��Ü���Ü3���������� 2�������1.+m�7� �*�7��¿3;���2m�7�z�������Ñ1.�����7�����z�7��,- ��0�� ���.�����e¿3;�� 2m�.+��º�/Ü��C���m��,-� ���/� 2m�h�����A�h����� 2��0� ��Ü���Ü��º�����`�/� 2����D�ü1.+��7� �*�7�9¿3;���2m�7�t��� ����1.�����7�Ð��� �7��,- ��0�� ���.� ���5¿�<ÔÝ ( � ��� 2m�9�"������ù�� �º��*+�� ��������m�.�±���/1.+m�7� �����m1������ �º���m���9�m�.����1����´�zÜ���*�*�9���91.+m�7��� ���m1� �1.��������7�������5��2m�� 2m�C�����º�1.�����7�������7��,- �´0�� ���.�/���e�.�m�;�A�7� � ��1��'���� 2m�;��� 2��7Ý( ���º�E�m���E�*�´��2����� �6Ü�+m�E�����º�E�A�����*� ���9�*�7�����.+��V�v�*�c� 2m���ô� 2����E� 2����E���������� ���.�~���Ñ�h��+����������7�´�E�*�

� 2m�M�.�m�MÜ���6�� ��ù_�����Ð2m�7��,-���*��~ �������� ���.��ë�Ý � 2�+��ù_���9��� ���C���,fc,+������*����Ü�� �����Ð��� � 2m�M�����1.�º������m�������� ���.�_ÝtøÑ+m���Ç���m���ã�A��������� ���'�����`�"� ,-�+�,�����Ç�Ñ�7�|�Ý îe����*�ùe�"�/��� �m���M �h��+�����c�*�7,+m����`���1.�������*��²���¥�����1������� 2���ù�Ü�+m�A�����*�*�7���¢��1.�������*�t¡�� £©p¡-��¨'����1������� 2��c�/�6�"�z� �É�[�A����/��Ü��.+m�M� 2��º��m�����E���7,-� +� ���VÝ"���7,-�.���gù��"�C�m�A�m���Ñ �h��+��� � � 2����Ñ� 2m�C����1������� 2��O�ª��������� �� ���"2��7���C���m��,-� ���~�h�´+����0� ��Ü���Ü��º����� ���7�����Ç1.+m�7��� ���m1~� 2m�±�A�7� ����1��±��� �7��,�2 ,���*�� S�"�A�.�����z �h��+�����'� 2����C� 2����Ü��A,���.�*�z�6�Ç�� �7���A�� �'��Ü��.+���2m�É�r,���.�*�h�m���|�����7,-� +m����VÝ � 2�+���

!

Page 5: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���$���������6���^�Q� � ¡-�3£7¤V¥V§�¨ª©¬«�� ­-£®|¡-¯c¡M«�°�¡-¤;¯c¡¦­V­-²7³�¡;­p§|².£¦¡9µ ©�­��*�7,+� �� p®�«�ÆÇ¡-°�¡-¤V·Ç�3«�¸�«��´³�¡-¤§m¡-¤6¶¡�£7¨ª¸Ò¥ ­¡ £7¼|¤*¡�&�© ¶V·�¶«�¤/²���¥Ð¨ªÆv« ¯±¡-­�­²7³´¡-­h¿3;ê*¿�<zÀ µ ²�� ¹ ²���¥��Jf�,���7�$�z²�¸û³$«�¤V©ª¨6®$¯�� ÆÇ¡®�²�°�¡B�

Ê 45ÉÈ �M���h�5Ì?¿3;ÉÊV��Ì^�ON��6¿3;V�`Ë8� 4v�È �M���h�5Ìx¿3;ÔÊV��Ì^�_N��6¿�<É�`Ë ÊO-��hP

#

Page 6: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�����k���J L¡Y�,¢'£¥¤�¦.§>¨# L£_©k¨��«ª¬ §®­ ¬�¯±°�²I³�´ �±¢� L¤_ª�µ«¶�¢Y�,¤5�·¢�¤ ¬  Y£_¸�¢�¤�¹� L¨#¸8º±£»p Y¤�¦T�h¡Y¡�¤� !¼�¤5��¨#¢Yº�¨��>½)¨#¢�¾

¿KÀ�ÁDÂ�Ã�Ä�À Å

Æ Ç ÈUÉ�ÊËÈ\ÌÎÍÏÊ.Ð`Ì�Ñ�ÒÓÑLÈ�ÌGÔgÈUÕ×Ö%Ê1ØpÙGÉÑ�ÒOÚ

Û ¢U¢Yº$�D�h��ª�¤�¦Ü¢Yº$�I©k¨�¡�¢d¶J©k¨�¡Y¡hÝ�Þ��D¹5¨B���I¢Yº��)¦�¤5©ß©,¤BÞU�ß�$¹Rª��Jà«���,¢Y�,¤5��¤�¦.¡��h¶Jµ$ @�,¢:£Há

âEã±ä.åÜæ�çFæTèHåMéKê\ë`ì�ë8íhî@ï@ðHñ�ò�ó#ë�ôJí�õOìJö`ì�ó#÷#ìJî1ö`ìøô@ô�ùhú_ì�ô�ð$ù5í@ìpû ò�ôy¡��h¶Jµ� Y�!ò ü@ý}üJó#îyù#þ�þFÿ�����ÿ�����ûù#ë�Eù#þTþ ì��íhò�ìJë«ñ�ù�þ ú{ó#î@ò�ñTõ±öRô �%ý�ñTõOì�üJó#þTþ ó��.ò�ë{ú¥õOó#þ��#ô��

� »p �� ���������Kÿ � � ��� �"!#�Tÿ � �%$&A»1 '� ����������ÿ � � ��� �#!#�Tÿ � �%$ �"(*),+

- º��k¡ ©,�h¨B���h¡pµ�¡ ÞU�,¢YºI¢:Þ�¤/.{µ��h¡�¢Y�,¤5��¡JáÜÞUº�¨#¢|ª$¤\Þ��10��h¨��32_£%��4`¶J�k�h�±¢65 Û ��ª�Ý5º�¤BÞ�¡70 ¨�©k©_¡Yº$¤5µ«©kª) 2Ó�'5

8:9%8 ;3<>=@?%ACBEDGFIH�JLKLMN?ODQPSR>TU µ$ y�$¤�¢Y�,¤5�¥¤�¦��4`¶J�,�h�{¢�Þ��k©k©@2Ó�U�,._µ�¨#¢��hª�ÞU�,¢Yº ð8î�óWVYùNV�òTþ ò�ôøñ�ò�ídðOó#þ ï}ëHó#öGò�ù#þÜñ�ò�ö`ìX��»�» - �6Y Û ¸Ó¤5©,£O�$¤[Z0 �k¨�©�¢Y�\0��\¨�©k¹�¤� L�,¢Yº]0^� �k¡1¤5����¦T¤� �Þ�º��k¶Lº`¢Yº$�J Y�\��_O�ß¡�¢Y¡�¨%¸Ó¤5©,£_��¤Q0 �k¨�©[`a�cbd�1¡Yµ�¶@º`¢Yº«¨#¢�¢Yº$�\ Lµ��«���k�$¹¢Y�e0��·¤�¦f� ¤5�b�ß�$¸«µ$¢hgi�kj�l"�,mon'p·�k¡G¨#¢X0�¤5¡�¢ `a� � g � �6Y Û ð8î�óWVLùWV�òTþ ò�ôøñ�ò�í�¨�©,¹�¤� L�,¢Yº]0e¨�©k¡Y¤3º«¨�¡I¢Yº��¨[2«�k©k�,¢'£ ¢�¤rqts«�k¸CuR L¨���ª$¤Q0 ¶�¤5�k��¡/�T¤� BÝ��,.{µ«�,�}¨�©k�h�±¢Y©,£�Ý"2«�k¢Y¡6��¨���ªEµ�¡��!¢Yº$�D Y�h¡Lµ�©,¢�¤�¦Ü¢Yº$�h¡Y�)¶�¤5�k�E¢�¤5¡Y¡��h¡�k�*�,¢Y¡R¶�¤Q0�¸«µ�¢Y¨#¢Y�,¤5�:Y U ¦d¶�¤5µ� L¡���Ý.�,¦�¢Yº$��¨�©,¹�¤� L�,¢Yº]0 �k¡I¸H¤5©k£_�$¤Q0`�k¨�©|¢Y�e0���Ý.�,¢R¶J¨��vs8�,¸*¨#¢�0�¤5¡�¢G¨¸H¤5©k£_�$¤Q0`�k¨�©Ó�{µ]0h2Ó�J �¤�¦w2«�,¢Y¡�Y- º$�G._µ$�h¡�¢Y�k¤5��Þ�¨�¡` L¨��k¡��hª�¨�¡`¢�¤AÞ�º$�J¢Yº$�J ¥�k¢ �k¡ �$�h¶��h¡Y¡L¨# Y£×¢�¤ ¶�¤5��¡Y�ߪ$�J  ¸� Y¤�28¨[2«�k©k�k¡Y¢Y�k¶`¨�©,¹�¤[Z

 L�,¢Yº]0 ¡,Y ´ �`¦�¨�¶�¢JÝ��,¢��k¡�¨30 ¨'x�¤� �¤�¸Ó�h�y.{µ��h¡�¢Y�,¤5���k�¥¶�¤Q0 ¸«©,��_$�,¢:£�¢Yº$�J¤� L£`¨�¡�¢�¤IÞUº��J¢Yº$�J � @¨���ª$¤Q0 �$�h¡L¡º$�h©,¸«¡�z��YV��Y,Ý�ÞUº$�J¢Yº$�J )¨�¸� Y¤�28¨[2«�k©k�k¡Y¢Y�k¶D¸Ó¤5©,£NZ�¢Y�e0��G¨�©,¹�¤� L�,¢Yº]0 �k¡!¨��{£{0 ¤� Y�G¸H¤FÞ��J L¦Tµ�© ¢Yº�¨���¨�¸Ó¤5©,£NZ¢Y�e0��·¨�©,¹�¤� L�,¢Yº]0�Y ¬ �J L¢Y¨��k��©,£�Ý| L¨���ª�¤Q0 �$�h¡Y¡I¡��J�,0 ¡G¢�¤3º$�h©,¸×µ�¡y�{ìøô@ò,ú�ë���4`¶J�,�h�{¢G¨�©k¹�¤� L�,¢Yº]0 ¡hÝ|¨��«ª L¨���ª$¤Q0 ���h¡Y¡�¢Yµ$ L�«¡�¤5µ$¢d¢�¤|2H�D�h¡L¡��h�±¢Y�ߨ�©¦T¤� U¡��h¶Jµ$ Y�I¶� Y£{¸«¢�¤�¹� L¨#¸«º{£�Y ´ �>¨��{£·�J���h�±¢JÝÓ¡Y�k��¶��DÞ��)Þy¨��±¢¢�¤R¸« Y¤�¢��h¶�¢�¨#¹5¨��k��¡�¢�¢Yº$�}2� Y¤5¨�ª$�h¡Y¢�¶J©k¨�¡L¡�¤�¦Ü¨�ª$���J L¡Y¨# Y£`¸H¤5¡L¡Y�~2«©,��Ý{¨���ª·¡Y�k��¶��\�,¢�¡Y�J�,0 ¡�¢Yº�¨#¢�¨�©k©,¤BÞ��k�$¹¢Yº$��¨�ª$���J @¡Y¨# Y£�¢�¤�s8�,¸  L¨���ª�¤Q0 ¶�¤5�k��¡�¡��J�,0 ¡1 Y�h¨�¡�¤5��¨[2«©k�3�TÞ���ª$¤R�,¢p¤5µ$ L¡Y�h©,���h¡�Y�Y�Y��`Ý_Þ��\¨�©k©,¤BÞ^»1» -¨�©,¹�¤� L�,¢Yº]0`¡d�k�·¤5µ$ Uª��Jà«���,¢Y�,¤5��¤�¦|¡��h¶Jµ$ L�k¢:£�Y�U¤FÞ)Ý.¨��$¤�¢Yº$�J 3.{µ$�h¡Y¢Y�,¤5��¢Yº�¨#¢%���J�hª�¡)¢�¤�2Ó� ¨���¡YÞ��J Y�hª*¦�¤� R¨E¸Ó¤5©,£O�$¤Q0 �k¨�©�Z�¢Y�\0��R¨�©k¹�¤� L�,¢Yº]0��ß¡Já

¸H¤5©k£_�$¤Q0`�k¨�©O�k���|õ_ù�ñ75 ¬ ¤5��¡Y�kª��J hÝ�¦�¤� ���_O¨�0 ¸«©,��Ý5¢Yº$�U¤5�$��Z�¢Y�e0��d¸8¨�ªG�h��¶� Y£_¸�¢Y�,¤5� ¡Y¶@º$�,0���Y����J Y��ݱ¢Yº��¶J�,¸«º$�J Y¢���_O¢U�k¡U¨�©,Þ�¨B£_¡U¢Yº$�%¡Y¨�0 �%©,�h�$¹�¢Yºr�T¦T¤� �¨`à@_O�hª�0��h¡Y¡Y¨#¹��R¡�¸«¨�¶��'�y¡Y¤`�,¢Uª$¤_�h¡d�$¤�¢f0 ¨[���h0Rµ�¶Lº¡��h��¡���¢�¤U¢Y¨�©~�D¨[2Ó¤5µ$¢�¨�¡�£"0�¸�¢�¤�¢Y�߶p Lµ��«���k�$¹y¢Y�e0��h¡�¤�¦�¨��%¨�©,¹�¤� @�,¢Yº]0�YE����¶J¨��Ià@_D¢Yº��k¡:2_£)�k�{¢� Y¤_ª«µ�¶J�k�$¹¨`ôhìYí,�_îøò�ñ�ï\ðOù#î�ù#ö`ìJñ:ì�î�Þ�º��k¶Lº¥ÞU�k©ß©��J���h�{¢Yµ�¨�©k©,£X2Ó�!¨�0��h¨�¡Yµ$ Y�\¤�¦Üº�¤BÞ ¡��h¶Jµ$ Y�!Þ��!Þ�¨��{¢y¢Yº$�D¡Y¶Lº$�,0 �¢�¤G2Ó��Y�� ´ ��¢Yº$�`¶J¨�¡��¥¤�¦y¢Yº$� ¤5�$��Z�¢Y�e0 � ¸«¨�ª�ÝÜÞUº��k¶@º��k¡D¸Ó�J Y¦T�h¶�¢Y©,£®¡Y�h¶Jµ$ Y��Ý ¢Yº��k¡D�k¡)�k Y Y�h©,�J�#¨��±¢�Y|��µ�¢Þ��·ÞU�k©ß©1¡��J�E©k¨#¢��J G��_$¨�0�¸«©,�h¡R¤�¦UÞ�º$�h� �k¢h2Ó�h¶�¤Q0��h¡G Y�h©k�J�}¨��{¢�Yd�����EÞU�k©k©1 Y�J¸� Y�h¡Y�h�±¢R¢Yº��k¡R¡��h¶Jµ$ L�k¢:£¸«¨# L¨�0��J¢��J �2_£�m ! ³ ¨3¡�¢� @�k�$¹®¤�¦ � ¤5�$�h¡�YI���E¢Yº$�h�× L�,.{µ��k Y�`¢Yº�¨#¢G¢Yº$�E¨�ª$���J @¡Y¨# Y£* @µ�� �ß� ¢Y�\0��¸H¤5©k£_�$¤Q0`�k¨�©Ó�k�G��Y ´ ¦|�,¢dº$�h©,¸«¡hÝ$¦�¤� U��¤BÞ ¤5���%¶J¨���¢Yº��ß�@�`¤�¦|¢Yº$�%¡Y�h¶Jµ$ L�,¢'£�¸8¨# L¨�0��J¢��J �¨�¡�2Ó�h�k�$¹�¨[�_�ß�¢�¤X���J£�©,�h�$¹�¢Yº ³ ¨�¡�¢Yº$�/���J£E©,�h�$¹�¢Yº>�k��¶� Y�h¨�¡��h¡h���k�$¦T¤� �0 ¨�©k©,£]�@Ý_¢Yº$�%¡Y¶@º$�,0��)¹��J¢Y¡f0�¤� Y�%¡��h¶Jµ� Y��Y

m

Page 7: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

8:9�� �rACJ�H%?%J�?���H�A�����BS=#DQ?�K�B�T- º$�`�$��_O¢h._µ$�h¡�¢Y�k¤5�AÞ��`¢Yµ$ L�A¢�¤®�k¡%º$¤BÞ ¡70 ¨�©k© ) ¡Yº�¤5µ�©kªI2Ó�'5 ­O�ß��¶�� Þ��¥ª$¤5�:�V¢%Þ�¨��{¢I¢�¤®¡��J¢I¨��{£ùRð8îøò�ó�î@ò���¶�¤5��¡�¢Y¨��{¢6� 2Ó¤5µ���ª>¤5�3¢Yº$��¡Yµ�¶J¶��h¡Y¡!¤�¦�¨���¨�ª$���J L¡Y¨# Y£�Ý�Þ��G¸� Y�J¦T�J D�k��¡�¢��h¨�ª®¢�¤E¢Y¨�©~�®¨[2Ó¤5µ$¢¨�¡�£#0 ¸�¢�¤�¢Y�k¶U¡Y�h¶Jµ$ L�,¢'£R¨���ª`¢�¤%©,�J¢p¢Yº��U¨�ª$���J L¡Y¨# L£� ¡�¨�ª$�#¨��±¢Y¨#¹��!ª$�J¸Ó�h��ªG¤5� ¢Yº��U¡��h¶Jµ$ L�k¢:£G¸«¨# L¨�0��J¢��J �YU �A¢Yº$�¥¤�¢Yº$�J Gº�¨���ª�ÝÜÞ��¥Þ�¨��{¢%¢�¤®¨�ª$���J L¡Y¨# Y£�� ¡R¨�ª$�}¨��{¢Y¨#¹��·¢�¤G2Ó����¨�¡�£"0�¸�¢�¤�¢Y�߶J¨�©k©,£]�!���J Y£�¡70 ¨�©ß©�Ý¡�¤¥º$¤FÞM¶J¨���Þ��I�h��¡Yµ$ L�)¢Yº��k¡65����Rª$¤`¢Yº«�k¡�2_£�¦T¤� L¶J�k��¹ ) ¢�¤y2Ó�)¨ ¦�µ���¶�¢Y�,¤5��¤�¦S��z«¦�µ$ Y¢Yº$�J �0 ¤� Y��Ý�Þ��ÞU�k©k©8 Y�,._µ��, Y��¢Yº�¨#¢ ) �O�]��¹� Y¤FÞ7¡70 ¨�©k©k�J �¢Yº«¨��E¨��±£��k�±���J @¡��!¸H¤5©k£_�$¤Q0`�k¨�©OY Û ¢:£_¸«�k¶J¨�©Ó��_$¨�0�¸«©k��¤�¦|¡Yµ�¶@º¨�¦�µ���¶�¢Y�,¤5���k¡)¢Yº$�`�k�{���J L¡�� ��_O¸Ó¤5�$�h�{¢Y�k¨�©.¦�µ���¶�¢Y�,¤5��á ) �O�]�/���N� ! Y|��¤�¢��`¢Yº�¨#¢��N� ! ���X�tm� [�"¡��\¦�¤� ¨��±£E¶�¤5��¡�¢Y¨��{¢f¢oY�­$�k��¶��D¦�µ���¶�¢Y�,¤5��¡y¤�¦ ¢Yº��k¡U¡�¤� Y¢d¶� Y¤�¸®µ$¸�¨ ©,¤�¢d�k�>¶� Y£{¸«¢�¤�¹� L¨#¸«º{£�Ý�Þ��%ÞU�k©k©H¦T¤� �0 ¨�©k©,£ª$�Jà«�$�D¢Yº$�,0�á

âEã±ä.åÜæ�çFæTèHå>£Mêbü��_ëHíhñ�ò�ó#ë ) �cbd�dòßô!�$�J¹5©k�,¹5�~28©,�)ò ü�ü�ó#î!ù#ë�ï ¢�¤¥l¥ñTõ$ì�îYì)ì7¦5òßô�ñTô�ù#ë�§ ¡ ¤¥l`ô6��íJõ�ñTõOù�ñTýü�ó#îGù#þTþC§¨¤*§ ¡ ��ì%õ_ù#÷#ì�� ) ��§I� ( m� �§ ¡,©

���I¶J¨����$¤BÞ7 L�J¸«º$ L¨�¡��/ª\�Jà«�{¢Y�,¤5��m%¨�¡�¦�¤5©ß©,¤BÞU¡há

âEã±ä.åÜæ�çFæTèHå « ê\ë·ìJë8íJîøï@ðHñ�ò�ó#ë`ôJí�õOìJö`ìUó#÷�ì�î�ö`ìøô@ô�ùhú_ìyô:ðOù5íøì�û òßôU¡��h¶Jµ� Y�)ò üHüJó#îUù#þTþ#ÿ�����ÿ�����ûù#ë�Eù#þTþ�¬S¬h­ ù#þVú±ó�î@ò�ñTõ±öRô �%ý1ñTõOì�îYì ì�¦�ò�ôøñTôIù¥ëHì�ú5þ ò,ú�ò�V�þkì1ü6�Oë8íBñ�ò�ó�ë ) �cbd�!ô���í�õ�ñTõ_ù�ñ��

®®® »p �� ���tm! �6�����Kÿ � � ��� �"!N�Tÿ � �%$&A»1 �� �3�tm ! �6�h���^ÿ � � �i� �"!N�Tÿ � �%$ ®®® (*) �O�]� +

�U¤�¢��R¢Yº�¨#¢�¦�¤� \µ�¡d¢�¤¥�J���J  2Ó�)¨[2«©k�)¢�¤�¡L¨#¢Y�k¡�¦T£�¢Yº«�k¡dª$�Jà«�«�,¢Y�,¤5��Ý$¢Yº$�����J£E¹��h�$�J L¨#¢Y�,¤5�®¨�©,¹�¤� L�,¢YºC00Iµ�¡Y¢Uª$�J¸Ó�h��ª�¤5��¢Yº$�)¡Y�h¶Jµ$ L�,¢'£¥¸«¨# @¨�0��J¢��J  ���TÞ�º±£]5'�6Y ´ �·¦�¨�¶�¢JÝ«¢Yº��ß¡�0 ¨[���h¡U¡��h�«¡���á�2Ó�J¦�¤� L�DÞ��%¶J¨��¹��h�$�J L¨#¢��I¨X���J£�Ý«Þ��%�$�J�hªE¢�¤X�O�$¤FÞ º$¤BÞk0Iµ«¶Lº�¡��h¶Jµ� L�,¢:£�Þ��)Þ�¨��{¢d¢�¤`¨�¶@º��,�J����Y

¯ °^È ±³²�ÙIÒµ´�ÒÓÑYÈ\Ì�´1¶{·�´�É:¸ ÌGÊpÔ8Ô

¹d�h¶J¨�©k©H¢Yº$�\ Y�h¨�¡�¤5��Þ���¹5¨B���D¢Yº$� 0 ¤_ª��kà��hª`ª$�Jà8���,¢Y�,¤5��Þy¨�¡y¢�¤Gª$�h¡Y�,¹5�y0�¤� Y�\��4`¶J�,�h�{¢�¡L¶Lº$�,0��h¡/����YV��Y,Ý¡Y¶Lº$�,0 �R�k�>ÞUº��k¶@º�Þ��Iª$¤·�$¤�¢\�$�J�hª>¢�¤·¡Yº�¨# L��º�2«�k¢Y¡U�k�>¤� Lª$�J \¢�¤¥�h��¶� L£{¸�¢�¨��GºµZ�2«�k¢�0��h¡Y¡Y¨#¹��'�6Y ´ �¤�¢Yº$�J �Þ�¤� Lª�¡JÝ#¨��{£!¡L¶Lº$�,0��1Þ���¶�¤5��¡�¢� Lµ�¶�¢�ÞU�k©k©o2Ó�»2� Y�h¨[�#¨[2«©,�»2_£D¨���q�¨�©k©�Z�¸Ó¤BÞ��J Y¦�µ�©eu�¨�©,¹�¤� @�,¢Yº]0¼��¡Y�ß��¶��¤5��©,£·¢Yº$�)¤5�$��Z�¢Y�\0��%¸«¨�ª>�k¡d¡��h¶Jµ$ L�I¨#¹5¨��k��¡Y¢Iù#þTþ|¨�©k¹�¤� L�,¢Yº]0 ¡��6z�Þ��R¶J¨���¤5��©k£·º�¤�¸H�D¢Yº«¨#¢U¢Yº$�R¡Y¶Lº$�,0 �ÞU�k©k©|�$¤�¢�2Ó��2� Y�h¨[�#¨[2«©,��2_£®¨>»1» - ¨�©k¹�¤� L�,¢Yº]0�Y - º«�k¡D¡Yµ$¹�¹��h¡�¢Y¡I¡�¤Q0�� �$¤�¢Y�k¤5��¤�¦3q�¶�¤Q0 ¸«µ$¢Y¨#¢Y�,¤5��¨�©º�¨# Lª��$�h¡L¡7u"z$Þ��I¨#¢�¢��,0�¸�¢d¢�¤ ¦T¤� �0 ¨�©ß�,¾J�D¢Yº��k¡��$¤�¢Y�k¤5��º$�J Y��Y���¥¶@º$¤{¤5¡Y�`¢�¤�¦T¤� �0 ¨�©k�k¾J��¢Yº$�`�$¤�¢Y�,¤5�A�_�k¨·¢Yº��`¶�¤5��¶��J¸�¢)¤�¦�¨*ó�ëHì�½c�1ù#ï%ü��Oë8íhñ�ò�ó#ë]Y ´ ��¦�¤� �0`¨�©k©,£�Ý

¢Yº��k¡d�k¡�¨ ¦Tµ��«¶�¢Y�,¤5�·ÞUº��k¶@ºE�k¡y�h¨�¡�£·¢�¤`¶�¤Q0�¸8µ$¢��)£��J¢�º«¨# Lª·¢�¤`�k�{���J Y¢�Y Û ¦�¤� 60 ¨�©ª$�Jà«�«�,¢Y�,¤5��¦�¤5©ß©,¤BÞU¡há

âEã±ä.åÜæ�çFæTèHå�¾gê�ü6�Oë8íBñ�ò�ó�ë�¿®á"j�l"�,mon p À j�l"�,mon p òßôR¤5�$��Z�Þy¨h£bò ü)ñTõOì�üJó#þTþ ó��.ò�ë{ú·õ_ó#þ~�Q�

Á ­�õOì�ü��Oë8íhñ�ò�ó#ë3ò�ôRìYù}ôøï�ñ�óEí@ó#ö�ð�$ñ:ì ©f ù#ö`ì�þ ï}ý�ñTõOìJî�ì ì�¦�ò�ôøñTôRù�¬�¬�­ ù�þ ú{ó#î@ò�ñTõ±öÃ���|õ{ò�íJõ5ýyó#ëòTë#ð�$ñEgbîYìJñ��Oîøë$ô ¿»�Äg:�!òTë3ñ�òTö`ì�ðOó#þ ï}ëHó#öGò�ù#þ|òTë � g � ©

Á ­�õOìGü6�Oë8íBñ�ò�ó#ë òßô�õ_ù#îÅ� ñ�ó×òTë�÷#ìJîøñ ©^ ù#ö`ìJþ ï}ýUüJó#î�ù#þ�þ1¬S¬h­ ù#þ ú{ó#îøòTñTõ±öRô�� ñTõOì�îYì*ì�¦�ò�ôøñTô3ùëHì:ú�þ ò ú5òOVJþ,ì1ü��Oë8íhñ�ò�ó#ë ) �cbd�Dô6��í�õ�ñTõ_ù5ñ��

»1 �� g�ÆÇj�l"�,mon ! zÅÈ��É¿»�Äg:�6zÅgCÊC�>���tm ! �ÅÈ@��áW¿»�ÄgÊ~�S� ÈW$�Ë ) �O�]� + �tm��

Page 8: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Û ¦T�JÞKÞ�¤� Lª«¡�¨# Y�D�k�¥¤� Lª$�J y¨[2Ó¤5µ$¢�¢Yº$�Dª$�Jà«���,¢Y�k¤5�:YEÌ|�, L¡Y¢JÝO©k�J¢�� ¡�©k¤{¤��`¨#¢�¢Yº$�D�$¤�¢Y¨#¢Y�,¤5� ´ µ«¡��hª�¢�¤ª$�Jà«�$��º�¨# Lª«�$�h¡Y¡\¤�¦��ß�±���J L¡L�,¤5�:Y�Í1���J Y£{¢Yº«�k�$¹�2Ó�J¦�¤� Y�G¢Yº$� ¶�¤5©k¤5�¥��q�áÎuW�� Y�J¸� Y�h¡��h�{¢Y¡)¨��3��_O¸Ó�J L�e0��h�{¢�z�J���J Y£{¢Yº«�k�$¹¥¨#¦T¢��J \¢Yº$�I¶�¤5©,¤5�� Y�J¸� L�h¡��h�±¢Y¡U¢Yº$�%�J���h�{¢!ÞUº$¤5¡��%¸� Y¤�2«¨[2«�ß©k�,¢:£¥Þ��G¨# Y�R�k�±¢��J L�h¡�¢��hª��ß�:Y�­O¤$Ý�k�×Þ�¤� @ª�¡JÝ �tm���0��h¨���¡G¢Yº$�E¦T¤5©k©,¤FÞU�k�$¹$á�Þ��>¸«�k¶7�b¨3 L¨���ª$¤Q0Ï�NZ�28�,¢ ¡�¢� L�k��¹�gEYÐ���>¢Yº$�h� ¶�¤Q0�¸«µ�¢��È��¼¿»�Äg:�6Y3����__¢JÝ�Þ���¹5�,����È®¨�¡)�k�$¸8µ$¢U¢�¤>¨�©k¹�¤� L�,¢Yº]0Ñ�ÏÞUº��k¶@º�¤5µ�¢�¸«µ$¢Y¡}g Ê Y��U¤FÞ)Ý¢Yº$���J���h�{¢)Þ��¨# Y���k�±¢��J L�h¡�¢��hª��k�3�k¡�ÞUº$�J¢Yº$�J �¿»�Äg Ê ���ÒÈ>¤� D�$¤�¢�Y Û ��ª®Þ��� L�,.{µ��k Y�%¢Yº�¨#¢!¢Yº$�G¸� L¤�2«¨[2«�k©k�k¢:£·¤�¦1¢Yº��ß¡�J���h�±¢f2H��q�¡�0 ¨�©k©eu"Y�U¤�¢��%¢Yº�¨#¢�¢Yº$�D¸� Y¤�28¨[2«�k©k�,¢'£�¤5�·¢Yº$�D©,�J¦T¢tZ:º�¨���ª"Z:¡Y�ߪ$��¤�¦��tm����k¡�¨R¦�µ���¶�¢Y�,¤5��¤�¦»� ³ ¢Yº$�D©,�h�$¹�¢Yº·¤�¦

g�Y�­_¤ �k¢10 ¨[���h¡U¡��h�«¡��)¢�¤`¶�¤Q0�¸8¨# Y�)¢Yº��k¡y¢�¤`¢Yº$�)¦�µ���¶�¢Y�,¤5� ) �O�]�6Y�U¤�¢��)¨�©ß¡�¤I¢Yº«¨#¢�Þ��!¹5�,���}�7¢Yº$�!¨�µ"_$�k©k�k¨# Y£��k��¸«µ$¢�m ! Y - º��k¡�¶J¨��y2Ó���O�,�JÞ��hª¥¨�¡�¡Y�\0�¸«©,£G¢��h©k©k�ß�$¹��

ÞUº�¨#¢U¢Yº��I©,�h�$¹�¢Yº>¤�¦|¢Yº$�%¤� L�k¹5�k��¨�©�k��¸«µ$¢d�k¡�YÓ�yµ$¢U¢Yº��% Y�h¨�¡�¤5��¦T¤� �ª$¤5�k��¹`¡�¤¥�ß¡10�¤� Y�I¦Tµ��«ª�¨�0��h�{¢Y¨�©OY´ 0 ¨#¹5�ß�$�R¨�¦�µ���¶�¢Y�,¤5��¢Yº�¨#¢�0 ¨#¸«¡\�J���J Y£®�k�$¸«µ�¢U¡�¢� L�k��¹`¢�¤·¢Yº$��m�Z�2«�,¢!¡�¢� L�k��¹|l"Y/��¤�¢���¢Yº�¨#¢D¨�©k©,¤BÞ��k�$¹�Ë¢�¤� Lµ��×�k� ¸Ó¤5©,£O�$¤Q0 �k¨�©�¢Y�e0��Eº$�J Y�·ª$¤_�h¡Y�:�V¢�º$�h©,¸��,¦UÞ��·¤5��©,£* Lµ��Ô�3�ÄÈ]� ³ ¡Y�k�«¶��|Èb�k¡G¨�©,Þ�¨B£_¡¹�¤5�k�$¹3¢�¤I2Ó��lOÝ�� ÞU�ß©k©�2Ó��¦�¤� L¶��hª×¢�¤A¨�©kÞ�¨B£_¡� @µ�� �ß�b¨�¶�¤5�«¡�¢Y¨��±¢ �_µ]0h2Ó�J R¤�¦\¡�¢��J¸«¡�Õr���>¨h��¤5�kª¡Yµ�¶Lº�µ«�$¸«©,�h¨�¡Y¨��{¢!¡Y¶��h��¨# L�,¤5¡}2_£>¹5�k�_�k��¹{m ! ¢�¤�� ¨���ª�¨�©k©k¤BÞU�k��¹y� ¢�¤E Lµ��3�k��¢Y�e0��G¸Ó¤5©,£_��¤Q0 �k¨�© �k�� m ! � �Ö��YÌ|�k��¨�©k©,£�ÝÜ�$¤�¢�� ¢Yº�¨#¢�� �k¡Gë8ó5ñ! Y�,.{µ«�, Y�hª3¢�¤�¤5µ$¢�¸8µ$¢!¢Yº$� ¤� L�,¹5�ß��¨�© �}¨�©kµ���g�Y Û ¡)©,¤5�$¹�¨�¡h�϶J¨��

à«��ª�¨��{£E�k�±���J @¡��)¤�¦»È�Þ��IÞU�k©k©Ó¶�¤5µ��{¢U¢Yº�¨#¢\�k���h� ¡d¦�¨h��¤� ,Y ´ ¢�0 �,¹5º{¢d���J Y£·Þ��h©k©:2Ó�)¢Yº$�I¶J¨�¡��%¢Yº�¨#¢}¿�k¡d�$¤�¢�¤5�$��Z�¢�¤[Z�¤5���R¨���ª>º$�h��¶��/0 ¨��{£Eª���×Ó�J Y�h�±¢��k��¸«µ$¢Y¡�0 ¨#¸�¢�¤`¢Yº$�)¡Y¨�0��)¤5µ�¢�¸«µ$¢�Y

´ ¢!�k¡UÞ�¤� Y¢Yº�¹5�,�O�k�$¹`¨¥¡Y�e0�¸8©k�,à��hªEª$�Jà8���,¢Y�,¤5�·¦T¤� �¢Yº��R¶J¨�¡��RÞUº$�h�{¿��k¡�¨`¸Ó�J �0Rµ$¢Y¨#¢Y�,¤5�Ð�TÞUº�¨#¢��ß¡0��h¨��±¢pº��J Y���k¡�¢Yº«¨#¢JÝ5ÞUº$�h�R Y�h¡�¢� @�k¶�¢Y�k�$¹\¢Yº$���k�$¸«µ�¢�¤�¦�¿`¢�¤3j�l"�,mon,ØÓÝ5¢Yº$�h��¿`�ß¡ �ß�%¦�¨�¶�¢p¨\¸H�J 60Iµ$¢Y¨#¢Y�k¤5�¤B���J hj�l"�,mon Ø �6Y

âEã±ä.åÜæ�çFæTèHå>ÙÛÚ�ì)ô�ù#ï�¿�ò�ô%ù·¤5�$��Z�Þy¨h£E¸Ó�J �0Rµ$¢Y¨#¢Y�,¤5� ò ü)ñTõ$ì�ü�ó#þTþ,ó[�.ò�ë_ú�õ_ó#þ~�Q�

Á ¿×ò�ôRìYù#ô@ï¥ñ�óEí@ó#ö�ð�$ñ:ìyÜ'ù}ôGùWVYó#÷�ìcÝ ©

Á¥Þ ìJñ�gÉ�Öj�l"�,mon,Ø®ü�ó�î·ù#î,V�ò�ñ�î�ù#î@ïGº © ­�õ$ì�ër¿»�Ägµ�X�Öj�l"�,mon,Ø ù#ë� ü6�Oî�ñTõOì�îøö ó#îYìß¿^Ü��|õOìJë îYì�½ô�ñ�î@ò�íhñ:ì��¥ñ�ó|j�l"�,mon Ø Ý�ò�ôIùDð�ì�îøö3�$ñ�ù�ñ�ò�ó#ë ó�÷#ì�î/j�l"�,mon Ø:©

Á ¿®ò�ô1õ_ù�ît�Rñ�óDòTë�÷�ì�î�ñ ©� ù#ö`ì�þ ï}ý5ü�ó#îUù�þ�þW¬S¬h­3ù#þVú±ó#îøòTñTõ{öIôS�gñTõOìJî�ì\ì7¦5òßô�ñTôUùDëHì:ú�þ ò ú5òOVJþ,ì�ü6�Oë8íBñ�ò�ó#ë) �cbd�!ô��$íJõ�ñTõ_ù�ñ��

»1 �� g�Æàj�l"�,mon ! zÅÈX�É¿»�Äg:��áW�3�ÄÈ]��� gC$�Ë ) �O�]� +

�U¤�¢Y�k¶��G¢Yº�¨#¢\¢Yº$�Rª$�Jà«���,¢Y�k¤5�E¤�¦�2Ó�h�k�$¹rq�º«¨# Lª>¢�¤·�k�{���J Y¢7u��k¡d�,._µ��,�#¨�©,�h�{¢U¢�¤�ÞUº�¨#¢UÞy¨�¡\¹5�,���h��¸� Y�J�O��Z¤5µ�¡Y©,£�z|Þ��}xYµ�¡�¢I¡Y�e0�¸«©ß�,à��hª®�,¢)¦T¤� %¢Yº$�`¶J¨�¡Y�¥¤�¦1¿Ð2Ó�h�k�$¹�¨>¸H�J 60Iµ$¢Y¨#¢Y�k¤5�� L¨#¢Yº$�J I¢Yº�¨��*¨E¦�µ���¶�¢Y�k¤5�:Y��§>¨[���G¡Yµ$ Y�!£�¤5µ�µ���ª$�J L¡Y¢Y¨���ª·ÞUº{£`Þ��%¶J¨��G0 ¨[���%¢Yº$�h¡��%¡Y�\0�¸«©k�,à8¶J¨#¢Y�,¤5��¡�Yd�

�w9%8 FIBk;há»â:R ã�H%Aä�J¢�� ¡�©,¤{¤���¨#¢1¨I¡Y�e0�¸8©,�y��_O¨�0�¸8©,�12Ó�J¦�¤� L�d¡��J�h�k�$¹%¡Y¤Q0���0�¤� Y�\¶�¤Q0�¸«©k�߶J¨#¢��hª���_O¨�0 ¸«©,�h¡p¦T Y¤Q0Ï�{µ]0h2Ó�J ¢Yº$�J¤� Y£�Y ¬ ¤5�«¡Y�kª$�J p¢Yº��U¦�µ���¶�¢Y�,¤5�y¿»�Äg��ÅÈ]�»� g/btÈGÞUº$�J Y��¢Yº$��ª$¤Q0`¨��k� ¤�¦�¿>�k¡p¸«¨��k L¡.¤�¦��k�{¢��J¹��J L¡�2Ó¤�¢Yº¡�¢� L�k¶�¢Y©,£E¹� L�h¨#¢��J )¢Yº�¨��Ôm[Y ´ ¡�¢Yº��k¡\¨�¤5�$��Z�Þ�¨B£�¦�µ���¶�¢Y�k¤5�5v���h©k©�Ý��$¤�¢��R¢Yº�¨#¢D�k�{���J Y¢Y�k�$¹¥¢Yº$�G¦Tµ��«¶�¢Y�,¤5� Y�,.{µ«�, Y�h¡Ü¦T¨�¶�¢�¤� L�ß�$¹$Ý5¡�¤Q0��J¢Yº��ß�$¹�Þ��y¨�©k©W�O�$¤FÞ �k¡fq�º«¨# Lª]u"Yw��µ�¢ ©,�J¢�� ¡|��_$¨�0 �k���1ÞUº$�J¢Yº��J  ¢Yº«�k¡E._µ�¨�©k�kà��h¡¢�¤X2Ó�D¤5�$��Z�Þ�¨B£�Y����R�$�J�hªE¢�¤`¶�¤5�«¡Y�kª$�J �¢Yº$�%¦�¤5©k©k¤BÞU�k��¹I¸� L¤�2«¨[2«�k©k�k¢:£Há

»1 �� g��ÅÈßÆàj�l"�,mon ! z7å�� gßb,È�z��Äg Ê �ÅÈ Ê �S�>�3��å#��áQg Ê b,È Ê �Éå[$ +

æ

Page 9: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

´ ¡�¢Yº«�k¡d�$�J¹5©k�k¹5�~2«©,�\¦�¤� �¨��±£�¸Ó¤5©,£_��¤Q0 �k¨�©�Z�¢Y�e0 �\¨�©,¹�¤� @�,¢Yº]0Ã��5r� - º��k�@��¨[2H¤5µ�¢d�,¢d¨�0 �k�_µ$¢���Yd�´ �A¦T¨�¶�¢JÝ ¨·���J Y£®¡L�e0�¸«©,�G¨�©,¹�¤� L�k¢Yº]0 ¡Yº$¤FÞU¡D¢Yº�¨#¢%�k¢!�ß¡Gë8ó�ñ\¤5�$��Z�Þy¨h£�Y ¬ ¤5��¡Y�kª$�J ��Ï¢Yº�¨#¢DÞ�¤� 7�_¡

¨�¡U¦T¤5©k©,¤FÞU¡JáS�g¶Lº��h¶7�O¡\�k¦ �,¢Y¡U�k�$¸«µ$¢�å¥�k¡d�J���h�:zH�,¦|¡�¤$Ý8�,¢�¤5µ�¢�¸«µ$¢Y¡h�O�#�7å# [�Q�6Y U ¢Yº$�J YÞU�ß¡���Ý]�g¹5�,���h¡Uµ�¸LY�^º�¨#¢��k¡�¢Yº$�D¸� L¤�2«¨[2«�k©k�k¢:£ ¢Yº�¨#¢f� ¡Lµ�¶J¶��J�hª�¡d�k�E�ß�±���J Y¢Y�ß�$¹Xå`�k�·¢Yº��D��__¸Ó�J L�e0 �h�±¢d¨[2Ó¤B���'5Ð� æQç ¯ ��U¤FÞ��J���J BÝ8Þ��)¶J¨���0 ¨[���)¢Yº��)¦�¤5©ß©,¤BÞU�ß�$¹I¸8©k¨�µ�¡Y�~28©,��¶�¤5�ox��h¶�¢Yµ$ Y�I¨[2H¤5µ�¢�¦�¨�¶�¢�¤� L�k��¹$áp¦�¤� �¨��±£·»1» -

¨�©,¹�¤� L�,¢Yº]0Ñ�Ï¢Yº$�J Y�G��_$�k¡�¢Y¡D¡�¤Q0��G¸Ó¤5©,£O�$¤Q0 �k¨�©C`a�cbd��¡Yµ�¶@º3¢Yº�¨#¢/�7üJù#òTþ ô�¢�¤E¦�¨�¶�¢�¤� �år���k�®¢Yº$��¨[2Ó¤B�����__¸Ó�J L�e0 �h�±¢6��¨#¢U©k�h¨�¡�¢/m� �`a�O�]�y¤�¦ ¢Yº$�D¢Y�e0 ��Y ´ ¢�¢Yµ� L��¡y¤5µ$¢�¢Yº�¨#¢UÞ��%¶J¨��>ª$�Jà«�$�)¨��$¤�¢Y�k¤5�E¤�¦ ¨���ìLùoè¤5�$��Z�Þ�¨B£E¦Tµ«��¶�¢Y�,¤5�E¨���ª·¢Yº�¨#¢ ¿>¹5�k���h��¨[2Ó¤B���%¡Y¨#¢Y�ß¡�à��h¡�¢Yº$�)¨#¸«¸� Y¤�¸� L�ߨ#¢��!ª��Jà«���,¢Y�,¤5�LYaÌ�µ� Y¢Yº$�J �0�¤� L��ÝÞ��%¡Y¢Y¨#¢��%¢Yº$�D¦T¤5©k©,¤FÞU�k�$¹�2Ó�h¨�µ$¢Y�,¦�µ�©H¢Yº$�J¤� Y�,0Eá

éhê ã{èëFã"ì éîíyëHì�½ï�1ù#ï�ü6�Oë8íBñ�ò�ó�ë$ôRì7¦5òßô�ñpò ðk��ìLùoè¥ó#ëHì�½ï�1ù�ïUü��_ëHíhñ�ò�ó#ë$ôRì7¦5òßô�ñ ©

- º$�D¢Yº$�J¤� L�,0 �k¡d�ß�·¦T¨�¶�¢U¶�¤5�«¡�¢� Lµ�¶�¢Y�,����Y - º{µ«¡JÝ��,¦�¦�¨�¶�¢�¤� L�k��¹`�k¡�0�¤Oª$�J L¨#¢��h©,£·º�¨# Lªv��¨�¡U¶J¨#¸�¢Yµ$ L�hª2{£�¢Yº��R�$¤�¢Y�k¤5��¤�¦S2H�h�ß�$¹`Þ��h¨[�>¤5�$��Z�Þy¨h£]�@Ý¢Yº$�h��Þ��G¶J¨��3¶�¤5��¡Y¢� Lµ�¶�¢D¨{��¡Y¢� Y¤5�$¹N�U¤5�$��Z�Þy¨h£®¦Tµ��«¶�¢Y�,¤5�¨�¡dÞ��h©k©OY�U��_O¢¥¶J©ß¨�¡Y¡�Þ��EÞU�k©k©y¡��J�>¡Y¤Q0���¡Y�e0�¸8©,�·��_O¨�0�¸8©,�h¡�¤�¦\¤5�$��Z�Þ�¨B£�¦Tµ«��¶�¢Y�,¤5��¡�¨���ª×¸Ó�J �0Rµ$¢Y¨#¢Y�,¤5�«¡

2«¨�¡��hªE¤5���{µ]0h2Ó�J �¢Yº$�J¤� Y£�Y

¯

Page 10: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ñfò]óeô�õ�ö�÷7ó~øïù|ú�û»üGý[ö�ù#þeý�òCÿ� ü�� ��������� òWøÅö�ú#ÿ� ���ø7ó~úQò�øÅú � ö7ù��]øÅú���ö�ý����Wù��ö7ú�ûÄõ,÷7÷Åú�ö���úQò]ý[ø��]ý�ò��/ý[ø��

�! #"%$'&)(' *

+ , -/.102-43�5637098�: ;=<?>#@#37A9BDCE-F37G

H ø�ø��@õ�õ,ò]ÿ�ú�û�þeý�÷7ø þ~õI��ø� ]ö7õ�J1K�õX÷E�@úLK�õ,ÿ�ø��]ý[ø%M Cþ~ø7óN�CþeóO��ý[ø7ó~úQòGóe÷QP�RTSfý�úQò@õVUWKÓý,ù{ûX ]ò���ø7ó~úQò1J:÷7ó\ò���õûÄý���øÅú�ö�ó\òY�Gò� �M[Zµõ�ö�÷�ó\÷]\tõ,ý�÷Åù�^�K/�@õ,ò�ø��@õ�ö�õßóe÷�ý�_T` � �Cö7ú�ZCý�ZCóeþ\ó~ø%ù�ø��Cý[ø/ø��@õXò� �M[Zµõ�ö/óe÷/õ�ô�õ,ò1acb û��úT @ö�÷Åõ�J]ó~ø�óe÷�ö7õ,ý�÷7úQò]ý�ZCþ~õ}øÅúXý�÷7÷� �M�õ}ø��]ý[ødM Cþ~ø7óN�CþeóO��ý[ø7ó~úQòßóe÷ÓýfehgEikj|úQò@õVUWKÓý,ù�ûX ]ò���ø7ó~úQòlJ#ý�ò]ÿ�ø��]õ,òø��@õ�ö7õ�óe÷Sý�ø��@õ�ú�ö7õIM^÷Åø7ý[ø7óeòY�/ø��Cý[ø�ø��]óe÷�óOM#�Cþeó~õ,÷aø��@õ�õVm@óe÷ÅøÅõ,ò���õ�ú�ûµ÷ÅúTM�õn�÷ÅøÅö�úQòY�powúQò@õVUWKÓý,ù�ûX ]ò���ø7óeúQò1qr� ]ø/��ý�òsK�õ/ÿ@útZµõ�øÅøÅõ�ö�ý�ò]ÿs�Qó~ô�õ�ý#M�ú�ö7õhò]ý[ø� @ö�ý�þ1��úQò]÷7øÅöE ���ø7ó~úQò�u

� ò*ûÄý���øvJwK�õx��ý�ò1qzyIõ�ÿ]úrò@ú�øßò@õ�õ,ÿ¥øÅúrö7õ,÷ÅøÅö6ó{��øXúT @ö�÷7õ,þ~ô�õ,÷�øÅúrûX ]ò���ø7óeúQò]÷#K/�@úQ÷7õ�ÿ@úTMXý�óeò¥ó\÷|L}�~I�k����� óeò]÷ÅøÅõ,ý�ÿ1J�K�õF��ý�òf��úQò]÷7óeÿ]õ�öSûX ]ò���ø7ó~úQò]÷�ú�ô�õ�ö1ý[ö�ZCóeøÅö�ý[ö7ù�ÿ@úTMXý�óeòC÷���� |L}�~I�k��� i��F�NR�Pp��i��w�� �gX��� � gvP�SX���7�Vi��F���{g�i��v�Ngvqhy!�]ý[ø'K�õ%M�õ,ý�òfZNù|ø��]óe÷Óóe÷Óø��@õ}ûÄúQþeþ~ú�K�óeòY�Y�»þ~õ�ø'�#�%�!��� |L}�~I�k� � n�óWq õ�qNJ�#�Xö7õv�]ö�õ,÷Åõ,òWø7÷}÷ÅøÅö6óeòY�Q÷ óeò�� ú�û�þ~õ,ò���ø�����o�q��/�@õ,ò�K�õ�ö7õI � ]ó~ö7õhø��]ý[ø�ó~ø4Z�õQ�µúQ÷7÷7óNZþ~õ/øÅú] ]ò]ó~ûÄú�öEMXþeù÷7ý�M#�Cþ~õ{ý�ò õ,þ~õIM�õ,òNøyú�û�#��óeò¥ø7óOM�õx�µúQþ~ù"ò@úTMXó\ý�þ1óeò!�¡q üGú�ö7õ{û�ú�ö�MXý�þeþ~ù�J1ø��]õ�ö7õ�õVm@óe÷Åø7÷�ý¢�£�h�ý�þN��ú�ö�ó~ø���M2¤�÷� �����ø��]ý[ø4¥dn � � o�ö�õ�ø� @ö�ò]÷�ýt Cò]ó~ûÄú�öEMXþ~ùXÿ]ó\÷ÅøÅö�óNZ� ]øÅõ,ÿ|õ,þeõIM�õ,òWøfóeò]�#��q

y óeø���ø��Cóe÷fó\òxMXó\ò]ÿ1J�K�õ#��ý�òIò@úLK ÿ@õv¦ò@õ3ý|úQò@õVUWKÓý,ù{û§ Cò���ø7ó~úQò{ý�÷�ý�ò�õv¨©��ó~õ,òWø7þeù���úTM#�� @ø7ý�Zþ~õû§ ]ò���ø7ó~úQòcK/�]ó{���yó\÷��Cý[ö�ÿyøÅúXóeòWô�õ�ö�øÓó\ò|ø��]õ�û�úQþeþeúLK�óeò��h÷Åõ,ò]÷7õ#nÄù�úT G÷��@úT ]þeÿc�E�]õI��ª�ø��]ý[ø�ø��]óe÷Óóe÷�õI � ]ó~ôpUý�þ~õ,òWø�øÅú{ø��@õf�]ö7õ�ô"ó~úT ]÷hÿ@õv¦Cò]ó~ø7óeúQò«K/�@õ,ò¢�?� |L}�~I�k��� o��XûÄú�ö�ý�þeþ£�h�£��ý�þN��ú�ö�óeø���MX÷Q¬�ø��@õ�ö7õ�óe÷3ýò@õv�QþeóN�QóNZþ~õfûX ]ò���ø7óeúQòs­Ln¯®°o�÷E ��E��ø��Cý[øv�

��ö�± ²]³´�#� ��µ �·¶=n§²Do � ²�¸��!¬7n µ ow�p¶=n§²�¸{o=� µp¹»º ­Ln¼��o�½

y óeø��|ø��]õ,÷Åõ%��úQò���õv�]ø7÷�óeò©MXóeò]ÿ1J#þ~õ�øv¾Î÷Ó÷Åõ�õ%�@úLK!K�õ%MXóN�T�Nø�ø� @ö6ò©MQ ]þ~ø7óN�þeó{��ý[ø7ó~úQòXó\òWøÅú�ý3úQò@õVUWK�ý�ùû§ ]ò���ø7ó~úQò1q � ò]÷ÅøÅõ,ý�ÿrú�û�ý�þeþ~úLKfóeòY��ø��@õßÿ]úTMXý�óeòIú�û/¶ÐøÅúxZ�õt¿ÁÀ¿ÃnÄú�öLJEõI � ]ó~ô[ý�þ~õ,òNø%ù�J |L}�~I�k��� ó~û�K�õ�Cý[ö�÷Åõ�ø��]ó\òY�Q÷}ý����]ö7ú��]ö6óeý[øÅõ,þ~ù�o�JLþ~õ�øv¾Î÷/÷Åõ�ø�ø��@õXÿ]úTMXý�óeòÂ�Ûú�û'¶vøÅú]Zµõ7�ÅÄÇÆWÈ�ÊÉËÀ�É#JÌKF�@õ�ö7õ7K�õXþeõ�øÉÍ�οzZµõfø��@õ�÷Åõ�øÓú�û»�]ö�ó{M�õ òp �M[Zµõ�ö�÷vq=ÏfúLKÐK�õ%��ý�ò�ô#ó~õvKÐ�#�hý�÷�ý�ý�ó~ö�n§² ~�µ o�ú�û»�]ö�ó{MXõ,÷�gEiT�VÑsRÓÒ�NgvPp�TS§Ñ©��Ô�Õ�q H ò]ÿ�óeò�ûÄý���ø ó~ø�óe÷1ýßô�õ�ö7ù�ö7õ,ý�÷ÅúQò]ý�Zþ~õ[��úQòkÖÅõI��ø� @ö7õhø��]ý[ø�û�ý���øÅú�ö�óeòY�yóeòNøÅõv��õ�ö�÷/×ØJ�K/�]õ�ö7õ×Çó\÷/ýs�]ö�ú#ÿ� ���ø�ú�û�øÙK�ú�õI p Cý�þÚU%þ~õ,òY��ø��Â�Cö�ó{M�õ,÷vJEóe÷s\��]ý[ö�ÿ�^�û�ú�ö3ý�òWùÂ�µúQþ~ù"ò@úTMXó\ý�þÚU�ø7ó{M�õ�ý�þN��ú�ö�ó~ø���M �M�ú�ö7õ1ûÄú�öEMXý�þeþeùcn�ÿ@õv¦CòCóeòY��É��}ý�÷Sø��@õ�÷Åõ�ø�ú�ûD�]ö6ó{M�õ,÷wú�û:þ~õ,òY��ø�����o��wûÄú�öSý�þeþ��h�£�¥ý�þ{��ú�ö�ó~ø���MX÷=¬ ø��@õ�ö�õóe÷1ý�ò@õv�Qþeó{�QóNZCþ~õ}ûX ]ò���ø7ó~úQòs­�n¯®°oÓ÷� �����ø��]ý[øv�

��ök± ² ~�µ ³´Éw� ��Û �!²©® µ ��¬7n Û o��Ün§² ~�µ o ¹»º ­Ln¼��o�½

nXÏ�ú�øÅõ/ø��Cý[ø'Z�õI��ý� C÷Åõ�²Gý�ò]ÿ µ ý[ö7õ%�Cö�ó{M�õ,÷vJ�MQ ]þ~ø7óN�Cþ\ó{��ý[ø7ó~úQòßó\÷�ò]úLKÉúQò@õVU�øÅú�U�úQò@õh÷Åú#K�õ�ÿ@ú�ò@ú�ø1ò]õ�õ,ÿøÅú���úQò]÷7óeÿ]õ�ö1ø��@õ���ý�÷ÅõK/�@õ,ò]¬kúT @ø��� @ø7÷�n§² ¸ ~�µ ¸ o4Ý�Ün§² ~�µ o�q°o

�d�@õ'��ý[ö7õ�ûX ]þ#ö7õ,ý�ÿ@õ�ö=K�ó\þeþWò@ú�øÅõÓø��]ý[ø=K�õÓúTMXó~øÅøÅõ,ÿ�úQò@õ�ó{Mt��ú�ö�ø7ý�òWøÞ��úQò]÷7óeÿ]õ�ö�ý[ø7ó~úQò1�Ì��ý�òtK�õ�óeòhûÄý���ø÷7ý�M#�Cþ~õhû�ö�úTMÃÉîõv¨���ó~õ,òNø7þ~ù�u!nXߣ � ]ó~ô[ý�þ~õ,òWø7þeù�J���ý�ò�K�õ3õv¨���ó~õ,òNø7þ~ù�÷7ý�Mt�Cþ~õ�ö6ý�ò]ÿ@úTMà�pUWZCó~ø/�Cö�ó{M�õ,÷�uko�d�@õÓý�ò]÷�K�õ�ö�óe÷aù�õ,÷ � K�õwK�óeþeþNò@ú�øÞ�Qó~ô�õ�ÿ@õ�ø7ý�óeþe÷aò@úLK[JTZ� @øaø��]óe÷»K�óeþeþ�Zµõ�ÿ]ó\÷��v ]÷7÷Åõ,ÿ3óeòhø��@õw�T @õ,÷Åø»þ~õI��ø� @ö�õúQò�üGúQò]ÿ]ý,ù�q

Page 11: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

á â2@#ãåä©0£.çæzè70£-/.D<

yIõ��Qý,ô�õ3ý7Z]ö�ó~õ�ûLö�õ�ô#ó~õvKÉú�û9M�ú"ÿ� ]þeý[ö�ý[ö�ó~ø���M�õ�ø7ó{��J@ý�ò]ÿfK/�]ý[ø1óeøwMXõ,ý�ò]÷ÓøÅú���úTM#�� ]øÅõs\�M�ú#ÿ� ]þ~úQél^�qyIõXý�þe÷Åú�óeòNøÅö7ú#ÿ� ���õ,ÿGø��]õ�ò]ú�ø7ý[ø7ó~úQò¿wê ÄÇÆWÈ� |L}�~I��~ ½v½v½ ~ éxë �k� q�yIõ�ø��]õ,òIÿ@õv¦Cò@õ,ÿ{ø��@õ�ò@ú�ø7ó~úQò{ú�û�ý��ö7úT Y�Øn�÷Åõ�õhý�þN��õvZ]ö�ýt�Cý�ò]ÿ@úT @ø�o�J]ý�ò]ÿGÿ@õv¦Cò@õ,ÿ�ø��@õh÷Åõ�øv�

¿ �ì ÄVƼÈ� | ²�� �%º ² º × ý�ò]ÿ)�T��ÿÌn§² ~ ×�oh� �k� ½

yIõ�ÿ@õv¦Cò@õ,ÿ·í�n§é»o]ÄVƼÈ� î ¿ �ì î°qÍÏfú�øÅõIø��]ý[øs¿ �ï � |���~ ½v½v½ ~Xð ë �k� K/�@õ,ò ð óe÷c�]ö�óOM�õ�ý�ò]ÿ!�@õ,ò���õíhn ð o�� ð ë � q£yIõ}ý�þ\÷Åúh÷��]úLK�õ,ÿ�ø��]ý[ø�óeûl×à� ð¡ñ óe÷�ø��@õF�Cö7ú#ÿ� ���øSú�ûLøÙK�ú�ÿ]ó\÷Åø7óeò���øh�]ö�ó{M�õ,÷ ðÌ~�ñ ø��]õ,òíhnX×Âo£�Íî ¿ �ì î��Ün ð ë � oVn ñ ë � o�q

yIõ ÷7ø7ý[øÅõ,ÿßø��@õ ó{M#�µú�ö7ø7ý�òNø»ûÄý���ø�ø��]ý[øvJ#û�ú�öÓý�òWù#×ØJWø��@õ ÷Åõ�øh¿ �ì ��ý�ò©Zµõ�ô#ó~õvK�õ,ÿßý�÷�ý[MQ ]þ~ø7ó{�Cþeó{��ýkUø7ó~ô�õ���ö7úT Y�Ìq

yIõhý�þe÷7úX÷Åø7ý[øÅõ,ÿGø��]õ�û�úQþeþeúLK�óeò���óOM#�µú�ö7ø7ý�òWø1þeõIMtMXý��

òhó�ôØôØõ÷ö!ø»gvS»ùú��gQS§Ñ�g7R�û�ü�gVû7RÓÒ�ýÿþ£P � S g � ��û�R���������ÌÑ�gvP� �!� � ÒVR�ûi�P��©P�R�P���gVû�R�������

�d�@õ�ö7õÓóe÷wýfòCó{��õ�n�÷�ó{M#�Cþ~õ�o1�]ö7ú#ú�û]ûÄú�öaø��Cóe÷ � ÷Åõ�õ�± � � ¹ û�ú�ö�ÿ]õ�ø7ý�óeþe÷vq9�d�]óe÷E÷7ó{M#�Cþeõ�þ~õIMtMßýF��ý�òQZµõ� C÷Åõ,ÿ3øÅúÿ@õIM�úQò]÷ÅøÅö6ý[øÅõ/ýhô�õ�ö�ù© ]÷Åõ�ûX ]þ�û�ý���ø'K/�]ó{����K�õ/÷Åø7ý[øÅõ/óeòyó~ø7÷ÓúLKfò�þ~õIMtMXýZµõI��ý� ]÷Åõ�ó~ø�óe÷Ó÷Åú�ó{M#�µú�ö7ø7ý�òNøv�

òhó�ôØôØõ�� ø»gvS��à�EgQiFþ�P � S g���û�R����¢RÓÒ[R�û�ü�gVû4ù��'øÌgvS������i�P�ü² �EgQi�P � P�S gW��gVû����ÌÑ�gVP��

������� �! Ä � ½

"$#&%'%�( )�õ�ø7²!��² ¸ M�ú"ÿ#ùÂq �d�]õ,òØK�õ]��ý�ò÷K1ö�ó~øÅõc²z� ��ù+*в ¸ q rw @ø�ò@úLK K�õs�]ý�ô�õ, � � � �.- ��/ �� � � ��/ �Ün0 � o � ��/ �Ün � o � ��/ �� ��/ q

13254 6�798;:=<?>@<BAC<?D�EF8;:9G9<IH�85:9J)Lõ�ø7× � ð�ñ Z�õyý��Cö7ú#ÿ� ���ø/ú�û1øÙK�úIÿ]óe÷7ø7óeò���ø%�]ö�óOM�õ,÷vq � �Cóeò@õ,÷ÅõXö�õIMXý�óeò]ÿ@õ�ö6óeòY��óe÷hý�òvõI � ]ó~ô[ý�þ~õ,òNøK�ý�ù�ú�û»ô#óeõvK�óeòY�t¿ �ì ý�÷4¿ �ï À�¿ �K q'�d�@õ[K�ý�ù�ø��]óe÷dK�ú�ö�ª"÷�óe÷fý�÷�ûÄúQþeþ~ú�K�÷v��û�ú�ö ý�òWù�õ,þ~õIMXõ,òWøF²C��¿ �ì JK�õ7��ý�òGô"ó~õvK6²Iý�÷Qn§² ï ~ ² K oL��¿ �ï À�¿ �K KF�@õ�ö7õ[² ï ��²cM�ú#ÿ ð ý�ò]ÿx² K ��²cM�ú#ÿ ñ qM)Lõ�øv¾Î÷}þ~ú#ú�ª�ý[øýs�Cý[ö7ø7óO�v ]þeý[ö�õVm@ý�M#�Cþ~õXûÄú�ö× � � � � _® � qfßSþ~õIMXõ,òWø�N���¿ � O5P ��ý�ò Zµõ#K1ö6ó~øÅøÅõ,ò�ý�÷fn ��~ ÕTo/÷7ó\ò���õN� � M�ú"ÿ�_Xý�òCÿ�NQ� Õ�M�ú#ÿ � qLQ úQóeò��ßø��]óe÷�û�ú�ö}ý�þeþ:õ,þeõIM�õ,òWø7÷fú�û»ø��@õ��ö7úT Y���Qó~ô�õ,÷�ø��]õ�ûÄúQþeþ~úLKfóeòY�ø7ý�ZCþ~õ��

�SR n ��~I� oÕ R n¼Õ ~ ÕTo� R n ��~ � oN R n ��~ ÕToT R n¼Õ ~ _To���UR n¼Õ ~I� o� _ R n ��~ _To� � R n¼Õ ~ � o

Õ

Page 12: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

nXÏ�ú�øÅõyý�÷�ý�÷7ý�ò]ó~øïù����@õI��ª�ø��]ý[ø�ø��@õXò� �MZµõ�ö ú�û�õ,þ~õIM�õ,òNø7÷�óeò¿ � O5P óe÷�óeòCÿ@õ�õ,ÿ��Qó~ô�õ,òÂZ#ù�Õ[® � � T q°oÏ�ú�øÅõ�ø��]ý[ø3õ,ý��E�Ðõ,þ~õIM�õ,òNø]n§�Cý�ó~öÇo�ú�ûd¿ �V À«¿ �P ý����µõ,ý[ö�÷húQò���õ�ý�ò]ÿ�úQòCþ~ùIúQò���õyúQòrø��@õ|ö�óN�T�Nø�]ý�òCÿ÷7óeÿ@õ�ý�Zµú'ô�õ�qF�d�]ó\÷1÷� Y����õ,÷Åø7÷ ø��]ý[øfø��@õ�ö7õ3óe÷�ýÂ� � W gE�IS � R�PÂZµõ�ø K�õ�õ,òx¿ � O5P ý�ò]ÿ�¿ �V À�¿ �P qd�d�]õ � �]ó\ò@õ,÷Åõö7õIMXý�óeò]ÿ]õ�öaø��]õ�ú�ö7õIM2n§K/�]ó{���QK�õ�ÿ@ú�ò@ú�øÞ�]ö7ú'ô�õ'�@õ�ö7õ�o»��ý�ò7Zµõ�ô"ó~õvK�õ,ÿ�ý�÷»÷Åø7ý[ø7óeòY�}ø��]ý[øwø��]óe÷wóe÷wóeòCÿ@õ�õ,ÿýsZCó ÖÅõI��ø7ó~úQòrû�ú�ö�ý�òWù«× K/�Có{�E�vóe÷�ý��]ö7ú"ÿ� ���ø�ú�û�ø K�ú��]ö�ó{M�õ,÷cn�ý���ø� ]ý�þeþ~ù�Jwø��@õ � �]óeò]õ,÷Åõ�ö7õIMXý�óeòCÿ@õ�öø��@õ�ú�ö7õIM�óe÷�M�ú�ö7õ4��õ,ò@õ�ö�ý�þµý�ò]ÿf��ý�òcZµõfõVm"øÅõ,ò]ÿ@õ,ÿyûÄú�öw×Ûú�ûLôoý[ö6ó~úT ]÷�ú�ø��@õ�öÓû�ú�öEMß÷�o�q�Ï�ú�øÅõ/ý�þ\÷Åúhø��Cý[øø��@õ � �]ó\ò@õ,÷Åõ}ö7õIMXý�óeòCÿ@õ�ö�ø��@õ�ú�ö7õIM �Qóeô�õ,÷�ý�ò�ý�þ~øÅõ�ö6ò]ý[øÅõ��]ö7ú#ú�ûaú�ûaø��]õ�ôoý�þ{ ]õ�ú�û�í�nX×Âo��

íhnX×Âo£�Íî ¿ �ì î��çî ¿ �ï Às¿ �K î��çî ¿ �ï î�®�î ¿ �K î��çn ð ë � o9®�n ñ ë � o�½

Ï�ú�K^úQò]õßó{M#�µú�ö7ø7ý�òNø/û�ý���ø�ý�ZµúT @ø/ø��]óe÷�ý�þ~øÅõ�ö�ò]ý[øÅõ|ö7õv�]ö7õ,÷Åõ,òNø7ý[ø7ó~úQòrú�ûw¿ �ì óe÷/ø��@õXûÄúQþeþ~úLKfóeòY�Y�/ó~û² ~�µ ��¿ �ì K�ó~ø��÷² R n§² ï ~ ² K o3ý�ò]ÿ µXR n µ ï ~�µ K o�J�ø��@õ,ò)²�® µ M�ú"ÿt× R n§² ï ® µ ï M�ú"ÿ ðÌ~ ² K ®µ K M�ú"ÿ ñ o�qs�d�]ó\÷�óe÷�ýGô�õ�ö7ù« ]÷Åõ�ûX ]þ»û�ý���øKF�@õ,òrÿ@úQó\òY�s��úTM#�� @ø7ý[ø7óeúQòØM�ú"ÿ� ]þeú�þeý[ö���õ|ò� �MZµõ�ö�÷�×Ø�óeò]÷ÅøÅõ,ý�ÿ�ú�û���úTM#�� @ø7óeò��©²f® µ ý�ò]ÿ{ø��@õ,ò�ö7õ,ÿ� ���óeòY�fM�ú"ÿ� ]þ~úc× J1K�õt��ý�òÂ��úQòWô�õ�ö�ø%²rý�ò]ÿ µ øÅú�ø��@õ,óeöý�þ~øÅõ�ö�ò]ý[øÅõXö�õv�]ö7õ,÷Åõ,òNø7ý[ø7ó~úQò]÷�n§² ï ~ ² K o ý�ò]ÿ�n µ ï ~�µ K o�JEÿ@ú�úT @ö[MQ ]þ~ø7ó{�Cþeó{��ý[ø7ó~úQò�M�ú"ÿ� ]þ~ú ð ý�ò]ÿ ñ J�ý�òCÿø��@õ,ò]��úQòWô�õ�ö�ø�ø��@õ/ý�ò]÷�K�õ�öwZCý���ªyøÅúXý�òyõ,þ~õIM�õ,òNø1óeò©¿ �ì qh�#ú�ó~û ð»~�ñ ý[ö7õ��pUWZCóeø��Cö�ó{M�õ,÷vJNø��@õ,ò�óeò]÷7øÅõ,ý�ÿú�ûSÿ@úQóeòY�ßúQò@õQMQ ]þ~ø7ó{�Cþeó{��ý[ø7ó~úQò]MXú#ÿ� Cþ~úßýfÕ��pUWZCó~øfò� �MZµõ�öF×ØJ�K�õ3ó\ò]÷ÅøÅõ,ý�ÿGÿ]ú|øÙK�úfM Cþ~ø7óN�CþeóO��ý[ø7ó~úQò]÷M�ú#ÿ� ]þ~úÂ�pUWZCó~ø�òp �MZµõ�ö�÷vqÂ�d�]ó\÷�õVm"øÅõ,ò]ÿ]÷�ûÄú�ö�ø��@õ]��ý�÷Åõ�ú�û�õVm��µúQò@õ,òNø7óeý[ø7ó~úQò1J�ý�÷7K�õ]K�óeþeþ�÷Åõ�õ�ò@õVm"øø7ó{M�õ�q

H ÷�ý�¦ò]ý�þ@ö7õIMXý[öEª�J#ò@ú�øÅõfø��Cý[ø�óe÷�ó\÷[\tõ,ý�÷Åù�^�øÅúQ��úQòWô�õ�ö�øh²C�]¿ �ì øÅúcn§² ï ~ ² K oY�s¿ �ï ÀQ¿ �K n�÷�ó{M#�Cþ~ù��úTM#�� @øÅõXø��]õßò@õI��õ,÷7÷7ý[ö�ù{ö�õIMXý�óeò]ÿ@õ�ö6÷�o�qZfúLK ��ý�ò K�õ���úGóeòIø��@õßú����µúQ÷7ó~øÅõXÿ]ó~ö�õI��ø7ó~úQò�u·yIõ,þeþ¼JE÷�ý,ùK�õ#�]ý�ô�õ�û�úT ]òCÿØn�óeò�ý�ÿ@ô[ý�ò���õ�o�ôoý�þO @õ,÷\[ ~^] �«¿ ì ÷� ����{ø��]ý[ø_[ R n ��~�} o}ý�òCÿ ]`R n }�~I� o�qcn§yIõý[ö7õ�ý�Z� ]÷7óeò��|ò@ú�ø7ý[ø7ó~úQò��@õ�ö�õ�Jµ÷�óeò���õ�óeòGû�ý���øtn ��~�} o#Ý�¿ �ï À�¿ �K ÷7óeò���õ } Ý�¿ �K q%�d�]ó\÷fóe÷FK/�NùsK�õ�÷�ý,ù[ ~^] �s¿ ì ý�ò]ÿ�ò@ú�øL[ ~^] �s¿ �ì q°o7�/�@õ,ò1JY�Qó~ô�õ,ò n§² ï ~ ² K o�K�õ��]ý,ô�õ��

n§² ï ~ ² K o��z² ï ®�n ��~�} o�*¢² K ®pn }�~I� oh�z² ï ®�[a*¢² K ® ]h~

ý�ò]ÿßø��Cóe÷£¦ò]ý�þ���úTM#�� @ø7ý[ø7ó~úQòc��ý�ò©Zµõ ÿ@úQò@õFMXú#ÿ� Cþ~ú[×Øq H ÷�ý�òyõVm"ý�M#�þ~õfóeò©¿ � O5P JpK�õ��]ý,ô�õtn ��~�} o R�v} n�÷�óeò���õ �v} � � MXú#ÿt_{ý�ò]ÿ �v} � } M�ú#ÿ � ohý�ò]ÿ!n }�~I� o R � qØ�#ú�K�õf��ý�ò)��úQòWô�õ�ö�ø]n ��~ _ToZ#ùÿ@úQóeòY�Y�

n ��~ _Toh� � ®�n ��~�} ob*)_4®pn }�~I� o£� � ® �v} *)_�® � M�ú#ÿ � � � � _ ~K/�]ó{����óe÷�ø��@õ���ú�ö7ö�õI��øfý�ò]÷�K�õ�öIq

_

Page 13: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

cMdIegfih�jlk^enm;o�piq�r�s�jlo�tgsid?uv r�w v.x�yiz|{~} d�m�jlp�uI�?��m^enp�d,m�p v j^o��Im�pi�ijls�����o�=j^piq�h�k^k�pij_�ip�dIs�m^�Isid��$s�m��

������������� �

� � ���'�a�\�~ �¡�¢l�£�9¤��a¥¦�¨§ª©L¢l�£«��9�¬¢l�£­

®¬¯�°£±~²´³¶µ ·3¸�¹I¯�º^¯$³�»^µ½¼�º^¯,¾I¿gÀ�°^¿gÁ?Â�°$ÃIºl¿gÄ�¯�À�ÅÇÆȯÉÀ^¼�¸`Êg¼iÀ^°|°^¿gÄ�¯°^¹Ë¯,ÁËÌi°^¿nÌ�ÁÍÌiÎ�ÏÑÐ�Ò�Ó¶ÔÖÕ�Ô�×^Ô�ØÙ�Ú Ò0Ó�Û Ô�×ÖÒ0Ó Ü�·=¸�¹Ë¯�ºl¯�Ý Þ�¸.¯,Â�¼iÁÍßà¿n¯�¸âáäã�åLæç´¼iÀÉè�á�éà»�á�ê�ë�ã�åLæé�ì åLæê Å�ÆȯɼiÊgÀ�̽Àl¼�¸a¹ËÌ&¸í°^¹I¿îÀº^¯�ÃIº^¯�À�¯�Á °^¼�°^¿nÌ�Á�Ä�¿nï�¹ °�À�ÃF¯�¯�¾�ðIÃ�Ä|ð?Ên°^¿nÃ?Êg¿îÂ�¼�°^¿nÌ�Á,¿gÁ�å æçLÅYñYðË°\¿n°�Â�¼iÁ�¼iÊgÀ�ÌÀ^ö¯�¯�¾½ðËÃǯ�òàÃFÌ�Á˯�Á�°^¿î¼@ó°^¿nÌ�Á'Å9ôËÌiº\Â�Ì�Ä�Ã?Ên¯�°�¯�ÁI¯�À^À�·Ë¸.¯õÀ�°^¼�°�¯õ°^¹I¯$Î0Ì�ÊîÊnÌö¸�¿îÁËï|ºl¯�À^ðIÊn°^À�÷

ø�ùIúiû½ü�ý Ô�þ9±½»ÿ³�»^µ Ú Õ Ú������ Ô�� ý Ô�þ� Û Ô�Ó � þ5Ô�þ�ÐàÔ�ÖÏÑÐ�Ò0ÓFÔÖÕ�Ô×^Ô Ù�Ú Ò�Ó�Û�Ô�× Ò�Ó Ü ��×^Ô��¶×�Ô�Õ�Ô�Ó�þ Ú þÿÒ � Ó ���Ú Ó�Ô��gÔ Ù Ô�Ó?þ¨Ò�Ó,å æç Ú Õ|Û�ÒîÕ�����ÕlÕ�Ô^Û Ú������ Ô�����ÐàÔ�Ó��� �!� á"� è�á�éà»�á�ê�ë Ú Ó�Û$#%� è&#�éà»'#�ê�ë½þ�ÐàÔ�Ó�á(#%� è�á�é�#�é\Ä�Ì�¾_³Ñ»�á�ê)#�ê¨Ä�Ìà¾�µ�ë+*&, � þ5ÔCþ�Ð Ú þ� ��Ù �-�Ëþ Ú þÿÒ � Ó�Ò0ÓÈþ�ÐàÔ.�gÔ � þ9Ð Ú � �/��� þ�ÐËÔ�þ0�����nÔ|Ò0Õ Ú �21 Ú�3 Õ£Û � ÓFÔ|Ò0ÓÉå æé Ú Ó�Û4� �iÙ ���àþ Ú þÿÒ � Ó�Ò0ÓÈþ�ÐàÔ

×ÖÒ ÜiÐ þMÐ Ú � �5��� þ�ÐàÔ,þ0�6�-�gÔ�Ò�Ó Ú �71 Ú�3 ÕÇÛ � Ó¶Ô,Ò0Ó�åLæê98 Õ � þ�ÐàÔ�Ó � þ Ú þÿÒ � Ó:�Ä�Ì�¾\³;� 8 ?Ä�Ìà¾�µ<��ÒîÕ×^Ô^Û��àÓ�Û Ú Ó?þ=�?>'× �iÙ Ó � 1 � Ó 8 1¨Ô ��Ù Ò�þ.Òÿþ=�A@� �!� á4� è�á éà»�á¶ê�ëõþ�ÐàÔ�Ó�áCBEDF� è�áGBEDé »�áCBEDê ë��

� �!� á4� è�á éà»�á¶ê�ë Ú Ó�ÛIH�ÒîÕ Ú Ó�Ò�Ó?þ5Ô5Ü Ô�× 8 þ�ÐàÔ�Ó�á�JK� è�á-Jé »�á-Jê ë��L ¹Ë¯�À�¯�Îÿ¼iÂ�°^ÀLÂ�¼iÁ�À^ö¯�¯�¾ÉðIÃ,Â�Ì�Ä�Ã?ðË°^¼�°^¿nÌ�ÁIÀ�ÅNMMÀ�¼iÁǯ�òà¼iÄ�Ã�Ên¯i·IÂ�Ì�ÁIÀ^¿g¾I¯�ºLÂ�Ì�Ä�Ã?ðË°^¿gÁIï/OPDRQ'S'T�Ä�Ì�¾4U�V�ÅW ¿gÁIÂ�¯XU�V�²ZY\[�V�·�¸.¯£Â�¼iÁ�º^¯�Ã?º^¯�À�¯�Á °;O,¼iÀ�è�U�»'O ë Å L ¹Ë¯�Á]O DRQ'S'T ²âè�U DRQ'S'T »'O DRQ'S'T ë.²âè�U�»öè�^_Uöë DRQ'S'T ëY²è�U�»�Uöë Å L Ì$ïi¯�°9Ì�ð˺F`?ÁI¼iÊI¼iÁ?À�¸¨¯�ºö·�¸.¯MÁËÌ&¸ba^ðIÀ�°=Á˯�¯�¾°�Ì|Â�Ì�Á ßi¯�º^°_è�U�»�Uöë=Ý?¼iÂ�c�°�Ì|¼iÁ�¯�Êg¯�Ä�¯�Á�°9Ìiάå æ DRS ÅÆȯ$ï�¼�ßi¯�¼£°�¯�Âl¹IÁI¿ed ð˯�Î�ÌiºL¾ËÌ�¿gÁIï�°^¹?¿gÀ�Êg¼iÀ�°Y°^¿gÄ�¯i·ËÝ�ðË°�¹Ë¯�º^¯�¸.¯$Â�¼iÁÉÌiÝ?À�¯�º^ßi¯õ°^¹I¼�°_U$ãÇåLæ DRS ¹I¼iÀY°^¹I¯ÃIº^ÌiÃF¯�º^°5Þ�°^¹I¼�°fU\²gU�Ä�Ìà¾XY�¼iÁI¾bUM²hULÄ�Ìà¾�Vji W Ì�Ì�ð˺;`?ÁI¼iÊ'¼iÁIÀ�¸.¯�º�¿gÀ\U�Å

ÆȯL¸�¿gÊîÊ À�¯�¯�ݶ¯�ÊgÌö¸�°^¹I¼�°FkY¹?¿gÁ˯�À�¯.º^¯�Ä�¼i¿gÁ?¾Ë¯�ºl¿gÁËï_¿gÀ�¼iÊgÀ^Ì_¼_ÃFÌ&¸¨¯�º^ÎÿðIÊà°^¹Ë¯�Ìiºl¯�°^¿gÂ�¼iÊË°�Ì�Ì�Ê ·�¯�ÁI¼�Ý?Êî¿gÁËïðIÀ�°�Ì�¯�¼iÀ^¿îÊnÞ�Ã?º^Ìößi¯õļiÁ�Þ,ðIÀ�¯�ÎÿðIÊF°^¹Ë¯�Ìiº^¯�ÄÀ�Å

l mon£©�«��'©qp'¢�r`¥¦�=¤F¢^«�n£�=¤L ¹Ë¯_ÁËÌi°^¿nÌ�Á�ÌiÎFs�� Ú Ûi× Ú þÿÒt�|×�Ô�Õ Ò Û��?ÔÖÕ_ÃFÌiÃ?À¨ðIÃ�ßi¯�ºlÞ�ÌiÎ0°�¯�ÁÉ¿îÁ�Â�º^Þ�ÃI°�Ìiïiºl¼�Ã�¹�ÞiÅuM\Á�¯�Ên¯�Ä�¯�Á °v�ãÇå æJ ¿îÀ¼wd ðI¼i¾Iºl¼�°^¿gÂ\º^¯�À^¿g¾?ð˯�¿nÎ�¼iÁI¾,Ì�ÁIÊgÞ�¿nÎb¿n°�¿gÀY¼�À�d�ðI¼�º^¯yxË¿ Å ¯iÅn·?¿nÎ�°^¹Ë¯�º^¯$¿gÀ�¼iÁ,¯�Ên¯�Ä�¯�Á�°�áCãÉåLæJ ÀlðIÂl¹É°^¹?¼�°á-z�²{v�Ä�Ìà¾XH¶Å�Æȯ�ÝF¯�ï�¿gÁ�Ý Þ�ÊnÌ Ìycà¿gÁËï�¼�°�°^¹Ë¯�Â�¼iÀ�¯fH²�³�·Ë¸M¹Ë¯�º^¯¨³�¿gÀY¼iÁ�Ìà¾I¾�ÃIºl¿îÄ�¯iÅ}|;°�¿gÀY¼�Î�¼iÂ�°°^¹I¼�°L¯�ßi¯�º^Þ,¯�Ên¯�Ä�¯�Á °�¿gÁÉåLæé ¹?¼iÀ�¯�¿n°^¹Ë¯�º�ÁIÌ�À�d ð?¼�º^¯�º^Ì�Ìi°^Àõèÿ¿ Å ¯iÅn·?¿gÀ�ÁËÌi°L¼�d�ðI¼i¾Ëºl¼�°^¿gÂ\º^¯�Àl¿g¾Ið˯&ë9ÌiºM¹?¼iÀ¯�òà¼iÂ�°^ÊnÞ,°;¸¨ÌË·¶¾I¿gÀ�°^¿gÁ?Â�°YÀ�d ðI¼�ºl¯�º^Ì Ìi°^À�·?¼iÁI¾É¸.¯õÁËÌ&¸ À�°^¼�°�¯|¼iÁ?¾ÉÃIº^Ì&ßi¯$°^¹I¿gÀYÎ�ÌiºlÄ�¼iÊgÊgÞiÅ~F�P�+� ù ü > � ×ѳ���Y Ú Ó � Û�Û?��×ÖÒ Ù Ô 8 Ô � Ô�× 3 Ô��gÔ Ù Ô�Ó?þ v�ãÇå æé Ð Ú Õ$Ô�Òÿþ�ÐàÔ�×\Ó � Õ�s�� Ú ×�Ô�× ��� þ�Õ � ×\þ01 �Û�ÒîÕ�þÿÒ0Ó���þ¨Õ�s�� Ú ×�Ô|× ��� þ�Õ�Ò0Ó,åLæé �

U

Page 14: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�_�6�E�-� ®�¯�°KvCã�å æé Å\| ÎNvǹ?¼iÀ\ÁIÌ�À�d�ðI¼�º^¯�º^Ì�Ìi°^À�·'¸.¯£¼�ºl¯£¾IÌ�Á˯iÅf�\°^¹Ë¯�º^¸�¿îÀ�¯i·¶Êg¯�°\áÈÝF¯£¼,À�d ð?¼�º^¯º^Ì Ìi°�ÌiÎqv?Å.��Ìi°�¯�°^¹I¼�°�^Lá�¿gÀ_¼iÊgÀ�Ìɼ,À�d ð?¼�º^¯£º^Ì Ìi°_ÌiÎqvÍè�¸�¹ Þ��@ë Å/�_Á�°^¹Ë¯�Ìi°^¹I¯�º$¹I¼iÁI¾'·Fá�¼iÁI¾�^�á¼�º^¯Ç¾I¿gÀ�°^¿îÁIÂ�°|Ä�Ìà¾IðIÊg̳ è�°^¹I¿gÀ�¿îÀ|¸M¹�Þ�¸.¯Éº^¯�d�ðI¿nº^¯�°^¹I¼�°_³��²���ë ·.À�Ìbv�¹I¼iÀ�¼�°�Ên¯�¼iÀ^°�°;¸¨ÌÍÀ�d ð?¼�º^¯º^Ì Ìi°^À�ÅqkY¼iÁÉ°^¹Ë¯�ºl¯�ݶ¯�Ä�Ìiºl¯6�ÇÆȯ�ÊîÊ ·IÊn¯�°#�ÝF¯$¼iÁËÌi°^¹I¯�º�À�d�ðI¼�º^¯_º^Ì�Ìi°LÌiÎ�v�Å L ¹Ë¯�Á,á z ²�# z ¼iÁ?¾É°^¹ ð?Àá-zK^�#jz�²��àÅ�MMÊgïi¯�ÝIºl¼,ï�¿nßi¯�À�÷Éè�á]^�#Ië�è�á5��#Ië�²��àÅ�ñYðË°$°^¹?¿gÀ$¹I¼iÀõ°^¹Ë¯�°5¸.̽À^Ì�ÊgðË°^¿nÌ�ÁIÀ_#C²��\áèÿ¿gÄ�ÃFÌiº^°^¼iÁ °MÁËÌi°�¯i÷�°^¹I¿îÀMÄ�¼�ci¯�À_ðIÀ^¯õÌiÎ�°^¹Ë¯�Î�¼iÂ�°\°^¹I¼�°\°^¹Ë¯õ¯�d�ðI¼�°^¿nÌ�Á��?�²o��Ä�Ì�¾_³È¹I¼iÀ\À�Ì�ÊgðË°^¿gÌ�ÁIÀÌ�ÁIÊnÞõ¿nÎ(�ä²{�\Ìiºu�|²��à·�Ìiº�ÝFÌi°^¹'Å L ¹I¿gÀb¿gÀ�°�º ð˯Y¸�¹Ë¯�ÁM³�¿gÀbÃIºl¿gÄ�¯¨Ý?ðË°3¿gÀ.Ó � þÑ°�ºlð˯�¿nÎi³�¿gÀ3Â�Ì�Ä�ÃFÌ�À^¿g°�¯i·¼iÀL¸.¯$¸�¿gÊgʶÀ�¯�¯$ÝF¯�ÊnÌ&¸\ë ÅL ¹I¿gÀ�Ên¯�ÄÄ�¼Ç¼iÊgÀ�ÌÉï�¿gßi¯�ÀõðIÀ$¼½Â�Ì�ðIÁ °$ÌiÎ�¹ËÌö¸âÄ�¼iÁ�Þ$d�ðI¼i¾Ëºl¼�°^¿gÂ�º^¯�À^¿g¾IðI¯�À_°^¹Ë¯�º^¯¼�º^¯�¿gÁÈå æé Å W ¿îÁIÂ�¯¯�ßi¯�º^Þ�À�d�ðI¼�º^¯\Ä�¼�Ã?À¨°�Ì|°;¸¨ÌË·I¾?¿gÀ�°^¿gÁIÂ�°9¯�Ên¯�Ä�¯�Á °^À¨ÌiÎ�°^¹Ë¯�ïiº^Ì�ðIì· ¯�ò˼iÂ�°^ÊnÞ¹?¼iÊnÎ'Ìiά°^¹Ë¯M¯�Ên¯�Ä�¯�Á °^À¨ÌiÎÑåLæéÄ|ðIÀ^°LÝF¯_À�d ðI¼�ºl¯�Àõèÿ¿ Å ¯iÅn·I°^¹Ë¯�ºl¯õ¼�º^¯�è ³�^�Uöë' ���À�d�ðI¼�º^¯�À ë Å

ÆȯÉÁËÌö¸`Â�Ì�ÁIÀ^¿î¾Ë¯�º|°^¹I¯�Â�¼iÀ�¯4H ²a±Í·�¸�¹Ë¯�º^¯�±~²¦³�µÇ¿gÀ�¼�Ã?º^Ì�¾?ðIÂ�°$ÌiÎL°;¸¨ÌË·9¾?¿gÀ�°^¿gÁIÂ�°,è�Ì�¾?¾�ëÃIºl¿gÄ�¯�À�Å<¡MÌö¸�Ä�¼iÁ ÞõÀ�d�ðI¼�º^¯.º^Ì Ìi°^ÀbÂ�¼iÁ�¯�Ên¯�Ä�¯�Á °^À¢v�ã�åLæç ¹I¼�ßi¯�ÁËÌ&¸£��Æȯ�À^¹ËÌ&¸B°^¹?¼�°Ñ¯�¼i ¹£¯�Ên¯�Ä�¯�Á °¹I¼iÀ�¯�¿n°^¹Ë¯�ºMÁËÌÀ�d�ðI¼�º^¯�º^Ì�Ìi°^À�ÌiºM¯�òà¼iÂ�°^ÊnÞ ��� �à×�¾?¿gÀ�°^¿gÁIÂ�°�À�d�ðI¼�º^¯�ºlÌ Ìi°^À�Ť.¥ ���-�6�P� üäý Ô�þѱ ²B³�µ Ú Õ Ú������ Ô��5��ÐàÔ�Ó Ú ÓÍÔ��nÔ Ù Ô�Ó?þ�v�ã½åLæç Ð Ú Õ�Ô�Òÿþ�ÐàÔ�×|Ó � Õ�s�� Ú ×^Ô�× ��� þ�Õ � ×��� �à×|ÛiÒîÕ�þÿÒ0Ó-�öþ¨Õ�s�� Ú ×^Ô|× ��� þ�Õ|Ò�Ó,å æç?��_�6�E�-� |;Î}v�ã�å æçX¹I¼iÀ�ÁËÌ�À�d�ðI¼�º^¯�º^Ì�Ìi°^À�·Ë¸.¯õ¼�º^¯$¾ËÌ�Á˯iÅ W ÌË·I¼iÀ^À^ðIÄ�¯_v�¹I¼iÀ�¼�°�Ên¯�¼iÀ�°�Ì�Á˯|À�d ð?¼�º^¯º^Ì Ìi°�áÑÅ�¦MÀ^¿îÁËï§kY¹I¿gÁ˯�À^¯�ºl¯�Ä�¼i¿gÁI¾Ë¯�º ¿gÁËïË·=Ên¯�°�v � è0v@éà»�v�ê�ë�¼iÁI¾ á%� è�á�éà»�á�ê�ë Å W ¿gÁIÂ�¯,á z ²�v�·¿n°õÄ�ðIÀ�°�ÝF¯�°^¹Ë¯Â�¼iÀ^¯°^¹I¼�°$á�zé ²�v&é�Ä�Ìà¾\³B¼iÁ?¾Èá-zê ²�v�ê¨Ä�Ìà¾�µÈè�Ý�Þ�ôI¼iÂ�°�Uöë Å,ñYðË°$°^¹Ë¯�Á+v�¹?¼iÀ°^¹Ëº^¯�¯�Ä�Ìiº^¯�À�d ð?¼�º^¯£º^Ì Ìi°^À�÷è�^�á é »�á ê ë ·Yè�á é »�^�á ê ë ·b¼iÁI¾ è�^�á é »�^Lá ê ë£èÿ¼iÁI¾È°^¹Ë¯�À�¯�¼�ºl¯¼iÊgÊ�¾I¿gÀ^°^¿gÁIÂ�°�·¼iÀ$¼�º^ï�ð˯�¾È¼�ÝFÌößi¯Î0Ìiº�°^¹Ë¯�Â�¼iÀ^¯�³�ÃIºl¿gÄ�¯&ë Å|ô3¿gÁI¼iÊgÊgÞi·F¿gÎNv½¹?¼i¾�¼iÁIÌi°^¹Ë¯�ºõÀ�d�ðI¼�º^¯£º^Ì�Ìi°è&#�éà»'#�ê�ëM°^¹I¯�Á#jzé ²{v@é�Ä�Ìà¾\³�¼iÁI¾�#Pzê ²�v êYÄ�Ìà¾�µ|À^Ì�°^¹I¼�°#�éõ²%�\á é�¼iÁI¾�#�ê�²%�\á¶ê|èÿ¼iÀL¼�º^ï�ðI¯�¾É¼�ݶÌ&ßi¯�Î0Ìiº�°^¹I¯Â�¼iÀ�¯L³�Ã?ºl¿gÄ�¯&ë Å W Ì�°^¹I¯�À�¯�Î0Ì�ðIº�À�d�ðI¼�º^¯�º^Ì�Ìi°^À�¼�º^¯�°^¹I¯ � Ó¨� 3 À�d ðI¼�ºl¯�º^Ì Ìi°^À�Ìi΢v?Å©M¯�`�Á˯]ª¬« ç ¼iÀ\°^¹Ë¯�À^¯�°MÌiÎNd�ðI¼i¾Ëº ¼�°^¿gÂ$º^¯�Àl¿g¾Ið˯�À\¿gÁ�å æç�ÅK��Ìi°�¯£°^¹I¼�°_°^¹Ë¯�°^¹Ë¯�Ìiº^¯�Ä ¼�ÝFÌößi¯�¿gÄ�Ã�Êg¿n¯�À°^¹I¼�°L¯�ò˼iÂ�°^ÊnÞbU�­6OÌiÎ3°^¹Ë¯�¯�Ên¯�Ä�¯�Á °^À�¿îÁÉåLæç ¼�º^¯.d ðI¼i¾Iºl¼�°^¿gÂ_º^¯�À^¿î¾Ið˯�À�xàÌiº/®eª¬« ç ®i²"® åLæç ®A 6OËÅ

è0MMÀM¼iÁɼiÀ^¿g¾Ë¯i·IÁIÌi°�¯�¸�¹�Þ�°^¹Ë¯�ÃIºlÌ ÌiÎÑ°^¹I¼�°�°^¹Ë¯�º^¯$¼�º^¯�Ì�Á?ÊnÞ°;¸¨ÌÀ�d ðI¼�ºl¯�º^Ì Ìi°^ÀLï�¿nßi¯�ÁÇ¿gÁ�°^¹Ë¯$Â�¼iÀ�¯ÌiÎ$åLæé ·3³ ÃIºl¿îÄ�¯i·¨Îÿ¼i¿gÊgÀ¹Ë¯�º^¯iÅ{| ÁXÃ?¼�º^°^¿gÂ�ð?Êg¼�º�·Y¿n°¿gÀ�ÁIÌi°°^¹Ë¯�Â�¼iÀ�¯�°^¹I¼�°�¿nÎ_á(#�²¯�õÄ�Ì�¾�±S°^¹I¯�Á¯�¿n°^¹Ë¯�º�á�²h�,Ìiº\#½²"�àÅ/M\À$¼iÁC¯�¼iÀ�ÞCÂ�Ì�ðIÁ °�¯�º^¯�òà¼iÄ�Ã?Ên¯i·�ÁËÌi°�¯�°^¹?¼�°�·¬Î0Ìiº$¼iÁ Þ�v�»�°|¸¨¯�¹?¼�ßi¯]± ðIÀ^¿îÁËïº^¯�ÃIº^¯�À�¯�Á °^¼�°^¿nÌ�ÁIÀ'² ÷Mè0v?»���ë¢[ è0�à»�°�ë=²¦è0�à»���ë.²{�àÅNM\ÊgÀ�ÌË·�³�µ$²{��Ä�Ì�¾�± ¼iÊn°^¹ËÌ�ðIï�¹£³�»^µ��²{�$Ä�Ìà¾�±ÍÅ ë|;°\¿gÀ�°^¹Ë¯|Â�¼iÀ�¯õ°^¹I¼�°MÀ�d ð?¼�º^¯$º^Ì�Ìi°^À�Ä�Ìà¾IðIÊgÌ£¼ÃIºl¿gÄ�¯¨³CÂ�¼iÁÇÝF¯$Â�Ì�Ä�Ã?ðI°�¯�¾�¿gÁÉÃFÌ�ÊnÞàÁËÌ�Ä�¿î¼iÊ ó °^¿gÄ�¯è�¸¨¯�Ä�¼�Þ�¾I¿gÀlÂ�ðIÀ^À�¹ËÌ&¸ °�Ì�¾ËÌ�°^¹I¿gÀ�Êg¼�°�¯�º�¿gÁ °^¹Ë¯�À^¯�Ä�¯�À�°�¯�ºÖë Å L ¹I¿gÀ�¼iÊîÊnÌö¸�À�¯�³Â�¿n¯�Á °Â�¼iÊgÂ�ðIÊg¼�°^¿gÌ�ÁÌiÎMÀ�d�ðI¼�º^¯�º^Ì�Ìi°^À�Ä�Ì�¾Ið?Ên̽±UÒ � °^¹Ë¯,Îÿ¼iÂ�°�ÌiºlÀ£ÌiÎ�± ¼�º^¯�càÁËÌö¸MÁ�è�Ý�Þ�¼�ÃIÃ�Êg¿gÂ�¼�°^¿nÌ�ÁÍÌiÎL°^¹Ë¯]kY¹I¿îÁ˯�À�¯º^¯�Ä�¼i¿gÁI¾I¯�º�°^¹Ë¯�Ìiº^¯�Ä ¼iÁI¾½ôI¼iÂ�°wUöë Å�Æȯ£¸�¿gÊîÊ'À�¯�¯|ÝF¯�ÊnÌ&¸ °^¹I¼�°_À�d�ðI¼�º^¯|º^Ì Ìi°^Àw� Ú ÓIÓ � þ�ÝF¯õÂ�Ì�Ä�Ã?ðË°�¯�¾¿gÁ�öÌ�ÊgÞ�ÁËÌ�Ä¿g¼iÊ�°^¿îÄ�¯�Ä�Ì�¾Ið?ÊnÌ�± ¸�¹Ë¯�Á�°^¹Ë¯£Îÿ¼iÂ�°�Ìiºl¿µ´�¼�°^¿nÌ�ÁÈÌiÎ.± ¿gÀ_ÁËÌi°\càÁËÌ&¸�Á'·¬ðIÁIÊg¯�À^À�Îÿ¼iÂ�°�Ìiºl¿gÁIïÂ�¼iÁÇÝF¯$¾ËÌ�Á˯$¿gÁ,ÃFÌ�ÊnÞàÁËÌ�Ä�¿î¼iÊ�°^¿îÄ�¯iÅ

¶}·R¸ ¹»º¨¼Cº(½N¾N¿Àº�Á ½F¾gÂ<ÁEÃPÄGÅqÆfÇ<ÈuÉ�ŬÄCÊ�Ë��Ìi°^¼�°^¿nÌ�Á¹I¼iÀ3¾Ë¯�ßi¯�ÊnÌiÃF¯�¾�Î0Ìiº�¾Ë¯�¼iÊg¿îÁËïM¸M¿n°^¹.d�ðI¼i¾Ëº ¼�°^¿gÂ9ºl¯�À^¿g¾Ið˯�Àb¿gÁ�Ä�Ìà¾IðIÊg¼�ºbïiº^Ì�ðËÃ?À�ÅbôËÌiº�¯�Êg¯�Ä�¯�Á�°^À¿gÁÉåLæé è ³�ÃIº ¿gÄ�¯&ë ·I¾Ë¯�`?ÁI¯\°^¹I¯õ®¬¯�ïi¯�ÁI¾Iº^¯$À�ÞàÄõÝFÌ�ʬ¼iÀ�Î�Ì�ÊgÊnÌ&¸�À�÷

Ì éIè&#Ië�²ÎÍ �IU ¿nÎ�#�¿îÀ�¼�d ð?¼i¾Ëºl¼�°^¿gÂ_º^¯�Àl¿g¾Ið˯_Ä�Ìà¾IðIÊnÌ\³^_U~Ìi°^¹Ë¯�º^¸M¿gÀ�¯iÅ

Page 15: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Æȯ½Â�¼iÁ�¯�òà°�¯�ÁI¾�°^¹I¿îÀ�¾Ë¯�`?ÁI¿n°^¿gÌ�ÁÍ°�Ì�°^¹Ë¯ÇÂ�¼iÀ�¯�± ² ³�µ�è ³�»^µC¾I¿gÀ�°^¿gÁ?Â�°�Ã?ºl¿gÄ�¯�À ë ·¨¼iÁI¾ ¾Ë¯�`?ÁI¯�°^¹I¯Ï�¼iÂ�ÌiÝ?¿¬À^Þ�Ä|ÝFÌ�ʬ¼iÀ�Î�Ì�ÊgÊnÌö¸MÀ�÷ Ðç è&#Ië=² Ì éËè&#Ëë}[ Ì ê@è&#Ëë)Ñ��Ìi°�¯½°^¹I¼�°�¿nΣ# ¿gÀ�¼�À�d�ðI¼�º^¯½¿gÁ�å æçL·Y°^¹Ë¯�ÁX¸¨¯½Ä|ðIÀ^°¹I¼�ßi¯ Ð ç è&#Ië,²Ò�wU�Å L ¹I¿îÀ�¿gÀ�À�Ì�ÝF¯�Â�¼iðIÀ^¯

¿nÎK# ¿gÀ¼ÍÀ�d ð?¼�º^¯½Ä�Ìà¾IðIÊgÌ�±�·L¼iÁI¾�#�� è&#�éà»'#�ê�ë ·�°^¹Ë¯�Á�#�é�Ä�ðIÀ�°ÝF¯�¼ÍÀ�d�ðI¼�º^¯½Ä�Ì�¾Ið?ÊnÌ,³�¼iÁ?¾#�ê�Ä|ð?À�°|ÝF¯,¼½À�d ðI¼�ºl¯�Ä�Ìà¾IðIÊg̽µÍèÿ¼iÁ?¾B¹I¯�ÁIÂ�¯�#Í¿gÀ|¼CÀ�d�ðI¼�º^¯,Ä�Ìà¾IðIÊnÌ�ݶÌi°^¹½³ ¼iÁI¾Bµ�ë Å+�_Á�°^¹I¯Ìi°^¹Ë¯�ºM¹I¼iÁ?¾'·à°^¹Ë¯�º^¯$¼�º^¯�¯�Ên¯�Ä�¯�Á�°^À;#�¸�¿g°^¹ Ð ç è&#Ië=²%�IU_¸M¹I¿gÂl¹É¼�ºl¯�Ó � þ¬d�ðI¼i¾Ëºl¼�°^¿gÂ\º^¯�Àl¿g¾Ið˯�À�xà°^¹Ë¯�À^¯¼�º^¯ÃIºl¯�Â�¿gÀ�¯�ÊnÞ�°^¹ËÌ�À�¯�¯�Ên¯�Ä�¯�Á °^Àf#§� è&#�éà»'#�ê�ë\¸�¹Ë¯�ºl¯Á˯�¿n°^¹Ë¯�º.#�éÇÁËÌiº.#�ê�¼�ºl¯�À�d�ðI¼�º^¯�À�èÿ¼iÁI¾�¹Ë¯�Á?Â�¯Ì é è&# é ë=² Ì ê è&# ê ë�²g^fUöë Å|;°�Â�¼iÁXÝF¯ÇÃ?º^Ìößi¯�¾X°^¹I¼�°�¯�ò˼iÂ�°^ÊnÞ ¹I¼iÊgÎM°^¹Ë¯½¯�Ên¯�Ä�¯�Á °^À�¿îÁ åLæç ¹I¼�ßi¯bÏ�¼iÂ�ÌiÝ?¿MÀ�ÞàÄõÝFÌ�Ê?�IU�·L¼iÁ?¾¯�òà¼iÂ�°^ÊnÞ�¹?¼iÊnÎ.¹?¼�ßi¯4Ï�¼iÂ�ÌiÝ?¿¨À�Þ�Ä|ÝFÌ�Ê»^_U�Å,ôIð˺^°^¹I¯�ºlÄ�Ìiº^¯i·bÌiÎY°^¹ËÌ�À�¯¯�Êg¯�Ä�¯�Á�°^À|¸�¿n°^¹+Ï�¼iÂ�ÌiÝ?¿¨À�Þ�Ä|ÝFÌ�Ê�IU�·I¯�ò˼iÂ�°^ÊnÞɹI¼iÊnÎ�ÌiÎ3°^¹ËÌ�À^¯õ¼�º^¯fd�ðI¼i¾Ëºl¼�°^¿îÂMºl¯�À^¿g¾Ið˯�À�ÅMMÁ�¿gÄ�ÃFÌiº^°^¼iÁ °�ºl¯�À^ðIÊn°�¿îÀ¬°^¹I¼�°b°^¹Ë¯qÏ�¼iÂ�ÌiÝ?¿�À�Þ�Ä|ÝFÌ�Ê�ÌiΨ#$Ä�Ì�¾Ið?ÊnÌL±`Â�¼iÁ�ݶ¯¨Â�Ì�Ä�Ã?ðË°�¯�¾�¯�³Â�¿n¯�Á °^ÊnÞèÿ¿ Å ¯iÅn·?¿gÁ�ÃFÌ�ÊnÞàÁËÌ�Ä�¿g¼iÊ�°^¿gÄ�¯&ë_Ô � Ô�Ó�Ò � þ�ÐàÔ ��Ú ��þ � × ÒÔÓ Ú þÿÒ � Ó ��� ± Ò0Õ�Ó � þuÕ&Ó � 1�Ó?Å L ¹�ðIÀ�·à¿nÎ�¸.¯$Â�Ì�Ä�Ã?ðI°�¯Ðç è�á'ë�²�^_U|¸¨¯/c�ÁIÌö¸´°^¹I¼�°_áÈÂ�¼iÁIÁËÌi°_ÝF¯|¼5d ðI¼i¾Iºl¼�°^¿gÂ$ºl¯�À^¿g¾Ið˯iÅ\�_Á½°^¹Ë¯|Ìi°^¹I¯�º_¹?¼iÁI¾'·¶ï�¿nßi¯�Á½¼iÁ¯�Ên¯�Ä�¯�Á °.á�Î0ÌiºY¸�¹I¿îÂl¹

Ðç è�á'ë=²%�IU\¼iÁI¾�¸�¿n°^¹IÌ�ðË°¨°^¹I¯MÎÿ¼iÂ�°�Ìiºl¿µ´�¼�°^¿nÌ�Á,ÌiÎb±�·ËÁËÌ�¯�³Â�¿n¯�Á °�¼iÊnïiÌiºl¿n°^¹?Ä

¿gÀNcàÁËÌ&¸�Á�°�Ì£¾I¯�°�¯�ºlÄ�¿gÁ˯M¸�¹Ë¯�°^¹Ë¯�ºYáÇ¿gÀ.¼/d ðI¼i¾Iºl¼�°^¿gÂMº^¯�À^¿g¾IðI¯LÌiºLÁËÌi°�Å�Æȯ_¸�¿gÊgÊ�ðIÀ�¯M°^¹I¿gÀ¨°�Ì�Ý�ðI¿gÊg¾�¼iÁ¯�ÁIÂ�º^Þ�ÃI°^¿nÌ�ÁÇÀ^ ¹Ë¯�Ä�¯õÊî¼�°�¯�º�¿gÁ,°^¹Ë¯õÂ�Ì�ðIºlÀ�¯iÅ

Ö × �£��Ø�Ù ©¢ÚÜÛwn��Ir�p'¢^�M�£¤ZÝ��'�M§ Þ�n�§:ß��=�"à�¡£�=�M� ÚL ¹Ë¯�º^¯�¼iÀ^Ì�Á�¸.¯¿gÁ °�º^Ìà¾IðIÂ�¯�¾�¼iÊgÊ=°^¹I¿gÀõÁ�ðIÄ|ÝF¯�º$°^¹Ë¯�Ìiº^ÞC¸Y¼iÀõ°�̽ÃIº^¯�À�¯�Á °õÀ�Ì�Ä�¯�ÁI¿gÂ�¯Â�Ì�Á?À�°�ºlðIÂ�°^¿nÌ�Á?ÀÌiÎ.èÿÂ�Ì�Ááa�¯�Â�°^ð˺^¯�¾�ë¨Ì�ÁI¯�ó ¸.¼öÞ�ÎÿðIÁIÂ�°^¿nÌ�Á?À¨¼iÁI¾�ÃF¯�ºlÄ�ðË°^¼�°^¿nÌ�ÁIÀ�Å�| Á�Î�¼iÂ�°�·I¼iÊîÊ(càÁËÌö¸MÁ�Ì�ÁI¯�ó ¸.¼öÞ�ÎÿðIÁIÂ�°^¿gÌ�ÁIÀ°^¹I¼�°.¼�ºl¯MðIÀ�¯�¾�¿gÁ�Ã?ºl¼iÂ�°^¿gÂ�¯M¼�º ¿gÀ�¯LÎ�º^Ì�ÄâÁ ðIÄ|ÝF¯�º=°^¹Ë¯�Ìiº^ÞiÅ L ¹Ë¯�º^¯�¼iÀ^Ì�ÁÎ�Ìiº9°^¹I¿îÀ9¿gÀ9°^¹Ë¯MÁI¿îÂ�¯M¼iÊnïi¯�ÝIº ¼i¿gÂÃIº^ÌiÃF¯�º^°^¿n¯�À£°^¹I¼�°�°^¹I¯�À�¯ÉÎ�ð?ÁIÂ�°^¿nÌ�ÁIÀ�¹I¼ößi¯iÅ L ¹I¿gÀ£¿gÀ£¸M¹�Þ͸¨¯ÉÃIºl¯�Î0¯�ºÉÓ � þ|°�ÌÍðIÀ�¯�â�Ä|ð?Ên°^¿nÃ?Êg¿îÂ�¼�°^¿nÌ�Á¨ã¼iÀ_Ì�ð˺�Ì�Á˯�ó ¸Y¼�Þ�Î�ðIÁ?Â�°^¿nÌ�Á�è�¸�¹I¿g ¹�¿îÀMÌ�ÁI¯�ó ¸.¼öÞi·Ñ¼iÀ�¾I¿gÀlÂ�ðIÀ^À�¯�¾�¿gÁ½°^¹Ë¯£ÃIº^¯�ßà¿nÌ�ðIÀ_Ên¯�Â�°^ð˺^¯&ë�Ý?¼iÀ^¯�¾½Ì�Á¹I¼�ºl¾IÁ˯�ÀlÀÑÌiÎ�Î�¼iÂ�°�Ìiº ¿gÁËïË÷�°^¹?¿gÀbÎ�ðIÁ?Â�°^¿nÌ�Á|¿îÀbÁIÌi°3¼\ÃF¯�ºlÄ�ðË°^¼�°^¿nÌ�Á'·�¼iÁ?¾�¾IÌ ¯�À3ÁËÌi°�¹I¼ößi¯�â�ÁI¿gÂ�¯�ã�¼iÊnïi¯�ÝIº ¼i¿gÂÃIº^ÌiÃF¯�º^°^¿n¯�À�ÅL ¹Ë¯?`IºlÀ�°=ÎÿðIÁIÂ�°^¿gÌ�Á�¸.¯M¸�¿îÊgÊË¿gÁ °�º^Ì�¾?ðIÂ�¯�¿gÀ.À�d�ðI¼�ºl¿gÁËïËÅ3ôËÌiº�¼_`Iò�¯�¾�Ä�Ìà¾IðIÊgðIÀ�± ²Í³�µ�·�¸�¹Ë¯�º^¯=³�»^µ¼�º^¯õ¾I¿gÀ^°^¿gÁIÂ�°YÃIºl¿gÄ�¯�À�·Ë¾Ë¯�`?Á˯_°^¹Ë¯$ÎÿðIÁIÂ�°^¿gÌ�Áåä ç ÷�åLæç�æ ª¬« ç Ý Þ4ä ç è�á'ë=²�á z Ä�Ì�¾�±ÍÅçD5�_ð˺L¼i¿îÄ¿gÀ�°�Ì�ÃIºlÌößi¯$°^¹Ë¯õÎ0Ì�ÊgÊgÌö¸�¿gÁIïË÷

¤.¥ ���-�6�P�éè �!�u��Ú ��þ � × Ò�Ó Ü,Ò0Õ�Ð Ú ×�Û 8 þ�ÐàÔ�ÓÈþ�ÐàÔ � �àÓ-�öþÿÒ � Óåä ç Ü�Ò � Ô�Ó Ú����y� Ô�Ò0Õ � Ó¶Ô�Ø�1 Ú�3 ��*&, � þ5Ô�þ�Ð Ú þÒ ���Ú ��þ � ×ÖÒ�Ó Ü�ÒîÕ£Ó � þLÐ Ú ×�Û 8 þ�ÐàÔ�Óbä ç Ò0Õ�Û�Ô���ÒÿÛ�ÔlÛP� 3 ÁËÌi° � ÓFÔ�Øê1 Ú�3 8 Õ Ò�Ó-� Ô�1¨Ô Ù Ô�Ó?þÿÒ � Ó¶ÔlÛw�¶×�Ô � Ò � ��Õ)� 3þ�Ð Ú þ¨Õ�s�� Ú ×�Ô|× ��� þ�Õw� Ú Ó � Ô�Ôtë���Ò Ô�Ó?þ0� 3 � ��Ù �-�Ëþ5Ô^Û�Ò � þ�ÐàÔ ��Ú ��þ � × ÒÔÓ Ú þÿÒ � Ó �'� ± ÒîÕKÕ@Ó � 1�Óy@y�L ¹I¿gÀ�¸�¿gÊîÊMÝF¯È¼�Â�Ì�Ì�Ê�º^¯�À^ðIÊg°Èè�Ì�Á?Â�¯�¸¨¯ÈÃ?º^Ìößi¯�¿g°�Å�Å�Å ë)i`Æȯ�ïi¯�°Ç°^¹Ë¯ÈÝF¯�Á˯�`?°^À,ÌiÎ|°^¹Ë¯ÈÎÿ¼iÂ�°�Ìiºl¿gÁIï¼iÀ^À^ðIÄ�Ã?°^¿nÌ�Á�èÿÁI¼iÄ�¯�ÊgÞi· °^¹I¼�°¨Î�¼iÂ�°�Ìiº ¿gÁËï|¿îÀ=Ì�Á˯MÌiÎ'°^¹Ë¯�ÃIºlÌiÝ?Ên¯�Ä�À=Ä�Ì�À�°9¸�¿î¾Ë¯�ÊnÞ ó ݶ¯�Êî¿n¯�ßi¯�¾�°�Ì|ÝF¯�¹I¼�º ¾�ëÝ?ðË°L¼iÊgÀ�Ì�ïi¯�°M°^¹Ë¯�ÝF¯�Á˯�`I°^À�ÌiÎb°^¹Ë¯]â�Á?¿gÂ�¯�ã�¼iÊnïi¯�ÝIº ¼i¿gÂ�À�°�ºlðIÂ�°^ðIº^¯�ÌiÎÑÌ�ðIºLÎ�ð?ÁIÂ�°^¿nÌ�Á'ÅW ¿gÁIÂ�¯|°^¹I¿gÀM¿gÀ�Ì�ðIº?`Iº À�°LÃIº^Ì�ÌiÎ�ÌiÎ=°^¹I¿îÀ�À�Ìiº^°�·¶¸.¯|¸�¿îÊgÊ'ï�¿nßi¯£À�Ì�Ä�¯£Ä�Ìiº^¯�¾Ë¯�°^¼i¿gÊb¼�ݶÌ�ðI°�¸�¹I¼�°\¿n°M¿îÀ°^¹I¼�°õ¸.¯¸�¿îÊgÊ3¼iÂ�°^ðI¼iÊgÊnÞCÃIº^Ì&ßi¯iÅ]�_ð˺õ¼i¿gÄ ¿gÀ�°�Ì�À^¹ËÌö¸í°^¹I¼�°|¿gΨ¸.¯�¹I¼ößi¯�¼iÁ�¯�³Â�¿n¯�Á °|¼iÊnïiÌiºl¿g°^¹IÄíì°^¹I¼�°LÂ�¼iÁÉ¿gÁ ßi¯�º^°ä ç è�°^¹I¼�°�¿gÀ�·ËÂ�¼iÁÉÂ�Ì�Ä�Ã?ðË°�¯�À�d�ðI¼�º^¯\º^Ì�Ìi°^À�¿gÁ+ª¬« ç ë ·Ë°^¹I¯�Á�¸.¯�Â�¼iÁ,Ý?ðI¿gÊî¾¼iÁËÌi°^¹Ë¯�ºî=ï!ð�ñ�ò9ó.ô�õ=ö÷yò�ø=ùáöêõ=ö'úf÷�ñ\ø=ùáöFðçô)ûêøuø=ùáô�øuô»üáý�ö'úfþoÿ����!ù�ô�õRú��GÿAõ=ö'ú��£ÿ��ø=ò� ����ñ�ò)ó.û�ô �/ûêò����tÿ ú�öêõuø=ùáöFð2ó��6ûêø=ÿ ò��������=þ���������� �¬ò�úqþ!�Rþ��#"?ï!ð�ñ�ò9ówô�õ=ö$�áò)øN÷yò�ø=ùáöêõ=ö'ú��jñ�ò)óIû'ô%�Iÿ�&��áò�õ=ö¬ø=ùáÿ��¢ðçò�ò)ø'�6ò)ø=ö»ðçò�õFø=ù6ö)(áó6õ'(yò �tö*�uò)ðEø=ù6ÿ�û,+7ô%�-�*"

Y

Page 16: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

¯�³Â�¿n¯�Á °9¼iÊnïiÌiºl¿n°^¹?Äoì/.à°^¹I¼�°=Â�¼iÁ�Îÿ¼iÂ�°�Ìiº¨Á�ðIÄ|ݶ¯�º À�Å<�MÌi°�¯L°^¹Ë¯�º^¯L¼�ºl¯�°5¸.ÌõÂ�Ì�Ä�ÃFÌ�Á˯�Á °^À�°�Ì$°^¹Ë¯�ÃIºlÌ ÌiÎ ÷¸¨¯_Á˯�¯�¾�°�Ì£Â�Ì�ÁIÀ�°�º ðIÂ�°¨¼iÁ�¼iÊnïiÌiºl¿n°^¹?Ä�ì . °^¹I¼�°9Îÿ¼iÂ�°�ÌiºlÀYÁ�ðIÄõÝF¯�ºlÀMè�ï�¿nßi¯�Á,¼iÁ¼�º^Ý?¿g°�ºl¼�º^Þ�¼iÊnïiÌiºl¿n°^¹?Ä�ì°^¹I¼�°M¿gÁ ßi¯�º^°^ÀKä ç ë�¼iÁ?¾Ç¸.¯|¼iÊgÀ�Ì�Á˯�¯�¾Ç°�Ì�¯�ÁIÀ^ð˺^¯õ°^¹I¼�°�Ì�ð˺\Â�Ì�ÁIÀ�°�ºlð?Â�°^¿nÌ�Á�¿gÀL¯�³Â�¿n¯�Á °�x�°^¹I¼�°\¿gÀ�·I°^¹?¼�°ì . ºlðIÁ?À�¿gÁ,ÃFÌ�ÊnÞ�ÁIÌ�Ä�¿g¼iʶ°^¿gÄ�¯èÿ¼iÀ^ÀlðIÄ�¿gÁË^¹I¼�°?ì ºlð?ÁIÀ�¿gÁ,ÃFÌ�ÊnÞàÁËÌ�Ä�¿g¼iʶ°^¿gÄ�¯&ë ÅM Êg¿n°�°^Êg¯�Ä�Ìiº^¯\Î0Ìiº Ä�¼iÊgÊnÞÇè�¸.¯M¸�¿îÊgÊIÀ�¯�¯\°^¹Ë¯\Î�ðIÊîÊËÃIº^Ì�ÌiάÁ˯�òà°9°^¿gÄ�¯&ë ·Ë¼iÀ^ÀlðIÄ�¯�¸.¯_¹I¼�ßi¯_¼iÁ�¯�³Â�¿n¯�Á °¼iÊnïiÌiºl¿n°^¹IÄ ì À^ð?Âl¹É°^¹I¼�°�÷

0=º�± ±21436587�9�5;:=< >,?�@A?;Bàè�U J ë)x�áC1 å æç x��|²�á z x'#D1 ì�è0�I»^±Cë.÷�# z ²{��²FEHGöètHIë » è�UöëÎ0ÌiºÀ^Ì�Ä�¯�èÿ¼�º^Ý?¿n°�º ¼�º^ÞËë|ÎÿðIÁIÂ�°^¿gÌ�ÁIG&ètHIë Å M Ý�¿n°�¼�ݶÌ�ðI°�°^¹Ë¯ÇÁËÌi°^¼�°^¿nÌ�Á'÷ɺl¯�Â�¼iÊgÊY°^¹I¼�° â%1�ãȺl¯�Î0¯�ºlÀ�°�Ì°^¹Ë¯£Ì�ðË°�Ã?ðË°\ÌiÎ9¼�ºl¼iÁI¾ËÌ�Ä ÃIºlÌ�Â�¯�À^À�·¶¸M¹I¿gÊn¯]âl²\ã�¯�¿g°^¹Ë¯�º�Ä�¯�¼iÁIÀ�è�Ì�ÁC°^¹Ë¯£Ên¯�Î0°_À^¿g¾I¯|ÌiÎ=°^¹I¯£Â�Ì�ÊgÌ�Á�ë�°�̼iÀ^À^¿nï�ÁIÄ�¯�Á�°.Ìiº$è�Ì�Á�°^¹I¯Mºl¿gï�¹�°.À^¿g¾Ë¯MÌiά°^¹Ë¯_Â�Ì�ÊnÌ�Á�ë9°�Ì|¯�d�ðI¼iÊg¿g°5Þ½èÿ¿ Å ¯iÅn·Ë¼�º^¯\°^¹Ë¯�À�¯\°5¸.Ì£°^¹I¿gÁËï�À¨¯�d�ðI¼iÊ!�@ë Å| ÁÉ°^¹I¯õ¼�ÝFÌößi¯|ÁIÌi°^¼�°^¿nÌ�Á'·636587�9�5;:=< >,?�@A?;B�º^¯�Î�¯�ºlÀL°�Ì�À�Ì�Ä�¯õ¼iÊnïiÌiº ¿n°^¹IÄ`¸�¹?¿gÂl¹'·àÌ�Á½¿gÁËÃ?ðË°\U�J ·IÌ�ðË°�Ã?ðI°^À¼5H ó Ý?¿g°MÂ�Ì�Ä�ÃFÌ�À^¿n°�¯�Á ð?ÄõÝF¯�º�¸M¹I¿gÂl¹½¿gÀ�¼�ÃIº^Ìà¾IðIÂ�°�ÌiÎ�°5¸.Ì�¾I¿îÀ�°^¿gÁIÂ�°LÃ?ºl¿gÄ�¯�À�Åõè L Þ�Ã?¿gÂ�¼iÊgÊnÞi·¶¿n°L¸�¿îÊgÊ'ÝF¯°^¹Ë¯�ÃIºlÌ�¾Ið?Â�°�ÌiÎ'°5¸.ÌIH¨ ��&ó Ý?¿g°9ÃIºl¿îÄ�¯�À�·iÝ?ðË°9°^¹Ë¯�¸.Ìiº�cà¿gÁËï�À�ÌiÎ'°^¹Ë¯\¼iÊnïiÌiºl¿n°^¹?Ä ¼�º^¯M¼iÂ�°^ð?¼iÊgÊnÞ�¿nº^º^¯�Êg¯�ß@¼iÁ °¹Ë¯�º^¯iÅ ëCÆȯ�À^°�º^¯�À^À\°^¹I¼�°�°^¹Ë¯�ÁI¼�°^ð˺l¯|ÌiÎJ3 587K9�5;:=< >*?�@A?;B¿gÀ_¿nº^ºl¯�Ên¯�ß@¼iÁ °\°�ÌÉ°^¹Ë¯£ÃIº^Ì�ÌiÎ=ÌiΨÀ�¯�Â�ðIºl¿n°5Þi·¬À^ÌÊn¯�°ML ÀLÎ�Ìiº�ÁËÌ&¸ ¼iÀ^À^ðIÄ�¯�¸¨¯õ¹?¼�ßi¯|ÀlðIÂl¹Ç¼iÁ�¼iÊnïiÌiºl¿n°^¹IÄ ¼iÀL¼�Ý?Êg¼iÂ�c�ÝFÌöò¶Å

Æȯõ¸Y¼iÁ�°�°�ÌÀ^¹ËÌ&¸ ¹ËÌ&¸ °�ÌðIÀ�¯_ì °�ÌÂ�Ì�Á?À�°�ºlðIÂ�°�¼iÁǯ�³Â�¿n¯�Á °�¼iÊnïiÌiºl¿n°^¹?Ä ìJ.FÎ�ÌiºL¸�¹I¿g ¹'÷0=º6± ±21N36587�9�5;:=< >*?�@A?;BËè�U J ë)xöè ³Ñ»^µiëO1 ì . èÿ±Èë¨÷�³�µ$²�±]²PEHG . ètHIë » èt��ë

¸�¹Ë¯�º^¯�GQ.ÿètHIë.¸M¿gÊgʶÝF¯\ºl¯�Êg¼�°�¯�¾Ç¿gÁÉÀ�Ì�Ä�¯�¸.¼öÞÉ°�ÌRGöètHIë ÅN| Á,¸.Ìiºl¾IÀ�÷uì/.F¸�¿gÊgÊ�ÝF¯�ï�¿nßi¯�ÁǼ£ºl¼iÁI¾IÌ�Ä`ß@¼iÊîð˯± Ì�ðI°�Ã?ðË°$Ý�ÞS3 587K9�5;:=< >*?�@A?;B�·}ì . ¸�¿îÊgÊbÌ�ðË°�Ã?ðË°�³Ñ»^µ ·b¼iÁI¾�°^¹I¯�ÃIº^ÌiÝ?¼�Ý�¿gÊg¿n°;Þ�°^¹I¼�°�³¶µ,² ±Sè�¸�¹Ë¯�ºl¯°^¹I¿gÀ�ÃIº^ÌiÝ?¼�Ý�¿gÊg¿n°;Þ�¿gÀ�°^¼�ci¯�Á�Ìößi¯�º�°^¹Ë¯,¯�Á�°^¿nºl¯�¯�òàÃF¯�ºl¿gÄ�¯�Á ° ë|¿gÀ£¼�°�Ên¯�¼iÀ�°TGU. ètHIë ÅBôËÌiº�Ì�ð˺�ÃIºlÌ ÌiÎ ·=¸.¯¸�¿gÊgÊ�¸Y¼iÁ °�¿n°|°�̽ÝF¯�°^¹Ë¯�Â�¼iÀ�¯�°^¹I¼�°£¿nÎJGöètHIëõ¿gÀÓ � þõÁI¯�ï�Êg¿nï�¿nÝ?Êg¯i·¬°^¹Ë¯�ÁIG . ètHIë�¸M¿gÊgÊ�ÁËÌi°�ÝF¯�Á˯�ï�Êg¿gï�¿nÝ?Ên¯¯�¿n°^¹Ë¯�º�Åu¡M¯�ÁIÂ�¯i·Ë¿nÎF°^¹Ë¯�º^¯\¯�òà¿îÀ�°^À¨¼iÁ�¯�³Â�¿n¯�Á °Y¼iÊnïiÌiºl¿n°^¹IÄ�ìäÀ^¼�°^¿gÀ^Î0Þà¿gÁËï,è�Uöë�¸�¿g°^¹VGöètHIë¨ÁËÌi°.Á˯�ï�Êg¿nï�¿gÝ?Ên¯i·°^¹Ë¯�ÁC°^¹Ë¯�º^¯�¯�òË¿gÀ�°^À�¼iÁC¯�³Â�¿g¯�Á�°$¼iÊnïiÌiº ¿n°^¹IÄ ìJ.3À^¼�°^¿gÀ^Î0Þà¿gÁËïÈèt��ë$W Ý?ðË°_°^¹I¿gÀ\ßà¿nÌ�Êg¼�°�¯�À�°^¹I¯�Îÿ¼iÂ�°�Ìiºl¿gÁIï¼iÀ^À^ðIÄ�Ã?°^¿nÌ�ÁEiLè0��Ìi°�¯i÷9¸�¹Ë¯�°^¹Ë¯�º.Ìiº�ÁËÌi°Y°^¹I¿gÀ.¼iÂ�°^ðI¼iÊgÊnÞ�ßà¿nÌ�Êg¼�°�¯�À.°^¹Ë¯\Î�¼iÂ�°�Ìiº ¿gÁËï�¼iÀ^À^ðIÄ�ÃI°^¿nÌ�Á¾I¯�ö¯�Á?¾IÀÌ�Á�Ì�ð˺�¾Ë¯�`?ÁI¿n°^¿gÌ�Á½ÌiÎ�36587�9�5;:=< >,?�@F?;B�Å W ÌǸ�¹I¼�°õ¸¨¯¼�ºl¯�º^¯�¼iÊgÊnÞC¼iÀ^À^ð?Ä�¿gÁËïÇ¿gÀ$°^¹I¼�°õ°^¹Ë¯�ºl¯¿gÀ$À�Ì�Ä�¯¼iÊnïiÌiºl¿n°^¹IÄ23 587K9�5;:=< >*?�@A?;B|Î�ÌiºL¸�¹?¿gÂl¹,Îÿ¼iÂ�°�Ìiºl¿gÁËï�°^¹I¯$Ì�ðË°�Ã?ðË°�ÌiÎO36587�9�5;:=< >*?�@A?;B£¿gÀL¹I¼�º ¾'Å ë

Æȯõ¸�¿îÊgÊFÀ�¯�¯$°^¹Ë¯�ÃIºlÌ ÌiÎ3Á˯�òà°�°^¿gÄ�¯iÅ�Å�Å

X}·R¸ Y Á Z¢ÆR½u¼\[{Á^]/º(¿�É`_�ayÁbaÀÆ=ÄC½MMÀ\¾Ë¯�`?Á˯�¾½¼�ÝFÌößi¯i·¶°^¹Ë¯|¾ËÌ�ļi¿gÁÇÌiÎNäÈ¿îÀL¼iÊgʬÌiÎ=å æç�·�¼iÁ?¾Ç°^¹Ë¯|ºl¼iÁËïi¯õÌiÎNä�¿gÀ|è�ÌiÎ=Â�Ì�ð˺ À�¯&ëIª¬« ç Ŭ| °¸¨Ì�ðIÊî¾ÝF¯_ÁI¿gÂ�¯_¿nÎ'¸¨¯�Â�Ì�ðIÊî¾�À�Ì�Ä�¯�¹ËÌö¸äº^¯�À^°�ºl¿gÂ�°.°^¹Ë¯�¾ËÌ�Ä�¼i¿gÁ�ÌiÎ<ä�°�Ì�ª¬« ç ¼iÁI¾�°^¹Ë¯�º^¯�Ý�ÞÄ�¼�ci¯_佼ö¯�º Ä|ðË°^¼�°^¿gÌ�Á'Å9ñYðË°�¹ËÌ&¸ Â�¼iÁǸ.¯õï�ð?¼�ºl¼iÁ�°�¯�¯|°^¹I¼�°?äC¿gÀLÌ�Á˯�ó °�Ì�ó Ì�Á˯õ¸M¹Ë¯�Áɸ.¯$º^¯�À�°�ºl¿îÂ�°L°^¹Ë¯$¾ËÌ�Ä�¼i¿îÁ¿gÁ�°^¹I¿gÀ¨¸.¼öÞ��dcNd�ðI¿nß�¼iÊn¯�Á °^ÊnÞi·à¹ËÌö¸ Â�¼iÁ,¸.¯_ï�ðI¼�ºl¼iÁ °�¯�¯$°^¹I¼�°YÎ0ÌiºL¼iÁ ÞX#,ã�ª¬« ç ·à¯�òà¼iÂ�°^ÊnÞ�Ì�Á˯_ÌiÎ�°^¹I¯Î0Ì�ð˺MÀ�d ð?¼�º^¯�º^Ì�Ìi°^ÀLÌiÎ�#�¿gÀ�¿n°^À�¯�ÊnÎb¿gÁ�ª¬« ç �

Æȯ�Â�¼iÁ�¾ËÌ�À�Ì�Ý�Þõºl¯�À�°�ºl¿gÂ�°^¿gÁIï_± °�Ì$ÝF¯.¼_Ã?º^Ì�¾?ðIÂ�°3ÌiÎ?°5¸.Ì$ÃIº ¿gÄ�¯�À�³�»^µM¸�¹Ë¯�º^¯b³,²�µ$²�Y$Ä�Ì�¾�OËÅÆȯ|ÁËÌ&¸ ÃIº^Ì&ßi¯$°^¹I¼�°L°^¹Ë¯|¼�ݶÌ&ßi¯|Â�Ì�ÁI¾?¿n°^¿nÌ�ÁɹËÌ�Êg¾?À¨Î�ÌiºMÀlðIÂl¹Ç±ÍÅ

~F�P�+� ù è ý Ô�þ�³ � Ô Ú �¶× Ò Ù Ô�Õ)�¨��Ð�þ�Ð Ú þ¶³�² Y_Ä�Ìà¾wO(�4��ÐàÔ�Ó Ì éËè�^_Uöë.²"^fU *5Ò0�ÿÔ�� 8 ^fU�Ò0Õ$Ó � þ Ús�� Ú Û�× Ú þÿÒt�£×�Ô�Õ Ò Û��?Ô�Ò�Ó�åLæé @y�¦MÀ^¿îÁËï�°^¹I¿gÀ�Ên¯�ÄÄ�¼à·Ë¸¨¯|ÁËÌö¸ ÃIº^Ìößi¯i÷O

Page 17: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

¤.¥ ���-�6�P�fe ý Ô�þY± ² ³�µ+13ÐàÔ�×^Ô\³Ñ»^µ Ú ×^Ô.�¶× Ò Ù ÔÖÕ Ú Ó�Û|³ ²aµ½²�Y$Ä�Ì�¾�O(�{��ÐËÔ�Ó ��� × Ú �!�<# 㪫 ç 8 Ô%g Ú �öþ0� 34� Ó¶Ô �'� þ�ÐàÔõÕ�s�� Ú ×�Ô�× ��� þ�Õ ��� #�Ò0Õ Ú � Õ � Ò0Ó�ª¬« ç ��_�6�E�-� hL¯�Â�¼iÊîÊ=°^¹I¼�°|°^¹Ë¯�Î�Ì�ð˺|À�d�ðI¼�º^¯º^Ì Ìi°^À|Ìiά#ÈÂ�¼iÁ�ÝF¯�ºl¯�ÃIº^¯�À�¯�Á °�¯�¾�¼iÀ,è�á éà»�á¶ê�ë ·Lè�^�á�é »�á¶ê�ë ·è�á éà»�^�á¶ê�ë ·?¼iÁ?¾�è�^�á éà»�^�á¶ê�ë¨Î�Ìiº�À�Ì�Ä�¯�á é�ãÇåLæé ¼iÁI¾Çá�ê\ãÇåLæê Å¢|;Îbá 飿gÀ�¼Xd ðI¼i¾Iºl¼�°^¿gÂ\º^¯�À^¿g¾?ð˯\¿îÁÉåLæé¼iÁI¾�á�ê.¿gÀ=¼\d�ðI¼i¾Ëº ¼�°^¿gÂYº^¯�À^¿g¾?ð˯�¿gÁ�å æê ·�°^¹Ë¯�Á½è�á éà»�á�ê�ë3¿gÀ�¼fd ðI¼i¾Iºl¼�°^¿gÂ.º^¯�À^¿g¾?ð˯�¿gÁ�å æçLÅ�ôIðIº^°^¹Ë¯�ºlÄ�Ìiºl¯i·À^¿gÁIÂ�¯4^_U£¿gÀ�ÁËÌi°�¼4d�ðI¼i¾Ëºl¼�°^¿îÂ|ºl¯�À^¿g¾Ið˯�¿gÁ�¯�¿n°^¹Ë¯�º$åLæé Ìiº$¿gÁCåLæê ·¬¿n°�Ä�ðIÀ�°_ÝF¯£°^¹Ë¯�Â�¼iÀ�¯�°^¹I¼�°�^�á�é�¿îÀÁËÌi°M¼�d�ðI¼i¾Ëº ¼�°^¿gÂ�º^¯�À^¿î¾Ið˯$¿gÁÇå æé ¼iÁI¾�À^¿îÄ�¿gÊgÊg¼�º ÊnÞå^�á�ê_¿gÀ�ÁËÌi°�¼Xd�ðI¼i¾Ëº ¼�°^¿gÂ�º^¯�À^¿î¾Ið˯�¿gÁÇå æê Å W ÌÁËÌ�ÁI¯ÌiÎ�è�^Lá é »�á ê ë ·Ñè�á é »�^�á ê ë ·IÌiº|è�^�á é »�^�á ê ëYÂ�¼iÁÉÝF¯_d�ðI¼i¾Ëºl¼�°^¿îÂ_º^¯�À^¿g¾IðI¯�À�¿gÁÉå æçLÅ

Æȯ�Êg¯�¼�ßi¯õ°^¹I¯|Â�¼iÀ�¯|¸�¹Ë¯�Á�è�á�éà»�á�ê�ëY¿gÀõÓ � þ�¼Xd�ðI¼i¾Ëº ¼�°^¿gÂ_º^¯�À^¿g¾?ð˯$¿gÁÇåLæç ¼iÀ�¼iÁ½¯�òà¯�ºlÂ�¿gÀ�¯$Î�Ìiº�°^¹I¯º^¯�¼i¾Ë¯�º�Å

V

Page 18: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

University of MarylandCMSC456 — Introduction to CryptographyProfessor Jonathan Katz

Lecture 8

1 A One-Way Permutation Equivalent to Factoring

Recall the theorem we wanted to prove from last time:

Theorem 1 (Informally:) The squaring permutation is hard to invert (i.e., one-way) ifand only if factoring is hard.

As a historical note, the squaring permutation is also known as the Rabin permutation afterMichael Rabin, who first noticed (in 1979 — one year after RSA was introduced) that theone-wayness of this function was equivalent to hardness of factoring.

Proof Let’s first show the easy direction: that that squaring permutation1 fN : QRN →QRN cannot be hard to invert if factoring is easy. If factoring is easy, we have the followingalgorithm for inverting fN : on input y ∈ QRN , first find the factors p, q of N . Thencompute the Chinese remaindering representation (yp, yq) of y. Find a square root xp of yp

modulo p and a square root xq of yq modulo q (recall we mentioned that for p prime, it isknown how to efficiently compute square roots in i ∗

p). Then (xp, xq) is a square root of y(verify this!).

We now turn to the more difficult direction. As we mentioned at the end of the lastlecture, we will take any algorithm that can invert fN and use it to factor N . Since this isour first proof of this sort, we will give some more detail about what it is that we will actuallyprove. Our aim is to show that if we have an efficient algorithm A that can invert fN (thatis, can compute square roots in QRN ), then we can build another efficient algorithm A′

that can factor numbers. Note there are two components to the proof: we need to constructan algorithm A′ that factors numbers (given an arbitrary algorithm A that inverts fN ) andwe also need to ensure that our construction is efficient; that is, that A′ runs in polynomialtime (assuming that A runs in polynomial time).

More formally, assume we have an efficient algorithm A such that:

Pr[N ← CompositeGen(1k); y ← QRN ; z = y2;x← A(z,N) : x2 = z] > ε(k), (1)

for some (arbitrary) function ε(k). A bit about the notation: recall that “←” refers tothe output of a random process, while “=” either means (on the left side of the colon) toassignment or (on the right side of the colon) to equality (i.e., are these two things equal?).In the above notation, CompositeGen refers to some algorithm which, on input 1k, outputsa k-bit composite number which is a product of two distinct primes. (Typically, it will be

1Note: we assume here that f is a permutation (and thus N = pq where p = q = 3 mod 4; see the notesfor Lecture 7) for convenience only. In fact, hardness of inverting f is equivalent to hardness of factoringeven when f is not a permutation.

1

Page 19: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

the product of two k/2-bit primes, but the workings of the algorithm are actually irrelevanthere.) We stress that the nature of CompositeGen is irrelevant to the proof of security, solet’s for now just assume we have such an algorithm as a black box.

We want to show how to use A to construct an efficient algorithm A′ for which:

Pr[N ← CompositeGen(1k); (p, q)← A′(N) : pq = N ] > ε′(k), (2)

where ε′(k) will be related in some way to ε(k). In words: A′ will be given a random valueN output by CompositeGen, A′ will output p, q, and the probability that pq = N (wherethis probability is taken over the entire experiment) is at least ε′(k). For our proof, wewill want it to be the case that if ε(k) is not negligible, then ε′(k) will not be negligibleeither. Hence, if there exists an efficient algorithm A satisfying (1) with ε(k) not negligible,then there exists an efficient algorithm A′ satisfying (2) — but this violates the factoringassumption! (Note: whether or not this actually violates the factoring assumption dependson our definition of CompositeGen. So what we are really assuming is that there is somealgorithm CompositeGen for which factoring the output of CompositeGen is hard.)

Before we present the actual algorithm, we first give the following lemma:

Lemma 1 Given two elements x, y ∈ i ∗

N such that x2 = y2 mod N and x 6= ±y, one canefficiently factor N .

The proof of this lemma is quite simple. Given that x2 = y2 mod N , we have:

x2 − y2 = (x− y)(x + y) = 0 mod N,

and furthermore x−y 6= 0 mod N and x+y 6= 0 mod N (why?). Since N divides (x+y)(x−y) but does not divide either of (x + y) or (x − y), it must be the case that gcd(x + y,N)gives a non-trivial divisor of N (and the gcd can be computed efficiently). Since N is theproduct of two primes, we are done.

Our algorithm A′ will work as follows: On input N , choose a random y ∈ i ∗

N andcompute z = y2 mod N . Run A(z,N) to get output x. if x2 = z and x 6= ±y, use Lemma1 to factor N . Otherwise, simply give up.

How can we represent the success probability of our algorithm? Note that, because ofLemma 1, our algorithm succeeds exactly when x2 = z and x 6= ±y. So,

Pr[N ← CompositeGen(1k); (p, q)← A′(N) : pq = N ] =

Pr[N ← CompositeGen(1k); y ← i ∗

N; z = y2 mod N ;x = A(z,N) : x2 = z ∧ x 6= ±y].

Let us represent the entire experiment to the left of the colon (in this last expression) byExpt. Conditioning (using basic probability) gives:

Pr[Expt : x2 = z ∧ x 6= ±y] = Pr[Expt : x2 = z] · Pr[Expt : x 6= ±y|x2 = z].

We now make the following two crucial observations. First, we can rewrite Pr[Expt : x2 = z]as follows:

Pr[Expt : x2 = z]def= Pr[N ← CompositeGen(1k); y ← i ∗

N; z = y2 mod N ;x = A(z,N) : x2 = z]

= Pr[N ← CompositeGen(1k); y ← QRN ; z = y2 mod N ;x = A(z,N) : x2 = z],

2

Page 20: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

where this is true because the distribution on z is exactly the same regardless of whether wefirst pick y from i ∗

N or from QRn (why?), and furthermore y does not appear any furtherin the experiment or in the event itself. Now, by assumption we have

Pr[N ← CompositeGen(1k); y ← QRN ; z = y2 mod N ;x = A(z,N) : x2 = z] > ε(k)

(compare to (1)).The second observation we make is that Pr[Expt : x 6= ±y|x2 = z] = 1/2. Why is this

true? Well, one way to see it is that algorithm A has no idea whatsoever which of the foursquare roots of z were picked by A′. So no matter what A does, as long as it outputs avalue x which is a square root of z, then with probability 1/4 we will have x = y and withprobability 1/4 we will have x = −y. And with probability 1/2 we will have x 6= ±y.

Putting everything together gives:

Pr[N ← CompositeGen(1k); (p, q)← A′(N) : pq = N ] > ε(k)/2.

In particular, if ε(k) is not negligible, then neither is the success probability of our algorithmA′ in factoring N . But this would violate the factoring assumption! So, we conclude thatε(k) must be negligible after all.

I presented the above proof in excruciating detail. As you become more familiar withthese types of proofs, they become much easier to read (and hence much shorter).

3

Page 21: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

j/kmlonqpMr%sQl�t#uVvqw�xzy;r%u�{oyqk}|~ x�� ~����q�K��� k�tUr%v�|m�}�=tQl�v�k�tUv ~ rQu��mtUvq�qr%y;����u��rQvqw�p�sQsUvqr��qv�kmy;tQ�myqkz��y;tU�

���D���)���)� �

  ¡ ¢J£6¤¦¥¨§©¤«ª�¬ ­�®N¯©°D§©±F²6³Q¢!§©´¶µ¸·¹¤�£¶º»°K²b­�²6³Q¢!§©´¼½p$¾$r%y;�D�¿�©v��¿r�|ml�nqpMr sQl�v�k©lok�tUv�k��mÀKÁ pMrFtQ�mpMvqrQuÃÂ�w�vqr�k¿v�¾�ÄAÁ�uK�mrQp�sUp�k�tQlok¿�!tž�v�ÀDvqrQp)¾JlÆ|¿p�{�u�ÇÅ�msUp�|ÂÈ�=v�k�É*p��=tQ�mrQp�|�ĸv�kmp=Ç'¾�y�uCw��mk}�=tQl�v�kmsMÊË6ÌÅË ÍVÎÐϹѹÒ�ÓÕÔ©ÏmÖ�×`Ø�ÙqÚ�Ù�ÛÅÜPÝÞ �¿p¸ßmr%sUt)ÂÈyqkm|à� pMr%�my;�}sAÁ p�sUt*Ç'á8k¿v�¾JkbĶp=â¿yqÀD�}{op�losFtQ�¿p)ã��8ä�� pMr%Àà�¿tQy;tQl�v�k6Ê«ä/s«Á pMw�vqrQpqå�{�pMt�æ2ç¹è�éÁbpKtQ�¿p��}rQv�|}�m�=tJvqw�t#¾OvC|mlosQtQlokm�=t)�mr%lÆÀDp�sMÊ$ã$p��Myq{Æ{PtQ�my;t�ê�ÂÈæëÄ)çíì î$ïð�ì8çñÂòè�ó^ô�Ä=ÂÈé�óHô�Ä Ê ~ �¿v�v�sQpyqkVy;rQÁ�l�tUr%y;rQuàp=â�� v�k¿p�k�tOõ!sQ�¿Á¿É*p��=t�tUvKtQ�¿p$rQp�sQtUr%lo�=tQl�v�kRtQ�my;tOõ!yqkm|Ãê�ÂÈæ½Ä�y;rQp/rQp�{oy;tQl�nqp�{ouà�}r%loÀDpqö�l'Ê÷pqÊ�å���M|AÂ-õqø ê�ÂÈæ½ÄUÄJçùô;Ê�úJv�¾�å tQ�¿pMrQpàp=â8lÆsUtQs�yVûdü½î$ïý�þ ð�ÿ sQ�m� �dtQ�}y;t�õ�ûÃçùô$ÀDv8|Rê�ÂÈæëÄàÂ�¾�p©¾Jlo{o{¶kmvqt�msUpàtQ�mlos!ûClokz|¿pMß�kmlok¿�Tv��mr/wÈ�mkm�=tQl�v�k¶å�Á}�mtJ¾�pK¾JlÆ{o{¶�msUp©l�t/tUvÃ�}rQv�nqp©tQ�mlokm��sJy;Á v��¿t!tQ�¿pKwÈ�mkm�=tQlov�k�Ä Ê�/pMß�k¿p�� ð�� ��� î$ïð� î$ïðHÁ�u� ð�� � Â�� ÄÐç � � ÀDv�|Dæ¹Ê��������� � ð�� ��������� �"!$#&% '(��')�+*-,.*-/��0! î$ïð21 ,3*�'4��'�56�7'�5 �0!8��9:�&;7*<,3*�':,=�>�>;?'(*@!A�0�B')!B�)CD' æFE �G��9:�;-�);IH"*-!�'�5 ���0J"%��-!B��,LK�� �0!B#&% '(��')�)*�,�M-N

Þ v©sUpMp!tQ�mlosMå�¾�p!�mrQv�nqp!tQ�my;t�� ð�� � losÐv�k¿p=Ç'tUv;Ç'v�k¿pTÂÈyqkm|V¾Op!|¿p�k¿vqtUpO� ð�� � Á�uP��w�vqrOÁmr%pMn�l�t#u¿Ä Ê�ä!sQsQ�}ÀDp��Â�� Ä�çQ��Â�RmÄ Ê Þ �¿p�kS� � çTR � ÀDv8|Tæ¹ÊVãJyqlÆsQlok¿��Á vqtQ� sQlo|mp�s�tUvztQ�¿pVûz��l�nqp�s � Â�� � ÄVURç¦Â�R � ÄVUqåPvqr� � U ç R � U Ê�ã$p��Myq{Æ{mw�rQv�Àñy�w pM¾�{�p��=tQ�¿rQp�s�y;�qvàtQ�my;tо�p/�MyqkVrQp�|m�m�=p!ÀDv8|m�m{�v�tQ�¿p/vqr%|¿pMrOvqw6tQ�¿pJ�qrQv��m�lok�tQ�¿p�p=â8� v�k¿p�k�tMʸ��v¿å � � U/ç � � UXWZY$[ ý�þ ðOÿ ç �]\�ç^�SÂÈyqkm| s%loÀTlo{o{Æy;r%{�uDw�vqr_R¿Ä å¿��lon�lokm�`��ç RbÊ

� t!los$�=v�k�ÉUp��=tQ�¿rQp�|ztQ�my;tJtQ�mpKã!�¿ä � pMr%Àà�¿tQy;tQl�v�kdlos)v�k¿p=Ç'¾¸y�uqÊ�ÂÈã��8䨾�yqs/v�k¿p�vqw�tQ�¿p�ßmr%sUt$v�kmp=Ǿ�y�u w��}km�=tQl�v�kms)�}rQvq� v�sUp�|6åmyqkm|dl�tJ�myqsQk]a÷t$Á pMp�k ÁmrQvqáqp�kdsQlokm�=p�tQ�¿p�kcbòÄ � kdÀTy;tQ�¿p�ÀTy;tQlo�Myq{Fk¿vqtQy;tQl�v�k �¾Op�yqs%sQ�mÀDp�tQ�my;tJw vqrJyq{Æ{�d�d�edyq{��qvqr%l�tQ�}ÀTs�f�å}tQ�¿p�w�v�{o{�v�¾/lok¿�Dlos)k¿pM��{olo��l�Á}{�p �

�Ðrhg�ÂÈæzø%õ�Äjilkcm n�oqp-r�Â*ô"sqÄ öA�ti î ïð öARRç � � ÀDv8|Dæ � fàÂÈædø%õqøARmÄÐç^��u(vÂwkXm�njoqp-rÃlos�yqk yq{��qvqr%lotQ�mÀ¦¾J�¿v�sUpD|¿pMtQyqlÆ{os�y;rQpT�mkmlÆÀD� vqrQtQyqk�t!r%l�����t�k¿v�¾�öAyq{o{«¾�pDá�kmv�¾ los!tQ�my;tKl�t�qp�k¿pMr%y;tUp�syx�Ç'Á}l�t$ÀDv8|m�m{o�}s�æ yqkm|zõ�rQp�{oy;tQl�nqp�{ouV�mr loÀDp�tUvÃê�ÂÈæëÄ ÊòÄ

~ yqkK¾OpOrQp�{oy;tUpÐtQ�mpO�my;r%|}k¿p�sQs6vqwmlok�nqpMrQtQlokm�Jã!�8äStUv/w�yq�=tUvqr%lÆk¿�Lzü½p�{o{-å�¾�pO�MyqkàsQy�u�tQ�mpÐw�v�{o{�v�¾/lok¿� �{:|~}.} ��^� HO��,�/��0!0')��,LK��y�6�T���F�(56�-!A;=��'�5~�",&H0�7CD'(*-!B��,LK�����+5~�-!w;X�0N

���h�c�3� ä!sQsQ�mÀTpRwÈyq�=tUvqr%lok¿��los��*p�yqsQu6�8Ê Þ �¿p�k ¾Op��=v�kmsUtUr �m�=tàyqk¹p"�R�Ml�p�k�t©yq{��qvqr%l�tQ�mÀ tUvëlok�nqpMr%tã!�8ä¦yqsTw�v�{o{�v�¾Js � ��l�nqp�k^lok¿�}�mtCÂÈædø%õqøARmÄ©¾�pzß}r%sUtDwÈyq�=tUvqrVæ tUv ß}k}|�è^yqkm|Hé�ÊHú/p=â8tV�=v�ÀD�}�mtUpêOÂÈæ½ÄJç Âòè�ó�ô�Ä=ÂÈé�ó�ô�Ä Ê�¼½pT�Myqkëk¿v�¾ sUv�{�nqp©w�vqr�ûCç õ��c\ÐÀDv8|RêOÂÈæ½Ä Ê&���¿tU�}�¿t��½ç�R~U$ÀDv8|Dæyqs�tQ�¿pDyqkmsU¾�pMr�Ê�ú/vqtUpDtQ�my;tO� � ç Â�R U Ä � ç�R U � ç�R \ ç�RàÀDv8|Dæ¹å¶sUv�tQ�mlos�los�lokm|¿pMp�|dtQ�¿pD�=vqrQrQp��=tyqkmsU¾�pMr�Ê

ô

Page 22: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

��kÃtQ�¿p!vqtQ�¿pMr$�myqkm|6å8l�t¸los�k¿vqt¸á�k¿v�¾JkV¾J�¿pMtQ�¿pMr¸tQ�¿p��=v�k�nqpMr%sUp�los�tUr%�¿p � l-Ê÷pqÊ�å8¾J�¿pMtQ�mpMr�wÈyq�=tUvqr%lokm�Ábp�lÆk¿�V�my;r |dloÀT�}{ol�p�s)tQ�}y;t�ã��8ä los!�my;r%|6Ê � kdwÈyq�=tMå tQ�¿pMrQpDlos/pMn�lÆ|¿p�km�=pàtUvVtQ�¿pD�=v�k�tUr%y;rQu¹ÂÈl'Ê÷pqÊ�å¶ã!�8äÀTl�����tOÁ p!p�yqsUuRpMnqp�k�tQ�¿v��m���RwÈyq�=tUvqr%lok¿�Dlos��my;r%|bÄ Ê Þ ���ms�å�tQ�¿p�ã��8ä\yqsQsQ�mÀT�mtQl�v�kÃlosOyCÂ�� vqtUp�k�tQloyq{o{�umÄ�B')!A*-,LKL�"! yqsQsQ�mÀT�mtQl�v�k�tQ�myqkCwÈyq�=tUvqr%lokm�TÁ p�lok¿�©�my;r%|6ÊË6Ì�� ÍVÎÐÏ�� Û4�-�8Ö�Ï¿ÙqÏ.�JÜ]�¶Ú Ö�Û'ÙqÎO× ��ØOÝ���Ù�Û'ÜPÝ�OpMw�vqrQp�|¿p�sQ�=r%loÁ}lok¿��tQ�¿pTwÈ�mkm�=tQlov�k6åP¾�pRtQy;áqpÃy¹Â�Ámr%lopMw Ä�|¿pMtUv��¿rKtQ�¿rQv��¿��� ÀTvqrQpR�qrQv��m� tQ�¿pMvqrQuqÊ�PpMt� Á p©y�ß�kml�tUpà�qrQv��¿�¶Ê&�¿vqr�yqk�udp�{op�ÀDp�k�ty  ü � åP|¿pMß}kmp?¡¢ �£ [0¤+¥ç§¦" L¨;øw �\�øw 6©;ø"v"v"v«ªRyqkm|½�Myq{o{FtQ�mlÆs'�5~���$%­¬(K7!w*-%h�®*wH � KL�",=�0!A��'4�8;S¬"¯  }ÊDúJvqtUpTtQ�}y;tMåAsQlok}�=p � los�ß}kml�tUpqå6tQ�mpTsUp±°��mp�km�=p& L¨;øw �\]v"v"v¶¾/lo{o{pMnqp�k�tQ�myq{Æ{�uRsUtQy;rQt¸rQpM� p�y;tQlokm�ÃÂÈ�=u8�M{olok¿��Ä Ê � kR��y;rQtQlo�M�m{oy;r�å�sQlok}�=pJ¾�p!�my�nqpI 3² ³´²�ç ô;å8tQ�¿p�sUp±°��mp�km�=p!�Myqk�my�nqp©y;t/ÀDv�sQt©ì � ì}|mlosUtQlÆkm�=t)tUpMr%ÀTs/lok l�tJyqk}| ¾�pK�Myqkd¾$r%l�tUpP¡µ £/綦" 6¨;ø"v"v"v�øw  ² ³´² �c\"ª�Ê·�!w��=v��¿r%sQpsUv�ÀDpI R¾Jlo{Æ{}�=u8�M{�p�ÁbpMw�vqrQp!tQ�mlosMÊ � w�¡ ?£\losÐtQ�mp!p�k�tQl�rQp��qrQv��m� � ¾�p�sQy�uÃtQ�my;t¸  ������K6�0,=�"!w��'(*�!�*wH� Ê � w�yV�qrQv��¿� � �myqs�yR�qp�kmpMr%y;tUvqr�å6¾�pàsQy�uztQ�my;t � ���&C±¯�C±¹º�+C Ê�ú/vqtUpàtQ�}y;t�ÉQ�msUt!Á p��Myq�msUp©yR�qrQv��m�� los$�=u8�M{olo��|mv�p�s$kmvqt$ÀDp�yqk tQ�my;tJpMnqpMrQu�p�{�p�ÀDp�k�tJlok � losJyD�qp�k¿pMr%y;tUvqr�Ê

�¶pMt"a�s!{�v�vqá�y;t!sUv�ÀDpKp=â8yqÀD��{�p�s$lok î$ï»�Ê�¼½p©�my�nqp<¡ ô�£/ç�¦�ô;ø�ô;ø"v"v"v�ª�綦�ôGª�Ê$sUvTtQ�mlosJlos$�mr%pMtUtÅu�mkmlok�tUpMrQp�sUtQlokm� b«ú/p=â8t)¾Op��my�nqp�¡µ¼`£/碦h¼-¨qø>¼L\�ø>¼-©;ø>¼-½�ø"v"v"v¾ª�翦�ô;ø>¼�ø � ø�ô;ø>¼�ø � ø"v"v"v+ª�翦�ô;ø>¼�ø � ª�Ê)�8v¼ losKk¿vqtKyz�qp�k¿pMr%y;tUvqr©vqw$î ï» ÊtÀJv�¾ùy;Á v��¿t<¡ÂÁS£/çæhÁ ¨ ø>Á \ ø>Á © ø"v"v"v¾ª�çæ�ô;ø>Á�ø>¼�ø � ø � ø � ø�ô;ø"v±v"v+ª ç¦�ô;ø>¼�ø>Á�ø � ø � ø � ª�Ê���v¿å­ÁDlÆs)yD�qp�k¿pMr%y;tUvqr!vqwFî$ï» å�yqkm| î$ï» los)�=u8�M{olo�;Ê

�«l â¹y��=u��M{olÆ�R�qrQv��m� � yqkm|S{�pMt�  ÁbpÃyz�qp�kmpMr%y;tUvqrDvqw � ÊS�¿vqrDyqk�u p�{�p�ÀDp�k�t`Ä�ü � ¾�pÃ�Myqk|¿pMß}k¿p�tQ�mp ;-���0C±!A�±'4�&¹¾*8KÅ�-!B�)'�5Å#Æ*wH Ä 9Z��'�5Ç!8�B�4� �8CD'�'(*  CyqsJtQ�¿pKlok�tUpM�qpMr�x Â�¾/l�tQ�ÇÈ<É x�¡ ì � ìqó^ô�Äw vqrà¾J�mlo� �S  s çÊÄ�ÂÈk¿vqtUpVtQ�}y;tKÁ p��Myq�msUpP ëlÆs�yd�qp�k¿pMr%y;tUvqrDsQ�m� ��x yq{�¾�y�u�s �8Ë7���0'�� yqkm|SlosKyq{�¾�y�u�s%6,­�)J±%�� Ä Ê��!p�k¿vqtUpTtQ�}los�Á�uë{�vq�LÌ�Ä çTxbÊ Þ �mlos�los�p�k�tQl�rQp�{�u�yqkmyq{�vq�qv��ms�tUv tQ�¿pT{�vq��y;r%lotQ�mÀTs�uqv�� y;r%pw�yqÀTlÆ{oloy;r�¾JlotQ�Dw r%v�À �Myq{o�M�m{Æ�msMå�p=â¿�=pM�mtÐtQ�}y;tO�¿pMrQp/¾Op!y;rQpJ¾�vqrQá8lok¿��v�nqpMr¸y�ß}kml�tUp)�qr%v��¿�Vyqkm|Vk¿vqtÐtQ�mprQp�yq{os�ÂÈ�¿p�k}�=p�tQ�¿p�kmyqÀTpt�U|mlosQ�=rQpMtUp±��Ä Ê

¼½pD�Myqkdk¿v�¾¨|mpMß}k¿p�v��¿r!v�k¿p=Ç'¾¸y�u wÈ�mkm�=tQlov�k6å}Á}yqsQp�| v�kdtQ�¿pà�}y;r%|mk¿p�sQs$vqw��=v�ÀD�}�¿tQlokm�V|mlos%�=rQpMtUp{�vq��y;r%l�tQ�mÀRsMÊq�PpMtmèKÁ p�yJ�mr%loÀDpqÊ � tFlos¶y$wÈyq�=tPtQ�}y;tAî ïÍ lÆs¶yJ�=u8�M{olo���qr%v��¿�ÃÂÈ�mk}|¿pMrPÀà�m{�tQl��}{Ælo�My;tQl�v�k�Ä Ê]�PpMt àÁ p$yqk�uà�qp�kmpMr%y;tUvqr)lokDî$ïÍ Â�¾�p/|¿pMw�pMrO|¿pMtQyqlÆ{os�tUv�{oy;tUpMr�lokDtQ�¿p/sUp�ÀDp�sUtUpMr�å�Á}�¿t�kmvqtUpJtQ�my;tÐlotÐlos�á8k¿v�¾/k�¿v�¾ tUv �)εC±�+�",­')¹Ï¯ ß}km|�yÃ�qp�k¿pMr%y;tUvqr�vqw�î$ïÍ Ä Ê��/pMß�k¿p�� Í � Ì � î Í �c\ � î$ïÍ yqs�� Í � Ì Â+xmÄJç�  s ÀDv�|!èAÊ� t©losà�=v�k�É*p��=tQ�mrQp�|StQ�my;t©tQ�mlos�lÆsKy�v�k¿p=Ç'¾¸y�u¹w��mk}�=tQl�v�km{ÐlÆk �}y;rQtQlÆ�M�m{oy;r�å�k¿v�yq{��qvqr%l�tQ�mÀ losà�M�¿rQrQp�k�tQ{�uá�k¿v�¾JkCw�vqr$p"�R�Ml�p�k�tQ{�u��=v�ÀD�}�¿tQlÆk¿�T|mlosQ�=r%pMtUp�{�vq��y;r%l�tQ�mÀRsMÊ

Ð Ñ ­)±]Ò ²b¢ÔÓ�¤Ð±�°©£6¤ÖÕS§©±�£ ®�×K² ³Q¢!§¼½p$¾Jlo{o{�k¿vqt�Á p)|¿p�yq{olok¿� �(���8C"� Ø:C$�-¹�¹Ï¯ ¾Jl�tQ�àtQ�¿p¸v�k¿p=Ç'¾¸y�u©w��}km�=tQl�v�kmsF¾�p$�my�nqp$sUpMp�kRsUv�w�y;rJÂ�wÈyq�=tUvqr%lok¿�¿ås8°��}y;r%lok¿�¿å6ã!�¿ä�å6|mlos%�=rQpMtUpD{�vq��y;r%l�tQ�mÀÃÄ)¾J�mp�kë¾OpD|mlos%�M�msQs!sQ�my;rQp�|8Ç'áqpMuë�=rQu��mtUvq�qr%y;�}��uqÊ � k��qp�k¿pMr%yq{'åsQ�my;rQp�|8Ç'áqpMuK�=rQu��}tUv�sUu�sQtUp�ÀTs«�Myqk©Ábp�|¿p�sQl���kmp�|�Á}yqsUp�|àv�k �-!±¬0�)')!w�-!B¯ v�kmp=Ç'¾�y�uKwÈ�mkm�=tQl�v�k}sMå�yqk}|�¾�pO¾/lo{o{k¿vqt�k¿pMp�|½yqk�u�sU� p��Mloyq{F�mr%vq�bpMr%tQl�p�sMÊ�À/v�¾�pMnqpMr�åP¾�pD¾Jlo{Æ{A�msQp©sQ�bp��MlÆyq{A�}rQvq� pMrQtQl�p�s/vqwOtQ�¿p�sQpDv�k¿p=Ç'¾�y�uw��mk}�=tQl�v�kms¸¾J�¿p�k�¾�p�|mlos%�M�msQs��}�¿Á}{Ælo��Ç'áqpMuV�=rQu��mtUv�sUu8sUtUp�ÀTs�Ê��¿vqrJk¿v�¾�å � ¾�yqk�tUp�| tUvRlÆk�tUrQv8|m�m�=p�tQ�¿p�sQpw��mk}�=tQl�v�kms)sUvTuqv��d�myq|CsUv�ÀDpMtQ�}lok¿�T�=v�km�=rQpMtUpKtUvTtQ�mlokmáVy;Á v��¿tMåml�wFtQ�mlos)�¿p�{o�}sMÊ

�¶pMt"a�sJrQp��Myq{o{ tQ�¿p�sUpMtUtQlÆk¿�Dw vqr/sQ�my;rQp�|8Ç'áqpMuÃp�km�=rQu��mtQl�v�k¶Ê�¼½p��my�nqpKä/{olÆ�=p�yqk}|�OvqÁPåm¾J�¿vDsQ�my;r%p�yáqpMu�Ùhx�lÆk�yq|¿n;yqkm�=pqÊ«¼��¿p�kKä/{Ælo�=p�¾¸yqk�tQsAtUv/sUp�k}|�yJÀDp�s%sQy;�qp�ÚñtUv���vqÁ¶å;sQ�¿pÐ�=v�ÀD���¿tUp�s´Û ç Ü Ý s Â�ÚdÄyqkm|HsUp�k}|ms`ÚÕtUv��OvqÁPÊ^¼��¿p�kÞ��vqÁ�rQp��=p�l�nqp�sÃ�Ml��}�¿pMrQtUp=â8t@Û©å)�¿pd�=v�ÀD�}�¿tUp�sPÚ çÆß�Ý s Â(Û�Ä Ê � k�}y;rQtQlo�M�m{Æy;r�å6w vqr�tQ�¿pDv�k¿p=Ç'tQloÀDpR�}yq|½sQ�%�mp�ÀDpqåPp�km�=r%u��mtQlov�kë¾�yqsK|¿v�k¿pDÁ�u��=v�ÀD���¿tQlok¿�ÇÛ çàÚ2á^ÙGxÂ�¾J�¿pMrQpàtQ�mlos!los!yq{�¾�y�u�s�Á�l�t*Ç'¾JlosUp�â�vqr=Ä å yqk}|d|¿p��=r%u��mtQlov�kz¾¸yqs�tQ�¿pàrQpMnqpMr%sUpqÊ�À/pMrQpqåÐì Ùhx¶ì¿ç ì Ú¹ìòÊ!¼½p

¼

Page 23: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

yq{osUvVs%y�¾ tQ�my;tJtQ�mp�v�k¿p=Ç'tQloÀTpK�}yq|z��l�nqp�s/�bpMr%w p��=t!sUp��=rQp��=uqå Á}�¿t!losJlokmp"�R�Ml�p�k�tJsQlokm�=p�¾Op©ÀK�}sUtJsQ�my;r%pyRáqpMuzyqs!{�v�k¿�ÃyqsJtQ�mpàÀDp�s%sQy;�qpqÊ��¸�¿tMåb¾Opà�Myqkmk¿vqt�|¿vRÁ pMtUtUpMr!tQ�myqkdtQ�¿pKv�kmp=Ç'tQloÀDp���yq|dl�w«¾�p�¾¸yqk�t�bpMr%w p��=tJsQp��=rQp��=uqÊ

¼½p/lok�tUrQv8|m�m�=p�|Dy�r%p�{oy�â�p�|V|¿pMß}kmlotQl�v�kàvqw sQp��M�¿r%l�t#uàlokDyqkRp0âbvqr%t�tUv��qpMtÐy�ÀDvqrQp$p"�R�Mlop�k�tÐs%�%�¿p�ÀDpqÊ� k½��y;rQtQlo�M�m{oy;r�å6¾OpRkmv�¾ ¾¸yqk�t�tQ�¿pTw�v�{o{�v�¾/lok¿� � tQ�my;t�w�vqr�yqk�uãd�d­e¹yq{o�qvqr%l�tQ�mÀäfùyqkm|½w�vqr�yqk�u�t#¾OvÀDp�sQsQy;�qp�s�Ú \ øAÚ © tQ�¿p�w v�{Æ{�v�¾JlÆk¿�DsQ�¿v��m{o|�Á p�k¿pM��{olo��l�Á}{�p �ååå �ÐrDgæÙhx�ièçTÂ*ô s Ä ö$Û¢iéÜ Ý s Â�Ú \ Ä � f©Â(Û�Ä�ç ôBubó ��rDgæÙGx�iÔçTÂ*ô s Ä ö$Û¢iêÜ~Ý s Â�Ú © Ä � f©Â(Û�ÄÐç ôBu ååå v

 � ¾JrQvqtUp�ÛëiéÜ Ý s Â�Ú \ Ä�Á p��Myq�msUp�p�k}�=rQu��}tQl�v�k ÀTl�����t$k¿v�¾ Á p !w�-,=;7*-#&�¾ìh�8; ÊòÄ��k¿pO¾�y�u�¾�pO�=v��m{o|KtUrQu�tUv�yq�%�mlopMnqp�tQ�}losPÀDvqr%pOp"�R�Mlop�k�tQ{�u�lÆs¶tUv!w v�{o{ov�¾StQ�¿pÐ�}y;r yq|ml���À vqw¿tQ�¿pOv�kmp=Ç

tQloÀDpà�}yq|6å Á}�¿t�lokmsUtUp�yq|dvqwOs%�my;r%lok¿�ÃyRtUr �m{�uCr%yqkm|mv�À»sQtUr%lok¿�?Ùhxbå6sQ�my;r%pàlok}sUtUp�yq|ëyV��sUp��m|¿vqr%yqk}|¿v�ÀsUtUr%lok¿�ÃÂ�¾Op!�my�nqp�k¿vqtÐuqpMt¸|¿pMß}kmp�|T¾J�my;tоOp!ÀDp�yqkRÁ�uD�}sUp��}|¿vqr%yqkm|¿v�À å�yqkm|TtQ�}los�¾Jlo{Æ{8rQp±°��}l�rQpJsUv�ÀTp�My;rQpMw��}{PtQ�¿v��m����t�k¿p=â8t/{�p��=tQ�¿r%p�bòÄ Ê � k �}y;r%tQlo�M�m{oy;r�åbloÀTy;��lokmp�l�w«¾Op©�myq|�y �3�"�0%�;7*�!w�-,=;7*-#íK6�0,=�"!w�7'(*-!� � ¦DÈ8ø�ôGª s �c\ �î¦DÈ8ø�ôGª s w�vqr�¾J�mlÆ�%��tQ�¿pDv��¿tU�}�mt!vqw � �U{�v�vqá�s�r yqkm|¿v�À���lokësUv�ÀDpTsQp�kmsUp©tQ�my;t�¾�p�my�nqp�uqpMt�tUvà|¿pMß�k¿pqÊ Þ �¿p�kRä!{olo�=p/yqkm|@��vqÁ��=v��m{o|VsQ�my;rQp!y�r%yqkm|mv�ÀñsUtUr lok¿��ÙGxDvqw¶{�p�km�qtQ�@x!ó½ô;å�yqk}|p�km�=rQu��mtJyqs)w�v�{o{�v�¾Js � Â*ô�Ä_ÙGx6ï}ç � Â+ÙGxmÄ åFÂ+¼�Ä_Û ç ÚðáñÙGx6ï yqkm| |¿p��=rQu��mt�Â�Á�u�rQpMnqpMr%sQlokm�DtQ�¿p�sUtUpM�}s�Ä �Ú ç�Û á � Â+ÙGxmÄ Ê��FpMr%�my;�}s!l�w�tQ�¿p©v��¿tU�}�mtJvqw � {�v�vqá8s/r yqkm|¿v�À p�k¿v��¿���6å tQ�mlos!�Myqk�Á pàsUp��M�mrQpKuqpMtä/{olÆ�=p�yqkm|t��vqÁdsQ�}y;rQp�v�km{�ux©ó`ô�Á}lotQs¸tUvVsUp�km|Cy@x�Ç'Á}l�t$ÀDp�sQs%y;�qp�b!Â'��vTtQ�mpMu��}y�nqp�sQy�nqp�|¹ô�Á}l�t¸n�s�ÊtQ�¿p�v�k¿p=Ç'tQloÀDp���yq|6Ê��!wA�=v��¿r sUpqå¿tQ�mlos¸los)k¿vqt)y©Á}lo�à|¿p�yq{'å¿Á}�¿t¸{�pMt"a�s$sUpMp�l�wP¾Op��MyqkC|¿vDtQ�}los�ÁbpMw�vqrQp�¾�psUpMp��¿v�¾ p"�R�Ml�p�k�t$¾�p��Myqk �qpMtMÊ�ÊMÊ Ä

��v¿å�v��¿r��qv�yq{bk¿v�¾�los�tUvà|¿pMtUpMr%ÀTlokmp)¾J�¿pMtQ�¿pMrOtQ�mlos�y;�m�}rQv�yq�%�Ãlos�w�p�yqsQl�Á}{opqÊ � kT�}y;r%tQlo�M�m{oy;r�å��MyqkV¾�p�=v�ÀDpD�¿�d¾/l�tQ�zyqkëy;�m�mr%vq�mr%loy;tUpà|¿pMß}kml�tQlov�k vqwÐyV�}sUp��m|mvqr%yqkm|¿v�À �qp�k¿pMr%y;tUvqr�sQ�m� ��tQ�my;t/tQ�mpày;Á v�nqpsQ�%�¿p�ÀTpz¾�v��m{o|`Á p sUp��M�mrQphz ä!km|`sUp��=v�k}|m{�uqå)�MyqkH¾�pztQ�mp�kH�=v�kmsUtUr �m�=tRy � ¾J�mlÆ�%�`sQy;tQlÆsUßmp�sTv��mr|¿pMß}kml�tQlov�k3zü½p�¾/lo{o{ sUpMp�lok�tQ�mp�k¿p=â�tJw pM¾ �M{ÆyqsQsUp�s$�¿v�¾ tQ�mlÆs)�Myqk Á p�|¿v�k¿pqÊ

Á

Page 24: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ò�ó�ô«õ�ö"÷>ø8ô¾ùûú@ü�ýZþÇÿ-÷>ú��«ÿ�ó��� þ�� ������ �� óÅùA÷>ü������0ù8ô¾ü7ó?ùAü � ÷8ú���ùAü���÷>ÿ����Åú�j÷8ü�ý�ö±ø8øAü�÷���ü7ó�ÿ-ù���ÿ�ó���ÿ-ù �

!#"%$'&)(+*)" ,.-

/ 021)3546187935:;4=<?>?@;A)BDCE7;FHGI7;4KJL ANM9O;@%7;PQ3RC�187� ��÷8ö"õ~ô¾üS��ø��«úT�U�«ÿ�ôWV`öX�µô«óY�U�«ÿ�ø8ø@ù���ÿ-ù<ù�� ö[Z·ÿ�\­ô«ó^]Aø�_���ÿ-÷>ô«ó`�SaFýb�­ó��0ù8ô¾ü7ó#cedgfihj=k lnm j ô«ø<ÿ�=ö"÷EV�� ù8ÿ-ù8ô«ü7ópoq� ö±ó?ô¾ù8øn� üSV�ÿ�ô«óô�ø:÷>ö±øAùA÷>ôr�0ùAöX�?ùAü[l)m jtsvu ��ô«ø ô«ø ÿ��0ù���ÿ��W�¾ú@ó ü�ù)_���ô¾ùAöyùA÷w� ö.x)ÿ�ø�ú�üS�ø8ÿXoTü7óy� üSV`öUo¸ü�÷�z+{�ô«óãù�� ö|��÷8üS�`�yfih} ~E� s Z�ÿ-ù�� ö"÷X�Kc�d�l)m j k l)m j ô«øOÿ��=ö"÷EV�� ù8ÿ-ù8ô«ü7ó��������oq� ö±ó��=������ÿ�ó��?�p�Y�?����V`ü�� � s �6ö"ö�ù�� ö8���5�­ÿ-ùAöX�Pó ü�ùAö±ø¸ý�ü�÷���öX�0ù�� ÷8öt�Oý�ü�÷:ý�� ÷8ù���ö"÷�� ö"ù8ÿ�ôr��ø s

� ����4�@;�%1gBRAi7;�%1g� ��4�7;4vBRA�3�1gBR�Z_öXV`öXV�\Xö"÷_ù���ÿ-ù_üS� ÷q��ü7ÿ���ô«ø ùAü|��ö±ø8ô �7ótÿ`øAöX�U��÷8ö�ö±ó��0÷8ú���ù8ô¾ü7óøw�w� öXV`ö?oq��ôr�E�p\=ö±ÿ-ù8ø_ù�� ö�ü7ó ö¢¡(ù8ôrV�ö�­ÿ��R£3ó­ÿ�V`öX�¾ú��­ù��­ÿ-ùIÿ��r�«ü¤o·øg��ø_ùAü<øAöX�U� ÷8öX�¾ú?ö±ó��0÷>ú¥��ù8V`ö±ø8ø8ÿ���ö±ø8�¾ü7ó`��ö"÷Iù���ÿ�óÇù�� ö�ø��­ÿ-÷8öX��z�ö"ú s8¦ ó ö��÷8ü��Xü7ø8ÿ���o:öp��ÿ��#x�÷8ö"õ6ô«öUo:öX�T\XöX�¾ü¤o � ��øAöX�.ù�� ö?ó ü�ù8ô¾ü7ó.ü�ý·ÿ��­øAöX����ü�÷>ÿ�ó�� üSV§��ö±ó ö"÷>ÿ-ùAü�÷�x���Zi¨ � dù���ô«ø·ô�ø_ÿ�ý���ó��0ù8ô¾ü7ó[©ªd�«¤¬�­X{¯® h k°«¤¬�­X{¯® h oq��ôW�w�±x�ý�ü�÷Ió�ü¤o � ��ÿ�ø_ù�� ö���÷8ü��Xö"÷8ùûú<ù��­ÿ-ù�ô¾ù8ø·üS� ù ��� ù·ô�ø{.\­ô¾ù��¾ü7ó`��ö"÷�ù��­ÿ�óFô¾ù8ø�ô«ó`��� ù s x²�6üÇô¾ý?³ ´N³D�2µ�¶�{��´ù�� ö±óY³·©%xb´ � ³¸�2µ s ��¹ ö?ó ö"öX��ù���öPý�üS�r�¾ü¤o�ô«ó`�ùºo¸ü��­÷8ü��Xö"÷8ù8ô¾ö±øOý�÷8üSV»©»x)ô«ó�ý�ü�÷wVPÿ��r�¾ú��]ý�ü�÷�ó�ü¤o � d�x¼{ � ©Öø���üS���r�½\XöPöU¾¿�"ô¾ö±óÅù��«ú[�0üSV.��� ù8ÿ�\��¾ö�£qxÁÀ �ù�� ö�üS� ù ����ù�ü�ýN©àø�� üS���r�+] �«üLü�z<÷>ÿ�ó�� üSV|a su � ö_ö±ó��0÷8ú¥�­ù8ô¾ü7ó�ø��E� öXV`öio¸ö�ø�������ö±øAùAöX�|o¸ü�÷�z�öX�@ÿ�øjý�üS�r�¾ü¤o�ø sN �«ôW�0ö_ÿ�ó��.øü�\<øw��ÿ-÷8ö_ÿtz�ö"ú.įµ&ü�ý�¾ö±ó`��ù��¿µq¶Å{ s ¹ � ö±ó  �«ôW�0öio¸ÿ�óLù8ø:ùAü��0üSV.VÆ��ó�ôr�"ÿ-ùAö8V`ö±ø8ø8ÿ���ögÇÉÈ[«¤¬�­X{¯®¤ÊOùAüÆÃ:ü�\¸�6ø���öq�0üSV%��� ùAö±øË �YǪÌT©.xÁÄ͵ � ÿ�ó��?øAö±ó���ø Ë ùAü.øü�\ svu ü.��öX�0÷8ú¥�­ùU��øü�\��0üSV.��� ùAö±ønÇÎ� Ë ÌT©%xÁÄ͵ � s�Ï ü�ùAöOù���öÿ�ó�ÿ��¾ü���útùAü?ù�� ö�ü7ó ö¢¡(ù8ôrV`öÆ�­ÿ�� ø��w� öXV�ö�d8©%xÁįµ � ÷8ö±øw���¾ù8øIô«óÇÿT]Aø���ÿ-÷8öX�Åz�ö"ú�a@ü�ýv�«ö±ó`��ù���µ�oq��ôW�w�Çô�øù�� ö±ó½��øAöX�Sÿ�ø�ÿ<ü7ó ö¢¡(ù8ôWV`ö%�­ÿ�� sÆu � ö%� ü��Xö`ô«øIù���ÿ-ù�ô¾ývo¸ö.� öUЭó ö|©Qÿ�����÷8ü���÷>ô�ÿ-ùAöX�¾ú��Xù�� ö±ó[o:ö.�"ÿ�ó��÷8üDõ�ö�øAöX�U� ÷>ô«ù4ú<ý�ü�÷_ù��­ô«ø)�0ü7ó�øAùA÷w���0ù8ô¾ü7ó s¦ õ�ö"÷>ÿ��r�²�_ù�� ö±óR�io¸öSó ö"öX� ùAü+� ü.ùÑo:üñù��­ô«ó`�7øUdyÒ´ô¾÷>ø8ùU�io:ö�o·ôr�W�q��ÿDõ�ö ùAüÓ��÷>ü��=ü7ø8ö ÿ.÷>ô ��ü�÷8üS�­ø� öUЭó�ô¾ù8ô«ü7óÇü�ý�oq��ÿ-ùOô¾ù�V`ö±ÿ�ó�øyý�ü�÷Iù���ö�üS��ù ��� ùyü�ýn©TùAü�] �¾ü6ü�zÇ÷$ÿ�ó�� üSV.a<ÿ�ó��ãù�� ö±ó õ�ö"÷>ô¾ý�útù���ÿ-ù�©ø8ÿ-ù8ô«øAý�ú6ô�ó`��ù��­ô«ø;� öUЭó�ô¾ù8ô«ü7ó±o·ôW�r��ô«ó�� ö"öX��÷8ö±ø����¾ù�ô«ó�ù�� öÿ�\XüDõ�ötö±ó��0÷>ú¥��ù8ô«ü7ó�øw�w� öXV`öp\Xö±ô«ó`�ãøAöX�U� ÷>ö s�6öX�0ü7ó��R�`o¸ö�ó ö"öX�?ùAü.�0ü7ó�øAùA÷E���0ù�ÿ%��Zi¨¶ø8ÿ-ù8ô«ø8ý�ú~ô«ó`��ù���ô«øn� öUЭó­ô¾ù8ô¾ü7óRÔ ¹ öt��ÿ�ó����«ö�ù�� ö'Э÷>øAù¸ü�ýqù�� ö±ø8ö�0ü7ó��0ö"÷>ó�ø·ùAü���ÿDú s

¹ �­ÿ-ùi� üLö±ø ô¾ùnV`ö±ÿ�ó?ý�ü�÷ ù�� öyüS� ù ��� ù¸ü�ýK©¿ùAü½] �¾ü6ü�zP÷>ÿ�ó�� üSV.aÕ ¦ ó ö��0üS���r�@÷8öX_¥�­ô¾÷8ö��~ø8ÿ±ú���ù��­ÿ-ùù�� öÖ�«ÿ�øAù;\­ô¾ù�ü�ý�©%xb´ � ��ÿ±õ�ööX_���ÿ�����÷8ü�\­ÿ�\3ôr�«ô¾ùûú ü�ýg\Xö±ô«ó`�Ŭ ü�÷�{��jü�÷&ù���ÿ-ù`ù���ö<ý�÷>ÿ��0ù8ô¾ü7ó�ü�ýÆ{"ø`ô«ó©%xb´ � ø�� üS���r�[\Xö`÷8üS���S���¾úÓ{¤×�À s Ãn� ù�ù���ö±øAöPÿ-÷8ö�Ø���øAù'�3ÿ-÷8ù8ôr�U���«ÿ-÷��0ü7ó���ô¾ù8ô¾ü7ó­øU�]ÿ�ó��±��ü7óRÙæù�øAö"öXV ùAü�"ÿ���ù�� ÷8ö�ö"õ�ö"÷8úLù��­ô«ó`�Pÿ�\XüS� ùiog��ÿ-ù·ô¾ùqV�ö±ÿ�ó�ø_ùAü¿\=ö�÷>ÿ�ó�� üSV s ¹ ��ÿ-ùqo¸ö?o¸üS���r����ô z�öOùAü@ø8ÿDútô«ø_ù��­ÿ-ù���p�Xü7ø8ø8ô \��¾ö|x�öU¾|�"ô¾ö±óLù � ùAö±øAù'�"ÿ�ó��­ô«øAù8ô«ó`�S�­ô«ø��Ö\Xö"ùºo¸ö"ö±óy©%xb´ � ÿ�ó��ãÿ@÷>ÿ�ó�� üSV õ-ÿ��r� ö s'u � ö�ý�ü�÷wVPÿ��o¸ÿDúÖo:ö?��ü`ù���ô«ø�ô«ø�ÿ�ø·ý�üS�r�«ü¤o·øUdÚÖÛÜNÝßÞáàÍÞbâ�Ýäã#©æåèç¯égêë¢é¢ìUíïî²ðwë�êÆçðw��ñ�ë¼òtîWéqçÆ�­øAöX��� ü�÷>ÿ�ó�� üSVä��ö±ó ö"÷>ÿ-ùAü�÷%å�ó�ôÆõßò'î önî�÷DîáéiëÁø�ìUî²ë¢��÷����ìw��ùgú�û`÷²çðU� ë|ç���êgö¢��í�ç��b��ü�ü�ý�ê�îWé¢÷�îá�¥þSû�îWéEÿîá��þÖç�� þ¥��íïîb÷bÿùÆé�� ÷bÿ�ë�öU���á� ���Nîb�¥þ�îáét��ë²þS� î þ�î²ð¢�rë��

��� �j÷�� ´9«¤¬�­X{¯® Ê��� £��%� ©%xb´ � d���x�� � � {���¶±�j÷�� �� «¤¬�­X{¯® Ê d��[x�� � �ª{�� �����{

Page 25: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

å��E������í¼ê�é��'îb�Å÷bÿ`ë���íwé¢÷në��ïú`ëUíEîbù|ë¢��÷���ënú�î�ì� .ç|í ç���êS��ù"!ÁéXëwëwê$#?�¼ö'� ëU�¥þS÷bÿ.µ�¶T{&%�çïúSú����;©6÷²�¿÷bÿ¥îWééUëwëwê�÷²�¿þ¥ëU÷'�+�¼öÆ� ëU�¥þS÷bÿÖµ(%qç���ê|þSîáñ�ë��Ó÷²�)�+*,�E�±÷bÿ`ëÆéUëwìw����ê�ë��ïú`ëUíEîbù|ë¢��÷-��ë�ú�îÁì� Öç�ìw��ùgú��rëU÷ºëU� �í¼ç���êS��ù.�p�¼ö�� ëU�¥þS÷bÿ'µÖç���êqþSîáñ�ën÷bÿîáé�÷²�/��*)0�ëní�ë21Uû�îbí ëi÷bÿ�ç�÷(�æ����÷Qðwë)çðU� ëq÷²�tê�îáéï÷�îb�¥þSû�îáéwÿ.ðwëX÷3��ëwë¢�÷bÿ�ëïéXët÷3���ÆéUìEëU��ç�íïî��¯é4�Nî�÷bÿ¿ù.��í�ët÷bÿ�ç�����ëºþ���î þSîÁðU� ë�ú�í¼�¥ð�çðUîá��îb÷���*/5���÷ºët÷bÿ�ç�÷6�Kÿ�ëU�7�»��û`÷ ú�û`÷bé+!98$#:��ëìwç��Óñ¤î²ë��I÷bÿîáé.ç�é;�=< é?þ�û�ëïéEé%÷bÿ�ç�÷��+îáé8ú�éUëUû�êS��í ç���êS��ù ç���ê>�Kÿ`ë¢�?� ��û`÷ ú�û`÷bé@!9A(#)��ë�ìwç���ñÍîÁë��÷bÿîWé;ç¯é/�=< éqþ�û�ëïéEét÷bÿ�ç��B�yîáé'í¼ç���êS��ù�*·ò

Ï üÍo6�¾ö"ùUÙ ø&÷8öX�"ÿ��W�:üS��÷;��öUЭó�ô¾ù8ô¾ü7óFü�ý�ÿãøAöX�U� ÷>ö<ö±ó��0÷8ú¥�­ù8ô¾ü7óø��w� öXV�ö s� ó.ö±ó��0÷8ú¥�­ù8ô¾ü7óø��w� öXV�öx3C;­�Dv­�E � ô�ø�ø8öX�U� ÷8ö?ô¾ýÑ�´ý�ü�÷`ÿ��r��ü�ü�ýÞÿ�� ��ü�÷$ô¾ù���V�ø,FÖÿ�ó��ÿ��r��V`ö±ø8ø8ÿ���ö±ø;ÇG¯­ Ç �Zù�� ö?ý�üS�r�¾üÍo·ô«ó`�Çô�øó öU�S�«ô �7ô \��¾ö�d��� ��÷��·Ä͵�HC.x¼{ Ê � £ Ë IDKJ Ê xbÇG � d�F;x Ë � �#¬&��¶y�j÷��·Ä¯µ�LC.x¼{ Ê � £ Ë MDNJ Ê xbÇ � d�F;x Ë � �#¬&� �����

x Ï ü�ùAö�d ù�� ö�ó ü�ù8ÿ-ù8ô«ü7ó�� ö"÷>öjô«ø]ÿq��ô¾ùAù��¾öv��ôPOXö"÷8ö±óLùcý�÷>üSV oq��ÿ-ù � ��ø8öX��ô«ó'�U�«ÿ�ø>ø ô�ót�U�«ÿ�ø8ø � ��ø8öX�'Ç ­ ÇQÿ-ù8Э÷>øAù \���ù�ü�ù�� ö"÷wo·ô«øAö�ô«ø·ù���ö�ø8ÿ�V�ö s � Z_ö±øAù8ÿ-ù8ô«ó��<ù��­ô«ø·ý�ü�÷Iù�� ö&ö±ó��0÷8ú���ù8ô¾ü7ó ø��w� öXV�öÆ� ö±øw�0÷>ô \XöX�ÿ�\=ühõ�ö?x)ô«ó�oq��ôr�E�;C.x¼{ Ê � ø8ôrV%���«úg�­ôW��z~ø ÿ�÷>ÿ�ó���üSVIįµ�ü�ý��¾ö±ó`��ù��ÖxÁµK¶.{ � ÿ�ó���ö±ó��0÷8ú���ù8ô¾ü7ó�ׯ��öX�0÷8ú¥�­ù8ô¾ü7óÿ-÷8ö�ÿ�ø�ø��=öX�"ôrÐ�öX��ÿ�\XüDõ�ö � ���7ô¾õ�ö±ø�ù�� öjý�üS�r�¾ü¤o�ô«ó`�`dRÒ ü�÷´ÿ��r�ü�ü�ý�ÿ�� ��ü�÷>ô«ù���V�øRFÿ�ó���ÿ��W�SV`ö±ø8ø8ÿ���ö±ø¸Ç G ­ Ç ù�� ö�ý�üS�r�¾ü¤o�ô«ó`�`ô«ø�ó öU�S�«ôr�7ô \��¾ö�d

��� ��÷��·Ä͵79«¤¬�­X{¯® Ê��� £��%� ©%xÁÄ͵ � £ Ë �YÇG�ÌS��d�F;x Ë � �#¬&�¶y�j÷��·Ä¯µ� «¤¬�­X{¯® Ê��� £��.� ©%xÁįµ � £ Ë �#Ç ÌS�¿d�F;x Ë � �#¬&� ����� x¼{ �

¹ ô¾ù��tù���ô«ø�ô«ó�VPô«ó��R��o:ö�ó ü¤o �­÷8üDõ�ö�ù�� öOý�üS�W�¾ü¤o·ô�ó`�&ù�� ö"ü�÷8öXVÖdTVU Û¥â(WÍÛNX ãY�áög©6îWé�ç%ó[Z�õä÷bÿ�ëU�[÷bÿ`ë%ë¢��ì�\E�Eú�÷�îÁ���[é¢ìUÿ�ë¢ù|ë,]ë¢é¢ì�\ïî²ðwë2]Öç¥ð���ñ�ëÆîWé�éXë�ìUûN\�ë�*^_WÍâRâ(` ¹ ö¿�­÷8üDõ�ö@ù���ô«øOõ~ô«ÿù�� ö@ý�üS�W�¾ü¤o·ô�ó`�ÖV`ö"ù�� ü�� üS�¾ü���ú�d  ø8ø���V�öÅx�ùAü¤o ÿ-÷w��ÿ��0ü7óLùA÷>ÿ���ôr�0ù8ô¾ü7ó �ù���ÿ-ù�ù�� ö@ö±ó��0÷8ú���ù8ô¾ü7ó�øw�w� öXV`ö¿�7ô¾õ�ö±óFÿ�\Xühõ�ö<ô«ø.����÷OøAöX�U� ÷>ö spu � ö±óSù�� ö"÷8ö@ö�a~ô�øAù8ø�øAüSV�öPÿ�� ��ü�÷>ô¾ù���Vü�ü�ý+F¶ù���ÿ-ùp]¼\�÷8ö±ÿ�z~ø�aPô«ùU£3ô s ö s �=ý�ü�÷8oq��ôW�w�±x¼{ � ô«ø·ó�ü�ù�ó öU�S�«ôr�7ô \��¾ö s ¹ ö�øw� ü¤oI� ü¤o�ùAü��­øAö�ÿ�óLúÇø����E�ÿ�� ��ü�÷>ô¾ù���V ùAüÖ�0ü7ó�øAùA÷w���0ùIÿÖü�ü�ýy��ô«øAù8ô«ó��S��ô«ø�� ö"÷b�Éoq��ôr�E�Å�"ÿ�ó[��ô«ø8ù8ô«ó`�S��ô«øw�?ù�� ö�üS��ù ��� ùIü�ý�©Tý�÷>üSVÿ?÷>ÿ�ó�� üSVäøAùA÷$ô«ó`�po·ô¾ù��ãó ü7ó�¡4ó�öU�S�«ô �7ô \��«ö?��÷8ü�\3ÿ�\­ôr�«ô¾ùûú s�u �­ô«ø8o·ôr�r�Q�0ü7óÅùA÷$ÿ���ôr�0ùOù�� ö`ý)ÿ��0ù�ù���ÿ-ù�©Qô�øOÿ��Zi¨Æ��� ö±ó��0ö�üS� ÷_ü�÷>ôr�7ô«ó�ÿ��Xÿ�ø8ø���V.��ù8ô¾ü7óô«ø�ý)ÿ��«øAö�ÿ�ó��?ù�� ö�ö±ó��0÷8ú���ù8ô¾ü7óøw�w� öXV`ö�V���ø8ùi\Xöyø8öX�U� ÷8ö s ��÷>üLü�ýjü�ý:ù���ô«øyøAü�÷8ùOô«øgz~ó ü¤o·óSÿ�øOÿ�\�ë2]�û�ì¤÷�î����Qd�o:ö&÷8öX�����0ö�ù���ö&ø8öX�U� ÷>ô¾ùûúÇü�ýjù�� ö&ö±ó��0÷>ú¥��ù8ô«ü7óø��w� öXV�ö�ùAü@ù���ÿ-ù�ü�ý�© s ¹ ö�ø8ÿ¤o�ÿ�ó�ü�ù�� ö"÷g�­÷8üLü�ý´ü�ýZù���ô�ø_ý�ü�÷wVæoq��ö±ó�o¸ö�÷8öX�����0öX�tù�� öÆ��ÿ-÷w��ó�ö±ø8ø�ü�ýô«óÅõ�ö"÷>ù8ô«ó`�`ù�� ötZ�ÿ�\­ô«óø�_���ÿ-÷>ô«ó���ý���ó��0ù8ô«ü7ó?ùAü�ù�� ö?��ÿ-÷E��ó ö±ø8ø ü�ý´ý)ÿ��0ùAü�÷>ô«ó`� s

�6ü`�­ÿ�ø8øw��V`ö'o¸ö?��ÿ±õ�ö�ÿ�ótÿ�� ��ü�÷>ô«ù���VcF¶ÿ�ó��ÖV`ö±ø8ø8ÿ���ö±øgÇG¯­ Ç ý�ü�÷qoq��ôr�E�Rd��� ��÷��·Ä¯µ�9«¤¬�­X{¯® Ê��� £��|� ©%xÁįµ � £ Ë �YÇ G ÌS��ddF;x Ë � �#¬&�

¶y�j÷��·Ä¯µ�9«¤¬�­X{¯® Ê��� £��|� ©%xÁįµ � £ Ë �YÇ ÌS�¿deFÆx Ë � �#¬&� ��� �gfNxÁµ � ­oq� ö"÷8ö7fKxÁµ � ô«ø�ó ü�ù`ó öU�S�«ôr�7ô \��¾ö s ¹ ö��0ü7ó�øAùA÷w���0ù&ÿ�óñÿ�� ��ü�÷>ô«ù���Vh� ÿ�ø&ý�üS�r�«ü¤o·øUd7� oq��ôr�E�±��ö"ù8øô«ó`��� ù[�Pÿ�ó�����ÿ�ø¸ùAü;�S� ö±ø8ønoq� ö"ù���ö"÷[�Pô�ø�÷>ÿ�ó�� üSVðü�÷��3øAöX��� ü�÷>ÿ�ó�� üSV Ð�÷$øAù��­ôW��z~ø¸ÿ�÷>ÿ�ó�� üSVe\3ô¾ùi ÈÓ«¤¬�­X{¯® s �äù���ö±ó ø8ö"ù8ø Ë �IÇjKÌk�Çÿ�ó���÷E��ó�ølFÆx Ë � ùAüp��ö"ùOÿ�\­ô¾ù inm stu �­ô«ø inm ÷8öU��÷8ö±øAö±óLù8ø�F?Ù ø�S� ö±ø8ø�ÿ�ø:ùAü?og��ÿ-ù�V`ö±ø8ø>ÿ���öqo ÿ�øjö±ó��0÷8ú���ùAöX� s � ý i � i�m x)ô s ö s �eFY�S� ö±ø8øAöX�¿�0ü�÷8÷8öX�0ù��¾ú � ù���ö±óo���S� ö±ø8øAö±ø

À

Page 26: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

]¼�­øAöX��� ü�÷$ÿ�ó�� üSV.a�xboq�­ôr�w�Öo¸ö?o�ôr�r�5� ö±ó ü�ùAöt\6úÖ��ÿ±õ~ô«ó`���íüS� ù ��� ù.]U{Ua � s � ý iqp� inm x)ô s ö s �rF ��ôr�ó�ü�ù�S� ö±ø8ø)�0ü�÷8÷8öX�0ù��«ú � ù�� ö±ós�6�S� ö±ø8øAö±ø%]w÷>ÿ�ó�� üSV.aÖxboq��ôr�E��o:ö�o·ôr�W��� ö±ó�ü�ùAö8\6ú¿�­ÿ±õ~ô«ó`�q�ÖüS� ù ����ùÆ] ¬�a � s

¹ ö'o¸ÿ�óLù�ùAü;z6ó�ü¤o � üÍo�o:öX�r�$�6��üLö±ø ÿ-ùi��ô«ø8ù8ô«ó`�S��ô«øw��ô«ó`�yüS� ù ����ù8ø:ü�ýN©¿ý�÷>üSVT÷$ÿ�ó�� üSVQøAùA÷>ô«ó��7ø su � öt_���ÿ�óÅù8ô«ù4úpo:ö�ÿ-÷8ö�ô«óLùAö"÷8ö±øAùAöX�Çô�óô«øUd��� �j÷�� ´9«¤¬�­X{¯® Ê��� £��.� ©%xb´ � d��[x�� � �ª{���¶y�j÷�� �� «¤¬�­X{¯® Ê d��[x�� � � {�� ����� xÁÀ �

x²�6ö"ö;tIöUЭó�ô¾ù8ô«ü7ó½{ s ��]ö"ùUÙ øt�«üLü�zÇÿ-ùyö±ÿ��E�Sü�ýjù�� ö±øAö&ùAö"÷wVPøOô«ó���ô¾õ~ôr���­ÿ��r�¾ú s �]ö"ù4u bv�wyx�»��÷�� ´? «¤¬�­X{¯® Ê��� £��Ö�e©%xb´ � d��x�� � � {�� s ���øAù)\6ú��«üLü�z~ô«ó`�`ÿ-ùqog��ÿ-ùb�Î��üLö±øU�`o¸ö?�"ÿ�ó�o_÷>ô¾ùAö�du �#��÷�� ´s9«¤¬�­X{¯® Ê��� £��|� ©.xb´ � £ i «¤¬�­X{¯®S£ i m LFÆxbÇj¸ÌS� � d i m � i �

\=öX�"ÿ��­øAö_�Öü7ó��«úPüS� ù ����ù8ø'{�oq� ö±ósF �S��ö±ø8øAö±ø�ù�� ö'\3ô¾ù i �0ü�÷8÷>öX�0ù��¾ú s � ü7ó��­ô¾ù8ô¾ü7ó�ô«ó���ü7ó?ù�� öOõ-ÿ��r� öyü�ýi �7ô¾õ�ö±øUdu � �j÷�� ´9«¤¬�­X{¯® Ê��� £��|� ©%xb´ � dzMFÆxbÇ G Ì{� � ��¬&�r|X�j÷�� i �#¬&�

} �j÷�� ´B9«¤¬�­X{¯® Ê��� £��|� ©%xb´ � ddF;xbÇ ÌS� � �ª{��r|X�j÷�� i � {�� �ò�ø8ô�ó`�`ù�� öOý)ÿ��0ù·ù���ÿ-ùq��÷�� i �#¬&�5�#��÷�� i � {���� {�~�ÀPÿ�ó��ù���ÿ-ù

��÷�� ´B «¤¬�­X{¯® Ê��� £��.� ©%xb´ � d�F;xbÇ ÌS� � � {��� {)¶y�j÷&� ´s9«¤¬�­X{¯® Ê��� £��|� ©%xb´ � ddF;xbÇ ÌS� � �#¬&�

�7ô¾õ�ö±øUdu

� {�~�À } {�~�À4|�������� ´9«¤¬�­X{¯® Ê��� £��%� ©%xb´ � d��ÆxbÇG�ÌS� � �#¬&�¶����&� ´s9«¤¬�­X{¯® Ê��� £��|� ©%xb´ � dd�;xbÇ ÌS� � �#¬&���q�

�����,���s��������������S�2�K�s�������d�2�����������r���2���e�2�K�������q���������2�� ?¡b�����V���V��aN��¢��2£¤�k¥:f'¦y§�¨©¦��2��¢���£�£[�2�r����2�K�;���r����£��K���_����£��K�4���'�2�K�;��aN���2�����2�¤���B���B�(���2���d�2���������/���-f'¦y§�¨�¨nª$�2�K�>«��2��¢�¢������b���2���(���r��£��¤�¬��­,���6��/�K��������2��¢�®�����¯7���K�/����¢��2�����2�¤����2¢n�K��°)��±�¡b�K�_®����s�������e�b�K���2�;���b�2�����b�/�K����2�K�;���K�(�K�³²7¯��¤��������>´h�����r������µK���n����µK��°Bª��2�K�����2�K���N�¤���¶�����·������aK��¢��2£¤�+�2�K�)�N�¤���¸�¹�����4�/�����?�����2��¢2®N���K¯>���������¢��2�����2�¤����2¢n�K��°)��±º?�,�����»£��e��®B���/�2���V����¢�����µ¼������°½���¾¦y¿�¨_¦��:�K���_ z¨n±-ÀR���luÁQ/ wÃxÄ ����Å ²7ÆÈÇ�ÉNÊ�Ë&Ì�Í)Îe´+¦�²K¨ Ä Ë�ÏñÐ������b���/�����Ñ���2��ªK���;¢����B��aN���2�����-�2�����b���s������°��-���'�K���»�[�V¢�������������¢�����µ>´>Î

uÒQ Ä ���&Å ²)ÆÓÇ�ÉNÊ�Ë&Ì ÍNÔ i ÆÕÇ�ÉNÊ�Ë&Ì Ô i m ÆM�,¦�ÖjØ×{²K¨�Î i m Ä i ÏÃ�Ð������b���/�����Ñ���2�o¦3�����2�4�[�;��°��¤�-�2�K�;µ����2����£��n¨nªK���_�������e�2����£�£¤�©¯�����ÎuÒQ Ä Ë�~�¿ } Ë�~�¿�| � ���&Å ²)ÆÓÇ�ÉNÊ�Ë&Ì Í Îe�,¦�Ö G ×S²�¨ Ä É&Ï(Ù¾����Å ²oÆÓÇ�ÉNÊ�Ë&Ì Í Îd�q¦�Ö+ÚÒ×�²K¨ Ä É&Ï��q�

Û ��µÜ���?���������2�����Y�2�K�+��a��r�2���2�2�¤���Ü���g�r���2���e�2�K��������$�������2�?��£����K Ý¡b�����s���;Þ2�����©�2�K�ß«��2��¢�¢����2����2���r���r��£��¤���N­���l�à�/�����S�����2��¢�®����K¯@á�âNã>ä�å$ã�æçá3è�é�ãqêNë�].¦3�2����¢��©²����q�K���à¢���°)�r£¤������£���������µK��°©¨n±ì

Page 27: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Û ��µ@�[�>®��K���½�2�r�����2�K�>������íÃ�2��°)��r��µ@���2������µ����)�����2�Ñ��¢��7����¢��2��¢���ª/���¾�2�����o�2�K�¼��a��r�2���2�2�¤���@����r���2���e�2�K�������/�r���-�&��£��K�4��aN��¢��2£��sÉNª(����µuÒQ Ä Ë�~�¿� ���K���2���K¯)�������2���2�����K¯)����¯����2�K���l���2��°î¦y¿�¨�¯��¤������Îïïï ����Å ðÆÓÇ�ÉNÊ�Ë&Ì Í�ñ Ú Ô ² Äßò ¦�ð�¨³Î�´�¦�²�¨ Ä Ë�Ï(Ù¾����Å ²oÆÓÇ�ÉNÊ�Ë&Ì Í Î�´+¦�²K¨ Ä Ë�Ï ïïïÄ ó u�Ú�Ù�uÁQ ó

Ä ó Ë�~�¿/¥�f6¦y§�¨�~�¿_ÙôË�~�¿ óÄ ó ¥:f6¦y§�¨�~�¿ óÄ f'¦y§�¨�~�¿��

õ9�s�r�����2��¢���£�����ªN�¤��f6¦y§�¨[�����-�����³����¯�£��¤¯��¤�r£��,¦3�y±ö��±¤ª��»����µB�K���Ní��K��¯�£��¤¯��¤�r£¤�:��µK�&���e�2��¯��;���{«ç��������®�����¯�­�2�K�B����¢��2�����2�¤���{�2¢n�K��°)��¨V�2�K���Sf6¦y§�¨�~�¿����q�K���)�K��¯�£���¯��¤�r£¤�©����µ{�2�K���2�������2�´÷�r���)�K���Ní��K��¯�£���¯��¤�r£¤���µK�&���e�2��¯��4����«ç��������®�����¯�­4�2�K�/�(������µK�������rµK��°¶¯����K�����������4¦3�y±ö��±¤ªNµ������2���K¯������2�r���K¯/�������r�K����� ò �Ñ����°������µK��°7¨n±��³�K�-�2�r���-¢����d���n��µ���¢��2�-�2�K�4�3��¢��-�2����� ò ���³�)��øbùà¦3�K�����4�2�r���b�¤�Á�ú���³�oû�û�ü>��£¤¯������¤�2�r°�2�����¼�2�7���b´B¨n±:ý��7���K�/������¯�������£R���2����°)���2�¤���>°V���2�b���_�b�2���K¯©����µ>���o�2��¢n�¼�¶¢����>��aN���2� Ô �K���r¢���ª�2�K�_����¢��2�����2�¤���B��¢��K��°)�V���b����¢��K����±

þ

Page 28: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ÿ������������� ����������������������� ��� �! #"�$&%(' �# )������*�+, ����-�. )� � ����/� )��0�����/21#�34�����5�6�)���87��-���� �1�����9:�� );

<>=@?BADCFED= G�G

H I�JLKNMPORQ&SUTNVXWPYNZ\[BW]MPZ^W`_aYbOdcfehgjilkma��) � ���no�Rp!�q��6pF �1��� ^�B3Rrqstp!����6���-*u0-1v )�:+,�-��) )��*�+, ����N�6��+,���w/� ����-�v�+�1u�6n@�Dp�1���+�1v��^)�6+*u������ yx`�6�� �y �1u�v�-�u�,z{ ���n@�N/����}| ' �~/2���� ���+*��������2���^p!�N1��6���@��3Rrqs� �1��� ��)� )��� �+�1u�6����� �����u/�*u qx���-�u�:x��� ����{|���|����-�~�wz{x��� q���u/�*u �6�u����� �*u����d���:�>��z{x��� d�-*u )/�*u U�R �1u�6��pR��+���~+,�-��) )��*�+, d�o)�6+*u����6��+,����/� ����-���+�1u�6no�8����pd1���+�1� �1u�8/����� ����6!�1������8�����yx��� �Rx�*u �+�����6��+,���w/� !n@�6����0��6��������6��0� �1�]|��

1������� �1������.0������P�y�� �p!�-*����>x`������6��x`� ) )������:pR��+,�-*����>��n@/��������� �1u�����+���6��+,�������+,�-��) )��*�+, ��6��+,���w/� ����-���+U1u�6n@�6�pd1������p!�B��6���Bn@�����y �1����: �*�) !�N����u0-����x��� |�¡��N���v)�u�u�� �)��6no �1��� &p�1��� :p!���u��6�¢��:��3£rqs¤ �1��� &) )��� �+�1u�6N�� �&���u/�*u Bx���n@������ �1����. �*�) v�������0-���@x��� ��5 �1���6��+,����/� ����-���+�1u�6n@�ypd������ �1u�6��*���d �1���£3£rqs¥�,¦���+, ����o��!�� 4p����*�)�6�����@ �1u�y����) R���6+, �*u���§�U|£���� �1��¨ *u�6� ����-��x]�6+,�-no�6©4+���.pR�8 ���ª��:�N3£rqs« �1��� �� )��� �+�1u�6D�� �!���u/�*u Rxw���v����u0-����x��� �������.*�)�8�� � )�x�*������.�@3RrDs� �1��� d) )��� �+�1u�6y�� �q����/�*u !����6��n@�����§¬

' �N+������-����*�n&x`������]�*u0�0��6) ����-���p!�����/�����/`�-)�6�&�5���^+,�-��) )��*�+, �����0y�y3£rqs�­® �1��� �) )��� �+�1u�6�� �8���u/�*u Dx��� �pR�.x��� �N��1u������a�¯©]°�±�²6�´³§µ�¶·°�±�²6�´³§µ&�����3£rqs� �1��� 8) )��� �+U1u�6B�� �y���u/�*u dxw���-���x��� U�U©��|�­¸�5¹`��º��o�{�@�5¹`�)�» |�­¸�5¹`��º��o�5¹@¼8���:�5pd1u����&¼N���6�u�� )�6d+,�-��+�� )�6���� ����-�2�U|½ |�­¸�5¹}¾�¼!¹]¿���º��@�5¹}¾��a¼q�@�5¹2¿��8�5p�1u�����À ¹a¾�À#ºÁÀ ¹2¿-À-ºl�� » À ¹a¾^¼!¹]¿�À��

��Ãd�� )�� �1��� N���f�6��+�1F+��)��­Ä) )��� �+�1u�6@�� �v���u/2*u :x��� Åp!��x2�� �|Æ� ' ���5��+, �4 �1���)�6+,�-����/�����/`�-������x]�§�����u���6BÇ2È�É!p!����ª]©4�@/��������a���^)�6+*u���� Å�����w�6q���� D0��@ �1u���-*u0-1.x`�6+��*�)�B �1��:���u/�*u ! )������!������-�u0���8�������u�-nÊ�5 �1u�@����) yx��� y���Ëu¦��6�]�D������ �1u�N)�6+*u���� Å�����£��3£rqs¯�u�/`�6����q�-���� �8)��6��x`�6���u0�������u�-n�|�Ì£���6�on@�����q)�u���� 4��^/`�-����x����£ )��0-�����d���~Í�Î�Ï2ÐÒÑ�Ó6Ñ5É^3RrDsÔ�Õ�*�+�1N �1��� £­Ö��q×#ÍØoÈ�ÇuÙ�É�Ú)Û#ÜÐÆÝ����)�6+*u���q��£�B3£rqs����Þ|���|��#����6�� �1u�-*u0-1�����4�:3Rrqsv�-p!�d+���o0-���������@�,¦�/2����+�� �����0������� �1�nÁß�pd1���+U1�àx����6��ª���o­��U| ' ���D�o����+,�:+U1��������6��0�����������no��0-1w D/`��/�*u/�����N )�6) ¨ *u�6) ����-�]�R )���1u�§p� �1���|¡q1u�v�� �1u��8 Åp!�./�����/`�-�����y�u�������6��������­ápd1���+�1�����.3£rqsv|]â4�-*������N��)ª��6�� )������������;�& �1�� �1������./�����/]�-����`�-�� �1u��1u�-no�pR����ª��50������~) �*��u�6�w �D�-�������U|

��&���������w;�� �1u�B���) D/�����/`�-����P�u��p�|

ã&ä�åwæ2ç§å�è éÕê Í6É��ë©�°�±�²6�´³§µ�¶ °�±�²6�´³§µ�ÜUÍ�Û�ì£ívîïÉ5ð�Û�É:Ù�É�Ú�ÍÉ{Ó,ðuÍ�Ù.Ñ5É5Ù.ÑñÇ�Ï2òuÉ�Ü,ÝfÈ�Ç]Í~Ü,Ñ�É:Û�Ç2××#Í�ó4Ç`Íy­¢�5¹P�4º��@�{�@�5¹P�)�,ô�õað�ÍÇ�­bÑ�Ù&Û@ì£ívîÔÉ5ð�Û�É£Ù�É�Ú�ÍÉ{Óð�Í�Ù�Ñ�É5Ù:Ñ5Ç�Ï2òuÉ�ÜÝ�É�ö£È�Ü,Ñ�É5Ù,ô÷ ç§æPæ2ø ¡q1u�B/��������ù�ñ�-������pdD�o� ��������������5����nL�5 �1��� q���-*��1��-*�����0�� �*���6�. )�w�U©��|�úy��*�no�@�5 )�§p!���U�~�o+,�-�w )��������+, ����-�2�R �1��� d­Ê��D���� d�o3£rqsv|

Page 29: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

» |�¡q1����n@�6����D �1��� d �1u����:�,¦u��) �q��û�û�ü~����0������� �1�nýߥ �1��� ��àx����6��ª���o­�þ��Þ|���|��ÿÿÿ 3£��� ¹�� °�±�²6�´³���þ��@º>­¸�5¹`��©#ßv�����4ºl����f34�� ��� °�±�²6�´³�� � ¿ ©-ßN���u�£ºl�� ÿÿÿ º����Þ���U² �à���

�����������Æ�D��q�u�� q�u�0-����0-��x�����|½ |��B*�)�8ßÕ )�@+,�-��) )�U*�+, !�@û�û�ü�����0������� �1�n�ß��� �1��� @�àx����6��ª���@�v|£¡d1���4p�������x`�y�v+,�-�w )��������+�z

����-�}�������+,�N�Á�����3RrDsN|2¡q1�*��2�-*u�d������0-�������P����*�n@/� ����-�~����) )�/¸��nv*�) qx`�:pd���-�u0u�2��������.����+, d­ ��q�@3RrDsN|�

�&�u�§p�0-�����: �1u�&�u� �������|� �-��) )�U*�+, 4����0������� �1�n«ß � �5pd1���+U1&0�� �R���u/�*� ��:�����@nv*�) ��u�6+���u��pd1u� �1u����B��^/2)�6*��u���������u�-n

� �Þ|���|��������-*� )/�*u D���4���a���d�U�����u�-n��!��D�5�-�����§pd©4ß��������D+,�-n@/�*� )�6��@º¥�@���w�D��������*���DßN�����U| ' �^ß�-*u )/�*u �@���5pd1���+�1�no�6��x`�&�����p!�6����:��0-*u�6�8x��~ßÁ �1��� �����y/�)�6*������������u�-n�� �Þ|���|��a�����-*u )/2*u ���£­��5�U�` �1u�6��ß��}0-*u�6�)�6y �1��� ��.��q/2)�6*��u���������u�-n�| ' �4ß��-*u )/�*� ��±��] �1u�6��ß��a0-*u�6�)�6� �1��� ������������u�-n�|

������ :��0-*u�6�:���v�à/���6*��u�����������-no�.+,�������6)/`�-���� )�������-*u )/�*u B���8�o��������0-*��6�B���

�à�������u�-no��+,�������6)/`�-���. )�������-*u )/�*u D����±�|m}� � �����������;�: �1u���*�++,�6�D����ß��P���F�àx����6��ª����u0-���v|

��v���������w )����6) )�6�����. �1u�:�5�-������p����u0u©

ÿÿÿ 3£�� ¹�� °�±�²6�´³���þ �vº��@�5¹P�!©#ß � �����£ºl��2��3£�� �!� °�±�²6�´³��"� ¾ ©#ß � ���w�4º��� ÿÿÿ$#m}� &%£¾�'"(*)ºý3£�� ¹�� °�±�²6�´³ � þ �vº��@�5¹P�!©#ß � �����£ºl��2�����o��� +%ù¿,'"(*)º 3£�� �!� °�±�²6�´³ � � ¾ ©wß � ���w�4ºl��{|��&+�������,z{pq���� )�-%£¾6��*������0v �1��:�u�Ë����� ����-����������0����U�� �1�n®ß��5����d�ñ�-������pd6©

% ¾ º>3£�� ¹�� °�±�²6�´³ � þ �vº��@�5¹P�Uþ���º��@���w��©#ßv�����4ºl���5 �1���.��� )��*u��x`�6+��*�)��ß � �-*u )/�*u �����,¦u��+, ����jpd1��6��ßL�u�w�6��U| Ãd�§p:�d �1����,¦�/`�����n@�6�w ¸�à¹.�°�±�²6�´³ � þ �vº��@�5¹P�Uþ���º��@���w� ������,¦u��+, ����. �1u�:�,¦�/]��U��n@�6�w N�à¹�� °�±�²6�´³ � þ���º>­¸�5¹`� ��þ�0-��������0u©

%£¾!º>34��� ¹�� °�±�²6�´³ � þ��oº>­¸�5¹`�!©#ßN�����4ºl�� #¡q1���D����-���:���^ �1u�B )���noq�����à���D)�op!�:nv*�) qx`�Bno��ª����u0@/�����0����6� /

m}� � ����w��ª.�� �%ù¿´|�ÿy����u0@ �1u�:���Ë����� ����-����������0������� �1�n®ß«0-�����6©%ù¿dº>3£�� �!� °�±�²6�´³ � � ¾ þ��oº��@������©-ßN���u�4º��� #

0�non�||6|) �1���4����u�� ¨ *��� )�Dpd1��� £pR�d����6�@x`�6+��*�)�q �1�����,¦�/����6�����-�@���w�6DÇ]È�É�+,�������6)/`�-�����������6+, ���� )�@�-�u�8���� �1u�8 )���n�D�����à���U|!m}� � D)��:1���pÕ )�@0�� q�����-*����� �1���|�¡q1u�8 )���nï )�@pd1���+�1�pR�:����6�. )����6���� )�1%ù¿B��q3£��� �2�b°�±�²6�´³ �"� ¿ ©-ßv�����4ºl��{| � �-���������! �1������43`����6��+,�Bx`� Åp!��6�� �1��6)�B )���no©

ÿÿÿ % ¿ ��34�� ��� °�±�²6�´³ � � ¿ ©-ßN���u�£ºl�� ÿÿÿº ÿÿÿ 3£�� �!� °�±�²6�´³ � � ¾ þ��oº��@������©-ßN���u�4º���2��34�5� �2� °�±�²6�´³ � � ¿ ©-ßv�����4ºl�� ÿÿÿ #

6!*� DpR�Bª��u��p¥pd1��� � �1���D��$/^¡d1������! �1u�:�*�++,�6�D/�����x���x2������ ��@���ùß«���¸�àx����6��ª����u0-�o�v|£úy���������+,�����q�@3Rrqsv�� �1���d���43`����6��+,�:��D�u�0-����0-��x�����þ�+�����}�� �7��Þ���U|

»

Page 30: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

rq�6+�����}pR�����������w )����6) )�6������ �1u�:���43`����6��+,��À % ¾ �8% ¿ ÀÆ|���1�������©

À9%4¾£�8%ù¿-Àº ÿÿÿ 34�� ¹�� °�±�²6�´³ � þ���º>­¸�5¹`�!©-ßN���u�£ºl��2��3£��� �!� °�±�²6�´³ � � ¾ þ���º��@������©-ßN���u�4º��� ÿÿÿº ÿÿÿ 34�� ¹�� °�±�²6�´³ � þ���º>­¸�5¹`�!©-ßN���u�£ºl��2��3£��� ��� °�±�²6�´³ �"� ¿ ©#ßv�����4ºl��

�¸34�5� �2� °�±�²6�´³ � � ¿ ©-ßv�����4ºl��2�f34��� �!� °�±�²6�´³ �"� ¾ þ��oº¥�@���w�!©#ßN�����4ºl�� ÿÿÿ: ÿÿÿ 34�� ¹�� °�±�²6�´³ � þ���º>­¸�5¹`�!©-ßN���u�£ºl��2��3£��� ��� °�±�²6�´³ �"� ¿ ©#ßv�����4ºl�� ÿÿÿ

� ÿÿÿ 3£��� �2�b°�±�²6�´³��"� ¿ ©-ßv�����4ºl��2�f34�� �;�b°�±�²6�´³�� � ¾ þ��@º¥�@���w��©-ßN�����4ºl�� ÿÿÿº �w�Þ���4�<7��Þ���U²¡q1w*��2ß��d�àx����6��ª���.�Áp��� �1�/�����x���x������� Å�=���Þ���£�>7§�Þ���U| ' �+�����Æ����d���� ��u�0-����0-��x������� �1��6�¢�������+,�;7����Æ�Ñ�Ù��u�0-����0-��x����§�:�u�6�� �1u��N��2�w���Æ�d�?7§���Æ�U| ' �¢�� �1u��@pR���U���£����ß �àx����6��ª����­Äp��� �1¸�u�-��zÅ�u�0-����0-��x����/�����x���x������� Å���� �1u�6��ß � �àx����6��ª������pd�� �1��u�-�uzÅ�u�0-����0-��x2���d/�����x���x������� Å���u�o+,�-�w )��������+, ����-�}|

½

Page 31: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

@�ACBEDGF HJI BLKNMPOGQSRUTVHJMXWETGAZY[ R]\ [_^a`Gb;ced AaK�HJOXYCfZg"K BLOhA�K�O [ H MXiCK�OGjGHJTVilkaMm�H OGQ�F$I I�OGH�nGOhACTVK kCTGAUopTVK�q

rts2u-v�wyx�s zP{

| } ~�����~����C������~&�;�J�����������-������������~��8���?���

��� O2�2OGHJF�g"Oh����F$AaK I,OhA=Ohf�H,H F$I fZWLK+Q�H Oh� WETGI�K_K BE�2F � kCBEg¡kPI kCO � F$Y=kCO � K�O�K TV¢GFpT;m&£�¤¦¥§K kZTVKI�K�H F K gJkCF$I�¨XMPOhACFp¨ZBLK�TGACY�g"OhAZI�K�HJfCg"K©Q�H Oh��BEK©T2m+£©¤¦ª«K kCTVK�I�K�HJF K gJk�F$I�¨¬M=K � O­¨lBLK I ®¯�°²±´³¬µ·¶¡¸�³X¹�ºC»¼�°�¹�½¿¾·»�À�»Á ¥!ÂÄÃ�f�H1g"OhACI K�HJfCg"K BLOhA � TGI1ªÆÅ�Ç�È-ÉÊ¥�ÅË¥2Å�Ç�È�È¡® � kCBÌI-g"OhACI�K�HJfCg"K BEOhAkCTGI�K k�FÍTGACA�O�MÎBEA�jÏiZH OGi�F H KÐMÆK kCTVK=¥�BEI�¨�F$BEA�j8TViZiZWEBLF$YÆK�O<BÌA�iZf�K I�OGQ-YCB4Ñ�F H F$A¬K­WLF$A�jGK kZI ®ÓÒ�DGF$AK k�Ohf�jhkPK kCBEI+BÌI�K�F$gJkZACBEg TGWEWLM2ÔlA�F;Å�I BEAZg"F�¥ � TGI_Y�F ÔZACF$Y�Q�OGH_BEACiZf�K I�BEA�Õ�ÖÎ×$Ø5ÙÚ�È+BEA�iCH¡TGg"K BEg"F � F��­T$M¨�F!jhBLDGF$AÏTPQ�fZACg"K BLOhA8¥ÛK kCTVK�OhACWLM � OGH ¢ÎI�Q²OGH�Ô�ÜÎF$YÄBEA�iZf�K�WLF$A�jGK kCI2Å � F � BEWEW·I F F!K kZTVK�K kZBEI�BEI�K kCFg TGI�F � k�F$A � F;YCBÌI g fCI I,¨ZWEOXgJ¢�g BEiZk�F HJI©WÌTVK�F H�BEA�K k�F;I�F$��F$I�K�F HÈ¡® d K���T�M�TGWEI�O­¨�F1��OGH FpF Ý­g BLF$A¬K�K�Og"OhACI BEY�F H�¥ÞK kCTVK©OGi�F HJTVK�F$I�OhA�Ô�ÜÎF$YÎßÐWLF$A�jGK k�BÌA�iZf�K I_OhACWLMG®

àÏF!g TGAÍ�2OÎYCBLQ�MPK k�F;g"OhACI�K�HJfCg"K BEOhAUTGI�Q²OhWEWEO � I á+Q²OGH�TGAaM�I�K�H¡BEA�j2Ç·âlWLF K�Ç·ã�ä9ä9ä å�Y�F$ACOGK�F1K k�FpÔZHJI�K,æ¨ZBLK ISOGQ�ÇPTGACY­WLF KSÇZç�Y�F$A�OGK�F�K k�F�è*é²ê-¨lBLK�OGQ�Ç�® d Q�ë Ç�ëGÉ�ì�â � F�g TGA�H F"ß � H¡BLK�F�K k�F©TV¨�O�DGF�g"OhACI�K�HJfCg"K BEOhATGI�ªÆÅ�Ç�È,ɦ¥2ÅË¥�Å�Ç�È ã�ä9ä9äîí4ï"ð�ã È�È¡®2ÅË\ÎBEACg"F�ë9¥2Å�Ç�È ë�ÉÞì!ñ�ØVâ � F�kCT�DGF�¥�Å�Ç�È ã�ä9ä9ä9íòï ð�ãNó ɦ¥2Å�Ç�È¡®ôÈ�õ,f�K � Fg TGAUTGWEI�OPg"OhACI BEY�F H�K kCF1TGWLK�F HJACTVK�F;g"OhACI�K�HJfCg"K BEOhA�ªUö*Å�Ç�È+É÷¥2ÅË¥2Å�Ç�ÈJã�ä9ä9ä ï5È·ø©¥2Å�Ç�È�ï ð�ã$®+ù�F H FGâ � FpÔCH¡I�Kg"Oh�2iZf�K�F�ú2É÷¥2Å�Ç�È+TGACY�K kCF$A­HJfZA2K k�F�ÔZHJI�K&ì�¨ZBLK I�OGQ�ú2K kCH Ohf�jhk=¥§T;I�F$g"OhACYPK BE�2FGâX¨Zf�K+WLF$T�DGF�K kCFWETGI�K,¨ZBLK&OGQ�¥2Å�Ç�È+fZACgJkCTGACjGF$Y�® d K,I k�OhfZWEY­¨�F�g WLF$TVH,K kCTVK,K k�F�iCH OXOGQ�OGQ�I�F$g fCHJBLKÐM � F�jhT$DGF;BEA­K kCF�WETGI KWLF$g"K f�H F � OhfCWÌY�TViCiZWLM�k�F H F!TGWEI�O�K�O�I k�O � K kCTVK�ªUö´BÌI�T­m&£©¤ûÅ�I�F F�BLQ¿MGOhfÍg TGAÍjhBLDGF;K k�F;A�F$g"F$IJI TVH M�2OXYZBLÔZg TVK BLOhACI,K�O­K k�FpiCH OXOGQ üôÈ¡®ý »�þ5³¬»�ÿ5¼¿°�¹·º�°�¹�½¿¾·»������Z¹��Cþ��¿°²»�þ��Cþ���� Ál¾¿¹�»  d KSI k�OhfCWÌY1TGWEI�O�¨�F,g WLF$TVH�K kCTVK�K k�F�g"OhACI�K�HJfCg"K BEOhATV¨�ODGF!�­T$M�¨�F1BLK�F H¡TVK�F$Y>Å�jhBLDÎBEA�j=ª>Å�Ç�È&É�¥2ÅË¥�Å �����"¥2Å�Ç�È������LÈ�È�È©K�O­OG¨CK TGBÌAÍTPm+£©¤ K kCTVK�I�K�HJF K gJk�F$IBLK ISBEA�iZfCK·F DGF$A2�2OGH FG®��?g TVH F Q�fZW¬iZH O¬OGQZOGQlI�F$g f�H¡BLKÐM!I k�O � I�K kCTVK+¥ g TGA2¨�F_BLK�F H¡TVK�F$Y2TGAaM���������������� ���A¬fC�;¨�F H�OGQ�K BE�2F$I&TGACY�ª � BÌWEWÎI�K BEWEW�¨�F©T;m+£©¤ Å�BEK+Y�OXF$I A"!9K�F DGF$A=��TV¢GF�I�F$ACI�F©K�O!BLK�F HJTVK�F��2OGH F©K kCTGAK kCBEI âCI BÌACg"FpBEA�K kCTVK�g TGI F1g"Oh�2iZf�K BÌA�j�ª�HJF$#¬fCBEH F$I,F"ÜXi�OhA�F$A¬K BETGW�K BE��FÈ¡®%´F K�!îI�TGg"K fCTGWEWLM�I�K TVK�F1K kZBEI�TGI�T2K kCF OGH F$��Q�OGH©Q�f�K f�H F�H F Q�F H F$ACg"FG®&;¼�³¬Álþ³� ')(+* �,.-0/21435�+68796:79;<�57=6>7 ?@1$7BA>;�1>6C�9796:�+�5��DE7GF>�H����1�F>� 79IG79;�1��J�,K-0/L1M35�+68796C79;<�57687 ?M1�7BA>;E186N�9796O�+����D�7.F����5�P����������������� ���Q���R��D���7BS�T9UH��?@1 * �5?V�R���9�W��I * ��?N�5�P�RXY1868�+?M1MXZT[���������������\���W]6V�_^`1 acbB687 ?M1�7BA�; * �dA`7B��?fe4I.gK1�AV���hA4���E6>7 ?VDPA`7i�j,K-k/);<��l��+�fm:79;��57K687 ?M1�7BA>; * �dA`7B��?>Sna

[ Oh��BEACj�Q�fCWEWÎg BLH¡g WLFGâaA�OGK�F�K kCTVK+K kCBEI�jhBLDGF$I�fZISK k�FRo�fCWLK BÌ��TVK�F$p-BE�2iCHJO�DGF$�2F$A¬KSO�DGF H_K k�F�OhA�F"ßËK BE��FiZTGY�ü d Qq��WÌBEg"F-TGACY�õ&OG¨�IJkCTVH F-T!H F$TGI�OhACTV¨ZWEF�AXfC�;¨�F H&OGQ�¨ZBLK I-Å�F$A�Ohf�jhk�I�O�K kCTVK�F"Ü�kCTGfCI K BLDGF-I�F$TVHJgJkBEIpBEA�Q�F$TGI BL¨lWLF5r·�2OGH F­Q²OGHJ�­TGWEWLMGâ´F$A�OhfCjhk<I�O�K kZTVKks�Å*ìCÈ c � k�F H F­ì8BEI-K k�F­AXfC�;¨�F H-OGQ_¨lBLK IpI kCTVHJF$YTGACYhs�Å �ôÈ�BEI�K k�F�Q�fCACg"K BLOhAÄg TViCK fCHJBEA�j=K k�F2TGY�DGF HJI TVHJM�!îIpTGYCD5TGA¬K TVjGF­BEAJo ¨CH F$TV¢ÎBEA�jdp=K k�F�m&£�¤ c BÌII ��TGWEW�F$A�OhfCjhklÈ+K k�F$A�K k�F M�g TGA�F$I I�F$A¬K BETGWEWEM­F$ACg"H MXiCK�TGA�TVH ¨lBLK�HJTVHJBEWEMaßÐWLOhACj;�2F$I IJTVjGFGâZTGACY�Y�O��!fCgJk´â�;fCg¡k�¨�F K�K�F H©K kCTGA�K k�FpOhA�F"ßËK BE��FpiZTGY�®

õ_F Q²OGH F � F�jGF K;K�OXOUF"Ü�g BLK�F$Y�â¿K k�Ohf�jhk´â¿WLF K�!îI;A�OGK BEg"F�K kZTVK1K k�F H F=BEIpI�K BÌWEW�iZWLF$A¬KÐMÄOGQ � OGH ¢ÄK�OU¨�FY�OhA�FG®ut�BLHJI�K â � F�A�F F$YUK�O=g"OhACI K�HJfCg"K�TPm+£©¤ K kCTVK�I K�H F K gJk�F$I�BLK I�BEA�ilf�K�¨¬M�F DGF$A]OhA�F;¨ZBLK;Å � F;jhBEDGF

Ø

Page 32: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

I fCgJk�T�g"OhACI K�HJfCg"K BLOhA�¨�F$WLO � È¡®�õ,f�K,F DGF$A�OhACg"F � F-Y�O�K kCBÌI â�A�OGK�F�K kCTVK©I�O�Q�TVH � F�OhACWLM­kZT$DGFpI�F$g f�HJFF$ACg"H MXiCK BLOhA � k�F$A Tv68�+�<m5�_1Ä�2F$I I TVjGFÆÅ�TGWL¨�F$BLK­T WLOhA�j8OhA�FÈ�BEI­I F$AaK ®÷Ã�¨CK TGBÌACBEA�j8I�F$g fCHJBLKÐM � kCF$A�;fCWEK BLiZWLF©�2F$IJI TVjGF$I,TVH F�I�F$A¬K�Å � kCBEgJkPg"OGH H F$I�i�OhACYZI�K�O�I�F$g f�HJBLKNM2TVjhTGBEACI�K_gJk�OhI�F$A�ßËiZWETGBEA¬K�F"ÜXK,TVK�K TGg ¢ÎI¡È� BEWEW�H F$#XfCBLH F-�2OGHJF � OGHJ¢l®

w x ~���yC���{z��¿���¡�������?���

��I � F=�2F$AaK BEOhA�F$Y<TV¨�O�DGFGâSTGWLK kCOhf�jhk � kCF OGH F$� Ø�IJk�O � I1K kCTVK;TUm+£©¤�I K�H F K gJkCBÌA�j�¨XM]OhACF­¨ZBLKpBÌIF$A�Ohf�jhk!K�O�g"OhAZI�K�HJfCg"K�T�m&£©¤ÓI K�H F K gJkCBÌA�j�¨XM1��TGA¬Mp¨ZBEK I â5TGAZY1K kXfCI¿BEI·F$ACOhf�jhk!K�O�I�OhWEDGF&K k�F_iCH OG¨lWLF$�OGQ·OhA�F"ßËK BE��F�F$ACg"HJM¬iCK BEOhA�â � F�kCT$DGFpA�OGK_MGF K�I k�O � A=K kCTVK_m&£©¤�I�I�K�H F K g¡kCBEA�j!¨¬M­OhA�F�¨lBLK+F"Ü�BEI�K ü � kZBEIBEI�K k�F-iCHJOG¨ZWLF$� � F1Y�F$TGW � BEK k�A�O � ®

� O!�2OGK BLDVTVK�F�K k�F�g"OhACI�K�HJfZg"K BLOhA�â¬WEF K�!îI�K TV¢GF�TGAPBEACQ²OGHJ�­TGW�Y�F K�Ohf�H&TGAZY­g"OhACI BÌY�F H�K k�F©iZH OGi�F H KÐM�OGQD<�5��?@14X��\A$7B�YF��+���97 �V®_àÏF5!îWEW�g TGWEW�T!Q�fCAZg"K BLOhA�¥¦áÎÕ�ÖÎ×$Ø5Ù Ú�| Õ�ÖÎ×$Ø5Ù Ú D<�5��?@14X��\A$7B�YF��}1;BLQ�â¬Q�OGH�TGAaM­æ5âÎjhBLDGF$A¥2Å�Ç�ÈJã�ä9ä9ä å�Å�Q²OGHPTÄHJTGACY�Oh� Ç�È¡â,BLK2BEI~o�kCTVH¡YPp]K�OÏiCH F$YCBEg"K­¥2Å�Ç�ÈËåNð�ã � BLK k>iCH OG¨ZTV¨lBEWEBLKNM8I BLjhACBLÔlg TGAaK WLM¨�F K�K�F H-K kZTGA>Ø`���X® d A]OGK k�F H � OGHJYCI$â�jhBEDGF$AÄK k�F;ÔZHJI�K�æ;¨ZBLK I�OGQ,¥�Å�Ç�È¡â·BLK�BEI:o�kCTVHJY�p­K�O�iCH F$YZBEg"K�K kCFŲæCñÄØ�ÈÐé�ê-¨ZBLK�OGQ´¥2Å�Ç�È�Å�OhACF�g TGA�K kCBÌA�¢pOGQ�K kCBEISTGI�TpjhTG�2FGâ¬BEA � kCBEgJk�¨lBLK I�OGQ�¥�Å�Ç�ÈSTVHJF�H F DGF$TGWLF$Y�OhACF"ߨ¬M¬ßËOhA�F1TGACYUTVK�I�Oh�2Fpi�OhBEA¬K�MGOhf���TV¢GF!T2jhf�F$I I�TGI©K�O�K k�F;A�F"ÜXK�¨lBLK�K�O�¨�F-H F DGF$TGWLF$YlÈ¡®���OhfUI k�OhfZWEY¨�F�TV¨ZWLF!K�O�K TV¢GF�K k�F2TV¨�ODGF2Y�F ÔZACBLK BEOhAUTGACYÄ��TV¢GF2BLK��2OGH F!Q²OGH¡��TGW*â�BEAÍK�F HJ��I�OGQK�P�P�8TGWLjGOGHJBLK kC�­ITGACY�A�F jhWEBEjhBL¨ZWLFN#¬fZTGAaK BLK BEF$I ®

d K-BEI�A�OGK�kCTVHJYÍK�O�I�F FGâ�TVK�WEF$TGI�K�BÌA�Q²OGH¡��TGWEWLM�I�i�F$TV¢XBÌA�j�âlK kZTVKp¥ÛBEI�T�m+£©¤ÛBLQ�TGACY]OhACWLM�BEQ&¥ BÌIfCA�iCH F$YZBEg"K TV¨ZWLFG®���g"K fZTGWEWLMGâ�I k�O � BEA�j!K kCTVK�T�m+£©¤�BEI_fCA�iCHJF$YCBEg"K TV¨ZWLF�BEI&H F$WETVK BLDGF$WLM­F$TGI�M�rCBEK_HJF$#¬fCBEH F$IT1¨ZBEK+�2OGH F � OGH ¢­K�O!I kCO � K kCTVK&TGA=fCA�iCHJF$YCBEg"K TV¨ZWLF�Q�fCACg"K BEOhA­BEI&BEA�Q�TGg"K_T!m+£©¤!® d A­TGA¬M�F DGF$AaK âÎK kCBÌIA�OGK BLOhA]OGQ+fCA�iZH F$YCBEg"K TV¨ZBÌWEBLKÐM=��BLjhk¬K�iZH Oh�2iCK�fCI�K�OPK�H MUK�O=K BEF!BEAÍK k�F�g"Oh�2iZf�K TVK BEOhACTGW�kZTVHJYCA�F$I I�OGQOhA�F"ß � T�MUi�F HJ�!f�K TVK BLOhACI�BEA�K kCF;Q�OhWEWLO � BEA�j­I OGH K�OGQ�g"OhACI K�HJfCg"K BLOhA�á�¥�Å�Ç�È,É��SÅ�Ç�È·ø�Ç·®©àtkCTVK�BEI©K kCFBEAaK fZBLK BLOhA�k�F HJF��ÏàÏF$WEWËâ�WLF K2ë Ç�ëXÉÞì�® [ F H K TGBEAZWLM8Å�I BEACg"Fk�<BEI�T­i�F HJ�;fCK TVK BLOhAlÈ¡âZMGOhfÄg TGACACOGK�iCH F$YZBEg"KK k�F�è*é�êp¨ZBLKSOGQ�¥�Å�Ç�È�jhBLDGF$A�K k�F�ÔCHJI�K+è��ÄØ�¨ZBLK ISQ�OGH+è��ÓìUÅ���TV¢GF�I f�HJF©MGOhfPI F F � k¬M�È¡®�tCfCH K k�F HJ�2OGHJFGâF DGF$A jhBEDGF$A<K k�FPÔCHJI�K�ìϨZBEK I1Y�OXF$I1A�OGK�TGWEWLO � MGOhf>K�O�o iZH F$YCBEg"KMpUK k�FPHJF$I�K1OGQ�K k�FPOhf�K�iZf�K;ÇqrSK kCBÌII�F F$��I©K�O­Q²OhWEWEO � Q�H Oh�ÊK k�FpYZBLÝ­g fCWLKNM­OGQ�BEA¬DGF H K BEA�j�T2OhACF"ß � T�M�i�F H¡�;f�K TVK BEOhA�®

@�ACQ²OGH K fZACTVK�F$WLMGâaK kZBEI+ACTGBEDGF�TViCiCH OhTGg¡k­BEI � H OhA�j�® [ WLF$TVH¡WLM2¥�Å�Ç�È�É)�SÅ�Ç�Èaø�Ç�BÌI����57ST;m+£©¤ Å�g TGAMGOhf>g"OhAZI�K�HJfCg"K!TGACY<TGACTGWEM¬q F�TGA<F"ÜÎiZWEBÌg BLKpTGWLjGOGHJBLK kZ��K kCTVK�YCBEI�K BÌA�jhfCBEI kCF$I�K kCFPOhf�K�ilf�KpOGQ�¥ûQ²HJOh�HJTGACY�Oh���5È¡®:��ACY<BEA8I�Oh�2FPI�F$ACI�FPK kCBEIpBEIp¨�F$g TGfZI�F­fCA�iZH F$YCBEg"K TV¨ZBÌWEBLKÐMUQ�TGBEWÌI�Q²OGH$â�I T�MGâ¿K kCF�ÔZACTGWS¨lBLK ájhBLDGF$A�K k�F�ÔCHJI�Ki�Gì��>Ø�¨ZBLK I âÎBLK,BEI+F$TGI M­K�O2iCH F$YZBEg"K+K k�F�ÔZAZTGWZ¨ZBEK+¨�F$g TGfCI�F�MGOhfk� fCI�K_K�H M�¨�OGK k�Ö!TGAZYØ-TGACY�I�F F � kCBEg¡k�I TVK BEI�ÔCF$I�oM�]TViCiZWÌBLF$Y­K�O�K k�FpI F$g"OhACY�kCTGWLQ´F$#XfCTGWEI_K k�F�ÔCH¡I�K�kCTGWLQMpή+\XO � FpACF F$Y�K�OY�O­T2WEBLK�K WEF-¨�F K�K�F H�®��g"K fZTGWEWLMGâÎT;WEBEK�K WLF©K k�Ohf�jhk¬K,I k�O � I+K kCTVK,fCA�iZH F$YCBEg"K TV¨ZBÌWEBLKÐM!��T$M­F DGF$APQ�TGBEWlQ²OGH_K k�F�Å*ì©ñ>Ø�ÈÐé²ê1¨lBLK�rK kCTVK©BEI âÎjhBLDGF$A��SÅ�Ç�È�Å�BË®9FG®Lâ�K k�F�ÔCHJI�K�ì=¨ZBLK I&OGQS¥�Å�Ç�È�È_BLK���T�MP¨�F:o F$TGI�M<p�K�O�YCF K�F HJ��BEA�F�K k�F�ÔCH¡I�K&¨lBLKOGQ+Ç·®kt�OGHpF"Ü�TG�2iZWLFGâ�WLF K�� ö ¨�F2T=OhA�F"ß � T�MÍi�F HJ�!f�K TVK BLOhAÄTGACYÄY�F ÔlA�F�� TGI á=�SÅ\�+ø©Ç�È�É��&ø�� ö Å�Ç�È¡®��Ohf8I k�OhfCWÌY�¨�F!TV¨lWLF�K�O=iZH O�DGF�K kCTVKG� BEI�TPOhACF"ß � T�M]i�F HJ�!f�K TVK BLOhA´â�MGF K-g WLF$TVH¡WLMUBLK-BEI�F$TGI�MGâ´jhBLDGF$AúlâCK�OPY�F K�F HJ��BEACF�K kCFpÔCHJI�K�¨lBLK,OGQ���� ã Å�úCÈ¡®t�OGH�TU�2OGH FPACTVK f�HJTGWSF"Ü�TG�2iZWEF�OGQ�k�O � OhA�F"ß � T�M8i�F HJ�!f�K TVK BLOhACIpYCO�A�OGK;A�F$g"F$IJI TVHJBEWLMÄkCBEYCF�BÌAÎßQ²OGHJ�­TVK BLOhA�TV¨�Ohf�K©K k�F-iZH F"ßÐBE��TVjGFGâZg"OhAZI BEY�F H�K k�F;£�\��§i�F HJ�;fCK TVK BLOhA��5�.� �Å�Ç�È+É Ç � �2OXYj� ®_\ÎBÌACg"F���1Å�Ç�È�É��.�1Å�Ç � È-Å � k�F$A�jhg Y·Å\�G×V�_Å �ÏÈ�È&É�Ø1TGI©BEI�K�HJfCF-Q²OGH©K kCF1g TGI�FpOGQS£�\E��È¡âCOhACFpg TGAUg"Oh�2iZfCK�Fc jhBEDGF$A­ú c K k�F�D5TGWEfCFN� � Å\� � ã�.� � Å�úCÈ�È¡âÎTGAZYPA�OGK_TGWEWliZTVH K BETGWlBEA�Q�OGHJ��TVK BLOhAPTV¨�OhfCK+K k�F�iCH F"ßÐBE�­TVjGF�OGQú=BEI�kCBEYCY�F$A´®

ù�OGi�F Q�fCWEWLMGâÎK kCBÌI©YCBEI g fZI I BLOhA=kCTGI©�2OGK BED5TVK�F$YUK k�FpQ�OhWEWLO � BEA�j2Y�F ÔZACBEK BLOhA�á

Page 33: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

� ³Y�S¹�°²»°�Á�¹�'¡  1$7Q�JF41��:����1�]¢g.�����E1>?8��DE7B�57 � �5�£����X¤�_1�7Q¥�á�Õ�ÖÎ×$Ø5ÙÚ | Õ�ÖÎ×$Ø5ÙF41����H1\¦cA��B1>��7 ���A4������DE7B�YF��}1 * D���A$7 �\����S¨§�1C6����£79;��57O¥��+6Z�ykCTVHJY�ßÐg"OGH F�¨ZBLK2OGQ���� * 79;�1 * ���9�[�5g��+�fm£;����[X�6>©Hªq��?���+�Q�P�P�¡���«mY��?8�979;f�k6�¬�I­79;E1���?®�fFM�YF��+���97 ��79;<�57�¬¯A4�5�°AV��?V?M1MA`7 �W�~A4�5�­��D�7±1�¥�Å�Ç�Èkm5�9l�1����SÅ�Ç�Èj�²6j���d76V�}m5�P� ³KA4�5��7 �W�jm5?M1M�d7±1>?k79;����8Ø`���ES­Uh�5?@1 * ��?8�j�5�+����I´´´ m+H`µ ÇC¶ Õ�ÖÎ×$Ø5Ù ï r�ú2É)�SÅ�Ç�È&áY¬�Å�ú�È�É�¥·Å�Ç�ȱ·�� Ø`��� ´´´ É)sÅ*ìCÈ¡×

* ��?G6>���¸1k��1±m5���[md�\F��}1 * D���A$7 �\����s�Å �ôÈ>S¹�OÍkCTVHJY�ßÐg"OGH F2¨ZBLK I�F"Ü�BEI�KpBEAÄjGF$ACF HJTGW+� � k�F2Q�OhWEWLO � BEA�j�BE�2i�OGH K TGA¬K�K k�F OGH F$� Å�YZf�F2K�O]Ã!®�¤�OhWLß

Y�H F$BEg¡k�TGACY�%+®P%´F DÎBEAlÈ_I�K TVK�F$I�K kCTVK©K k�F M�Y�O�á&;¼�³¬Álþ³�»º�¼ l51>?8������1�]¢g.�5���E1�?V��DE7B�57 �\����½;���60�¸;<��?@X�]@A4��?M1¸F��97�¥qSÅ � k�F1K kCF OGH F$� TGWEI�O�I k�O � I�k�O � K�O=jGO�TV¨�OhfCK�ÔZACYZBEA�j c � BLK kUkZBLjhk�iCHJOG¨ZTV¨ZBEWEBEKÐM c T=kCTVHJYÎßÐg"OGHJF¨ZBLK�Q�OGH-TGAaM��¿â�I�O�BLK�BEI�T=g"OhACI�K�H¡fCg"K BLDGF!H F$I fCWLK ®ôÈ�àÏF � BEWEW´ACOGK�iCHJO�DGF�K k�F!K k�F OGH F$� K kCBEI�I�F$�2F$I�K�F H$â¨Zf�K-BLQ�MGOhf8TVHJF�BÌAaK�F H F$I K�F$Y8gJk�F$gJ¢ÍK kCF2H F Q²F HJF$ACg"F$I�OhAÄK k�F � F ¨ÏiZTVjGF2OGH-K TV¢GF�jGHJTGYCfZTVK�F�g"H M¬iZK�O�BEAK k�F1I�iCH¡BEA�j�ü

Ã�f�HPiCH OG¨ZWLF$� kCTGI­ACO � ¨�F F$A I BE��iZWEBLÔCF$YÆI�Oh��F � kCTVK áÏk�O � Y�O � F]g"OhACI�K�HJfCg"K�T<m&£©¤�K kZTVKI�K�H F K gJkCF$Ip¨XMUOhA�F2¨ZBEK â�jhBEDGF$A8T=OhA�F"ß � T�MÄi�F H¡�;f�K TVK BEOhAh� � BEK k]kZTVHJYÎßÐg"OGH F2¨ZBEKN¥��tù�OGi�F Q�fCWEWLM�K kCFYCBEI g fCIJI BLOhA>K�O8K kCBEI�i�OhBÌAaK2��TV¢GF$I­K k�F�Q�OhWEWLO � BEA�jÏg"OhACI�K�HJfZg"K BLOhA �2OGH F�BÌAaK fCBEK BLDGFGá�Y�F ÔZA�FU¥2Å�Ç�È�É�SÅ�Ç�È�øO¥·Å�Ç�È¡®:¾�OGK BEg"F­K kCTVK â�BEAÄK�F HJ�­IpOGQ�fCA�iCHJF$YCBEg"K TV¨ZBEWÌBLKÐMGâ�BLKpTGWE��OhI�K1BE����F$YCBETVK�F$WLMÍQ²OhWEWEO � I-K kZTVK¥�BEI1fCA�iCH F$YZBEg"K TV¨ZWLF�Å�WLF K�K BEACj>ë Ç�ë´É�ì�â�K k�F­ÔCHJI�K;ìϨZBLK I-OGQ�¥�TVH FPg"F H K TGBÌACWLMÄfCA�iCH F$YZBEg"K TV¨ZWLFGâ·TGAZYfCA�iCH F$YZBEg"K TV¨ZBEWEBEKÐMUOGQ�K k�F=WÌTGI�K1¨ZBEK1Q�OhWEWLO � I1Q�H Oh�«K k�FPY�F ÔlACBLK BLOhA<OGQ©TÍkCTVH¡YÎßÐg"OGH F=¨ZBLK üôÈ¡®Íù�O � F DGF H$âI BEACg"F � F,YCBEYpACOGK�iCHJO�DGF,k�F H F+K kZTVK¿fCACiCH F$YCBEg"K TV¨lBEWEBLKNM�BE�2ilWEBLF$I�iZI�F$fZY�OGHJTGACY�Oh�­A�F$I I â � F&jhBLDGF,T�YCBEH F$g"KiCH OXOGQ¿k�F HJFG®

&;¼�³¬Álþ³�À¿Á  1$7��½F41O�k�5��1>] g.�����P1>?8�0DE7B�57 �\����g�� 79;�;<��?@X�]@A4��?M1�F>� 7�¥�SjÂq;�1>��¥2Å�Ç�È+É)�SÅ�Ç�È5ø�¥�Å�Ç�È�²6k�j,.-0/�Sà þÁ�Á�Ä Ã�f�H©iCHJO¬OGQ � BEWEW�Q�OhWEWLO � K k�F-¨¬M¬ßÐA�O � I�K TGACYZTVHJY�Q²OGH¡��TVK á

ØV®���I I fC��F�K�O � TVH¡YÍT�g"OhA¬K�HJTGYCBÌg"K BLOhA�K kCTVK�¥.BEI�A�OGK�T2m&£©¤!®�X® � kZBEI,�2F$TGACI�K kCTVK�K k�F H F1BEI�T��P�P�]TGWLjGOGHJBLK kZ�Ŭ§Q�OGH � kZBEgJk�á

´´´ m�H`µ Ç�¶ Õ�ÖÎ×$Ø5Ù ï r�ú2É÷¥2Å�Ç�È_áf¬!Å�úCÈ�É�Ø8·��<m�H`µ úR¶ Õ�ÖÎ×$Ø5Ù ï ð�ã ád¬�Å�ú�È�ɦØ8· ´´´ É)ÆXÅ*ìCÈ¡× Å Ø�È

� kCF H F�ƬŠ�ôÈ�BEI�A�OGK�ACF jhWEBLjhBL¨ZWEFG®Ç ®�àÏF­fZI�Fj¬ K�OUg"OhAZI�K�HJfCg"KpT��P���<TGWLjGOGH¡BLK kC�Ȭ�ö � kCBEgJk¡o ¨CHJF$TV¢XIMp�K k�F�kCTVHJYÎßÐg"OGHJF2¨ZBLK�¥ÏOGQ��

Å�BEA�K kCF1TViCiCH OGiZHJBETVK�F-I�F$ACI�FÈ¡®àÏF1g"OhACI�H¡fCg"K�¬�ö�TGI�Q²OhWÌWLO � I$rXHJF$g TGWEW�K kCTVKu¬�öljGF K I©BEA�ilf�K+ú=TGACY � TGA¬K I�K�O2g"Oh�2iZfCK�FG¥·Å\� � ã Å�ú�È�ȡᬠö Å�úCÈ�iZBEg ¢ÎIpT�H¡TGACY�Oh� ¨ZBLK���TGACYÄH¡fCACI=¬�Å�ú�ø��"È¡® d Qi¬ OhfCK�iZf�K I�Ø�Å�H F iZH F$I�F$A¬K BEA�j�T�jhf�F$IJI-¨¬M�¬

K kCTVK�K kCF;I�K�HJBÌA�j­ú1øu�pBEI�TGAÍOhf�K�iZf�K�OGQ&¥pÈ¡â�K k�F$A�¬�ö�Ohf�K�ilf�K IO�pTGI�BLK I�jhf�F$I I�Q�OGHN¥�Å\� � ã Å�úCÈ�È¡® d Q�¬Ohf�K�iZf�K I©ÖÎâZK k�F$A�¬ ö OhfCK�iZf�K I­É�®

Ç

Page 34: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

%´F K�!îI,ÔCH¡I�K&HJF"ß � HJBLK�F�Å Ø�È¡®�àÏF-kCT$DGF­Å�K kCBÌI&Y�F H¡BLD5TVK BEOhAPBEI&iCH F$I F$AaK�F$Y�BEAPQ�fZWEWZY�F K TGBÌW*âXI O!��TV¢GF-I fCH FMGOhfUfCACYCF HJI�K TGACY�F$TGg¡kUI�K�F i�È¡á

m�H�µ új¶ Õ�ÖÎ×$Ø5Ù ï ð�ã áf¬!Å�úCÈ�É�Ø8·É m+H`µ ú ö ¶ Õ�ÖÎ×$Ø5Ù ï r4�u¶ Õ�ÖÎ×$Ø5Ù;áY¬�Å�ú ö øi� È�É�Ø8·É m+H`µ Ç ö ¶ Õ�ÖÎ×$Ø5Ù ï r4�u¶ Õ�ÖÎ×$Ø5Ùdr�ú ö É)�SÅ�Ç ö È&áf¬!Å�ú ö øi�"È+É�Ø8·É m+H`µ Ç ö ¶ Õ�ÖÎ×$Ø5Ù ï r4�u¶ Õ�ÖÎ×$Ø5Ùdr�ú ö É)�SÅ�Ç ö È&áf¬!Å�ú ö øi�"È+É�Ø1ë5��É)¥·Å�Ç ö ȱ·��hØ`���

ñym�H`µ Ç ö ¶ Õ�ÖÎ×$Ø5Ù ï r4�u¶ Õ�ÖÎ×$Ø5Ùdr�ú ö É��SÅ�Ç ö È_ád¬!Å�ú ö ø�� ÈSÉ�Ø;ë��0ÊÉ)¥�Å�Ç ö ȱ·��hØ`���É Ø`���=��m�H`µ Ç�¶ Õ�ÖÎ×$Ø5Ù ï r�ú�É ¥2Å�Ç�È�ád¬!Å�úCÈ�É�Ø8·

ñ�Ø`���=�$m�H`µ Ç�¶ Õ�ÖÎ×$Ø5Ù ï r�ú�É)�SÅ�Ç�È,ád¬!Å�ú1ø ¥·Å�Ç�È È�É�Ø8·BËm+WEf�jGjhBÌA�j�K kCBEI©BEA�K�O]Å Ø�È_jhBLDGF$I á

ƬÅ*ìCÈ É ´´´ Ø`�����$m�H�µ ÇC¶ Õ�ÖÎ×$Ø5Ù ï r�ú­É ¥2Å�Ç�È,ád¬!Å�úCÈ�É�Ø8·� Ø`���=��m�H`µ Ç�¶ Õ�ÖÎ×$Ø5Ù ï r�ú�É)�SÅ�Ç�È_ád¬�Å�ú1ø ¥·Å�Ç�È È�É�Ø8· ´´´ Ë Å\�hÈ

¾�O � â � F � TGA¬K2K�OÏ¢ÎA�O � K k�F�iCH OG¨lTV¨ZBEWEBLKNMÄK kCTVKR¬�ö_g"OGH H F$g"K WLM>g"Oh�2iZf�K�F$IR¥�Å\� � ã Å�ú�È�È¡®>@�I BÌA�jK k�F-Y�F ÔZACBLK BEOhA­OGQ¿TGWLjGOGHJBLK kC�̬�ö¿Å�TGACY�g"OhACYZBLK BLOhACBEACj1OhA � k�F K kCF H_K kCF=��K kCTVK�BEI,gJkCOhI�F$A�BEI_F$#¬fZTGW�K�O¥·Å\��� ã Å�Ç�È�È,OGH�A�OGK¡È¡â � FpkCT�DGFGá´´´ m�H`µ Ç�¶ Õ�ÖÎ×$Ø5Ù ï r�ú�É)�SÅ�Ç�È&áY¬ ö Å�ú�È�É�¥·Å�Ç�ȱ·�� Ø`��� ´´´

É ´´´ m+H`µ Ç:¶ Õ�ÖÎ×$Ø5Ù ï r�ú�É)�SÅ�Ç�ÈVr4�u¶ Õ�ÖÎ×$Ø5Ù;áY¬�Å�úpø�� È�É�Ø1ë���É)¥·Å�Ç�ȱ·��$m+H�µ«��É)¥·Å�Ç�ȱ·ñÆm+H�µ Ç:¶ Õ�ÖÎ×$Ø5Ù ï r�ú­É)�SÅ�Ç�ÈVr4��¶ Õ�ÖÎ×$Ø5Ù;ád¬�Å�ú1ø�� ÈSÉtÖPë��0ÊÉ)¥�Å�Ç�ȱ·��$m+H`µ«��ÊÉ)¥·Å�Ç�ȱ·�� Ø`��� ´´´

É ´´´ Ø`���N�m�H`µ Ç:¶�Õ�ÖÎ×$Ø5Ù ï r�ú2É��SÅ�Ç�È&áY¬�Å�ú-øu¥·Å�Ç�È�È&É�Ø8·ñtØ`�����$m�H�µ ÇC¶ Õ�ÖÎ×$Ø5Ù ï r�ú­É)�SÅ�Ç�È&ád¬�Å�ú1ø ¥·Å�Ç�È ÈSÉtÖ�·�� Ø`��� ´´´

É ´´´ Ø`���N�m�H`µ Ç:¶�Õ�ÖÎ×$Ø5Ù ï r�ú2É��SÅ�Ç�È&áY¬�Å�ú-øu¥·Å�Ç�È�È&É�Ø8·ñtØ`���kͬح�8m�H`µ Ç�¶ Õ�ÖÎ×$Ø5Ù ï r�ú�É)�SÅ�Ç�È&áY¬�Å�úpø ¥·Å�Ç�È È�É�Ø8·²ÎC� Ø`��� ´´´

É ´´´ Ø`���N�m�H`µ Ç:¶�Õ�ÖÎ×$Ø5Ù ï r�ú2É÷¥2Å�Ç�È_áf¬!Å�úCÈ�É�Ø8·� Ø`�����$m�H�µ ÇC¶ Õ�ÖÎ×$Ø5Ù ï r�ú­É)�SÅ�Ç�È&ád¬�Å�ú1ø ¥·Å�Ç�È ÈSÉ�Ø8· ´´´ Ë

��ACY­K kCBEI�BÌISF"ÜÎTGg"K WEM¸Æ¬Å*ìCÈ�Å�g"QN®�Å\�hÈ�È¡® � kXfCI â¬BEQ�¬tg TGA½o ¨CH F$TV¢�p!¥.Å�BË®9FG®Lâ�ÆXÅ*ìCÈ�BÌI�A�OGK+A�F jhWÌBLjhBL¨ZWLFÈ�K kCF$A¬ ö g TGA2g"OGH H F$g"K WLM�g"Oh�2iZf�K�Fu¥�Å�Ç�È·jhBLDGF$AR�SÅ�Ç�È � BEK k1iCHJOG¨ZTV¨ZBEWEBEKÐM-A�OhAÎßÐA�F jhWÌBLjhBL¨ZWLM-YCBLÑ�F H F$A¬K¿Q²H Oh�ÊØ`Ï��X®��ACY�K kCBEI�BEI�OhfCH©Y�F$I BLH F$Y�g"OhA¬K�HJTGYCBÌg"K BLOhA�âC¨�F$g TGfCI�FG¥ÍBEI�T�kZTVHJYÎßÐg"OGH Fp¨lBLK,OGQ��¿®

^

Page 35: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ð�ÑPÒ_Ó5Ô�Õ4ÖMÒ}×¢Ø�Ù5Ú�ÛZÜ�Õ4Ø<Ý_Ü5Ñ�Þß Û�à ß�áYâ5ãkä»å ÑY×@Õ4Ù<ÞPæ�ç>×MÒ}ÙdÑ:×@Ù ß ÕMØ<èP×@Ù5é5Õ4Ü�è�êYØë�ÕMÙ5Ú9Ô$ÖMÖ@Ù5ÕNì5ÙdÑPÜ�×MêPÜ5ÑZí�Ü�×@î

ï¡ðjñ=ò�óvô�ð õ¸ö÷ øJùOúüûjý4þ�ÿ������¯ÿ���þ��"ùGù ���$ù��ÿ���þ������ ø¡ù�����ÿ�����Qÿ��4ù���uÔ$ç�Ü5Ý_Ý�×MêEÔ­ë�����ç>ÙdÑPÖ@×@Õ4æPç>×MÒ_ÙdÑ���Ô�édÒ_ÑPÑPÒ²ÑEé� �Ò}×MêRÜ5ÑfØ0ÙdÑPÔ"!# �Ü`Øjè�Ô�Õ�$kæE×MÜ�×MÒ_ÙdÑ&%' ­Ò}×Mêjê�Ü�Õ4Þ(!±ç>Ù5ÕMÔ���Ò}×)�* KÔjÞEÔ,+�ÑEÔ.-�/10�2435%6/10�287 ) /10�2�9;:~ÔRç>Ù<$jè�Ý}Ô�×@Ô�×MêEÔ�èPÕ4ÙfÙ5Ú�Ò_ÑÝ_Ô$ç>×MæEÕMÔ�×@Ù<Þ�Ü$Ø�=�Ö@Ô�Ô�×MêEÔjÝ}Ô$ç>×MæEÕ4ÔÑEÙ5×@Ô$ÖuÚ9Ù5Õ >{Ô$ç>×MæPÕMÔ&?A@�Ú9Ù5Õu×MêEÔ�Ú æPÝ_Ý�èPÕ4ÙfÙ5ÚB9C DE���6"þGFH���I���üÿ���þ�JLK0û�F������Mù4�å Ñk×MêEÔièPÕMÔ�Ó�Ò}ÙdæPÖ�Ý}Ô$ç>×MæEÕMÔM KÔiÞPÒ_Ö4ç�æPÖMÖ@Ô$ÞkêEÙN °×@Ù=ç>ÙdÑYÓ5Ô�Õ4×�ÜOë�����×MêPÜ�×QÔ"O�è�Ü5ÑPÞPÖQÒ}×MÖ�Ò_ÑEè�æP×��fØ�ÜNÖMÒ²ÑEédÝ}Ô��Ò}×q×@Ù=ÜOë�����×MêPÜ�×QÔ"O<è�Ü5ÑPÖ�Ò}×MÖ�Ò²ÑEè�æE×��<ØGÜOÝ_Ü�ÕMé5Ô�Õ�ÜP$jÙdæPÑf×,9RQEÙ5Õ�ç>Ù<$jè�Ý}Ô�×@Ô$ÑPÔ$ÖMÖ,SP �Ô�Ö@×MÜ�×@Ôi×MêEÔ�ÕMÔ$ÖMæ�Ý}×Ù5Ú�×MêPÒ_Ö�×@ÕVÜ5ÑPÖ@Ú9Ù5ÕI$RÜ�×MÒ}ÙdÑ ä Ü�èPè�ݲÒ}Ô$Þ~×@Ù�ÙdæEÕGè�Ü�Õ4×MÒ_ç�æPÝ_Ü�ÕGç>ÙdÑ�Ö@×@Õ4æPç>×MÒ}ÙdÑ ä êPÔ�ÕMÔP9T�uÔ$ç�Ü5Ý_ÝK×MêPÜ�×G×MêPÔÒ_ÞEÔ$ÜRÒ_Ö­×@Ù¸Ú9Ù5ÕI$UV/10�2W3X-�/BY,Y,Y"-./10�2ZY,Y,Y[2�SZ ­êEÔ�ÕMÔ;- ç�Ü5Ñ\��ÔGÒ_×@Ô�Õ4Ü�×@Ô$ÞÜ5Ö�$RÜ5ÑfØC×MÒ]$jÔ$Ö­Ü5ÖOÞEÔ$ÖMÒ_ÕMÔ$Þ�9^ Ý_ÖMÙ_S` �ÔOÖM×MÜ�×@Ô$Þ:×MêPÜ�×�Ò_×KÒ_Ö�ÑEÙ5×iÑEÔ$ç>Ô$ÖMÖMÜ�ÕMظ×@Ù0Ü�èPè�Ý}Ø&-�×@Ù0×MêEÔ�a"bdcfe1gha=Ö@×@Õ4Ò²ÑEékÜ�×�Ô$Ü5ç4êCÒ_Ñf×@Ô�ÕI$jÔ$Þ�Ò_Ü�×@ÔÖ@×MÜ�é5ÔP=KÒ_Ñ�Ö@×@Ô$Ü5Þ�S8 KÔ:ç�Ü5Ñ£Ü�è�è�Ý}ØL-Å×@Ù�ÙdÑPÝ}Ø�×MêEÔi+PÕ4Ö@×,S�ÖMÜ`ØPS�jk��Ò}×MÖGÙ5Ú�Ô�Ó5Ô�ÕMØhÒ_Ñf×@Ô�ÕI$jÔ$ÞPÒ²Ü�×@Ô�Ö@×@Õ4Ò²ÑEé_9^ èPè�Ý}Ø<Ò²ÑEék×Mê�Ò_Ö�$jÔ�×MêEÙ�ÞEÙdÝ}Ù5é5ؤ×@ÙRÙdæEÕ�ç>ÙdÑPÖ@×@Õ4æPç>×MÒ_ÙdÑCÚ+Ù5Õl-;S�Üm��Ù`Ó5ÔPS�édÒ}Ó5Ô$Ö *

Un/10�2G3o%dpA/10�2�7 ) /q%dpsr�tA/10�2s2�7 ) /q%dpurdvP/10�2s2�74Y,Y,Yw7 ) /q%6/10�2s2�7 ) /10�2�x ­êEÔ�ÕMÔyU Ò_ÖuÑEÙz ÜRëW�H� ×MêPÜ�×­Ò_Ñ�ç>ÕMÔ$Ü5Ö@Ô$ÖuÒ}×MÖ­Ò_ÑEè�æE×M�<Ø&{|��Ò}×MÖA9} ���5�~�XF���������þ8��� FG���.�.���Qÿ��Mù4���:~Ô�êPÜ`Ó5Ô­ÞPÒ_ÖMç�æ�ÖMÖ@Ô$ÞjÒ_Ñjç�Ý_Ü5Ö4Ö�×MêEÔ­ç>ÙdÑPÖM×@Õ4æPç>×MÒ}ÙdÑjÙ5Ú�Ü�ëW�H�)Ú+Õ4Ù<$�ÜNÙdÑEÔ"!# iÜ$Øjè�Ô�ÕI$0æE×MÜ�×MÒ}ÙdÑ�SYÖ4êEÙN ­Ò²ÑEé×MêPÜ�×�ë����NÖ�Ô"OEÒ_Ö@×�Ò}ÚPÙdÑEÔ"!# iÜ$Ø0è�Ô�ÕI$kæP×MÜ�×MÒ}ÙdÑPÖ�Ô"O�Ò_ÖM×,9 ^ ç>×MæPÜ5Ý_Ý}ØPSP �Ô�ç�Ü5ÑjÖ@×MÜ�×@Ô�Ül$jÙ5ÕMÔié5Ô$ÑEÔ�Õ4Ü5Ý<ÕMÔ$ÖMæ�Ý}×,9�����`�Z�z�(� �o���;�W� aI�Pe � cGe ���mb����PbE����e ���mb�a"�B ��m��¡B¢`£z¤"¥�¦ §(£N¨�ah�<e � c#©ª ÑEÔkÞ�Ò}ÕMÔ$ç>×MÒ}ÙdÑCÙ5Ú�×MêPÔG×MêEÔ�Ù5ÕMÔA$ Ò_Ö­Ô$Ü5Ö@ØL/ å KÙdæ�Ý_ÞCÔ"O�è�Ô$ç>×OØ5Ùdæ×@Ù&��ÔGÜm��Ý}Ôk×@Ù�ÖMêPÙN yS�ÙdÑÜ5ÑZÔ"O�ÜP$TSêEÙN Á×MêEÔ�Ô"OEÒ_Ö@×@Ô$ÑPç>Ô�Ù5Ú.ܤë���� Ò]$jè�Ý_Ò}Ô$Ö�×MêEÔ�Ô"OEÒ_Ö@×@Ô$ÑPç>Ô0Ù5ÚKÜ�ÙdÑEÔ"!# iÜ$Ø�Ú9æ�ÑPç>×MÒ}ÙdÑZ2�9�à�êEÙz ­Ò_ÑEé�×MêPÜ�×=ÜÙdÑEÔ"!# �Ü`Øy«Ea"g­¬;®_c#�Pcfeq�mb¤Ò]$jè�Ý_Ò_Ô$Ö{Ü�ëW��� Ú9ÙdÝ_Ý}Ùz ­Ö�×MêEÔKÙdæP×MÝ_Ò_ÑEÔ�Ù5Úd ­êPÜ�ׯ �Ô�ÖMêEÙz KÔ$Þ0Ò_Ñ0ç�Ý_Ü5ÖMÖ,S5Ü5Ý}×MêEÙdæPédê×MêEÔOç>ÙdÑ�Ö@×@Õ4æPç>×MÒ}ÙdÑ�ÕMÔA°<æPÒ}ÕMÔ$Þ¸ÜkÞPÒ]±¸ç�æPÝ}×�×MêEÔ�Ù5Õ4ÔA$~�<Øi�OÙdÝ_ÞPÕMÔ$Ò_ç4ê�Ü5ÑPÞ�>{Ô�Ó�Ò_ѸÜm��ÙdæE×K×MêEÔ­Ô"OEÒ_Ö@×@Ô$ÑPç>ÔOÙ5ÚêPÜ�Õ4Þ(!±ç>Ù5ÕMÔl��Ò_×MÖ,9Kà�êEÙz ­Ò_ÑEék×MêPÜ�×uÜ0ÙdÑEÔ"!# �Ü`Ø���®(bZ²Ncfef�mb�ÖMæ_±¸ç>Ô$Ö�×@ÙRç>ÙdÑPÖM×@Õ4æPç>×�Üjë�����Ò_Ö³$0æPç4êT$jÙ5Õ4ÔÞPÒ[±¸ç�æPÝ_×,S�Ü5Ñ�ÞT KÔkÞEÙRÑEÙ5×uÞPÒ_Ö4ç�æPÖMÖ�Ò}×uêEÔ�Õ4ÔP9´ ����hµHF�ÿ�þ8�`¶Åþ6�·Jn���6��0ûkÿ��Mù4�T¸�� þ6µ;�h���®ÿ�þ��ë����NÖ=Ö@ÙdÝ}Ó5Ôjç>Ù<$jè�Ý}Ô�×@Ô$Ý}Ø�×MêEÔ�èPÕ4ÙP��Ý}ÔA$LÙ5ÚKèPÕ4Ò}Ó�Ü�×@Ô"!#¹5Ô�ØZÔ$Ñ�ç>ÕMØfè�×MÒ}ÙdÑL Rº_a"bn�PbE���»� � e1b`¼P�]a�¬�a �I� �A¼`ae � a"bZ²Ag���«�c½aI��=PÔ�Ó5Ô$Ѥ×MêEÔ$Ñ�SY×MêPÔOÖMçVêEÔA$jÔ4 KÔOédÜ$Ó5Ô=Ò_Ö�Ö@Ô$ç�æEÕ4Ô­ÙdÑPÝ}ØRÜ�édÜ5Ò²ÑPÖ@×KÜ0ç�Ò}è�êPÔ�ÕM×@Ô"O<×KÙdÑPÝ}ØRÜ�×@×MÜ5çI¹Z9

?

Page 36: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

à�Ò_ÑPç>Ô�Ô$ÑPç>ÕMØ<èP×MÒ}ÙdÑ~æ�Ö@Ô$ÖN×MêEÔRë���� ×@ÙZé5Ô$ÑEÔ�Õ4Ü�×@Ô¸ÜCÖ@×@ÕVÒ_ÑEé' ­êPÒ_çVê~Ò_ÖNæPÖ@Ô$Þ~Ü5Ö�Ò²Ñ�×MêEÔjÙdÑEÔ"!B×MÒ]$jÔjè�Ü5ÞÖMç4êEÔA$RÔPS­×MêEÔ�ÖMç4êEÔA$RÔ�Ò²Ö¸Ò_ÑPÖMÔ$ç�æEÕMÔZÒ}Ú�Ò_×�Ò_Ö�æPÖ@Ô$Þ¾$jÙ5ÕMÔ�×MêPÜ5ÑcÙdÑ�ç>Ô * Ò}Ú�¿ t 3ÁÀ t� -./qÃwjE2¸Ü5Ñ�Þ¿ v 3ÄÀ vM -./qÃzjE2�Ü�ÕMÔ0×½ �ÙCç�Ò}è�êEÔ�Õ4×@Ô"O<×MÖ�×MêPÜ�×NÜ�ÕMÔ0Ò²ÑY×@Ô�Õ4ç>Ô�è�×@Ô$ÞÅ�<ØZÜ5Ñ�Ü5ÞEÓ5Ô�Õ4ÖMÜ�Õ4ØPS�×MêPÔ0Ü5ÞEÓ5Ô�ÕVÖMÜ�ÕMØÝ}Ô$Ü�Õ4ÑPÖi×MêPÜ�× À t¯Â À v 3o¿ t¯Â ¿ v 9 ^ Ñ�Þ�S�Ù5Ú�ç>ÙdæEÕ4ÖMÔPSP×MêEÔ=ÖMç4êPÔA$jÔGÒ_Öiç>Ù<$jè�Ý_Ô�×@Ô$Ý}ؤҲÑPÖ@Ô$ç�æEÕMÔNÜ�édÜ5Ò_ÑPÖM×ÜL¹�ÑEÙN �Ñcè�ݲÜ5Ò_ÑY×@Ô"O�×:Ü�×@×MÜ5çh¹ * Ò}Ú=×MêEÔ�Ü5ÞEÓ5Ô�Õ4Ö4Ü�ÕMØƹ<ÑPÙN ­Ö¤×MêPÜ�×\¿ t Ò_Ö¤Ü5Ñ�Ô$ÑPç>ÕMØ<èP×MÒ}ÙdÑ�Ù5Ú�À t Su×MêPÔÜ5ÞEÓ5Ô�Õ4ÖMÜ�ÕMØ�Ý}Ô$Ü�Õ4ÑPÖ|-./qÃwjE2;/ ÑEÙ5×@Ô�×MêPÜ�×N×MêEÔ0Ü5ÞPÓ5Ô�Õ4ÖMÜ�ÕMØV�<�Na � bZ�PcOÝ}Ô$Ü�Õ4ÑÇÃwj�=���æE×�×Mê�Ò_Ö�Ò²Ö�ÑEÙ5×=ÑEÔ�Ô$ÞPÔ$Þ×@ÙT�PÕMÔ$Üm¹�×MêEÔjÖMçVêEÔA$jÔPÈ�2�Ü5Ñ�Þ�ç�Ü5Ñ�×MêEÔ$Ñ~ÞPÔ$ç>ÕMØfè�×OÜ5ÑfØÖMæE��Ö@ÔA°<æEÔ$ÑY×Oç�Ò}è�êEÔ�Õ4×@Ô"O<×MÖO×MêPÜ�×�Ü�ÕMÔ�Ö@Ô$Ñf×,9 å ÑèPÕ4Ü5ç>×MÒ_ç>ÔPSE �Ô| �ÙdæPÝ_Þ:Ý_Ò[¹5Ô=æPÖ@Ô�ÕVÖi×@Ù���ÔNÜm��Ý}ÔN×@ÙjÔ$ÑPç>ÕMØ<èP×�$kæ�Ý}×MÒ}è�Ý}Ô4$jÔ$ÖMÖMÜ�é5Ô$ÖA=�ÖMÒ_ÑPç>ÔN×MêEÔ$ÖMÔy$jÔ$ÖMÖMÜ�é5Ô$Ö$RÒ}édêf×6��Ô³¹�ÑEÙN �ÑÉ/9Ù5Õ�Ô�Ó5Ô$ÑRçVêEÙdÖ@Ô$Ñ�È�2��<Ø0Ü5ѸÜ5ÞEÓ5Ô�Õ4ÖMÜ�ÕMØPS` KÔH �ÙdæPÝ_ÞjÜ5ݲÖ@Ù�Ý_Ò[¹5Ô�Ü5ÑjÔ$ÑPç>Õ4ØfèP×MÒ_ÙdÑjÖMç4êEÔA$RÔ×MêPÜ�×­Ò_ÖuÖMÔ$ç�æEÕMÔ�Ü�édÜ5Ò_ÑPÖM×­ç4êEÙdÖMÔ$Ñ�è�Ý_Ü5Ò²ÑY×@Ô"O�×uÜ�×@×MÜ5çh¹�Ö,9

Ê Ùz �Ü�ÕVÞ�×MêPÒ_Öué5ÙdÜ5ÝqSE �ÔGÜ�ÕMÔ=é5ÙdÒ_ÑEé¸×@ÙRÔ�Ó5Ô$ÑY×Mæ�Ü5Ý_Ý}Ø:êPÜ$Ó5Ôk×@ÙRÕ4Ò}é5Ù5ÕMÙdæ�ÖMÝ}ظÞEÔ,+�ÑEÔ�ÜRÑEÙ5×MÒ}ÙdÑ:Ù5Ú�Ö@Ô$ç�æ_!Õ4Ò}×±Ø�Ü�édÜ5Ò_ÑPÖ@×�çVêEÙdÖ@Ô$Ñ~è�Ý_Ü5Ò_Ñf×@Ô"O<×NÜ�×@×MÜ5çh¹�/ Ò}×N×MæEÕVÑPÖOÙdæE×=×MêPÜ�×=×MêPÒ_ÖNÒ]$Rè�Ý_Ò}Ô$Ö�Ö@Ô$ç�æEÕ4Ò}×¢ØÉ ­êPÔ$Ñk$kæPÝ_×MÒ}è�Ý}Ô$jÔ$ÖMÖMÜ�é5Ô$ÖiÜ�ÕMÔuÔ$Ñ�ç>ÕMØfè�×@Ô$ÞZ2�9 å Ú� �Ô­ÕMÔA$jÔA$���Ô�Õ�ÙdæEÕ�ÞEÔ,+�ÑPÒ}×MÒ_ÙdÑ�Ù5Ú6/ ç>Ù<$Rè�æE×MÜ�×MÒ}ÙdÑPÜ5Ý12�Ö@Ô$ç�æEÕVÒ}×±ØjÜ�édÜ5Ò_ÑPÖM×ç�Ò}è�êEÔ�ÕM×@Ô"O�×u!BÙdÑPÝ}Ø£Ü�×@×MÜ5çh¹�Ö,SW �ÔCêPÜ5Þ * Ú+Ù5ÕjÜ5Ý_ÝHËEËEÌ Ü5ÞEÓ5Ô�Õ4ÖMÜ�Õ4Ò_Ô$Ö.ÍLÜ5ÑPÞ°Ü5Ý_ݳ$jÔ$ÖMÖMÜ�é5Ô$Ö�Î t xsÎ v S�×MêPÔ°fæPÜ5Ñf×MÒ}×¢ØÏÏÏ ë.ÕNÐÑÃzj�ÒÔÓNÕ(xA?wÖN×(=�¿ t 3¾Ø(Ù × /1Î t 2 * Í;/#¿ t 2G3�?­Ú�Ûhë�ÕNÐÑÃwj&ÒÜÓNÕ(xA?wÖN×(=�¿ v 3¾Ø_Ù × /1Î v 2 * Í�/#¿ v 263�?­Ú

ÏÏÏÖMêEÙdæPÝ_Þn��Ô�ÑEÔ�édÝ_Ò_édÒ[��Ý}ÔP9 å ×RÖ@Ô�ÔA$¸Ö�¹�Ò_ÑPÞ½Ù5Ú4$RÔ$ÖMÖ@ؽ×@ÙL$jÙ�ÞPÒ}Ú9Øh×MêPÒ_ÖjÞEÔ,+�ÑPÒ}×MÒ}ÙdÑ£×@ÙHêPÜ5ÑPÞ�Ý}ÔCç4êEÙdÖ@Ô$Ñè�Ý_Ü5Ò_Ñf×@Ô"O<×�Ü�×@×MÜ5çh¹�Ö,9Oà<Ù_SZ ­êPÜ�×� �Ô�+PÕVÖ@×­ÑEÔ�Ô$ÞZ×@Ù�ÞEÙ¤Ò_Ö�×@Ùi$RÙ<ÞPÒ_Ú+ؤ×MêPÒ_ÖuÞPÔ,+�ÑPÒ}×MÒ}ÙdÑCÖMÙ¸×MêPÜ�×OÒ}×­êPÜ5Ö�ÜÚ+Ù5ÕI$$jÙ5ÕMÔGç>ÙdÑfÓ5Ô$ÑPÒ}Ô$Ñf×u×@Ù� �Ù5Õh¹' ­Ò}×MêCÜ5ÑPÞ�×@ÙRÔ"O<×@Ô$Ñ�Þ�9Ê ÙRÞPÙj×MêPÒ_Ö,S( �Ô�ÞEÔ,+�ÑEÔN×MêEÔ=ÑEÙ5×MÒ}ÙdÑ:Ù5Ú�Ü5ÑL�mgs�<²A�[a,9 ^ Ñ�Ù5Õ4Ü5ç�Ý}Ô�Ò_ÖiÖ@Ù<$jÔ�×Mê�Ò_ÑEé�×MêPÜ�×­Ü5ÑCÜ5ÞEÓ5Ô�ÕVÖMÜ�ÕMØÒ_ÑY×@Ô�ÕVÜ5ç>×MÖ. ­Ò}×Mê�=M ­êEÔ$ѽ×MêEÔCÜ5ÞEÓ5Ô�ÕVÖMÜ�ÕMØ°Ö@Ô$ÑPÞPÖ�Ö@Ù<$jÔ�Ò²ÑEè�æE×'/ ÜÞÝ,®da"g­�(2k×@Ù~×MêEÔ:Ù5Õ4Ü5ç�Ý_ÔPS.×MêEÔCÙ5Õ4Ü5ç�Ý_ÔÕMÔ�è�Ý_Ò}Ô$Ö& ­Ò}×MêJÖ@Ù<$jÔ�ÙdæE×@è�æE×Ç/ Ü5Ñ5�mb �  Wa,g�S�Ù5ÕÇgsa � «(�mb � am2�9 Ê êEÔÙ5Õ4Ü5ç�Ý}Ô~Ò_Ö¸Ü�ßu��ݲÜ5çh¹`!#��ÙNO(à½Ü5Ö¤Ú9Ü�ÕÜ5Ök×MêPÔ¤Ü5ÞEÓ5Ô�ÕVÖMÜ�ÕMØhÒ_Ö0ç>ÙdÑPç>Ô�Õ4ÑEÔ$Þ * ×MêEÔ�Ü5ÞEÓ5Ô�ÕVÖMÜ�ÕMØHÞEÙ<Ô$Ö0ÑEÙ5×0ÖMÔ�Ô¤êEÙz ×MêPÔ�ÙdæE×@è�æE×MÖGÙ5Ú�×MêPÔ�Ù5Õ4Ü5ç�Ý_ÔÜ�ÕMÔ�èPÕMÙ�ÞPæPç>Ô$Þ�S­Ü5Ñ�ÞJç�Ü5Ñ�ÑEÙ5×C×MÜP$jè�Ô�ÕCÒ_Ñ�Ü5Ñfؾ �Ü`Ø� ­Ò}×MêJ×MêEÔ�Ú æPÑPç>×MÒ}ÙdÑ�Ò_ÑEéhÙ5Úk×MêEÔ�Ù5Õ4Ü5ç�Ý}ÔP9 Ê êEÔÜ5ÞEÓ5Ô�Õ4ÖMÜ�ÕMØ�ánÖ�Ó�Ò}Ô, ¡Ù5Úq×MêPÒ²Ö.Ù5Õ4Ü5ç�Ý_Ô�Ò_Ö�ÕMÔ$Ö@×@Õ4Ò_ç>×@Ô$Þ¤×@Ù4âMæPÖ@×M ­êPÜ�×@Ô�Ó5Ô�Õ °<æEÔ�Õ4Ò_Ô$ÖK×MêPÔNÜ5ÞPÓ5Ô�Õ4ÖMÜ�ÕMؤÜ5Ös¹<Ö�Ü5Ñ�Þ×MêEÔCç>Ù5ÕMÕMÔ$Ö@è�ÙdÑPÞPÒ²ÑEéZÜ5ÑPÖs �Ô�Õ4Ö�Ò}×0ÕMÔ$ç>Ô$Ò}Ó5Ô$Ö,9�/fã�Ù5×@ÔC×MêPÜ�×� �Ô¤Ú9ÕMÔA°<æEÔ$Ñf×MÝ}ØHÞEÙ�ÑEÙ5×jÑEÔ�Ô$Þ½×@Ù~Ü5ç>×MæPÜ5Ý_Ý}ØÒ]$jè�Ý}ÔA$RÔ$ÑY×i×MêEÔ�Ù5ÕVÜ5ç�Ý}ÔP=�Ò}×uÒ_Ö�Ö4Ò]$jè�Ý}Ø�Ü5Ñ�Üm��Ö@×@Õ4Ü5ç>×MÒ_ÙdÑ�×MêPÜ�×� �ÔGæPÖMÔ�Ò_Ñ:ÙdæEÕ­ÞEÔ,+�ÑPÒ}×MÒ}ÙdÑPÖA9�2ª Õ4Ü5ç�Ý}Ô$ÖCç�Ü5Ñ)ê�Ü$Ó5ÔHÜ5ÑfØcÚ æPÑPç>×MÒ}ÙdÑ�Ü5Ý_Ò}×±ØvÙdÑPÔhÝ_Ò[¹5Ô$Ö ä ×MêPÜ�×,ánÖ:è�Ü�ÕM×:Ù5Ú� ­êPÜ�×T$RÜm¹5Ô$ÖC×MêEÔA$ ÖMÙæPÖ@Ô�Ú æPÝqÈ Ê êEÔ�Ù5Õ4Ü5ç�Ý}Ô� �Ôk ­Ò_Ý²Ý ��ÔhÒ_Ñf×@Ô�ÕMÔ$Ö@×@Ô$Þ�Ò²Ñ�Ú9Ù5ÕZÑEÙz üÒ_ÖCÜä�]af�"cf�u�Pg��Bg�e[¼mºåc�a"b�²,g­�I«�cfef�mb5�mgs�<²,�]aÞEÔ$ÑEÙ5×@Ô$Þ»æåç�èBé Ù × /BY[x,Y�2�S( ­êEÙdÖMÔuÔ"O�Ü5ç>×iÖ@è�Ô$ç�Ò[+�ç�Ü�×MÒ_ÙdÑRÞEÔ�è�Ô$ÑPÞPÖ�ÙdѸ×MêEÔ­Ô$Ñ�ç>ÕMØfè�×MÒ}ÙdÑRÖMçVêEÔA$jÔ­Ù5Ú{Ò²ÑY×@Ô�ÕMÔ$ÖM×,9

Ê êEÔ¤Ù5Õ4Ü5ç�Ý}Ô:Ò_ÖkÒ²ÑPÞEÔ"O�Ô$Þ��<ØhÜÉ��Ò_×;êRÜ5Ñ�Þ£ÜŹ5Ô�ØnÃwj�9 Ê êPÔ¸Ù5Õ4Ü5ç�Ý}Ô¤×MÜm¹5Ô$Öj×ë KÙ~Ò_ÑEè�æE×MÖ|Î\ìmxsÎ t Ü5Ñ�ÞÕMÔ�×MæEÕ4ÑPÖk×MêEÔ¤ÙdæE×@è�æE×�Ø(Ù × /1Î\èI2�=�Ò}Úi×MêEÔ¤Ô$ÑPç>ÕMØ<èP×MÒ}ÙdѽÖMç4êEÔA$RÔ¤Ò_ÖkÕ4Ü5ÑPÞPÙ<$RÒ}î�Ô$Þ�S�×MêEÔ$Ñ£Ù5Õ4Ü5ç�Ý}Ô:ç4êEÙ<ÙdÖ@Ô$ÖÑEÔ, Õ4Ü5Ñ�ÞEÙ<$í��Ò}×MÖ­Ô$Ü5ç4ê×MÒ]$jÔGÒ_×uÕMÔ$Ö@è�ÙdÑPÞPÖi×@Ù¤Ü5ÑZÒ_ÑPè�æE×H°fæPÔ�ÕMØP9 Ê êEÔGÑPÜP$jÔÅß@Ý}Ô�Ú+×u!BÙ5Õh!BÕ4Ò}édêf×hà¸ç>Ù<$jÔ$ÖÚ+ÕMÙ<$ ×MêEÔOÚ Ü5ç>×K×MêPÜ�×�Ò}Ú¯êH3äÕk×MêEÔ$Ѥ×MêEÔOÙ5Õ4Ü5ç�Ý}ÔNÜ5Ý[ iÜ$Ø�ÖKÕ4Ô�×MæEÕ4ÑPÖ�Ü5ѤÔ$ÑPç>ÕMØ<èP×MÒ}ÙdÑRÙ5Úq×MêEÔOÝ_Ô�Ú+׳$jÔ$ÖMÖMÜ�é5ÔPSÜ5ÑPÞ�Ò}ÚRê�3�?N×MêEÔ$ÑC×MêEÔGÙ5Õ4Ü5ç�Ý}ÔGÜ5Ý[ iÜ$Ø�ÖuÕ4Ô�×MæEÕ4ÑPÖ�Ü5ÑCÔ$Ñ�ç>ÕMØfè�×MÒ}ÙdÑCÙ5ÚQ×MêEÔ�Õ4Ò}édêf×�$jÔ$ÖMÖ4Ü�é5ÔP9ª æEÕ4+PÕ4Ö@×OÑEÙ5×MÒ}ÙdÑÙ5Ú.Ö@Ô$ç�æEÕVÒ}×±ØL/1 ­ê�Ò_ç4êÅ KÔ0Ú9Ù5ÕI$RÜ5Ý_Ò}î�ÔkÕ4Ò}é5Ù5ÕMÙdæ�ÖMÝ}ØCÒ_Ñ�Üi$jÙ<$RÔ$ÑY×�2�ç>ÙdÑPÖMÒ²ÞEÔ�Õ4Öu×MêPÔÚ+ÙdÝ_Ý_ÙN ­Ò_ÑPé~édÜP$jÔ\��Ü5Ö@Ô$ÞcÙdÑ Ü5ÑfؽÔ$Ñ�ç>ÕMØfè�×MÒ}ÙdÑ ÖMçVêEÔA$jÔ * ÜhÕ4Ü5ÑPÞEÙ<$Á¹5Ô�Ø�Ãwj°Ü5ÑPÞ ÜHÕ4Ü5ÑPÞEÙ<$î��Ò}×�êÜ�ÕMÔCç4êEÙdÖ@Ô$Ñ�9 Ê êPÒ_Ö0ÞEÔ,+�ÑEÔ$ÖkÙ5Õ4Ü5ç�Ý}ÔÇæ(ç�èBé Ù × /BY[x,Y�2'/9×MêPÔ¤Ô$Ñ�ç>ÕMØfè�×MÒ}ÙdÑ£Ö4ç4êEÔA$jÔ:Ò_Ö�Ò]$jè�Ý_Ò_ç�Ò}×�êEÔ�Õ4Ôz2�9 Ê êEÔÜ5ÞEÓ5Ô�Õ4ÖMÜ�ÕMØvç�Ü5Ñ�ÑEÙN ÖMæ_�d$RÒ_×�Ü � e1b`¼P�]a»°fæPÔ�ÕMØä/1Î\ìwxsÎ t 20×@Ù£×MêPÒ_Ö�Ù5Õ4Ü5ç�Ý_ÔPSM �êPÒ_ç4ê ÕMÔ�×MæEÕ4ÑPÖjÜ5Ñ�Ös KÔ�Õ

Ø(Ù × /1Î\èh2�9L�NÒ}Ó5Ô$ѽ×MêPÒ²Ö,S�×MêEÔ¤Ü5ÞPÓ5Ô�Õ4ÖMÜ�ÕMؽÑEÙN Ì×@Õ4Ò}Ô$Ök×@Ù�èPÕMÔ$ÞPÒ²ç>×G×MêEÔ¤Ó�Ü5ݲæEÔ�Ù5Ú4êj×MêPÜ�×� iÜ5Ö0æ�Ö@Ô$ÞL�<Ø×MêEÔ¸Ù5Õ4Ü5ç�Ý_ÔP9T:~Ô�ÖMÜ`Ø~×MêPÜ�×k×MêPÔRÔ$ÑPç>ÕMØ<èP×MÒ}ÙdÑ£ÖMç4êEÔA$RÔ�Ò_Ö�ÖMÔ$ç�æEÕMÔ�Ò_Ú�×MêPÔ¸Ü5ÞEÓ5Ô�Õ4ÖMÜ�Õ4ØZánÖkèPÕMÙP��Üm��Ò_Ý_Ò}×¢ØZÙ5Úç>Ù5ÕMÕMÔ$ç>×MÝ}Ø:édæEÔ$ÖMÖ4Ò_ÑEé&êOÒ_Ö�ÑEÙ5×���Ô�×@×@Ô�Õu×Mê�Ü5ÑL?Nïm@��<ØT$jÙ5ÕMÔ=×MêPÜ5Ñ�ܸÑEÔ�édÝ_Ò}édÒ[��Ý}Ô4°<æPÜ5Ñf×MÒ}×±ØP9Û�Ù5Õ4ÔGÚ9Ù5ÕI$RÜ5Ý_Ý}ØPS�édÒ}Ó5Ô$ÑÜ5ÑYؤÔ$ÑPç>ÕMØ<èP×MÒ}ÙdÑZÖMç4êEÔA$RÔ�ðä3ñ/1ØGxsò.2y/ Ö@Ô$ç>ÕMÔ�× ¹5Ô�Ø�Ö­Ü�Õ4ÔGÜ5ÖMÖMæE$RÔ$ÞC×@Ù���Ôç4êEÙdÖ@Ô$Ñ0æPÑPÒ_Ú+Ù5ÕI$¸Ý}Ø�Ü�×�Õ4Ü5ÑPÞEÙ<$ Ú+ÕMÙ<$5ÓNÕ(xA?wÖ × 2"Ü5ÑPÞkÜ5ÑYØ�Ü5ÞEÓ5Ô�ÕVÖMÜ�ÕMØ|Í�S5ÞEÔ,+�ÑEÔ�×MêEÔ � ®E²I²�a �I� «Zgu�`óh��ó,eô�õe1cf�@

Page 37: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�u� Ííe1bÇ�<cqc#�<²"öweôb`¼�ð�Ü5Ö *÷ ¢z¤,¤,ø é ù /qjE28ú"ûqü3 ë�ÕNÐÑÃwj&ÒÜÓNÕ(xA?wÖ × =IêHÒÜÓNÕ(xA?wÖ * Í�ý�þmÿ�� ������� é � �h/u? × 2�3oêIÚ#x

­êEÔ�ÕMÔ�×MêEÔ�Ü5ÞEÓ5Ô�ÕVÖMÜ�ÕMØHÒ_ÖkÜ5Ý_Ý}Ùz KÔ$Þ£×@Ù»°<æEÔ�ÕMØ~×MêPÔÅæ(çvÙ5Õ4Ü5ç�Ý}Ô¤ÙdÑPÝ}ØÞ�mb�²�a"9 ^ édÜ5Ò²Ñ�S�×MêPÒ_ÖGÒ²Ö³âMæPÖ@×G×MêPÔèPÕMÙP��Üm��Ò²Ý_Ò}ױذ×MêPÜ�×:×MêEÔ~Ü5ÞEÓ5Ô�ÕVÖMÜ�ÕMØcç>Ù5ÕMÕ4Ô$ç>×MÝ}ØcédæEÔ$Ö4Ö@Ô$Ö\êz9í/fã�Ù5×@Ô~×MêEÔ~ÑEÙ5×MÜ�×MÒ}ÙdÑ¡êEÔ�Õ4Ô * Í ý�þ ÿ� �� ��� é � �ÞEÔ$ÑEÙ5×@Ô$Ö.Ü5Ñ�Ü5ÞEÓ5Ô�Õ4ÖMÜ�Õ4ØkédÒ}Ó5Ô$ѸÙ5Õ4Ü5ç�Ý}Ô­Ü5ç�ç>Ô$Ö4Ö.×@Ù&æ(ç�èBé Ù × /BY[x,Y�2�9�2�Q�Ò_ÑPÜ5ݲÝ}ØPSP KÔ�ÖMÜ$Øj×MêEÔuÖ4ç4êEÔA$jÔOÒ_Ö�Ö@Ô$ç�æEÕ4ÔÒ}Ú�×MêPÔ=Ú+ÙdÝ_Ý_ÙN ­Ò_ÑPé�Ò²Ö�ÑEÔ�édÝ_Ò}édÒ[��Ý}Ô *� ÷ ¢w¤,¤ ø é ù /qjE2RÛÞ?Nïm@ ��

��ÙdæÖMêPÙdæPÝ_ÞCç>ÙdÑfÓ<Ò_Ñ�ç>Ô�Ø5ÙdæEÕ4Ö@Ô$Ý_Ú�×MêPÜ�×­×MêPÒ_ÖuÞEÔ,+�ÑPÒ}×MÒ}ÙdÑCÒ_ÖuÔ"O�Ü5ç>×MÝ}Ø:ÔA°<æPÒ}Ó�Ü5Ý}Ô$Ñf×u×@ÙRÙdæEÕ�ÙdÝ_Þ�ÞEÔ,+�Ñ�Ò}×MÒ}ÙdÑÙ5Ú�ç>Ù<$Rè�æE×MÜ�×MÒ}ÙdÑPÜ5ÝqÖ@Ô$ç�æEÕ4Ò}×¢ØP9����� �����������! "��#$�&%!�(')�+*,�� ��& -.�^ ÑPÒ_ç>Ô�Ú+Ô$Ü�×MæPÕMÔKÙ5Ú�×MêPÒ_ÖQÑEÔ, ½ÞPÔ,+�ÑPÒ}×MÒ}ÙdÑkÒ_Ö�×MêPÜ�×�Ò_Ö{Ô$Ü5ÖMÒ²Ý}Ø�Ô"O�×@Ô$ÑPÞPÖ�×@Ùl$RÙ<ÞEÔ$Ý<Ö@Ô$ç�æEÕVÒ}×±ØGÜ�édÜ5Ò²ÑPÖ@×�ç4êEÙdÖ@Ô$Ñè�Ý_Ü5Ò_Ñf×@Ô"O<×NÜ�×@×MÜ5çh¹�Ö,9 å Ñ×MêEÔ0ÞPÔ,+�ÑPÒ}×MÒ}ÙdÑZ×Mêfæ�Ö�Ú Ü�ÕAS� �ÔkÕMÔ$ÖM×@Õ4Ò_ç>×@Ô$Þ�×MêEÔ�Ü5ÞEÓ5Ô�Õ4ÖMÜ�ÕMØZ×@Ù'$¸Üm¹<Ò_ÑPé�ÙdÑPÝ_Ø�Ü� e1b`¼P�]a'°<æEÔ�ÕMØ~×@Ù�×MêEÔÅæ(çvÙ5Õ4Ü5ç�Ý_ÔP=.×MêPÒ_Ö� �Ü5Ö�$jÔ$Ü5Ñf×0×@Ù»$jÙ�ÞEÔ$ÝKç�Ò}è�êEÔ�ÕM×@Ô"O�×GÙdÑ�Ý}Ø~Ü�×@×MÜ5çh¹�Ö,90/iæE×; �Ôç�Ü5ÑHÖ@×@ÕMÔ$ÑPé5×MêEÔ$Ñ~×MêEÔRÞEÔ,+�ÑPÒ}×MÒ}ÙdÑ»�<Ø�Ü5Ý_Ý_ÙN ­Ò_ÑPé¤×MêPÔ¸Ü5ÞEÓ5Ô�Õ4ÖMÜ�Õ4Ø×@ÙÉ°<æEÔ�ÕMØ�×MêEÔjÙ5Õ4Ü5ç�Ý_Ô�«(�m�õ�mbZ�m¬�ef�P�ô�õ�m�¬.�PbE�:×MÒ]$jÔ$Öy/ Òq9«ÔP9[S�Ü5Ö�$RÜ5ÑfØ�×MÒ]$RÔ$ÖuÜ5Ö�×MêEÔ�è�ÙdÝ}Ø�ÑEÙ<$RÒ²Ü5Ý_Ý}Ø`!#��Ùdæ�ÑPÞEÔ$ÞRÜ5ÞEÓ5Ô�ÕVÖMÜ�ÕMØ:Ý_Ò[¹5Ô$Ö�2�9 å Ñ:Ú Ü5ç>×,SPÚ+Ù5Õç>Ù<$jè�Ý}Ô�×@Ô$ÑEÔ$Ö4Öl KÔ�èPÕMÔ$ÖMÔ$ÑY×N×MêPÒ_ÖNÞEÔ,+�ÑPÒ}×MÒ}ÙdÑ�Ò_Ñ�Ò}×MÖOÔ$ÑY×MÒ}Õ4Ô�×±ØPSqÜ5ÑPÞ�ç�Ü5Ý_Ý�×MêPÔ�ÑPÔ, ÁÑPÙ5×MÒ}ÙdÑ � ah²A®(g�efcf�Teôbc21(a � a"b � a;�43leôb��me � cfeôbå¼P®(e � 1å��ó,eô�õe1cf�k/ Ü5ÑPÞCç�Ü5Ý_Ý�Ü�ÖMçVêEÔA$jÔ�ÖMÜ�×MÒ_ÖMÚ+Ø�Ò_ÑEé0×MêEÔ=ÑEÙ5×MÒ}ÙdÑÇe1bZ�me � cfe1b`¼P®(e � 1å�`ó"�]aw2�95 �7698;:�<=:1�>8X�@? bVa,bZ²,g­��«�cfeq�mb � ²�1_a"¬&a�ð~3/1Ø�xsò.2.e � Ö@Ô$ç�æEÕMÔjÒ_Ñ�×MêEÔRÖMÔ$ÑPÖ@ÔjÙ5ÚKÝ_Ô�Ú+×u!BÙ5Õs!BÕVÒ}édêY×�Ò²Ñ(!ÞPÒ_Ö@×MÒ_ÑPédæPÒ_ÖMêPÜm��Ò_Ý_Ò}×¢Ø(A½ef©faA©CB � aI²,®(gsa��A¼`�me1b � c4²�1å� � a"b\«Z� �Peôbdc½aI�<cl�Pcqc#�<²,ö ��D e 3EBF3,�mg&�m�1�¯ËEËEÌä�<�HGPa"g � �mg­e#a �Í�c21(a�3,�m�ô�[�m 6e1b`¼'e � b�a#¼<��e[¼Pe#ó"�]aJI

ÏÏÏ ë�ÕwÐÑÃzj�ÒÜÓNÕ(xA?wÖ × =Iê�Ò·ÓNÕ(xA?wÖ * ÍLK þ ÿ�� ��� ��� é � � /u? × 263oêIÚZÛÞ?Nïm@ ÏÏÏ

ANMÅa|bZ�.�[�mb`¼åa"glgsa � cfg­ef²NcGc21(aybE®(¬&óIa"g|�43|cfe1¬�a � c21(a��<�HGPa"g � �mg­��¬��m�&ÝA®Ea,g��.æ(ç;O6�P3|²��m®(g � aQB � e1bZ²­aHÍe � ��ËEËEÌ��m� ¼`�mg­e1c21�¬ efcM²��mb�Ý,®da"g­�iæ(ç��PcW¬�� � c¯«(�P���mbZ�m¬�eq�m�1���m�B¬.�PbE�\cfeô¬&a � © D��ÙdæZÖMêEÙdæ�Ý_ÞCç>ÙdÑYÓ�Ò_ÑPç>ÔGØ5ÙdæEÕ4Ö@Ô$Ý}ÚQ×MêPÜ�×u×Mê�Ò_ÖuÞEÔ,+�Ñ�Ò}×MÒ}ÙdÑ�S�Ò_Ñ:è�Ü�ÕM×MÒ²ç�æPÝ_Ü�ÕASPÒ]$Rè�Ý_Ò}Ô$ÖiÖ@Ô$ç�æEÕVÒ}ױؤÔ�Ó5Ô$Ñ\ ­êPÔ$Ñ$kæPÝ_×MÒ}è�Ý}Ô4$jÔ$ÖMÖMÜ�é5Ô$Ö�Ü�ÕMÔNÔ$ÑPç>Õ4ØfèP×@Ô$Þ�S�Ü5Ñ�Þ¤Ü5ݲÖ@ÙjÒ]$jè�Ý_Ò_Ô$Ö�Ö@Ô$ç�æEÕ4Ò_×±Ø�Ü�édÜ5Ò²ÑPÖ@×�ç4êEÙdÖMÔ$Ñ:è�Ý_Ü5Ò_Ñf×@Ô"O<×iÜ�×@×MÜ5çh¹�ÖÜ5ÑPÞL/9Ù5Ú�ç>ÙdæEÕ4Ö@Ôz2³¹<ÑEÙz ­ÑCè�ݲÜ5Ò_ÑY×@Ô"O�×uÜ�×@×MÜ5çh¹�Ö,9

��Ùdæ�ÖMêEÙdæPÝ_ÞCÜ5ݲÖ@Ù&��ÔGÜm��Ý_Ô�×@Ù¤ÖMÔ�ÔG×MêPÜ�×�bZ�T��a,c½a,g�¬�e1bEe � cfeq²RA � c#�Pc½a"�]a ����D a"bZ²Ag���«�cfeq�mb � ²�1(a,¬�a&²I�mb� �Pcfe � 3­�Lc21�e � ��aSGbde1cfeq�mb5/1 ­êYØUTw2�9và<Ù�ÙdæEÕRÑPÔ"O<×RçVêPÜ5Ý_Ý}Ô$ÑEé5Ô' ­Ò²Ý_Ý���Ô:×@Ùhç>ÙdÑPÖMÒ²ÞEÔ�Õ�Õ4Ü5ÑPÞEÙ<$RÒ_î�Ô$Þ£Ô$Ñ_!ç>ÕMØfè�×MÒ}ÙdÑ�ÖMç4êEÔA$RÔ$Ö,9 å ×�Ü5Ý_Ö@Ù:Ö@Ô�ÔA$RÖ=×Mê�Ü�×NëW���NÖ�Ü�ÕMÔjÑEÙ5×|°<æPÒ}×@Ô�Ô$ÑEÙdæEédê�×@ÙCÙP�P×MÜ5Ò_Ñ�Ü5Ñ�Ô$ÑPç>Õ4ØfèP×MÒ_ÙdÑÖMç4êEÔA$RÔkÖMÜ�×MÒ_ÖMÚ+Ø�Ò_ÑEéj×MêPÒ²ÖuÞEÔ,+�ÑPÒ_×MÒ}ÙdÑ�=EÒ_Ñ:×MêEÔ0ÑEÔ"O<×�Ý}Ô$ç>×MæEÕMÔy �Ôy ­Ò²Ý_Ý�Ò_ÑY×@Õ4Ù<ÞPæ�ç>Ô�Ö@Ù<$jÔ�$RÙ5ÕMÔ�è�ÙN �Ô�ÕMÚ æPÝ$RÜ5ç4êPÒ²ÑEÔ�ÕMؤ×MêPÜ�×� ­Ò_ݲÝ�Ô$Ñ�Üm��Ý}Ô=ÜRÖ@ÙdÝ_æP×MÒ}ÙdÑ�9

V

Page 38: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

WYXUZC[�\�]E^_Za`"bdc�e9fhgH]EbjiCg�X�kl f0m l,n7o�prqts X7`P]EcjkUu�v�`_Zac&Xw`Pc l ]_bjxU`Pc�y�]EgHx{z7b|�]_c�e2\J^_^Pc�]~}�c&XUgH`_zUg�Xh��gH`P�

�@����������� ���

� �����_���,�>�������9��� �¢¡9�£��¤r�£�_¥¦�¨§ª©«�9���_¬£�4�>�!­

® c¯]_\Jv�gHx°`_z+\±\JXUk²c�e�iCg�^P`LiC\Jv�`_u+]_\�³.´,\±y&gN[�\µghk+\�¶�XUZa`_ZCc&X·c�eY¸J¹_ºJ»j¼Q½¾¿h½2À ¾2Á¹±¸J¹�À+¸J¹wÃ4Ī½2À{ÅH½�¸Q¾½2ÀUÆÇ »Â½�¸ÈÁjÉ7Ê�½�Ë̽2¾¿²Í2c�]Yg�X)½�À{Å�½�¸�¾½�À Ç »Â½�¸ÈÁjÉ7Ê�ËC¹�\JXUv�]Eb�xU`_ZCc&X¨^_vEz+\JΪ\=ÏÈÐ

ѨÒ7Ó9Ô;Õ�Ö=Õ2×>ÔÙØ@Ú¦À(¹�À{º�¼Q¿ÈÛ>¾½�ÃHÀ ¸�º�Á+¹�ܱ¹rÝßÞàÍ2á!âPã�Ï�½�¸w^P\Jv�u+]_\�ZCX·`_z+\ª^_\JXU^P\�c�e�iC\�e�`4äc�]Pä]ÈZay&z7`�Z�XÂäkUZC^P`_ZCXUy&uUZC^_zUgHå{ZCiCZa`"bR½ ÄÈæUÄ�ÃH¼rÉHË2Ë{çUçUè�É&ÅHéH¹�¼E¸�ÉH¼Q½�¹�¸Yêë¾2Á¹�Ä�ÃHË2ËìÃHí9½2À Ç ½�¸�À>¹ Ç Ëî½ Ç ½Ê�ËC¹Jï

ððð |�]òñôó=õªöø÷NùÂâJúòûNüÂýEþÿö�÷NùÂâJúòû��7ê������� � ������� �_Í4ú�ü&Ï9Þ þ����$ú���� ððð����� ¹�À{Ã�ËaÃHÀ Ç ¹�¼~¼P¹�¸Q¾¼Q½ºN¾�¾2Á¹�ÀU»ÂܱÊE¹�¼�Ã4Ä�¾½2ܪ¹�¸¦¾2Á¹rÉ&ÅHé�¹�¼È¸�ÉH¼Q¿ªÜªÉH¿ �J»U¹�¼È¿"!$#&%9ÃPÄ�ºÈÃH»Â¼E¸J¹Qæ ¸È½2À{ºQ¹�ê½�¸rÉ¢çUçUè�ÉHË Ç ÃH¼Q½2¾2Á7Ü ½¾,ºÈÉHÀ'��»�¹�¼Q¿(!$#�É�¾�ܪÃò¸Q¾.ÛÂÃ�Ëî¿HÀ{ÃHÜ¢½�ÉHË2Ëî¿HÆ Ü�É�ÀU¿R¾½�ܱ¹�¸*),+

-²\ªg�iC^PcwX+c�`P\Jk·`_zUgH`¦`_zUZC^~k+\�¶�XUZC`_Zac&XhZCΪx�iCZa\J^ÿ^_\Jv�u+]EZa`"bR´ z+\JX0Îru�ia`_Zax�ia\rÎ�\J^_^_gHy�\J^�gH]E\r\JXUv�]Eb�xU`P\Jk�³g�XUk·ZCÎ�x�iCZC\J^ ^P\Jv�u+]ÈZa`�b0gHy&g�ZCXU^P`�vEzUc&^P\JX·x�iCg�ZCX�`P\*.Â`¦gH`P`_g�v�/j^�Ð10�Ã�»¯¸ÈÁjÃH»ÂËaÅ0Êȹ±É7Ê�Ëa¹±¾Ã�Û>¼4ÃHé�¹wÊ_Ã�¾2Á)Ã4ľ2Á¹Q¸J¹L¸Q¾É�¾�¹�ܪ¹�À�¾2¸LܪÃH¼_¹,Ä�ÃH¼ÈܪÉHË2ËÌ¿2)

s `9´YZCiCijå£\ Z�XU^P`P]EuUv�`_ZC[�\�`PcL^_\�\�\*.Âg�Î�x{ia\J^9c�e£^_vEz+\JΪ\J^9`_zUgH`�gH]_\�À>Ã�¾ ^P\Jv�u+]_\ÿuUX�k+\�] `_zUZC^9k+\�¶�X�Za`_Zac&XÍ2`Pc�y�\�`�gre�\�\Ji{e2c�]�`_z+\~k+\�¶�X�Za`_Zac&XFÐQÐJÐÈÏÈÐ43+c�]�\*.+g�Î�x�ia\�³j`_g�/�\~`_z+\¦c&X+\�ä`_ZCÎ�\¦x�g�kd\JXUv�]_bjxU`_Zac&X±^EvEz+\JÎ�\�Ðl g�Xµb�c&u¨^_z+cN´@`_zUgH` `_zUZ�^�^_vÈz+\JÎ�\�k+cj\J^ X+c�`,^EgH`_ZC^Pe2bª`_z+\~gHå>c=[�\�k+\�¶�XUZC`_Zac&X�åjb±^_z+c=´ ZCX+y¢g�Xµ\*.jx�i�ZCv�Za`g�k+[�\�]E^_gH]_b¨êÙe�c�]Y´ zUZCvÈz

|�]=ñôóòõªöø÷NùÂâJúòû ü ýEþÿöø÷NùÂâJúòû5�7ê ��� �6� 7� �8����� � Í4ú ü Ï9Þ þ9�

ZC^ÿg�ia´ gJbÂ^LúH³�g�X�kw`_z+\Lg�k+[�\�]E^EgH]_bwc&XUiabdαg�/�\J^ÿ`�´,c(:ju+\�]EZC\J^ `Pc;!$#=<>�c&u±^_zUc&uUiCk�g�iC^Pc�å£\ÿgHå�iC\�`PcL^_z+c=´�`_zUgH`ÿÀ>Ã�Å�¹�¾�¹�¼QÜ¢½�À�½�¸�¾½º � ¸Q¾É�¾�¹�Ëa¹�¸E¸?+�¹�À>º�¼Q¿EÛ£¾½ÃHÀw¸�º�Á¹�ܱ¹�ºEÉHÀ

¸�É�¾½�¸�ÄQ¿¯¾2Á¹µÉ7ÊEÃHéH¹µÅ7¹@�ÀU½¾½Ã�À@Í2´ z�bA<=ÏÈбmjcwc&u+]�X+\*.Â`�vEz�g�iCia\JX+y�\�´ Z�iCi.å£\¢`PcRv�c&XU^_Z�k+\�]~]Eg�XUk+c&ΪZC��\Jk\JXUv�]_bjxU`_Zac&Xµ^_vEz+\JΪ\J^�Ð s `,g�iC^_cr^P\�\JΪ^ `_z�gH`,|CBED�^,gH]E\~XUc�`F:juUZa`P\Y\JX+c&u+y&zµ`Pcrc�å�`_g�ZCXwg�X±\JXUv�]Eb�xU`_ZCc&X^_vEz+\JΪ\r^_gH`_ZC^_e�bÂZCX+y�`_zUZ�^ÿk+\�¶�XUZC`_Zac&XFý+ZCXw`_z+\�X+\*.j`Yia\Jv�`_u+]_\�´,\�´ Z�iCi£ZCX7`P]EcjkUu�v�\�^Pc&Î�\rΪc�]_\�x£cN´,\�]_euUiΪg�vEzUZ�X+\�]_bµ`_zUgH`ÿ´ ZCi�i{\JX�gHå�ia\�gª^Pc&iCuU`_Zac&XFÐ

G ©«�ÿ�¢­�¥IH �9JLK�­ � �¢¡9�£�r¤r�F�_¥¦�NML¡&OPKQHRK�¬

SYX�`_ZCi.XUcN´�³>g�iCi�\*.+g�Î�x�ia\J^ c�e9\JXUv�]_bjxU`_Zac&XR^EvEz+\JÎ�\J^¦`_zUgH`Y´�\�zUgJ[�\¢^P\�\JX(Í*TPv�iCg�^_^_Z�v�g�iVU±v�Zax{z+\�]E^�³Uc&XU\�ä`_ZCÎ�\µx�g�kF³9|QBWD v�c&XU^P`P]Eu�v�`_Zac&X{ÏLzUgJ[�\wå£\�\JX kU\�`P\�]EΪZCXUZ�^P`_ZCvHÐX>�\�`¢´,\µ^_gN´�gHå>c=[�\w`_zUgH`�XUc0k+\�`P\�]_äΪZCXUZC^_`_ZCv²Í^P`_gH`P\Jia\J^_^ÈϪ\JXUv�]_bjxU`_Zac&X ^_vEzU\JÎ�\¯v�g�X å£\h^P\Jv�u+]_\¯ZCX�`_z+\0^P\JXU^P\Rc�e�Z�XUkUZC^P`_Z�X+y&uUZC^_z�gHå�ZCiCZì`"b�ÐY�\�` Z ^YåU]EZa\ [UbµZCX�`P]_cjk�uUv�\~`_z+\LX+c�`_Zac&XRc�e�]Èg�XUk+c&ΪZa��\JkR\JXUv�]_bjxU`_Zac&X¨^_vÈz+\JÎ�\J^�Ð]\ \�]E\�³U`_z+\�\JXUv�]Eb�xU`_ZCc&Xg�iay�c�]EZa`_zUΠΪZCy&z7`�vEzUc�c&^P\¨]Èg�XUk+c&Î å�Za`_^�´ z+\JX \JXUv�]Eb�xU`_Z�X+y0g²Îª\J^_^_gHy�\�Ð s X x�gH]_`_Z�v�uUiCgH]¨ÍZae `_z+\J^_\]Eg�XUk+c&Î å�Za`_^�gH]_\µ`Pc¯å£\wuU^P\�euUi�Ïȳ�`_z+\�]_\µ´ ZCiCi�XUcN´ å£\hܪÉHÀU¿)v�Zax{z+\�]_`P\*.Â`_^�v�c�]E]_\J^Px£c&XUkUZCXUy¨`Pc·g�X�b

ú

Page 39: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

^_ZCX+y&ia\~Î�\J^_^EgHy�\�³�g�XUkw\JXUv�]_bjxU`_Zac&XwZC^ X+c�iac&XUy�\�]�c&X+\�ä`PcHäc&X+\�ÐC>�\�`�`_z+\�k+\Jv�]_bjxU`_Zac&Xwg�iay�c�]EZC`_zUÎ ÎruU^_`^P`_ZCiCiF^Pc&Ϊ\Jz+cN´ ]_\Jv�cN[�\�]~`_z+\Lc�]EZay&ZCXUg�iFΪ\J^_^_gHy�\rg�i�iF`_z+\�`_ZCΪ\�Ð,|�\�]EzUgHx{^�`_zUZC^ÿ^_c&uUXUkU^ÿΪc�]_\LkUZ_^dv�uUia``_zUg�XRZa`�]_\Jg�i�iabµZ�^�ý´�\�y&ZC[�\Lgª^_Ϊg�i�i£\*.Âg�Î�x{ia\�z+\�]_\�Ð

l c&XU^_Z�k+\�]�`_z+\ªe2c&iCiac=´ ZCX+yw\JXUv�]_bjxU`_Zac&X°^_vEz+\JΪ\2�L`_z+\±^EzUgH]_\Jk`/�\�b²ZC^LóµÞ ó2aFb&ó�cH³;´ z+\�]_\ed ófagdFÞd ó�chdjÞ õ>Ð ® cd\JX�v�]_b�x�`YÎ�\J^_^EgHy�\�i²³£vEz+cjc&^P\¢g±]Èg�XUk+c&Î å�Za`Wj±g�X�kh`_z+\JX¯c&u+`Px�uU`lk«Þ Ímj=ânipo$ó�qJÏÈЮ c·k+\Jv�]_bjxU`Lv�Zax{z+\�]_`P\*.Â`Pklr!Þ Ímj�r2â9str�Ïȳ ^EZCÎ�x�iab·v�c&Î�x�uU`P\ iXr!ÞustrAo@ó qwv Ð;xYc�`P\µ`_zUgH`rk+\Jv�]Eb�xU`_ZCc&Xg�ia´,gNbj^Y^_uUv�v�\�\JkU^JÐy ´�c�]Ek�c&X$X+c�`_gH`_Zac&Xz�¯ZCX�k+\J^Ev�]EZaå�ZCXUy·g�X�\*.jx£\�]EZ�Î�\JX7`RÍZCX�^_ZCk+\¨g²xU]Ec�å�gHå�ZCiCZC`�b°\*.ÂxU]_\J^E^_Zac&X{Ï s

zUgJ[�\µå£\�\JX°v�gH]_\�euUi9`Pc·kUZC^P`_Z�X+y&uUZC^_z¯å£\�`�´,\�\JX kU\�`P\�]EΪZCXUZ�^P`_ZCv�\�[�\JX7`_^¢g�XUk°xU]_c�å�gHå{ZCiCZC^P`_Z�vLc&X+\J^JÐRmÂc+³\JXUv�]_bjxU`_ZCX+y·g·Î�\J^_^_gHy�\Xi uU^_ZCXUy0g·k+\�`P\�]EαZCXUZC^P`_Z�v�\JXUv�]_bjxU`_Zac&X(^_vÈz+\JÎ�\¨´,c&uUiCk å>\µ´ÿ]EZC`P`P\JX1k Þá${=Ími¯Ïµ´ zUZCiC\¯\JXUv�]Eb�xU`_ZCc&X@`_zUZC^¨Î�\J^_^EgHy�\°uU^_ZCXUy g�x�]_c�å�gHå�ZCi�ZC^P`_ZCv¨\JX�v�]_b�x�`_Zac&X ^_vEzU\JÎ�\)´,c&uUiCk@å£\´ÿ]EZa`P`P\JXek ö á|{JÍmi¯ÏÈÐ- z�gH` ΪZay&z�` `_z+\�g�k+[Hg�X7`_gHy�\rc�e;`_zUZ�^ \JXUv�]_bjxU`_Zac&Xw^_vEzU\JÎ�\�å£\~cN[�\�] `_zU\�c&X+\�ä`_ZCÎ�\�x�g�k}<°Í y e�`P\�]

g�iCi�³+´,\L^_zUgH]_\���õ¨å�Za`_^ å�u+`�c&X�iabd^P\JX�khõ�äå�Za`ÿÎ�\J^E^_gHy�\J^ ZCXw`_zU\LgHå£cN[�\L^_vÈz+\JÎ�\�ÐîÏ"-²\JiCi�³+`_zUZC^ ^_vEz+\JΪ\Tw/jZCX�k¨c�e9U�y&Za[�\J^Y^P\Jv�u+]EZa`"bd´Yz+\JX¨`�´,cdΪ\J^_^_gHy�\J^¦gH]_\L^P\JX�`�Ð s X¨x�gH]_`_Z�v�uUiCgH]J³+Zae~j�ZC^LÅH½ �¦¹�¼_¹�À{¾�e2c�] `"´�cÎ�\J^_^_gHy�\J^ `_zUgH`ÿy�\�` \JX�v�]_b�x�`P\Jk)Í2´ zUZCvÈz¨cjv�v�uU]E^�´ Za`_zµxU]Ec�å�gHå�ZCiCZC`�bhú����&Ï�`_z+\JX¨`_z+\�^_vÈz+\JÎ�\Lg�vÈzUZa\�[�\J^T4x>\�]Ee�\Jv�`,^P\Jv�]_\Jv�b$ULe2c�]�`_z+\J^P\Y`�´,c�Ϊ\J^_^_gHy�\J^�³Â^_ZCXUv�\�`_zU\�bªgH]_\ÿå£c�`_zª\JXUv�]Eb�xU`P\Jk±´ Za`_zªk�Z��>\�]E\JX7`�T4c&X+\�ä`_ZCÎ�\¦x�g�kU^9UÂÐC�¦e;v�c&u+]E^P\�³Â`_z+\�^_vÈz+\JÎ�\�g�^ gr´Yz+c&ia\YZ�^YÀ>Ã�¾!x£\�]_e2\Jv�`_iabª^P\Jv�]E\�`�³U^_ZCXUv�\¦´ Za`_zdxU]_c�å�gHå{ZCiCZa`"bú����µ`_z+\ªþÈ^~gH]_\¢`_z+\�^_g�Ϊ\�ZCX0´ z�ZCvEz0v�g�^P\�ZCX+e2c�]EΪgH`_Zac&X0gHå£c&u+`¦`_z+\¢`�´,cRÎ�\J^_^_gHy�\J^�ZC^~ia\Jg�/�\Jk�Ð��,uU`^_ZCXUv�\ª´,\dgH]E\�³ Z�X²y�\JX+\�]Eg�i�³ ZCX�`P\�]_\J^P`P\Jk ZCX)v�c&Î�x�uU`_gH`_Zac&XUg�i!^_\Jv�]_\Jv�b)c&XUiab�Í2´ z+\�]E\±´,\±]_\�:juUZa]_\�`_z�gH``_z+\Yg�k+[�\�]È^_gH]_b�X+c�`!å£\ÿgHå�iC\ÿ`PcrkUZ�^P`_ZCX+y&uUZ�^_z�å>\�`"´�\�\JXd\JXUv�]_bjxU`_Zac&X�^9c�e£`�´,c�kUZ_�>\�]_\JX�`�Î�\J^_^EgHy�\J^�´YZa`_zzUZay&zwxU]_c�å{gHå�ZCiCZa`"b+Ï!`_zUZ�^�gHxUxU]_c&g�vÈzhΪZay&z�`�å£\~x�iCg�uU^_ZCå�ia\�Ð

s `,´ ZCi�i+å£\ ´,c�]_`_z7´YzUZCia\ÿ`Pc¢ZCX�[�\J^P`_Zay&gH`P\~´ zUgH`,^Pc�]_`,c�e.^P\Jv�u+]ÈZa`�b�y&uUgH]Èg�X7`P\�\J^ ´,\¦v�g�Xµy&Za[�\¦e2c�]�`_zUZ�^^_vEz+\JΪ\�Ð]D~Za[�\JXhg�XRg�k+[�\�]E^_gH]Ebwêr³�ia\�` Z ^ iac�c2/wgH` �

ððð |�]Nñôó~öø÷NùÂâJúòû c ü+ýEþ�öø÷NùÂâJúòûP�&ê���� �� ������� �_Í4ú�ü�Ï�Þ þ����$ú���� ððð â

´ z+\�]_\¦´�\�X+c=´ ]_\J^P`P]EZ�v�`�c&u+]�g�k+[�\�]È^_gH]_b±`Pc�Ϊg�/ÂZCX+y�c&XUiab±`�´,c�:�u+\�]ÈZa\J^!`PcX!$#�Ð|x c�`P\�`_zUgH` `_z+\�]E\~gH]E\`�´,cdx£c&^_^_ZCå�ZCiCZa`_ZC\J^!e2c�] `_z+\r`�´,c�:ju+\�]EZC\J^�`Pce!$#~��\JZa`_zU\�] `_z+\L^Eg�Î�\�jªZC^YuU^P\Jk¨\Jg�vÈz0`_ZCÎ�\�c�]~X+c�`�³{g�X�k\Jg�vEz0v�g�^P\�cÂv�v�u+]E^�´ ZC`_zwxU]_c�å�gHå�Z�iCZa`�bRú����jÐFY�\�`Wjfa�kU\JX+c�`P\�`_z+\�å{Za`�uU^P\Jkw`Pc±g�XU^_´�\�]ÿ`_zU\�¶U]E^P`W:ju+\�]_b`Pc;!$#²g�XUk¨ia\�`Wj�c�kU\JX+c�`P\�`_z+\�å�ZC` u�^P\Jk¨`Pc±g�XU^_´�\�]ÿ`_zU\L^P\Jv�c&XUk�:ju+\�]_b�Ð ® z�u�^ �

ððð |!]=ñôó¦ö�÷NùÂâJúòû c ü ýEþÿöø÷NùÂâJúòû��7ê �t� �� �8����� � Í4ú ü Ï�Þ þ����$ú���� ðððÞ ððð ú�����bJ|�]Nñôó�öø÷NùÂâJúòû c ü ýEþÿö�÷NùÂâJúòû��7ê ������ 9������� � Í4ú ü Ï9Þ þ�dgj2a,Þ�j�ct�

� ú����lbN|!]Nñôó¦ö ÷NùÂâJúòû c ü ýEþ ö ÷NùÂâJúòû��7ê �t� �� �8����� � Í4ú ü Ï9Þ þ�dgj2a��Þ�jgc��}�$ú���� ððð �® z+\~ZCÎ�x£c�]_`_g�X�`�`_zUZ�X+yL`Pc�X+c�`P\�ZC^�`_zUgH`�³+´ z+\JX(j2a��Þ�jgc=³Â`_z+\�g�k+[�\�]È^_gH]_b¨ÁjÉH¸~À>Ã�½2ÀJÄ�Ã�¼ÈܪÉ�¾½�ÃHÀ)É7Ê_Ã�»Â¾¾2Á¹�éHÉHËÌ»U¹dÃ4Ä�þ±Í2å£\Jv�g�uU^_\�`"´�cRk�Z��>\�]E\JX7`¦c&X+\�ä`_ZCÎ�\¢x�g�k�^¦gH]E\¢u�^P\Jk{ÏÈÐ ® zjuU^�³F|�]Nñôó�öt÷NùÂâJúòû c ü ýEþ�ö÷NùÂâJúòû5�&ê ������ ��8����� � Í4ú ü Ï�Þ þ�dgj2a��Þ�jgc��FÞ ú����j³�g�XUk

ððð |�]=ñôó~öø÷NùÂâJúòû c ü ýEþÿöø÷NùÂâJúòû5�7ê ��� �6� ������� � Í4ú ü Ï�Þ þ����$ú���� ðððÞ ððð ú�����bJ|�]Nñôó�öø÷NùÂâJúòû c ü ýEþÿöø÷NùÂâJúòû5�7ê �t�g�6� ��8����� � Í4ú ü Ï9Þ þldgjfa,Þ�j�ct���$ú�� n ððð� ú�� n â

Page 40: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

´ z+\�]_\¨`_z+\¨¶{XUg�i�ZCX+\�:juUg�iCZa`"b z+c&i�kU^¢å£\Jv�g�uU^P\¨xU]_c�å{gHå�ZCiCZa`_ZC\J^rgH]_\Rg�iC´,gNbj^±å>\�`"´�\�\JX ù²g�XUk úHÐ ® z+\x>c&Z�X7`�ZC^,`_zUgH`�`_z+\�g�kU[�\�]E^_gH]_bµk+c�\J^�XUc�`�y&u+\J^_^ÿþYv�c�]_]_\Jv�`_iabµ´ Za`_zµxU]_c�å{gHå�ZCiCZa`"bRúªÍg�^�´ g�^ `_z+\�v�g�^P\�Zae´�\�g�iCiac=´ `_z+\�g�kU[�\�]E^_gH]_b¯`Pc�:ju+\�]_b'!$#°`�´ ZCv�\�ZCX¯`_z+\¢c&X+\�ä`_ZCΪ\rx�g�k·^_vÈz+\JÎ�\=Ïȳ�^Pc¨gH`�ia\Jg�^P`¦´,\¢gH]E\Ϊg�/jZ�X+yª^Pc&Î�\�x�]_c�y�]_\J^_^ ��¦e{v�c&uU]E^P\�³&`_zUZ�^�Z�^�XUc�` XU\�y&iCZay&Zaå�iC\�³òå{u+`�iC\�` Z ^9^P\�\ÿz+cN´�egH]9´,\ÿv�g�X¢\*.j`P\JXUkª`_zUZC^ gHxUxU]_c&g�vÈzFÐ�- z�gH`

Zae"³ ZCXU^_`P\Jg�k)c�eÿ^_zUgH]ÈZCX+yR`"´�c�/�\�bj^¢ó a âEó c `_zU\±x�gH]_`_ZC\J^L^_zUgH]E\Jk���/�\�bÂ^�ó a â � � � âEó�� ÍXUcN´ `_z+\±`Pc�`_g�i^_Za��\ c�eF`_zU\Y^_zUgH]E\Jk"/�\�bdó¦ZC^Q�·õ�å{Za`_^ÈÏn<�xYcN´�³j`_z+\YxU]_c�å�gHå�Z�iCZa`�br`_zUgH`,`_z+\Y^_g�Ϊ\�/�\�bªZC^�uU^P\Jk±Z�X�å£c�`_z\JXUv�]_bjxU`_Zac&XU^,ZC^�]_\JkUuUv�\Jkd`Pcµú��t� ÐL>�c&uw^_z+c&u�iCk±vÈz+\Jv9/d`_zUgH`�`_z�ZC^!y&ZC[�\J^~Íu�^_ZCX+yr`_z+\Y^_g�Ϊ\~kU\�]EZa[HgH`_Zac&Xg�^ gHå£cN[�\=Ï��

ððð |!]Nñôó¦öø÷NùÂâJúòû � üÂýEþÿöø÷NùÂâJúòûP�&ê��t� �6� ������� �_Í4ú�üHÏ!Þ þ����$ú���� ððð � ú������ �® z�u�^�³+`Pc±Îªg�/�\�`_z�ZC^ÿX+\�y&iCZCy&Zaå�ia\�ÍZCXRõUÏ,´,\]�_uU^P`ÿ^P\�`W� Þ�� ü Ð y X�k¨´�\rgH]_\Lk+c&X+\�Ð�Ð�ÐEc�] gH]_\�´,\g<

® z+\�¶�]E^P`�`_zUZCXUy�`Pc±X+c�`_ZCv�\�ZC^ `_zUgH`Y^_\�`P`_ZCX+y � Þ�� ü ZC^�XUc�` c&XUiabwZCÎ�x�]Eg�v�`_ZCv�g�i�³Âå�u+`YZC^�X+c�` \�[�\JX`P\JvEzUXUZ�v�g�iCiabRg�iCiac=´�\Jk�Ð ® z+\L\JXUv�]Eb�xU`_ZCc&X¯g�XUk0k+\Jv�]_bjxU`_Zac&X0g�iay�c�]EZC`_zUΪ^¦X+\�\Jkh`Pcw]Eu�XhZCX¯x£c&iabjXUc&ΪZCg�i`_ZCÎ�\dÍgHe2`P\�]¦g�i�i�³U´ z�bµ^Ez+c&uUiCkw`_z+\�^P\JXUk+\�]Yg�XUkR]_\Jv�\JZC[�\�] å£\LÎ�c�]_\Lx>c=´�\�]Ee2uUi�`_zUg�X¨`_z+\�g�k+[�\�]E^_gH]_bA<òÏg�XUk)`_zjuU^Lv�g�X�X+c�`L\�[�\JX(^P`Pc�]_\µg�/�\�b²c�e�ia\JX+y�`_z õ�bf� ü Ðhmjch´�\µzUgJ[�\µ`Pc¯ZCΪxU]_cN[�\±`_zUZC^L^_c&Î�\Jz+cN´LÐ3Uu+]_`_z+\�]ÈÎ�c�]_\�³�´,\L^P\�\JÎ `Pc±zUgN[�\ry�c&XU\L`_z+]_c&uUy&zRg�XhgN´ÿe2u�i�g�Î�c&uUX�` c�e ´,c�]9/¨g�XUk¨´,\r^P`_ZCi�iFzUgJ[�\JX=Zô`g�vEzUZa\�[�\Jkhc&u+] y�c&g�i��`_z+\�^EvEz+\JÎ�\�gHå£c=[�\LΪgJbµxU]_c�`P\Jv�`ÿgHy&g�Z�XU^P`ÿg�X¨g�k+[�\�]E^EgH]_bµÎ±g�/jZCXUy�`"´�c�:ju+\�]EZC\J^`Pc�`_z+\�!$#µc�]Eg�v�ia\�³�å�u+` ´�\ ´,g�X�`!^P\Jv�u+]ÈZa`�b¢gHy&g�ZCXU^P`�g�X±g�k+[�\�]E^_gH]_b�Ϊg�/ÂZCX+y�g�X�b¢x>c&iCbjX+c&αZCg�iÂX�uUÎrå£\�]c�e�:ju+\�]EZC\J^ `Pc;!$#&�

® c�]_\Jv�gHx�³j´ zUgH`�´�\2Z kµiCZ_/�\Y`Pc�kUcrZC^!`PcLå£\YgHå�iC\ÿ`Pc�g�v�v�\J^E^E���*��Þ�� ü kUZ��£\�]_\JX�`YÍ^_zUgH]_\Jk>Ï ]Eg�XUkUc&Î/�\�bj^�³.b�\�`LX+c�`�zUgJ[�\±`PcR^P`Pc�]_\�`_z+\JÎ g�iCiÐr|�\�]EzUgHx�^~´,\�v�g�X)k+cwå£\�`P`P\�]�åjb¯^_zUgH]ÈZCX+y±Û�¸�¹�»+Å7ÃH¼4É�À{Å&ÃHÜ/�\�bj^~ZCXU^_`P\Jg�kRc�eÿ¼PÉHÀ{Å&Ã�Ü�/�\�bj^�Р\ c=´«ÎªZay&z�` `_zUZC^Yiacjc2/|<'-²\JiCi�³>ia\�` Z ^~ZCΪgHy&ZCX+\L´�\�zUgJ[�\�g±|CBWD¢¡`_zUgH`r^P`P]E\�`_vEz+\J^¢õ�äå�Za`�^P\�\Jk�^�`Pc0õ$�häå�Za`�x�^_\JuUk+c�]Eg�XUkUc&Î ^P`P]EZCX+y&^JÐ ® z+\JX²`_zU\ªx�gH]_`_Za\J^Lv�c&uUiCk²^_zUgH]E\g0^_\�\Jk(ó¨c�eYia\JX+y�`_z(õ g�XUk(uU^P\�¡à`Pc·kU\�]EZa[�\�/�\�bj^ªófa�â � � � âEó � Í2´ z+\JX(X+\�\Jk+\Jk{ÏÈÐ ® zU\�]_\µgH]E\µ`"´�cxU]_c�å�ia\Jα^~´ Za`_z²`_zUZ�^�gHxUxU]_c&g�vÈz=�µÍ4úNÏ�Za`�´ ZCiCi�]_\�:juUZa]E\�\*.Âx£c&X+\JX�`_ZCg�i9`_ZCÎ�\�`Pc0v�c&Î�x�u+`P\�¡�Í�óòϪÍgHe�`P\�]g�iCi�³£¡�Ím¤£ÏYzUg�^¦\*.Âx£c&X+\JX7`_Z�g�iCiab�ä�Ϊg�X7bhå�Za`_^ �îϯg�XUk0`_zUZC^¦ZC^~X+c�`~g�iCiac=´�\JkFý!Í�&ÏY´,\�^P`_gH`P\Jk·\JgH]EiCZC\�]¦`_z�gH`c&u+]�|QBWD�v�c&X�^P`P]EuUv�`_Zac&X�^~´�\�]_\�y&u�gH]Eg�X7`P\�\Jk°`Pc¨å£\¢^_\Jv�u+]_\hÃHÀUËÌ¿0´ z+\JX0`_z+\�b·^P`P]_\�`_vEz°`_z+\JZa]�ZCX+x{u+`_^å�b¨grÛÂÃHËÌ¿HÀ{ÃHÜ¢½�ÉHË!g�Î�c&uUX�`�ÐQ� u+` zU\�]_\�´�\rX+\�\Jk¨`Pc±^P`P]E\�`_vEzRåjb¨g�X¨\*.Âx>c&XU\JX7`_ZCg�iFg�Ϊc&uUX7` �

s XRe2g�v�`�³{´�\�´ Z�iCi£X+\�\JkRg�xU]ÈZCΪZa`_Za[�\¦´ zUZ�vEz¨ZC^ÿ^_`P]_c&X+y�\�]ÿ`_zUg�Xhgª|QBWD�Ð

¥ ��¬}KQ¦¢­�¥Y�F�ÿ�¢­�¥IH §"¦��¢¡��F�_¥Y�¢¬Ù�ÿ�¢­ �¨K!�zH©¦r�£�,�£�_¥¦�¢¬

y |QBWDà´,g�^�XUZCv�\�³ å£\Jv�g�uU^P\µZa`rg�i�iacN´,\Jk u�^L`Pc·^_zUgH]_\d]Eg�XUk+c&Ϊ/�\�bÂ^rc�eÿ^EΪg�iCi!^EZa��\dg�XUk y�\JX+\�]EgH`P\^_zUgH]_\Jk(x�^_\JuUk+c�]Eg�XUkUc&Ϊ/�\�bÂ^¢c�e~iCgH]_y�\¨^_Za��\�Ы�,uU`�\�[�\JX$XUZCv�\�]¢`_zUg�X�zUgN[jZ�X+y²^_zUgH]_\Jk(g�v�v�\J^_^±`Pc)gx�^P\JuUk+c�]Èg�XUk+c&Î ¸�¾¼È½2À Ç ´�c&u�iCk)å£\±^_z�gH]_\Jk g�v�v�\J^E^r`Pc·g¯x�^P\JuUk+c�]Èg�XUk+c&Î�ÄQ»ÂÀ{ºJ¾½�ÃHÀUÐ�Y�\�` Z ^¢iacjc2/)gH`´ zUgH`ÿ`_zUZ�^�ΪZay&z�`ÿÎ�\Jg�XFÐ3+c�]ª^_ZCΪx�iCZCv�Za`"b�³.´,\µv�c&X�^_ZCk+\�]re2uUX�v�`_Zac&XU^�´YZa`_z)¶|.Â\Jk ZCX+x�u+`rg�XUk°c&uU`Px�u+`riC\JX+y�`_zFÐ l c&X�^_ZCk+\�]�g

e2uUX�v�`_Zac&X1¬­� ÷NùÂâJúòû ü;® ÷NùÂâJúòû�¯ª° ÷NùÂâJúòû�±�ZCX(´ zUZCvÈz(`_z+\¨¶U]E^_`¢Z�X+x�u+`�`Pc�¬ ZC^�v�g�i�ia\Jk(`_z+\`²&¹�¿g�XUk²`_z+\±^_\Jv�c&XUk°ZCXUx�u+`¦`Pce¬ëZC^L^_ZCÎ�x�iCbhv�g�iCia\Jk²`_zU\R½�À�Û{»+¾Ðdõ²ZC^�`_zU\�²�¹�¿hËa¹�À Ç ¾2Á7³~i ZC^�`_zU\R½�À�Û{»+¾Ëa¹�À Ç ¾2ÁRg�XUk�³²ZC^ `_z+\ªÃH»+¾îÛ{»+¾�Ëa¹�À Ç ¾2Á¨c�e~¬�ÐQ¬ ZC^ ZCXweg�v�`ÿg�²�¹�¿7¹_ÅYÄÈ»ÂÀ{ºN¾½Ã�À0ZCXµ`_z+\�e2c&iCiac=´ ZCX+y¢^P\JXU^_\2�¶|.ÂZ�X+y)g²x�gH]_`_Z�v�uUiCgH]"/�\�b$óµ´ ÷NùÂâJúòû ü kU\�¶�X+\J^ªg²euUXUv�`_ZCc&X¶¬~{e�Y÷NùÂâJúòû�¯·° ÷NùÂâJúòû�±$g�^ªe2c&iCiacN´Y^ �¬~{òÍm¤£Ï9Þ¸¬dÍ�ó&ân¤FÏÈÐ�mjcYb�c&u¢v�g�X¢[ÂZa\�´�vEz+cjc&^_ZCXUy¦g¦x�gH]_`_Z�v�uUiCgH]£/�\�b�g�^9vEz+cjc&^_ZCXUy¦g¦x�gH]_`_Z�v�uUiCgH]�e2uUX�v�`_Zac&Xe�]_c&Îà^Pc&Î�\LiCgH]Ey�\�^P\�`ÿc�e e2u�XUv�`_Zac&XU^�Ð

¹

Page 41: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

x c=´�³FZCX+e2c�]EΪg�iCiCbd^Px£\Jg�/ÂZCX+y+³�c&uU]Y|QBWD~^~zUg�kh`_z+\Lx�]_c�x£\�]_`�b¨`_zUgH`Y`_z+\JZa]Yc&u+`Px�u+`_^�TPiacjc2/�\Jkh]Eg�X+äk+c&Î�U¯´ z+\JX)\�[Hg�iCuUgH`P\Jk c&X(gR]Eg�X�k+c&Î ^_\�\JkFÐ ® z+\µg�XUg�iac�y�c&uU^�xU]Ec�x>\�]E`�b·´�\µ´,c&uUiCk°i�Z_/�\ªe2]_c&Ϊ¬ZC^�`_zUgH`�Za`�^_z+c&u�iCk�TPiacjc2/²iCZ_/�\±g¨]Èg�XUk+c&Î euUXUv�`_ZCc&XAUw´ z+\JX)gh]Eg�XUk+c&κ/�\�b)ZC^�vÈz+c&^P\JXFÐX� u+`�´ z�gH`\*.Âg�v�`_iab¨ZC^ g�]Eg�XUk+c&Î euUXUv�`_ZCc&X�<y ]Eg�XUk+c&ΫeuUXUv�`_ZCc&X�v�g�X�å£\�[jZa\�´,\JkªZCX¢`_zU\ÿe�c&i�iacN´ Z�X+y~´,gNb��9Zae{b�c&uªy&Za[�\ÿ`_zU\ÿe2u�XUv�`_Zac&X�g�XªZCXUx�u+`

¤�´ ÷NùÂâJúòû ¯ `_z�gH`¢ZC`rzUg�^�X+c�`¢^P\�\JX å>\�e2c�]_\�³9`_z+\µeuUXUv�`_Zac&X°x�ZCv9/Â^�gh]Eg�X�k+c&κ»¼´ ÷NùÂâJúòû ± ³�^P`Pc�]_\J^Ím¤.ân»UÏ,e�c�]Ye2uU`_u+]_\~]_\�e2\�]_\JXUv�\�³�g�XUkRc&u+`Px�u+`_^]»>Ð s e;b�c&uRi�gH`P\�]Yg�^n/µ`_z+\�euUXUv�`_Zac&X¨g�XhZCX+x�u+`F¤0´ zUZCvÈz¨Za`ÁjÉò¸±^P\�\JX²å£\�e2c�]_\�³�Za`~¶�XUk�^Y`_zU\ªgHxUxU]_c�xU]ÈZCgH`P\¢x�g�Za]ªÍm¤.ân»UϦg�XUk·]_\�`_u+]ÈXU^ »(Í^Pc¨`_zU\�e2uUX�v�`_Zac&X·g�ia´,gNbj^]_\�x£c�]_`_^�`_z+\L^Eg�Î�\�g�XU^P´,\�]YZae;b�c&uhg�^n/wZa`�`_z+\r^_g�Î�\�:ju+\�]_bUÏÈÐ-²\Yv�g�Xµg�iC^Pc�[ÂZa\�´�g�iCiU`_z+\J^P\ vÈz+c&ZCv�\J^,g�^�å£\JZCX+y�Ϊg�k+\YZCXªg�k+[Hg�XUv�\�³�Z�XU^P`P\Jg�kªc�eFu+x£c&X¢]_\Jv�\JZC[jZCXUy�g

:�u+\�]Eb�Ð ® zjuU^�³Ne�c�];\�[�\�]_b~x>c&^E^_Zaå�ia\ ZCX+x{u+`½¤�¾�´·÷NùÂâJúòû�¯(`_z+\�euUXUv�`_Zac&X�vÈz+c�c&^_\J^;g�]Èg�XUk+c&θ»2¾�´·÷NùÂâJúòû�±g�XUk(^P`Pc�]_\J^RÍm¤�¾4ân»2¾Ï�Z�X ^_c&Î�\µzju+y�\w`_gHå�ia\�Ð s X e2g�v�`�³�Zaeÿ´,\µc�]Èk+\�]r`_zU\�¤>^¢ia\*.ÂZ�v�c�y�]EgHx�zUZCv�g�i�iab�³9Za`�Z�^\JX+c&u+y&z¨`Pc��PuU^_` ^_`Pc�]_\�`_z+\�c�]Èk+\�]_\JkRiCZ�^P`]» a â � � � ân» ± Ð\ c=´ Î�uUvEz·^Px�g�v�\�´,c&uUiCk¯`_zUZC^~g�v�`_uUg�i�iabR`_g�/�\ªZae�´,\¢´�\�]E\�`Pc¨ZCΪx�ia\JÎ�\JX�` `_zUZ�^�<µ-²\JiCi³F\Jg�vEz`»2¾]_\�:�u�Za]_\J^I³°å�Za`_^¦g�XUk0`_z+\�]_\¢gH]_\�� ¯ c�e�`_z+\JÎ `Pcw^P`Pc�]_\�³Fe2c�]�g±`Pc�`_g�i c�e4³eb�� ¯ å�ZC`_^ c�e!^_`Pc�]EgHy�\ �Pu�^P``Pch]_\�xU]E\J^P\JX7`rg¨]Eg�XUk+c&Î e2uUX�v�`_Zac&XFÐ l iC\JgH]Eiab�³.Za`rZC^�ZCX+e2\Jg�^_Zaå�iC\�`Pc¯^P`Pc�]_\dgR`P]EuUiab0]Eg�XUk+c&Î e2uUX�v�`_Zac&Xe�c�]µÎ�cjkU\�]EgH`P\h[Hg�iCu+\J^±c�e i²ân³9Ð�� u+`ª´,\¯v�g�X v�\�]_`_g�ZCX�iab Z�ΪgHy&ZCX+\R^_u�vEz�g°`_zUZCX+y²\*.+ZC^P`_Z�X+y°g�^±g�XÃH¼4É7º�ËC¹�Ð�Í s X e2g�v�`�³9b�c&u(v�c&uUi�k°ZCΪgHy&Z�X+\±ZCÎ�x{ia\JÎ�\JX�`_ZCX+yRg¯]Eg�XUk+c&Î e2uUX�v�`_Zac&X°Zae `_z+\wg�XU^P´,\�]E^r´�\�]E\vEz+c&^P\JX�T4c&XÂä`_z+\�ä6[�b¿UdZCXU^P`P\Jg�k¨c�e9ZCXhg�k+[òg�X�v�\�ÐC� u+` `_zU\�]_\�´,c&uUiCkwå£\L^Pc&Î�\Lå>c&u�XUkwc&X¨`_z+\�X�uUÎrå£\�]c�e�kUZ��£\�]_\JX7`I:�u+\�]ÈZa\J^�b�c&u0v�c&uUiCkhg�^9/wZC`�³{k+\�x£\JXUkUZ�X+y¢c&XR`_z+\�g�Î�c&uUX�` c�e9^P`Pc�]EgHy�\�`_zUgH`YZ�^ gJ[Hg�ZCiCgHå{ia\�Ðs eÿb�c&u ´ g�X7`¢`Pc ¸EÁÂÉH¼P¹¯g¯]Eg�XUk+c&Î euUXUv�`_Zac&X�³ zUcN´,\�[�\�]J³�`_z+\¨g�XU^P´,\�]E^�Îru�^P`rå£\wk+\Jv�ZCk+\Jk u+x£c&X ZCXg�k+[òg�X�v�\�Ð y XUk¨^_ZCXUv�\~b�c&u¯k+c&X=Zô`E/ÂX+cN´ ´YzUgH`ÿ`_z+\�:ju+\�]EZa\J^�gH]E\�y�c&ZCX+y�`Pcªå£\�³+`_zUZ�^ ]_\�:�u�Za]_\J^�^_zUgH]ÈZCX+y`_z+\ euUiCi|³5bn��¯ å{Za`_^�ÐîÏ�-²\¦kU\JX+c�`P\ `_z+\¦^P\�`!c�e�g�i�iUeuUXUv�`_Zac&XU^!e�]Ec&Î�÷NùÂâJúòû�¯ `Pcd÷NùÂâJúòû�±ªå�b�#=ÀÂÁgà ¯FÄl± Ð-²\ ´ Z�iCi�XUcN´�k+\�¶{X+\ÿg~e�c�]ÈΪg�iUX+c�`_Zac&X¢c�eFg~x�^_\JuUk+c�]Eg�XUkUc&Î e2uUX�v�`_Zac&XRÍ|QBI39ÏÈÐÂBÿ\Jv�g�iCi+c&uU]�ZCX+e2c�]Pä

Ϊg�i�k+\�¶�X�Za`_Zac&Xµ`_zUgH` gª|CB�3ÅTPiacjc2/j^ iCZ_/�\�U�g�]Eg�X�k+c&Î�euUXUv�`_Zac&XFý+Î�c�]E\�e�c�]Eαg�iCiab�³Ug±çUç�èhg�k+[�\�]È^_gH]_bv�g�XUX+c�`¢`P\JiCi�`_z+\JÎ gHx{gH]_`r´YZa`_z°`Pcjc·zUZay&z ghxU]Ec�å�gHå�ZCiCZC`�b�Ð s X°´YzUgH`L´ gJbA<Æ-²\JiCi³ ^EgJb°´,\µy&Za[�\µg�Xg�k+[�\�]E^_gH]_b$g)å�iCg�v9/(å£c�.$g�XUk�`P\JiCiÿZa`±`_zUgH`ª`_z+\¯å>c�.$ZCÎ�x�ia\JΪ\JX7`_^¢\JZa`_zU\�]dg²]Eg�X�k+c&Î euUXUv�`_Zac&X�c�]ghx�^_\JuUk+c�]Eg�XUkUc&Î euUXUv�`_Zac&X g�XUk g�^n/°Za`�`Pc0kU\�`P\�]EΪZCX+\d´ zUZCvÈz)c&X+\�Ð ® zU\JX)`_z+\µxU]_c�å�gHå{ZCiCZa`"b0`_z�gH``_z+\±g�k+[�\�]E^EgH]_b²y&u+\J^_^P\J^eT4x�^P\JuUkUc�]Eg�XUk+c&Î�U¨^Ez+c&uUiCk·å£\ª`_z+\d^_g�Î�\ª´ zU\�`_z+\�]�`_z+\de2uUX�v�`_Zac&X²ZC^�g�v�`_uÂäg�iCiabRx�^_\JuUk+c�]Eg�XUkUc&Î c�]�XUc�`�Тfhc�]_\�e2c�]EΪg�iCiab�³FgweuUXUv�`_Zac&XǬ���÷NùÂâJúòû ü ® ÷NùÂâJúòû ¯ ° ÷NùÂâJúòû ± ZC^~gÍmÈÈâ�ÉQÏ�ä�|QB�3�Zae.e2c�] g�iCi�g�iCy�c�]EZa`_zUΪ^�êÙ]EuUX�XUZCX+y¢ZCXw`_ZCÎ�\�gH`YÎ�c&^P`WÈ,`_z+\�e2c&iCiac=´ ZCX+y�z+c&iCk�^ �

ððð |�]Nñôó~ö�÷NùÂâJúòû ü �&ê Ê ��� � Þ út�}�)|�]Nñ ¬ ö�#=ÀhÁ�à ¯]Äl± �&ê Ê �8� � Þ út� ððð � É �Í7x c�`P\r`_zUgH`ÿz+\�]_\~´,\�zUgJ[�\rõwg¢¶A.j\Jkw[Hg�iCu+\�³+^Pc�`_z+\�gHå£c=[�\LZC^ X+c�`ÿg¢euUXUv�`_Zac&Xµc�e g¢x�gH]Eg�Î�\�`P\�]Yõwg�^´,g�^L`_z+\ªv�g�^P\dZ�X·xU]_\�[ÂZac&uU^~ia\Jv�`_u+]_\J^JÐdmÂZCΪZCi�gH]Eiab�³FZCXU^P`P\Jg�k·c�e Ϊg�/ÂZCX+yRêß]EuUX²ZCX·x£c&iabÂX+c&ΪZ�g�i;`_Z�Î�\´�\wzUgJ[�\wZCXU^_`P\Jg�k)¶|.Â\Jk°g�X u+xUx£\�]�å£c&uUXUk`È~e2c�]r`_zU\dg�k+[�\�]E^EgH]_b�Z ^r]EuUX�XUZCX+yw`_ZCÎ�\�Ð ® z+\±ZCX�`P\JX7`_ZCc&X´,g�^L`Pc¯^_ZCΪx�iCZae2b¨`_zUZCX+y&^�ghiCZa`P`_ia\�å�ZC` �îÏ s X)´�c�]Ek�^�³�`_z+\±iC\�e�`�\*.ÂxU]_\J^_^EZac&X)v�c&XU^_Z�k+\�]�`_z+\ª\*.Âx£\�]EZCÎ�\JX�`ZCX¨´ zUZ�vEzRg±]Eg�XUk+c&Î�/�\�bhóLZC^YvEz+c&^_\JXF³{g�XUk¯ê ZC^ÿy&Za[�\JXhc�]Eg�v�ia\�g�v�v�\J^_^ `Pcd`_z+\LeuUXUv�`_ZCc&X�¬~{JÐ s XR`_zU\]EZay&z�` \*.Âx>\�]ÈZCÎ�\JX�`�³{gµv�c&Î�x�ia\�`P\Jiab¨]Èg�XUk+c&ÎëeuUXUv�`_ZCc&X;¬ße�]_c&Ωi å{Za`_^ÿ`Pc�³°å�ZC`_^ ZC^YvEz+c&^P\JX�³£g�X�khêZC^�y&Za[�\JX²c�]Èg�v�ia\±g�v�v�\J^_^r`Pc;¬�Ð�-²\dv�iCg�ZCÎ `_zUgH`rêßv�g�XUXUc�`LkUZC^P`_Z�X+y&uUZC^_z¯å£\�`�´,\�\JX)`_z+\J^_\ª`�´,chv�g�^P\J^´ Za`_zwxU]_c�å{gHå�ZCiCZa`"bªå£\�`P`P\�] `_zUg�XeÉ�Ð-²\µg�iC´,gNbj^�ZCÎ�x�i�ZCv�Za`_iabRg�^_^EuUÎ�\ª`_zUgH`�Za`�ZC^�T4\Jg�^_b¿U¯`Pc¯v�c&Î�x{u+`P\(¬ { Ím¤FÏȳ;y&Za[�\JX°å£c�`_z óµg�XUk`¤;Ð

�,uU` `_z+\�gHå>c=[�\rk+\�¶{XUZa`_Zac&X¨^_gNbj^J³>ZCXRx�gH]_`_ZCv�uUi�gH]J³+`_zUgH`YZC` ZC^�TPzUgH]ÈkAUª`PcdxU]_\JkUZCv�`�`_zU\L[Hg�iCu+\�c�eL¬�{=Ím¤FÏZae órZC^�uUXA/jX+c=´ X¨g�XUk¨]Èg�XUk+c&ΪiabdvEz+c&^P\JX Íg�XUk¨\�[�\JX0Zae£¤0ZC^E/ÂX+cN´YX{ÏÈÐ

n

Page 42: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

y iC`_z+c&u+y&z�´,\ kUZCk¢X+c�`!y&Za[�\ g~e2c�]EΪg�i�³7v�c&Ϊx�ia\*.+Za`�b�ä`_z+\�c�]_\�`_ZCv k+\�¶�XUZa`_ZCc&Xrc�e£c&X+\�ä´ gJb¢euUXUv�`_Zac&X�^�³´�\¢z+c�x£\L`_z+\�]_\Jg�k+\�]¦´ ZCi�i�^P\�\¢z+cN´ ^_uUvÈz¯gµk+\�¶�XUZC`_Zac&X¨´,c&uUiCkRx�]_cjv�\�\Jk�Ð a y XUkR´,\r`_zU\�]_\�e�c�]E\�^P`_gH`P\`_z+\�e2c&iCiacN´YZCX+y¢`_z+\�c�]_\JÎà´ Za`_z¨iCZa`P`_iC\Q�_uU^P`_ZC¶�v�gH`_Zac&X¨g�XUkRXUc�xU]_cjc�e"Ð

Ë�Ì Ò�×�Í=Ò$Î Ø�Ï,¸J¹�»UÅ&ÃH¼PÉHÀ{Å7ÃHÜ ÄÈ»ÂÀ{ºN¾½ÃHÀU¸�¹9Ð�½�¸Q¾�½ ÄLÉHÀ{ŨÃHÀUËÌ¿±½ ÄrÃHÀ£¹�Æ í!ÉH¿ ÄQ»ÂÀ{ºJ¾½�ÃHÀ+¸�¹�Ð�½�¸Q¾6)

Ñ�ÒÔÓ Õ ÖÇÕ(×-`Ø�v�g2Ù·k|Ø�¶�ÙAØLg2Ù¯g2Ù�g�i_Ú�y2Ú&uAÛ Ù|Ú2Ü9Ý_ÚfÙeÚ2Þ�gªx�ÛnØJu�k|Ú2ßEg2ÙUk|Ú&Îëx}Ø ßÈÎru|Ü_g�Ü9ÝVÚfÙ Í|CB |,ÏÈÐ}\�Ø ß�Ø�³{´FØ�v*ÚfÙ+äÛ9ÝCk|Ø ß�ÞuAÙUv*Ü9Ý_ÚfÙAÛLàá�Â÷NùÂâJúòû ü ® ÷NùÂâJúòû ¯ °ø÷NùÂâJúòû ¯ Û_uUvÈzPÜ_zUg�Ü�³hÞmÚ2ß�g�iCiUó�´·÷NùÂâJúòû ü ³&´FØÿzUg�â2ØWÜ_z�g�Ü4à~{ÝVÛ�gµx}Ø ßÈÎru|Ü_g�Ü9ÝVÚfÙeÚgâ2Ø ßª÷NùÂâJúòû�¯ Íg2ÙUk²v�g�iCi~ÞuAÙUv*Ü9Ý_ÚfÙAÛ¦´�ÝVÜ_zeÜ_z�ÝVÛ¦xAß9Ú�x½Ø ß9ÜÔã'²�¹�¿7¹EÅ¢Û+¹�¼ÈÜ¢»+¾É�¾½�ÃHÀ+¸=ÏÈÐy iVÛ9Ú+³F´CØ�k|Ø�¶�ÙAØ�ä|å�æ6ç ¯ g2ÛlÜ_z|ØPÛnØ Ü�Ú2Þ!g�iCi�x½Ø ßEÎ�u|Ü_g�Ü9Ý_ÚfÙAÛIÚfÙ ÷NùÂâJúòû ¯ Ð s Ù·gwΪg2ÙAÙAØ ß~v*Ú&Ϊx�i_Ø ÜnØJi_ãg2ÙUg�i_Ú�y2Ú&uAÛ]ÜnÚ5Ü_z|Ø~gHå}Úgâ2Ø�³+´CØlÛ_g�ã Ü_zUg�Ü]àÅÝVÛ�gwÍmÈQâ�ÉÈÏ�ä�|QBY|¶Ý_ÞzÝ_ÜFÝVÛ,g�/2Ø ã2ØJkwx½Ø ßEÎ�u|Ü_g�Ü9Ý_ÚfÙ�Û_uUvÈz(Ü_z�g�ÜÞ�Ú2ß g�i�i�g�iay2Ú2ß�Ý_Ü_z�Î�Û�êèßÈuAÙAÙAÝVÙUy5ÝVÙ�Ü9Ý�Î"Ø�g�Ü Î"ÚfÛnÜ�È�´FØLzUg�â2Ø2�

ððð |4ßNñôó�öø÷NùÂâJúòûNüP�&ê é �8� �.Þ út�}�)|4ß=ñ à ö�ä|å�æ6ç ¯ �7ê é~�8� ��Þ út� ððð � É �3�ÝVÙUg�iCi_ã�³{´CØ�Î"Ø�ÙÂÜ9Ý_ÚfÙ;Ü_z�g�Ü�ÚfÙUv*Ø�´CØ�zUg�â2Ø�g /2Ø ã2ØJk¯x½Ø ßEÎ�u|Ü_g�Ü9Ý_ÚfÙeÝ_ÜIÝ8Û�ÙUg�Ü_u|ßÈg�izÜnÚ�Ü_g�i_/RgHå½Ú&u|Ü

Ü_g�/¿ÝVÙUyXÜ_z|Ø ÝVÙÂâ2Ø ß�ÛnØ"Ú2Þ]à { ³£ÞmÚ2ßLgwx�g�ß9Ü9Ý�v�uUiCg�ßLó7Ð y y&g2ÝVÙ�³£ÝVÙÇÜ_zAÝVÛ�v�g2Û9ت´Cر´CÚ&uUi�k²z|Ú�x½Ø5Ü_z�g�Ü�Ý_Ü�å½ØTwØJg2Ûnã¿U�ÜnÚRv*Ú&Ϊx�u|ÜnØ"àPê a{ Ím»+ÏȳFyfÝVâ2Ø�Ù)ó�g2ÙUk�» Í7Ù|Ú2ÜnØ"Ü_zUg�ÜlÜ_zAÝ8ÛIÝVÛlÙ|Ú2Ü�Ù|ØJv*Ø�Û�Û_g�ß�ÝCi_ãXÜ_zAØ¢v�g2Û9Ø q g2Û´CØ�zUg�â2ØPÛ9Ø Ø�Ù0g�iVß9ØJg�k|ã�ÝVÙeÜ_zAÝVÛYv�iCg2Û9Û�³½Ü_z|Ø ß9Ø¢g�ß9Ø�x�i_Ø�ÙÂÜÔã�Ú2Þ!x}Ø ßÈÎru|Ü_g�Ü9ÝVÚfÙAÛ�ÞmÚ2ߦ´ zAÝCvÈz;Ü_z|Ø5ÞmÚ2ß_´,g�ßÈkkAÝ_ß9ØJv*Ü9Ý_ÚfÙµv�g2Ùwå½Ø¦v*Ú&Ϊx�u|ÜnØJk�ØJg2Û9ÝCi_ã±å�u|ÜFÜ_z|Ø Ý8Ùhâ2Ø ß�Û9Ø~v�g2Ù�Ù|Ú2Ü å}Ø=ÏÈÐL� u|Ü]Ú2Þ;v*Ú&uAß�ÛnØ�³Â´CØ�Î�ÝCy&zhÜ z|Ú�x½ØÜ_zUg�Ü�Ý_Ü�å½Ø�TPkAÝ_^dv�uUi_Ü9UPÜnÚªxAß�ØJkAÝCv*ÜEà ê a{ Ím»UÏ,´ z|Ø�ÙRó�ÝVÛÿuAÙA/¿Ù|Ú=´�ÙFÐ s ÙXÞ2g�v*Ü�³�Ü_z|Ø ß9Ø�g�ß9Ø�ÛnØJv�u|ß9ØL|CB |CÛÞ�Ú2ß�´ z�ÝCvEz�Ý_Ü]ÝVÛPTwØJg2Ûnã¿UPÜnÚ¢xAß�ØJkAÝCv*ÜQÜ_zAØlÝ8Ùhâ2Ø ß�Û9Ø Ú2Þ&àPê a{ Ím»+Ïȳ$ÞmÚ2ß�grx{g�ß9Ü9ÝCv�uUiCg�ß vEzAÚfÝCv*Ø�Ú2Þz»dg2Ù�k(Ø â2Ø�ÙÞ�Ú2ß g�ßEg2ÙUk|Ú&ΪiVãdvEzAÚfÛnØ�Ù¯ó7Ð

m¿Ú"Ü_zAÝVÛFß9Ø�:juAÝ_ß�Ø�Û,g2ÙAÚ2Ü_z|Ø ß k|Ø�¶�Ù�Ý_Ü9Ý_ÚfÙFÐ~-`Ø�Û_g�ã(Ü_zUg�ÜIàëÝVÛ�gµ¸Q¾¼PÃHÀ Ç ÍmÈQâ�ÉÈÏ�ä�|CB |¼Ý_Þ£Ý_ÜWÝVÛ gP/2Ø ã2ØJkx}Ø ßÈÎru|Ü_g�Ü9ÝVÚfÙ�Û_uUvÈzXÜ_zUg�ÜWÞmÚ2ß g�iCiFg�iay2Ú2ß�Ý_Ü_zUÎ�Û�êìßEuAÙ�ÙAÝVÙ+yPÝVÙ�Ü9ÝCÎ"Ø�g�ÜYÎ"ÚfÛnÜWÈ,´CØrzUg�â2Ø2�

ððð |4ßNñôó~ö�÷NùÂâJúòû ü �&ê é ��� �7� é~í¿î �8� � Þ út�}�)|4ß=ñ à ö�ä|å�æ6ç ¯ �7ê é~�8� �� é íÂî �8� � Þ út� ððð � É �x�Ú2ÜnØPÜ_zUg�Ü�Í7ÝVÙ�ØJg�vÈz�Ø*.jx½Ø ß�ÝCÎ�Ø�ÙhÜÈÏ�´CØ5Ù|ÚN´ yfÝ_â2Ø�ê g�v�v*Ø�Û9Û ÜnÚ(Ü"´CÚ�Ú2ßEg�v�i_Ø�Û��EÚfÙ|Ø�ß�Ø�xAß9Ø�ÛnØ�ÙÂÜ9ÝVÙ+y�Ü_zAØÞ�Ú2ß_´ g�ßEkXØ âHg�iCuUg�Ü9ÝVÚfÙXÚ2Þ�Ü_z|Ø�x½Ø ßEÎruAÜ_g�Ü9Ý_ÚfÙRg2ÙUk�ÚfÙ|Ø�ß9Ø�xAß9Ø�ÛnØ�ÙÂÜ9ÝVÙ+y"Ü_zAØ�ÝVÙhâ2Ø ß�ÛnØ�Ðy Û�Î�Ø�ÙhÜ9Ý_ÚfÙAØJk(ØJg�ßÈiVÝ_Ø ßJ³$Ù|Ú2Ü]Ø â2Ø ß9ãd|CB |¶Ý8Û�g�u|ÜnÚ&αg�Ü9ÝCv�g�iCi_ãµg5Û9Ünß9ÚfÙ+y¢|QB |�Ð$-`Ø�ÛnÜ_g�ÜnØ�Ü_zAÝ8Û�z|Ø ß�Ø

g2Û g"Ü_z|Ø Ú2ß�ØJÎX�Ë�Ì Ò�×�Í=Ò$Îðï Ú�¸E¸È»ÂÜ¢½2À Ç ¾2Á¹d¹9Ð�½�¸Q¾�¹�À{ºÈ¹ªÃ4Ä�ÃHÀ£¹�Æ í!ÉH¿~ÄÈ»ÂÀ{ºN¾½ÃHÀU¸Eæ�¾2Á¹�¼_¹±¹�Ð�½�¸Q¾2¸¢É�ÏCñWϸà í Á7½�º�Á¯½�¸Å7¹7@�À�½2¾�¹�ËÀ{Ã�¾ ɱ¸Q¾¼PÃHÀ Ç ÏCñWÏW)3~Ý8ÙUg�iCi_ã�³Â´FØ�ÛnÜ_g�ÜnØ�Ü_zAØ�Þ�Ú&iCiVÚN´�ÝVÙUy5Ü_zAØ Ú2ß9ØJÎpÞmÚ2ßWß9Ø ÞmØ ß9Ø�ÙUv*Ø2�Ë�Ì Ò�×�Í=Ò$Îóòõô�¾¼PÃHÀ Ç ÏCñWÏ,¸r¹�Ð�½�¸Q¾�½ ÄLÉHÀ>ŨÃHÀUËÌ¿±½ ÄrÃHÀ>¹�Æ"í!ÉH¿YÄÈ»ÂÀ{ºN¾½Ã�À+¸�¹9Ð&½�¸�¾6)

îÔö�÷�ø�ù6ú�û�ü*û?øý�þ,ÿ ù6û ø6û � ù6û���ø6û����gû?ø���}û�2÷�þ,ÿ�ùI÷���ùIù6ú ��ùWù6ú�û��w÷��� �,û���þ�ùmý��Vù6ú�ûw÷�ø6û ù6þ����gû���ÿ�þ�ù6þ,÷tÿ�����ÿXÿ�÷��,÷�ÿ �tû?ø� ÷tÿ � þ���û ø����ÿ � ù6þ,÷tÿ �"! �zþ�ù6úlþ,ÿ �gù#�,ûwÿ ��ù6ú%$gþ,ÿ � ù6û����&��ù6ú�û�þ,ÿ �gù#�,ûwÿ ��ù6ú'�(� � ù#��û�2ûwÿ �I÷tÿ�ù6ú�û*)*û?ý+� û ÿ ��ù6ú&,

o

Page 43: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ñ�Ò.- /1032*46587�93:(;=<?>h×- zUg�ÜeÝVÛRk|ÚfÙ|Ø'Ý8Ù@xAßEg�v*Ü9ÝCv*Øg< s Ù xAßEg�v*Ü9ÝCv*Ø�³�x½Ø Ú�x�iVØ·u�ÛnØ Ê�ËaÃJº*²�ºJ½ Û�Á+¹�¼È¸ÈÐ ® z|Ø�Ûnزå�i_ÚÂv9/@v Ýax�zAØ ß�Ûg�ß9ØX/2Ø ã2ØJk¶ÞuAÙUv*Ü9Ý_ÚfÙ�Û�³}�Pu�ÛnÜ�g2Ûªk|Ø�Û_v*ß�ÝCå}ØJk gHå½Ú�â2Ø�ж3Ug�Î�Ú&uAÛ"Ø*.+g�Î�x�i_Ø�ÛPÚ2Þ¦å�i_ÚÂv9/ v Ýax�z|Ø ß�Û5ÝVÙUv�i�uUk|Ø@BA m>³ s @'A y ³]Ünß�Ýax�iVØ�äC@BA m>³�g2Ù�k y A,m>Ð y Ûµg2Ù�Ø*.+g�Î�x�i_Ø�³D@BA m¼ÝVÛ ß�ØJg�iCi_ã$g¨/2Ø ã2ØJk@x}Ø ßÈÎru|Ü_g�Ü9ÝVÚfÙE+FHG �>÷NùÂâJúòûJI�K ® ÷NùÂâJúòûJK�L�° ÷NùÂâJúòûJK�L7Ð ® zUg�ÜIÝ8Û�³�Ü_z|Ø5/2Ø ãhi_Ø�ÙUy2Ü_z�Ú2Þ=@'A,m�ÝVÛ o�p å�ÝVÜ9Û�³�g2ÙUkeÚfÙUv*Ø¢g/2Ø ã·ó"Ý8Û ¶|.$ØJkF³ E+FHG {lÝVÛ~gµx½Ø ßEÎ�u|Ü_g�Ü9Ý_ÚfÙeÚ�â2Ø ß�÷NùÂâJúòû ü Ð�3UuAß9Ü_z|Ø ßEÎ"Ú2ß�Ø�³zÝ_Ü Ý8ÛIØ ^dv Ý_Ø�ÙhÜlÜnÚ¨v*Ú&Î�x�uAÜnØg2ÙUkXÜnÚ ÝVÙÂâ2Ø ß9ÜM@'A m�Ý_Þ£ã2Ú&u;/¿ÙAÚN´�Ü_z|Ø�/2Ø ã�Ð- zAØ�Ù·å�i_ÚÂv9/¯v ÝCx�z|Ø ß�Û¦g�ß9Ø�uAÛnØJk`ÝVÙ0g2Ù²gHxUx�i8ÝCv�g�Ü9Ý_ÚfÙF³=Ý_ÜlÝVÛlÝCÎ�x½Ú2ß9Ü_g2ÙÂÜIÜnÚRg2Û9/h´ zUg�Ü~xAß�Ú�x}Ø ß�Ü9Ý_Ø�Û

Ú2ÞLÜ_z|Ørv Ýax{z|Ø ß g�ß9Ø�g2Û9Û_uUÎ"ØJk�ÐF3|Ú2ß Ø*.+g�Î�x�i_Ø�³>k|ÚÂØ�ÛIÛnØJv�u|ß�ÝVÜÔã�ß9ØJi_ã�ÚfÙ�Ü_zAØ�Þg�v*Ü�Ü_zUg�Ü Ü_z|ØLv Ýax{z|Ø ß�ÝVÛYg|QB |�Ú2ß�k|ÚÂØ�Û�ÝVÜ ß�Ø�:�uAÝVß9Ø�Ü_z|Ø�Û9Ünß9ÚfÙ+y2Ø ß�g2Û9Û_u�Î�xAÜ9Ý_ÚfÙ�Ü_zUg�ÜlÜ_z|تv Ýax{z|Ø ß Ý8Û¦gXÛ9Ünß9ÚfÙ+y¨|QBY|F< ��ÙeÜ_zAØÚfÙ|ØÿzUg2ÙUkF³2Ý_ÜLÝVÛ~ß9ØJg2Û9ÚfÙUgHå�i_Ø]ÜnÚ�g2Û9Û_u�Î"Ø]Ü_zUg�ÜN@'A,m�Ú2ß y A,m�ÝVÛ g�ÛnÜnß9ÚfÙ+y�|CB |�Ív Ýax{z|Ø ß�Û g�ß9Ø�k|Ø�Û9ÝayfÙAØJk´�Ý_Ü_z�Ü_z�ÝVÛ4ÝV٪ΠÝVÙUk{ÏÈÐ4�lÙ�Ü_zAØWÚ2Ü_z|Ø ß zUg2ÙUkF³ÂÜ_zAÝVÛ4ÝVÛ�g�ÛnÜnß�ÚfÙ+y2Ø ß�g2Û9ÛEuUÎ�xAÜ9Ý_ÚfÙ�³hÛnÚLå{g2Û9ÝVÙ+y�Ü_zAØ�ÛnØJv�u|ß�ÝVÜÔãÚ2ÞLã2Ú&u|ß ÛEvEz|ØJÎ"Ø�ÚfÙ¯g±´CØJg�/2Ø ß�g2Û9ÛEuUÎ�xAÜ9Ý_ÚfÙ;ÝVÛÿxAß�Ø Þ�Ø ßEgHå{i_Ø�ÐF3�ÝVÙ�g�iCi_ã�³�ÛnÚ&Î�ØeTwÚ��>ä6Ü_z|Ø�äÔÛ_z|ØJi_Þ9Uµv Ýax�zAØ ß�ÛΪg�ãwå½Ø�|QBY|QÛ�å�uAÜWÙ|Ú2ÜWÛnÜnß�ÝVÙUy¢|CB |CÛ�Ð3�ÝVÙUg�iCi_ã�³7´CئÎ"Ø�ÙhÜ9ÝVÚfÙ"Ü_zUg�ÜQÜ_zAØIÛnØJv�u|ß�Ý_ÜÔã5Ú2ÞFå�i_ÚÂv9/¢v Ýax�z|Ø ß�Û Íg2ÛQÚ�x�x}ÚfÛ9ØJk"ÜnÚL´ z�g�Ü�´CئzUg�â2ØYå½Ø Ø�Ù

g2ÝCÎ�ÝVÙ+y�ÞmÚ2ßlÝ8ÙeÜ_z�ÝVÛYv�i�g2Û9ÛÈÏIÝVÛ~iCg�ß_y2ØJi_ã¯z|ØJu|ß�ÝVÛ9Ü9ÝCvHÐ s Ù�Ú2Ü_z|Ø ß�´FÚ2ßEkAÛJ³>´FØ�zUg�â2Ø�ÙAÚµx{g�ß9Ü9ÝCv�uUiCg�ß ß9ØJg2ÛnÚfÙÜnÚ²å}ØJi8Ý_Ø â2Ø Ü_zUg�Ü y A,m¨ÝVÛ¢g0y2ÚÂÚÂk å�i_ÚÂv9/°v ÝCx�z|Ø ßJ³�Ú2Ü_z|Ø ß"Ü_zUg2Ù«Ü_z|Ø�Þg�v*ÜPÜ_zUg�Ü"Ý_Ür´ g2Û�k|Ø�Û9ÝCyfÙ|ØJk°å¿ãØ*.jx½Ø ß9Ü9Û�g2ÙUk�ÛnÜ_u�kAÝ_ØJk�ÝVÙhÜnØ�Ù�Û9Ý_â2ØJi_ã�å¿ãdv*ß9ãjxAÜnÚ�y2ßEgHx{z|Ø ß�Û]ÞmÚ2ßFÜ_zAئx{g2ÛnÜW��ã2ØJg�ß�Û�ÐF�lÙ�Ü_z|ØlÚ2Ü_z|Ø ßÿzUg2Ù�kF³´CØLv*Ú&u�iCk°Í2å¿ã ® z|Ø Ú2ß9ØJÎ ¹ Ïÿv*ÚfÙAÛnÜnßEuUv*ÜYg�ÛnÜnß9ÚfÙ+yª|CB | å�g2ÛnØJkXÚfÙ�Ü_z|ØLzUg�ßEk�Ù|Ø�Û9Û]Ú2Þ�Þ2g�v*ÜnÚ2ß�ÝVÙ+y+Ð ® z|ØzUg�ßEkAÙ|Ø�Û�Û�Ú2Þ4Þg�v*ÜnÚ2ß�ÝVÙUy�ÝVÛYå�g�v�/2ØJk²u+x¯å¿ã ¹ ù�ù�ã2ØJg�ß�ÛlÚ2ÞQÝVÙÂÜnØ�ÙAÛnØPÛ_v*ßEuAÜ9ÝVÙhãwå¿ãhΪg2ÙÂã�³FΪg2Ùhãhx}Ø Ú�x{i_Ø´ z|Ú�´CØ ß9ØlØ*.Âx½Ø ß9Ü9ÛCÝ8Ùdg5âHg�ß�Ý_Ø ÜÔã Ú2Þ.¶AØJiCk�Û�Ð ® z|Ø ÌPOQ Ò±kAÝVÛ_g�k|âHg2ÙÂÜ_gHy2Ø Ú2Þ&Ü_zAÝVÛ,gHxUxAß9Ú&g�vÈz�ÝVÛQÜ_z�g�ÜFÜ_zAØß9Ø�Û_uUi_Ü9Ý8Ù+yXÞ2u�ÙUv*Ü9Ý_ÚfÙ²´FÚ&uUiCk°å}Ø"Ú2ßÈk|Ø ß�Û�Ú2ÞÿαgHyfÙAÝ_Ü_uUk|Ø(Û_i_ÚN´FØ ß�Ü_z�g2ÙR@'A,m`Ú2ß y A,m£³.g2ÙUk'Ü_zAØ ß9Ø Þ�Ú2ß�ØÙ|Ú2ÜÿxAßEg�v*Ü9Ý�v�g�i=Þ�Ú2ßYuAÛnØ�Ð

S �����_���,�}K����uK9��� �¢¡9�£��¤r�£�_¥¦�UT߬£�E�WV ��©ë§²¬

-`ØIÛnÜnß9Ø�Û�Û4Ü_z?X�Ü!|CB�3�Û=X2ÙZYª|QBY|QÛNX\[_ÚfÙAØMX�ß9Ø�Ù|Ú2ÜQØ�Ù|Ú&u+y&z"ÜnÚ�Ý�ΪÎ"Ø�YAÝ]X�ÜnØ�[VãLyfÝ_â2Ø ÛnØJv�u|ß9ØEØ�ÙUv*ß�ã�xAÜ9ÝVÚfÙÛ_vEz|ØJÎ�Ø�Û�ýLÛnÚ&Î"رv^X�ß9Ø(ÝVÛ�Ù|Ø Ø�Y|Ø�Y�Ð�3|Ú2ß�Ø*.HX�Î�xZ[_Ø�³�v*ÚfÙAÛ9Ý]YAØ ß�Ü_z|Ø�TnÙ?X�Ü_u|ß�X\[VU�Ø�ÙUv*ß9ãjxAÜ9Ý_ÚfÙ`Û_vÈz|ØJÎ"Ø�ÝVÙ´ zAÝCvÈz¼à ÝVÛ_X(|QBY| Í2´IÝ_Ü_z�Ø ^±v Ý_Ø�ÙÂÜ�Ý8Ùhâ2Ø ß�Û�Ý_ÚfÙ{Ï�X2ÙZY¼Ü_zAØ�ÛnØ�ÙZY|Ø ß1X2Ù?Y ß�ØJv*Ø�Ý_â2Ø ß�Û_z?X�ß9Ø`X«/2Ø ã óÝVÙaX\Y|âbX2ÙUv*Ø�Ð ® Ú`Ø�ÙUv*ß�ã�xAÜ�Î"Ø�Û�ÛcXHy2Ø�i²³4Ü_z|Ø�Û9Ø�Ù?Y|Ø ßªv*Ú&Î�x�uAÜnØ�Û(k Þ à�{NÍmi¯ÏdX2ÙZY1ÛnØ�Ù?YAÛPÜ_zAÝ8ÛPÜnÚÜ_z|Ø ß9ØJv*Ø�ÝVâ2Ø ßJÐ ® ÚeY|ØJv*ß9ãjxAÜ�³~Ü_z|Ø�ß9ØJv*Ø�Ý_â2Ø ß¢v*Ú&Î�x�uAÜnØ�Û�i�ÞuàPê a{ Í6kLϪÍ7Û�ÝVÙUv*Ø à©ÝVÛ�Xhx}Ø ßÈÎru|ÜcX�Ü9ÝVÚfÙF³Y|ØJv*ß9ãjxAÜ9Ý_ÚfÙfX\[C´gX�ã¿ÛlÛ_uUv�v*Ø Ø�Y�ÛÈÏÈÐ�\IÚN´FØ â2Ø ßJ³F´ zAÝ][VØ�Ü_zAÝVÛ�Ý8Û�ÛnØJv�u|ß�ØhXHy6X2ÝVÙAÛ9ÜYv Ýax�zAØ ß9ÜnØ*.¿ÜIÚfÙ?[_ã1X�ÜnÜcX�v9/$ÛÍ2´ z|Ø�ÙiX�Û9ÝVÙ+y6[VØ�Î"Ø�Û�ÛcXHy2Ø�ÝVÛ Ø�ÙUv*ß9ãjxAÜnØ�Y{ÏIÝ_Ü'Y|Ú¿Ø�Û¢À{Ã�¾ ÛcX�Ü9ÝVÛnÞmã;Ú&u|ß�Ù|Ú2Ü9ÝVÚfÙÇÚ2ÞCÝVÙ?Y�ÝVÛnÜ9ÝVÙ+y&u�ÝVÛ_z?XHå�Ýj[_Ý_ÜÔãÍ7ÝVÙXÞkX�v*Ü�³�Ý_Üÿv^X2ÙAÙ|Ú2ÜIÛ9ÝVÙUv*Ø�Ü_zAÝ8Û]Ø�ÙUv*ß9ãjxAÜ9Ý_ÚfÙXÛ_vÈz|ØJÎ"Ø�Ý8Û�Y|Ø ÜnØ ßEÎ�ÝVÙ�ÝVÛnÜ9ÝCvNÏÈÐ

l X2ÙXã2Ú&umY|Ø â2Ø�[_Ú�xfX2ÙeÝVÙ?YAÝVÛ9Ü9ÝVÙ+y&uAÝVÛEz?XHåZ[_ØEØ�ÙUv*ß�ã�xAÜ9ÝVÚfÙXÛ_vEz|ØJÎ�Ø�åZX2ÛnØ�YXÚfÙh|QB�3�Û�<�<

p

Page 44: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

nDo?p]q\r^sutcpwvyx{z\|*}f~bsux��]~\oZ�� }e� �g�&�\�h��� o&v�suz��?�Z��vcpwz6o�v�z � scx��?v�z\�\su~b���&x�Nscz\|�r�tct�z\s'�\z6o?~bvc�?~\of��~bv��

���������a��� �{�

  ¡£¢¥¤=¦W§�¨D©Pª+«W§�¨D¬ ­�¦�«W®°¯P±c¨D«W¢²ª+«W§³¡R¤=©#¬´¦h¯µªg¯µ±c¨B«W¢

¶ �=·+¸º¹g~\tdo?p]��r\»½¼µr��^~\�?t�r_pwvh~\�j�wzJ¹gr��R�Zt�v�z`tc�?~bscr{su~\o?�Hz6¾À¿\r^xÁthz\|+tu¾Â~\�]�=tupw�^r{~\o?�R�\r�oHr^su~bv�rtc�?~bscr��Ã�Ztcr��?�Hz\su~\o?�?z6¾À¿\r^xÁtWz\|'�]~bsc�\r1tcpw�^r\ÄÆÅg�?v�r^q\r�oÇo?p]��r^sWvc�?~\oa�?~Jq�pjoH�itc�?~bscr��Ã~\�^��r�tctÈv�zÉ~�Zt�r��?�Hz\s�~\o?�Hz6¾ËÊ�ÌkÍ�Î�Ï%Ðѹ(z6�Z�]�ɼµrÈtc�Z~bscr��R~\�^��r�tuthv�z`~m�Zt�r��?�Hz\s�~\o?�Hz6¾ÓÒ�ÔÁÏ�Õ�ÌkÎ×ÖbÏ?ÄeØ#r^v^Ù�tW�wz�z\¿É~bv¹M�?~bv+vc�?pjt�¾Âpw�6�%v+¾�r�~\oPÄ

ÚHz\sÂtcp]¾Â�Z�]p]�^pwvyx\»"¹gr_��z6oZtcp]�Hr^sh|��?oZ��vcpwz6o?t�¹Dpwvc�ÉÛHÜÁr��Æp]oH�Z�Hvh~\o?�Ñz6�?v��Z�Hvh�]r�oH�\vc�PÄ � z6oZtcp]�Hr^sd~|��?oZ��vcpwz6oaÝßÞMàJáÁâ�ã äJåfæaàJáÁâ�ã ä�çÀè àJáÁâ�ã ä�éap]oùM�?p]���Ãvc�Hr1Û?sutcvWpjoH�Z�Hvdv�zÑÝêp]t��^~\�j�wr��Ãvc�Hrië6ì�í~\o?�ivc�HrÈtcr���z6o?�Ñp]o?�Z�HvBv�zmÝîp]t�tcp]¾��Z�]xf�^~\�]�wr��ivc�?rïÎ.Ï\ð�ÔHÌkÄ{ñip]t�vc�?r�ë\ì�ífòwì^Ï%Ð6Ì�ó&»°ôßp]t�vc�?rïÎ.Ï\ð�ÔHÌòwì^Ï%Ð6Ì�óÈ~\o?�Èõfp]t*vc�?rdÖbÔHÌöð�ÔH̽ò]ì�Ï%Ð&Ì�óÂz\|µÝÂÄNÝ÷p]t(�^~\�]�wr��Â~Âë\ì�í&ìuøùÒ�ÔÁÏ�ÕJÌkÎkÖ\Ï_|�z\s=vc�?rM|�z6�]�wzJ¹Dp]oH��scr�~\t�z6o#ÞÛHÜÁpjoH�É~i�Z~bscvcpj�^�?�]~bs�¿\r^xÇúÃûüàJáÁâ�ã äJåi�?r^ÛZoHr�tÂ~i|k�?o?��vcp]z6oÇÝ°ýmÞDàJáÁâ�ã ä�çþè àJáÁâ�ã ä�éÇ~\tÂ|�z6�]�wzJ¹Dt^ÞÝ ý ÿ������ Ý ÿ ú6â ��� Ä ��zÃx\z6� �^~\o�q�p]r^¹ ���Hz�z6tcp]oH�Ã~ÿ\r^x ~bv�su~\o?�?z6¾ ~\t1���Hz�z6tcp]oH�a~Æ�Z~bscvcp]�^�Z�]~bs|��?oZ��vcpwz6o1~bv+su~\o?�?z6¾Ó|.suz6¾ît�z6¾�r��j~bsc�\r�t�r^v+z\|½|k�?o?��vcpwz6o?t�Ä

�Mz ¹�»Pp]oH|�z\su¾Â~\�]�]x{t��µr�~b¿Áp]oH�H»Zz6�?sD�=·+¸'t'�?~\�fvc�Hr��Zscz\�µr^scvCx1vc�?~bvDvc�Hr�pwsDz6�Hv��Z�Hvct���wz�z\¿\r��fsu~\o��Hz6¾ �m¹M�Hr�oÉr^qb~\�]�?~bv�r��Rz6oÃ~ïsu~\oZ�Hz6¾Àtcr^r��PÄ��+�Hr_~\o?~\�wz\�\z6�?td�?suz\�¥r^suvCx`¹(r_¹gz6�?�]�Ñ�jpw¿\rÂ|�scz6¾ÀÝp]t�vc�?~bvdpwv�tc�Hz6�Z�]�����wz�z\¿i�]pw¿\rÈ~1s�~\o?�Hz6¾ê|k�?o?��vcp]z6o���¹M�Hr�oÉ~fsu~\o?�Hz6¾ ¿\r^xÉp]t����Hz6t�r�oPÄ1Åù�Hv�¹M�Z~bvr�ÜÁ~\��vc�wx1p]tM~�su~\o?�Hz6¾ |k�?o?��vcp]z6o��

¶ su~\o?�Hz6¾ |k�?o?��vcp]z6o��^~\o�¼µr�q�pwr^¹gr��Âp]oWvc�?r+|.z6�j�wzJ¹MpjoH�'¹g~Jx�Þ*pw|�x\z6�Â�6pwq\r+vc�?r+|��Zo?��vcpwz6o�~\oÂp]o?�Z�Hv� û àJáÁâ�ã ä ç vc�Z~bvWp]vh�?~\tdoHz\vWt�r^r�oƼ¥r^|�z\scr\»*vc�Hr_|k�?o?��vcpwz6oÑ�Zp]�c¿Átd~fsu~\oZ�Hz6¾�� û àJáÁâ�ã ä é »Nt�v�z\scr�tÿ�� â�� � |.z\sD|��?vc�Hscr'scr^|�r^scr�o?��r\»Z~\o?�ïz6�Hv��Z�Hvct��¥Ä � |x\z6�ï�j~bv�r^sD~\t�¿_vc�Hr�|k�?o?��vcpwz6o1~\ofp]oH�Z�Hv � ¹M�?p]���1pwvó�� ÊÈt�r^r�oi¼µr^|�z\scr\»#pwv'ÛZo?�ZtDvc�?rÂ~b�?�?scz\�?s�p]~bv�rW�Z~\pws ÿ�� â�� � ~\o?�`scr^vc�Hs�o?t�� ÿ t�z1vc�?r�|��?oZ��vcpwz6o`~\�w¹g~Jx�tscr^�µz\scvct�vc�Hr�tu~\¾�r�~\o?t�¹gr^sDpw|x\z6�f~\t�¿�pwv�vc�Hrhtc~\¾�r����Hr^scx � Ä

�irD�^~\o_~\�]t�z�qÁpwr^¹£~\�]�?vc�Hr�t�rM���Hz6p]��r�tg~\tN¼µr�p]oH��¾Â~\�HrDp]oÂ~\�Hqb~\o?��r\»%pjo?t�v�r�~\�Âz\|P�H�µz6oWscr���r�p]q�p]o?��~�%�Hr^sux\Ä��+���?t^»J|.z\sr^q\r^scx'�¥z6tutcpw¼Z�wr½p]oH���Hv � � û`àJáÁâ�ã ä ç vc�HrN|k�?o?��vcpwz6o����Hz%z6tcr�t~�s�~\o?�Hz6¾!� � û`àJáÁâ�ã ä é~\o?�Ãt�v�z\scr�t ÿ�� � â�� �"� pjoÆtcz6¾�r_���H�\r�vc~b¼Z�wr\Ä � oÆ|�~\��v^»(pw|+¹gr_z\s��Hr^shvc�?r � tW�wr�ÜÁpj��z\�\su~b�Z�?p]�^~\�j�wx\»*pwvdpjtr�oHz6�H�6�1v�z�#��?tcvùtcv�z\scr�vc�Hr�z\s��Hr^scr��ï�]pjt�v��%$�â'&'&'&Jâ�� é Ä(Mz ¹ ¾d�?�u�`t��Z~\��r�¹gz6�?�]�mvc�?p]t'~\��vc�?~\�j�wxïvc~b¿\rÂpw|N¹grW¹(r^surdv�z1p]¾Â�Z�wr�¾�r�o%vMvc�?pjt)�*�ir��]�3»Pr�~\�u�+� �scr,�%�Zpwscr�tDõѼZpwvctB~\o?�evc�Hr^scrW~bscr - ç z\|Nvc�Hr�¾ v�z�t�v�z\scr\»P|�z\s�~Èv�z\vc~\�½z\|Nõ/.0- ç ¼Zp]vctMz\|=tcv�z\su~b�\r�#��Zt�vv�zfscr^�?sur�t�r�o&vh~1su~\o?�Hz6¾ê|��?oZ��vcpwz6oPÄ � �]r�~bsu�wx\»"pwvhp]t�p]oH|�r�~\tcpw¼Z�]rdv�zmt�v�z\scr{~ïv�su�?�wxesu~\o?�Hz6¾ê|��?oZ��vcpwz6o|.z\s_¾�z��?r^su~bv�rfqb~\�]�Hr�tÈz\|Bôiâ�õ*Ä Åù�Hv¹grm�^~\o ��r^scvc~\p]oZ�wxÆpj¾Â~b�6p]oHrïtc�Z�u�£~Ñvc�?p]oH�ir�ÜHp]t�vcpjoH�Ñ~\tÈ~\oÖbÍ1�&Õ^ò]ì�Ä ÿ � oR|�~\��v^»*x\z6�Ã��z6�?�j�Ñp]¾Â~b�6pjoHrÈp]¾����wr�¾�r�o%vcp]oH�ï~msu~\o?�Hz6¾ |��?oZ��vcpwz6oÑpw|ùvc�Hr�~\o?t�¹gr^suth¹(r^sur�u�Hz6t�r�o2��z6o%3vc�Hr3"4Zx��{p]o?t�v�r�~\�1z\|*p]of~\�Hq ~\oZ��r\Ä(Åù�HvMvc�?r^scr�¹gz6�?�]��¼µr�t�z6¾�r�¼¥z6�Zo?��z6o1vc�Hrdo%�?¾h¼µr^sz\|N�?p65µr^scr�o&v7�%�Hr^s�pwr�t�x\z6�e��z6�?�]�f~\tc¿�p]v^»��Hr^�µr�o?�?pjoH�Wz6oïvc�Hrd~\¾�z6�?o%vMz\|*t�v�z\su~b�\rdvc�?~bvDpjtM~�qb~\p]�]~b¼��wr\Ä� |+x\z6�ƹù~\o&vWv�zÆÊuó%�bÍ�ìm~msu~\o?�Hz6¾þ|k�?o?��vcpwz6o#»½�?zJ¹gr^q\r^s�»Nvc�Hr1~\o?t�¹gr^sutd¾h�Zt�vh¼µr��Hr��^p]�Hr��Æ�H�µz6oÆp]o~\�Hq ~\oZ��r\Ä ¶ o?�1tcp]o?��r'x\z6�m�Hz6oPÙ v�¿ÁoHzJ¹ ¹D�?~bv+vc�Hr����Hr^supwr�t�~bsur��\z6p]oH��v�z¼µr\»Hvc�?pjtùscr,�%�Zpwscr�t�tc�?~bs�p]oH�vc�HrM|k�?�]�Hõ8.�- ç ¼�pwvct^Ä � �irB�?r�oHz\v�rMvc�HrBt�r^v=z\|#~\�j�?|k�?o?��vcpwz6o?t=|.suz6¾ÓàJáÁâ�ã ä ç v�z{àJáÁâ�ã ä é ¼%x:9�;=<?> çA@'é Ä

�irM¹Mpj�]�%o?zJ¹£�Hr^Û�oHr+~'|.z\s�¾Â~\�?oHz\vcpwz6oWz\|P~'�Ztcr��?�Hz\su~\o?�?z6¾ |��?oZ��vcpwz6o ÿ �=·DÚ � Ä%·+r��^~\�]�Hz6�?sNp]oH|�z\s�¾Â~\�#�Hr^ÛZoZpwvcpwz6o_vc�?~bvM~Â�(·MÚB���wz�z\¿�tM�]pw¿\r,��~�su~\oZ�Hz6¾Ó|k�?o?��vcpwz6o�CH¾�z\sur�|.z\su¾È~\�]�wx\»?~ED�D Ff~\�Hq\r^s�tc~bscx

ã

Page 45: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�^~\o?oHz\vWv�r��]�(vc�Hr�¾À~b��~bscvh¹Dpwvc�Ñv�z�z`�?pw�6�Æ~f�?suz\¼Z~b¼Zp]�]p]vCx\Ä � oѹD�?~bv�¹ù~�x��G�ir��]�3»½tu~�xѹgr_�6pwq\r_~\o~\�Hq\r^sutc~bscxi~1¼��]~\�c¿`¼µz�ÜÉ~\o?�iv�r��j�Npwv�vc�?~bv�vc�?r¼µz�ÜÉp]¾��Z�]r�¾�r�o&vct'r�p]vc�Hr^s�~1su~\oZ�Hz6¾ |k�?o?��vcpwz6o`z\sd~�Zt�r��?�Hz\s�~\o?�Hz6¾Ë|k�?o?��vcpwz6o $ ~\oZ�Ç~\t�¿apwv�v�zR�?r^v�r^su¾Âp]oHr1¹D�?p]�u�Çz6oHr\ÄH�+�?r�oavc�Hrï�?suz\¼Z~b¼Zp]�]p]vCxÑvc�Z~bvvc�HrÈ~\�Hq\r^sutu~bscxi�6�Hr�tct�r�t/���Zt�r��?�?z\su~\o?�Hz6¾ �1tu�Hz6�?�]�`¼µrÂvc�Hr{tc~\¾�r¹M�?r^vc�Hr^s�vc�Hr{|��?oZ��vcpwz6oip]t�~\��vc�%~\�]�wxï�Ztcr��?�Hz\su~\o?�?z6¾´z\s�o?z\v^ÄW}fz\scr�|�z\su¾Â~\�]�wx\»P~�|k�?o?��vcpwz6o`ݺÞ#àJáÁâ�ã äJå_æÉàJáÁâ�ã ä�ç è àJáÁâ�ã ä�éip]t'~ÿ�I âKJ � C�=·MÚ£pw|"|�z\sM~\�]�#~\�]�\z\supwvc�?¾ÂtML²su�?oZo?p]oH�Wp]o�vcp]¾�r�~bvD¾�z6t�v I vc�Hr�|�z6�]�wz ¹Mp]oH���Hz6�]�Zt^Þ

NNN �NsPO úRQ àJáÁâ�ã ä å ÞSL�TVUXWZY [ � ã]\_^É�NsPO ÝBQ`9�;a<0> ç�@'é ÞSL�T�WbY [ � ã]\ NNN�c J,&ÿ �Mz\v�rhvc�?~bv+�Hr^scr'¹gr��?~�q\rhñ�~WÛ?Ü�r���qb~\�]�Hr\»Ht�z�vc�Hr�~b¼µz q\r�p]tùoHz\v+~W|k�?o?��vcpwz6o_z\|½~W�Z~bsu~\¾�r^v�r^sDñ�~\t¹g~\t�vc�HrÂ�^~\t�r{pjo`�?scr^qÁpwz6�?t'�wr���vc�Hscr�t�Ä{�Áp]¾Âp]�j~bsu�wx\»Pp]o?t�v�r�~\�`z\|ù¾Â~b¿Áp]oH�:L su�?oip]o`�µz6�wxÁoHz6¾Âpj~\�vcpj¾�r¹(r��?~�q\r�p]o?tcv�r�~\�ÉÛHÜÁr��Ñ~\oR�H�?�µr^s�¼µz6�?o?� I |�z\shvc�?r{~\�Hq\r^sutu~bscx�Ù�thsu�?oZo?p]oH��vcp]¾�r\Ä:�M�HrÈp]o%v�r�o&vcp]z6o¹g~\t�v�zmtcp]¾Â�Z�]pw|�x1vc�?p]oH�6t�~f�]pwv�vc�wr�¼Zp]v'd � � oɹ(z\su�Zt^»#vc�HrÈ�]r^|.v�r�ÜÁ�?scr�tctupwz6oÉ��z6o?tcpj�Hr^s�vc�HrÂr�ÜÁ�µr^sup]¾�r�o%vp]o1¹M�?pj�u�ï~Èsu~\o?�Hz6¾º¿\r^xfú�p]tD�u�Hz6tcr�oP»�~\o?�/L p]t+�6pwq\r�ofz\su~\�^�wrd~\�^��r�tctMv�z{vc�Hr�|k�?o?��vcp]z6oïÝ ý Ä � oïvc�?rsupw�6�%vMr�ÜÁ�¥r^s�p]¾�r�o%v^»�~_��z6¾��Z�wr^v�r��wx1s�~\o?�Hz6¾î|k�?o?��vcp]z6ofÝ |.scz6¾´ô ¼�pwvct+v�z_õѼZp]vctMp]tD�u�Hz6t�r�o#»µ~\oZ��Lp]t��6pwq\r�oiz\s�~\�^�wrÈ~\�^��r�tcthv�zfÝ�Ä �ir{�^�]~\p]¾ vc�?~bveL �^~\o?o?z\v��?p]t�vcpjoH�6�?p]tc�m¼µr^vC¹gr^r�oÉvc�Hr�tcrÂvC¹gzf�^~\t�r�t¹Mpwvc���?scz\¼�~b¼Zp]�]pwvyx¼µr^v�v�r^sMvc�?~\o/J^Ä

�ir_~\�]¹g~Jx�tdp]¾��Z�jp]�^pwvc�wxï~\tctu�?¾�rÂvc�?~bvdpwv�p]t:��r�~\tcx��mv�zm��z6¾����Hv�r{Ý ýJÿ���� »�6pwq\r�oѼµz\vc�Rú_~\o?� � ÄÅg�?vMvc�Hrd~b¼¥z q\rh�Hr^Û�o?pwvcpwz6o1tc~Jx�t�»¥p]oï�Z~bscvcp]�^�?�j~bs�»Hvc�?~bvDp]vMp]t ���?~bs����Âv�z{�?scr��?p]��v�vc�?r�qb~\�]�Hr�z\|*Ý ý ÿ����� r^q\r�oï¹D�Hr�o � p]t+¿�oHz ¹Mo � p]|°úhp]t��?oH¿ÁoHzJ¹Do1~\o?�1su~\o?�?z6¾Â�wx{���Hz6t�r�oPÄ

¶ �]vc�Hz6�H�6��¹grM�?p]�WoHz\v=�6pwq\rM~'|�z\su¾Â~\�×»&��z6¾Â�Z�wr�ÜHpwvCx=3vc�Hr^z\scr^vcp]�M�Hr^ÛZo?pwvcp]z6ohz\|µz6oHr33¹ù~�xW|k�?o?��vcpwz6oZt^»¹(rW�Hz\�µr�vc�Hrdscr�~\�Hr^sB¹Mp]�j�#t�r^rW�HzJ¹ tc�?���m~_�Hr^ÛZo?p]vcpwz6o1¹gz6�?�]�ï�Zscz���r^r��#Ägf ¶ o?�ï¹grhvc�?r^scr^|.z\surdt�vc~bv�rvc�Hr�|�z6�]�wzJ¹Dp]oH�Wvc�Hr^z\scr�¾º¹Mpwvc�1�]pwv�vc�]rh#c�?t�vcp]ÛZ�^~bvcpwz6o1~\o?�ïo?z��?scz�z\|yÄ

iejlk=m�n?k%o prq Ê�ì�Ô?ø6ÖbÍ��bÏ�ø&Öts Ò�ÔÁÏ�ÕJÌkÎkÖbÏ?ÊdìXu\Î.Ê�Ì(Î Òv�bÏ�ø1ÖbÏ?ò�íÈÎ ÒhÖbϵì3wyxh�bíMÒ�ÔÁÏ�Õ�ÌkÎ×ÖbÏHÊdìKu\Î.Ê�Ì"z

{�|}{ ~E��~���ird�^~\o`�Hr^ÛZo?r�~\om~\oZ~\�wz\�\z6�?tBoHz\vcpwz6omz\|N~Â�Zt�r��Z�Hz\su~\o?�Hz6¾î�¥r^s�¾h�Hvc~bvcp]z6o ÿ �(·M� � Ä_(Mr^sur\»�¹grd��z6o�tcp]�Hr^s½|k�?o?��vcpwz6o?t��üÞÁàJáÁâ�ã äJågædàJáÁâ�ã ä�ç è àJáÁâ�ã ä�ç tc�?���Wvc�?~bv^»&|�z\sN~\�]�?ú�û`àJáÁâ�ã äJåÁ»6¹gr+�?~�q\r+vc�Z~bv��°ýp]t�~_�¥r^s�¾h�Hvc~bvcp]z6omz q\r^sÂàJáÁâ�ã ä ç ÿ ~\o?�i�^~\�]�°|k�?o?��vcpwz6o?tB¹Mp]vc�mvc�Zp]tB�?scz\�µr^scvCxÉë\ì�í&ìuøWðHì^Í)sWÔHÌ"�\ÌkÎ×ÖbÏHÊ � Ķ �]tczH»6¹(rD�Hr^ÛZoHr����t�"� ç ~\t*vc�?r+t�r^vNz\|µ~\�]�Á�µr^su¾h�?vc~bvcpwz6o?t½z6o_àJáÁâ�ã ä�çWÄ ÿ ¶ tN~\oÂr�Ü�r^su�^pj��r\Þ½�?zJ¹£¾Â~\o%x¼ZpwvctB�Hz%r�t'pwvBvc~b¿\rWv�z�scr^�?scr�t�r�o%v'~_��z6¾Â�Z�wr^v�r��wxïs�~\o?�Hz6¾´�¥r^s�¾h�Hvc~bvcp]z6o�� � Ä r\Äw»µ�?zJ¹ü¾Â~\o%xf�?p65µr^scr�o%v�¥r^s�¾h�Hvc~bvcp]z6o?t�zJq\r^shàJáÁâ�ã ä�çü~bscr�vc�?r^scr?� � � oï~ȾÂ~\oZoHr^sM��z6¾��Z�]r^v�r��wx_~\oZ~\�wz\�\z6�?t+v�z{vc�Hr�~b¼µzJq\r\»¥¹grtc~�xWvc�?~bvh�÷p]t*~ ÿ�I âKJ � C�=·D�Rp]|�pwvNp]t*~'¿\r^x\r��{�µr^su¾d�Hvc~bvcpwz6o�tc�?����vc�?~bvN|�z\s=~\�]�H~\�]�\z\supwvc�?¾Ât�L su�?oZo?p]oH�p]o�vcp]¾�r�~bvB¾�z6t�v I ¹(r��Z~�q\r\Þ

NNN �NsPO ú�Q àJáÁâ�ã ä å ÞSL��VU)WbY [ � ã]\_^É�Ns?O ��Q`���t�"� ç ÞaL���WbY [ � ã]\ NNN�c J,&Ú½p]o?~\�]�wx\»�¹(rd¾�r�o%vcpwz6ofvc�Z~bvMz6o?��rd¹(rd�?~�q\r�~È¿\r^x\r��m�µr^su¾d�Hvc~bvcpwz6ompwvDpjtMo?~bvc�Hs�~\�#v�z_vc~\�w¿ï~b¼µz6�Hv

vc~b¿�p]o?�1vc�HrÈp]o%q\r^sut�r�z\|�� ý »"|�z\s�~��Z~bscvcpj�^�?�]~bs�ú&Ä ¶ �6~\p]o#»"p]o`vc�?p]t��^~\tcr¹(rȹ(z6�?�j�i�Hz\�µrdvc�Z~bv�pwv�¼µr�}���1�}�0�0�g���K�g�g�v���V���K�,���?�?�a�"�0� �)¡"�g�]�g�t�K��¢£�g���¤�0���1¥0¦?�)¡}�K�t¦P�]§H¨©¥0�?�y�"�g�]�8ª?¥?���K�'���t�X¡��"���y¥0���K¡ ¢ U¤« �­¬ �]¬g�=�]�?�1���3�y�® ���°¯0±P��¦0²³�g���?�)��¬l´Z¨��'�)¥��P�?��µ��"�0�h¶)�)�g¥0���)¨ ® �t�"�0��¨6¥?�0�y�"�g�)�v�?�·���)�0�)�y¡R¸Z���'�)�,�¹¡}�)�t¦P�]§MºP¬¹»,�·���?�¼�"���]¦��K¨_�"�1�P�g�0��"�0�K�¹�"�?�AªV��±½¸Z��§M�?�g�1§M�1�'�"�¹�·�0���1¥0¦?�K¡}�)�t¦P�]§¾¨6¥0�?�y�"�g�]�?º�´°���0�]¥?��¦��"�1�}�0�?�g���)�g�¿���"�1���"�0�AªV��±½¸Z�g§M�V�1�g§M�1�'�"�¹¢ U ¨6�K¡¡}�K�t¦P�]§M�¿�,ÀZ�}�0�)���1� ® ºP¬ Á_¥?�°´�µ_�]�a �l�K�gµ��1�?�°ªV�¹�"�0�g�����X¡"�y¨6¥?�ìÄ1Å �)¡¤�"�0��¶3�Æ¡���g�'�"�y¡"�1�¼�"��¦¡"�1�]¦?�Æ¡��lµ_�v�V�)�g�,�7�]¥P�7�"�0�K�·�"�0���1�)§M�0�g�y±P�¿���,À­�"�0�1�K¡"�y�"�g��¦?�y¯0�0�¿�"�g�]�Ç�1�)�½�0�È�g�)�0�]�Æ¡�y�]�0����¦P�y¡l¨©¥0�?�y�"�g�]�0�°¢�µ³�¿�"�R�g�0�?¥?�³�g�1�?�)�"�=É?�g�0�¼�"���)¦a�P�"�0�¹�g�?�0¥?�³�g�1�?�)�"�R§ ¥0�¼�³¦P�1�V�1�0¦7�]�¤�"�0���3�Æ�·���y�0�)�"�a¬

-

Page 46: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

��r�~\t�x���v�zï��z6¾Â�Z�Hv�rÈ�ËÊ $ý ÿ � � »P�6p]q\r�oÉú�~\o?��� ÿ oHz\v�r�vc�?~bv'vc�?pjtDp]t'oHz\v�oHr���r�tutc~bsup]�wx1vc�?rW�^~\tcr � ~\t¹(r��?~Jq\rWtcr^r�oe~\�]scr�~\�Hxïp]omvc�?p]tD�^�]~\tct^»µvc�Hr^scrW~bscrd�Z�wr�o%vCxïz\|=�¥r^s�¾h�Hvc~bvcp]z6o?tM|�z\sB¹M�?p]���fvc�Hrd|�z\sc¹g~bs���?pwscr���vcpwz6o_�^~\o�¼µrB��z6¾Â�Z�Hv�r��_r�~\tcp]�wxȼZ�Hvgvc�HrBpjo&q\r^sutcr'�^~\oZoHz\vù¼¥r � Ä*Åù�Hvùz\|��z6�?sut�r\»Á¹(r�¾Âp]�6�&vù�Hz\�µrvc�?~bvMpwv�¼µr:���?pÃÌ{�^�?�wvX�Wv�zÂ�?sur��?p]��vM� Ê $ý ÿ � � ¹M�Hr�oïúhp]t+�?o?¿�oHz ¹MoPÄ � o1|�~\��v^»�vc�Hr^scr�~bscrht�r��^�Hscr��(·M�(t|.z\s�¹M�Zp]�u�_pwvùp]tË��r�~\t�x��Wv�zW�?sur��?p]��v=vc�?r'pjo&q\r^sutcrBz\|l�ËÊ $ý ÿ � � »Á|�z\s�~h��~bscvcp]�^�?�]~bsù�u�?z6p]��r�z\|³�{~\oZ�{r^q\r�o|.z\sM~Âsu~\o?�Hz6¾Â�]x{�u�?z6t�r�omú&Ä

��z�vc�?p]tgscr,���?pwsur�tg~\o?z\vc�Hr^sM�Hr^ÛZoZpwvcpwz6oPÄ��ir�tc~Jx{vc�?~bv7� p]t�~_Ê�ÌkÍ�ÖbÏ%Ð ÿ�I âKJ � C�(·M� pw|"pwv+p]tù~W¿\r^x\r���¥r^s�¾h�Hvc~bvcp]z6oïtc�?���1vc�?~bv+|�z\sM~\�]�P~\�w�\z\s�pwvc�?¾ÂtML8su�?oZo?p]oH�Wp]o�vcp]¾�r�~bvD¾�z6t�v I ¹(rh�?~�q\r\Þ

NNN �NsPO úRQ àJáÁâ�ã ä å ÞSL � U WZY [¼Í ��Î �U WbY [°� ã]\_^É�Ns?O ��Q`���t�"� ç ÞaL ��WbY [�Í � Î � WbY [�� ã]\ NNN c J,&�Mz\v�rWvc�?~bv ÿ p]ofr�~\���er�Ü��µr^sup]¾Âr�o&v � ¹grWoHzJ¹8�6pwq\rËL ~\�^��r�tctBv�ze̼x=Ö_z\su~\�^�]r�t^Þùz6oHrhsur^�?scr�t�r�o%vcp]oH�Âvc�?r|.z\sc¹ù~bsu�1r^qb~\�]�?~bvcp]z6o1z\|½vc�Hr��µr^su¾h�?vc~bvcpwz6oï~\o?��z6oHr�scr^�?scr�t�r�o%vcp]oH��vc�?r�p]o&q\r^s�t�r\Ä

¶ t�¾Âr�o&vcpwz6o?r��{r�~bs��]pwr^s�»ÁoHz\vùr^q\r^scx{�(·M�Çpjt(~\�Hv�z6¾È~bvcp]�^~\�]�wx_~dtcv�scz6oH�W�=·M�°Ä%�ir�t�vc~bv�r�vc�?pjt(�Hr^sur~\tM~�vc�Hr^z\sur�¾1Þiejlk=m�n?k%oÐÏ�Ñ ÊuÊ�Ô%sWÎ�Ï%ÐdÌ�óÁì'ìXu\Î.Ê�ÌCì^Ï�Õ�ìBÖ�ÒMÖ\Ï¥ì3wyxh�bí=Ò�ÔÁÏ�Õ�ÌkÎ×ÖbÏHÊ)ÒÌ�óÁì^Í�ì�ìKu\Î.Ê�Ì�ʤ��ë\ì�í&ìuøùðHì^Í)sWÔHÌ"�\ÌkÎ×ÖbÏ�Óx½ó%ÎkÕ^ó�Î.Êe� qhÔ·qBÕ ÔÁÌ=Ï�Ö\ÌM�ÈÊ�ÌkÍ�ÖbÏ%Ð q Ô·q z� vMpjt�~Â�?t�r^|k�?�µr�ÜÁr^su�^p]t�r�v�zÂv�sux_v�z{��z6o?t�v�su�?��v+~\ofr�ÜÁ~\¾Â�Z�wr�z\|½tc�?�u�f~ ��ÄÚ°pjo?~\�]�wx\»Á¹gr�t�vc~bv�r�vc�?r�|.z6�]�]zJ¹Mp]o?�dvc�?r^z\scr�¾î|�z\s+scr^|�r^scr�o?��r\Þiejlk=m�n?k%o×ÖÙØ ÌkÍ�ÖbÏ%Ð q Ô·q ÊhìKu\Î.Ê�Ì(Î Òv�bÏ¥ø1ÖbÏ?ò�íÈÎ ÒhÖbÏ¥ì'wÆxh�bíDÒ�ÔÁÏ�ÕJÌkÎkÖ\ÏHÊdìXu6ÎjÊ�Ì"z{�|ZÚ Û½Ü"Ý�ÞSß�àÇá"â ãhä�åa�� �?~bvdp]t��?z6oHrÈp]o`�?s�~\��vcp]��r?� � oi�?su~\��vcpj��r\»°�µr^z\�Z�]rÂ�?t�r Õ ò Ö�Õ^ëmÕ^Î�ðZóÁì^ÍuÊ�ÄÇ�M�Hr�t�r¼Z�]z��u¿`�^p]�Z�Hr^sut�~bsur¿\r^x\r��i|��Zo?��vcpwz6o?t^»a#c�?t�v'~\t��Hr�tc��supw¼µr��m~b¼¥z q\r\Ä�Ú?~\¾�z6�Zt'r�ÜÁ~\¾����wr�tBz\|=¼Z�wzÁ�c¿m�^pw�Z�Hr^sutBp]o?�^�j�?�HrvæRçg�µ»� æRç ¶ »+v�sup]�Z�wr3}æRçg�¥»�~\o?� ¶ çù�¥Ä ¶ tm~\o r�ÜH~\¾��Z�wr\»�æRçg��pjt8#��Zt�v1~a¿\r^x\r��÷�µr^su¾d�Hvc~bvcpwz6oéè¤ê%ëÓÞàJáÁâ�ã äPì�ídæÉàJáÁâ�ã äPí1îWè àJáÁâ�ã äPí1î6ÄR�+�?~bv'p]t^»�vc�Hrd¿\r^xm�wr�oH�\vc�mz\|hæRçg�fp]t �\� ¼�pwvct^»µ~\o?�fz6oZ��rd~{¿\r^xeúp]tgÛHÜ�r��#»_è·ê�ë ý pjt(~d�µr^su¾d�Hvc~bvcpwz6o_zJq\r^s'àJáÁâ�ã ä í1î ÄNÚ?�Hscvc�Hr^s�¾�z\scr\»Hpwvùp]tgr'ÌÈ�^pwr�o%vgv�z���z6¾����Hv�r�~\o?�_v�zp]o&q\r^suv¤æRçg��pw|x\z6�ï¿ÁoHzJ¹÷vc�Hr�¿\r^x\Ä

� �?r�o`¼Z�wzÁ�c¿m�^p]�Z�Hr^sutB~bscr��?t�r��ip]oe~\oi~b�?�Z�jp]�^~bvcpwz6oP»Ppwv'p]t'p]¾��µz\scvc~\o%vDv�zï~\tc¿f¹M�?~bv'�?suz\�¥r^suvcpwr�tz\|*vc�Hrh�^pw���Hr^sM~bscrd~\tctc�?¾�r��#ÄgÚHz\sBr�ÜH~\¾��Z�wr\»¥�Hz%r�tDt�r��^�Hsup]vCx�scr��wx�z6oïvc�?r�|k~\��vMvc�?~bvBvc�Hr��^pw���Hr^sMp]tD~�=·M� z\s��Hz%r�t�p]vBsur,�%�?p]scr�vc�HrÂtcv�scz6oH�\r^s�~\tctc�Z¾��?vcpwz6oevc�?~bv'vc�HrÂ�^pw���Hr^sBpjtB~1tcv�scz6oH�1�=·D�A�Eï�omvc�?rz6oHr+�?~\o?�P»\pwv*p]t°scr�~\tcz6o?~b¼Z�wrùv�z�~\tctc�Z¾�rùvc�?~bv�æRçg��z\s ¶ çg�hp]t½~�t�v�scz6oH���(·M� ÿ �^pw���Hr^sut½~bscr��Hr�tcpw�6o?r��¹Mpwvc�Âvc�Zp]tNp]o¾Èp]o?� � Ä�ï'oÂvc�?r+z\vc�Hr^sù�?~\o?�P»%vc�?p]tNp]t(~�t�v�suz6oH�\r^s(~\tctu�?¾��?vcpwz6o#»&t�z�¼�~\tcp]oH��vc�?rMt�r��^�Hsup]vCxz\|*x\z6�HsBtu�u�Hr�¾�rhz6om~ȹ(r�~b¿\r^s�~\tctu�?¾��?vcpwz6ofp]t+�?sur^|.r^su~b¼��wr\ÄgÚ½p]oZ~\�]�wx\»Zt�z6¾Âr/��zt5_3vc�Hr3Ctc�Hr��w|X�_�^pw�Z�?r^sut¾Â~�x�¼µr��=·D�=t�¼Z�?v+oHz\v+t�v�scz6oH�{�=·M�(t^Ä

Ú½p]o?~\�]�wx\»&¹(rB¾�r�o&vcp]z6o�vc�?~bv=vc�?rDt�r��^�Hs�pwvCxdz\|P¼Z�wzÁ�c¿W�^pw�Z�Hr^s�t ÿ ~\t=z\�Z�¥z6tcr���v�z�¹M�Z~bvN¹(rB�?~�q\rD¼µr^r�o~\p]¾Âp]oH�_|�z\s'pjomvc�Zp]tD�^�j~\tct � p]t'�]~bsc�\r��wxm�Hr��Hsup]tcvcp]�bÄ � oez\vc�Hr^s�¹gz\su�?t�»¥¹gr��?~�q\rÂo?z_��~bscvcp]�^�?�]~bsBscr�~\t�z6ov�zi¼¥r��jpwr^q\rÈvc�?~bv ¶ çg�Rp]tW~e�\z%zÁ�ƼZ�wzÁ�c¿Ñ�^p]�Z�Hr^s�»½z\vc�Hr^s�vc�?~\oÆvc�Hr�|k~\��vWvc�?~bv�pwvh¹ù~\t��Hr�tcp]�6oHr��Ѽ�xr�Ü��µr^scvct�~\o?��t�vc�Z�?pwr��_p]o&v�r�oZtcpwq\r��wx�¼�x{��scx��?v�z\�\su~b���Hr^sutù|�z\sgvc�?rB��~\t�v·-�x\r�~bs�t^ÄAï'o_vc�Hr'z\vc�Hr^s+�?~\oZ�P»¹(r���z6�Z�]� ÿ ¼�xÇ�M�Hr^z\scr�¾ñð � ��z6o?t�v�su�?��vD~Ât�v�scz6oH�Â�(·M� ¼Z~\t�r��1z6oïvc�Hr��?~bsu�ZoHr�tctùz\|½|�~\��v�z\s�p]oH�HÄ �M�Hr�?~bsu�?oHr�tutMz\|N|k~\��v�z\sup]o?�_p]tD¼Z~\�u¿\r��i�H�m¼�x�ðbá\á_x\r�~bsut'z\|=p]o%v�r�o?t�rWtc��su�?vcp]o&x�¼�xf¾Â~\o%x\»P¾Â~\o&xf�¥r^z\���wr¹M�Hzd¹(r^scr'r�ÜÁ�µr^scvct(pjo{~dqb~bsupwr^vCxÈz\|"Û?r��]�Zt^Ä��M�Hr j�òló k �?p]tc~\�Hqb~\o%vc~b�\rBz\|vc�?p]tg~b�?�?scz6~\����p]t=vc�Z~bvgvc�?rscr�tc�?�wvcpjoH�1|��Zo?��vcpwz6oi¹gz6�?�]�Ѽ¥r�z\s��Hr^sut�z\|+¾È~b�6o?pwvc�?�Hr{tc�wzJ¹gr^s�vc�Z~\oôæRçg�iz\s ¶ çg�µ»"~\o?�Évc�?r^scr^|.z\suroHz\v+�?su~\��vcpj�^~\�P|.z\sD�?t�r\Ä

ð

Page 47: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

õ7ö�÷­øVù'úKûX÷ÃüÆý�þVÿ����túKý����Vö�� �� ��������� öaü�úKþ������3üX÷ÃþSöÇü�þ úXý���ü�þ��Vú������aý��úXþVÿ�ù,ûXû�þVú! VþSö��tü"���Vö�#$�tü&%

')(+*-,/.10/( 243

5 687:9";/< =?>A@CBD>�EGFIHKJ�7LENM�=L9"OPHRQTSL9�HKUV68WYXZS

�\[^]¹û_�Vö��+�\[`�hûa���ÃþSö�ù^�túXù¤ö�þVü�ù,ö�þb�c�b�Èü�þv÷�d4dÈùe��÷��tü�ùe�Ãý��S÷ÃøVù¤û�ùe�f��úXù·ù,ö��3úXý���üX÷ÃþSöÈû�����ùedÈù,ûeg�û�þbd ù�f�túXù¤÷­û�ö�ù'ùe��ùe�ih�]�þVúhùkjc��d+���Ãù�lC�3þSö�ûX÷���ù'ú�ü"��ùnm�ö��tü"��ú�����oRù,ö��3úXýC� üX÷ÃþSöÈû"����ùedÈù¤÷­öNp^��÷q���+r�÷­û��s�\[^�t p¤÷Ãü"�½ùfu4�'÷­ù,öaü¤÷böaøVù'úKûK÷ÃþSö�v �Vö��½ü"��ù�û�ù,ö���ù'ú^�Vö��½úXùe�3ù,÷ÃøVù'ú�û"���túXùs�nwVù'ýyxe÷­öz����ø{�Vö��3ù�h_|°þ�ù,ö��3úXý���üdÈù,ûXû"���Vù~}Zl°ü"��ùû�ù,ö���ù'ú��3þbd+����ü�ù,ûK����r�� t }�v-�Vö�� û�ù,ö�� û�ü"��÷­û�ü�þ/ü"��ù úXùe�3ù,÷­øVù'úehy|°þ���ùe�3úXýC� üflü"��ù¤úXùe�3ù,÷­øVù'ú �3þbd+����ü�ù,ûa}��)r+�i�� t ��v t ûX÷bö��3ù^r ÷bûa�$��ù'ú�dN��ü"�tüX÷ÃþSöilC��ùe�3úXý���üX÷ÃþSö4����p �Pý�û û"���f�3ù'ùe��û�v�h�¤þ�p ù'øVù'úelPp^��÷q�Ãù½ü"��÷­û�÷­û�ûXùe�f��úXù����b�V÷bö�û�ü��'÷�����ù'úXü�ùkj%üEþSö��Ãý1�tü�ü"���"w%û t p^��ù,ö��ôûX÷böc�b�Ãù�dÈù,ûKû"���Vù�÷bûù,ö��3úXý���ü�ùe��v�÷­ü_��þ�ù,û^�?����û"�tüX÷­ûXÿZýÈþb��úAö�þVüX÷ÃþSö�þVÿ³÷­ö���÷bû�üX÷­öc�b��÷bû"����� ÷��b÷6ü}ý t ÷­öEÿ����3üfl�÷Ãü\�f�Vö ö�þVüAûK÷­ö��3ù·ü"��÷bûù,ö��3úXý���üX÷ÃþSö:û"����ùedÈùv÷bû/��ù'ü�ù'ú�d ÷­ö ÷­û�üX÷���v�h

�i��� �Z�����\�_�����b���\ ¢¡\�£�{¤\¥L¦_§©¨«ª+�\¬®­�¯�°a�b��±¢�Y²�¬�¤\¨�³´¨�¤ù'úXùô÷­û½þSö�ùµ��þSûXûK÷����Ãù�û�þb����üX÷­þSöi¶1·³ù'ü�¸ ¶+¹�º®»e¼{½�¾Z¿)¹�º®»e¼{½eÀÂÁ ¹�º®»e¼{½eÃLhÅÄ+ùÆ�3þSö û�ü�ú����3ü��Vöù,ö��3úXý���üX÷ÃþSö�û�����ùedÈùeÿZþVúPdÈù,ûXû"���Vù,û7þVÿa�Ãù,öc�Vü"��ǵp^��ù'úKù�ü"��ùeû"���túXùe�zwVù'ý:÷­û·þVÿa�Ãù,öc�Vü"��È t ö�þVü�ùeü"���tüù'øVù,ö�÷­ÿ\È�É´Ç«p ùK�túXù��3þSö û�ü�ú����3üX÷­ö��Ê�û"����ùed ùeü"���tü!���VûsËÌ�ÎÍ&���CÏ�ÐkÍ�ûXùe�f��úK÷ÃüÆýy��úXþ���ù'úXüX÷Ãù,û¤ü"���Vö�ü"��ùþSö�ùkÑ"üX÷�dÈùÒ������v�h�|Ó��ù�û�ù,ö���ù'úA�Vö��8úXùe�3ù,÷­øVù'ú�û"���túXùÒ��ú��Vö���þbdÔwVù'ýNx�ÕÖ¹�º®»e¼{½�¾®h�|°þRù,ö��3úXý���ü�dÈù,ûXû"���Vù×Øl_ü"��ùËû�ù,ö���ù'úP����þ�þSû�ù,û!�Eú��Vö���þbdÅÙ~Õ«¹�º®»e¼{½ À lL�3þbd+����ü�ù,û$�Ú��×ÜÛ1¸�� t Ù�v�lL�Vö��/ûXù,ö���û�Ý�Ù{»��sÞ�h|°þn��ùe�3úKýC��üÒ�'÷�����ù'úXü�ùkj�ü$Ý�Ù{»��sÞ�l�ü"��ù�úKùe�3ù,÷ÃøVù'úÓ�3þbd+����ü�ù,ûÓ×ß�à�áÛƸ � t Ù�v t ö�þVü�ù�ü"���tü/p ù�ö�ù'ùe�½ö�þVü�_ù$�����Ãù�ü�þE÷­ö=øVù'úXüÓ¸ � ÷­öÇÿ����3üfl?¸A�¤ö�ù'ùe�:ö�þVü¤ù'øVù,öz��ù$�K��ù'ú�d���ü"�tüX÷ÃþSö�v�h

â¤þVü�ù½ü"��ù~���tú������Ãùe�_p¤÷Ãü"�£ü"��ùù,ö��3úXý���üX÷ÃþSö*û"����ùedÈùÇÿZúKþbd ü"��ù���úXù'ø%÷Ãþb��ûN�Ãùe�3ü"��úXùÇ÷­öIp^��÷���� ü"��ùû"���túXùe�ÊwVù'ýÊxPpÒ�Vû ø%÷ÃùfpAùe�Ê�Vû/x � »fãfãfãe»�x�äsh�·³ù'ü�üX÷­ö��Kx�å�ækçéè�ëê�� t�ì v�l�÷ÃüA÷­û �Vû ÷Ãÿ�ü"��ù7û�ù,ö���ù'ú �Vö���úXùe�3ù,÷ÃøVù'ú�túXù�û"���túK÷­öc��í{À û�ùe�3úXù'üXûKx � »fãfãfã�»�x�î&ï t ûK÷­ö��3ù ü"��ù'úKùÊ�túXù�í{À���÷�ð_ù'úXù,ö=ü$��þSûXûX÷����ÃùËø{���q��ù,û�ÿ�þVúsÙ�v-�Vö����ûX÷­öc�½ü"��ùed�÷­öµ���3þbd+���Ãù'ü�ùe�­ýÖ�Vö����Ãþ��Vþb��ûsd4�Vö�ö�ù'úeh��¤þ�p ù'øVù'ú�lÓ��ñ®ÐkòZÐ"óbôfñ«����õöò�Ëk�©��Í&ÐnÈ´÷fø���ËÊ�Vö����ñ®Ðkòzô�ó��IÐ"óbôfñ4ù�úIû�ü�ù�ý?þ{ÿ�� ô��������c��Ð$x�å�f��ÍNó���ò ì � ü"��ù,û�ù���Xþ���ù�XüX÷Ãù,û`��Xùvùkjc���3ü"�Ãýzp^���tü^�\[`]�û�����Ãþ�psh� Rö�ù4��÷�ð_ù�Xù,ö��3ùnp¤÷Ãü"�+ü"��ù�û"����ùedÈùÿ�Xþbd�ü"��ù4��Xù'ø�÷­þb��û$�Ãùe�3ü"��Kù4pÒ�Vûvü"���tüfl���Kù'ø�÷Ãþb� û"�Ãý����q�ü"��ù`wVù'ý%û p ù�KùP�3þbd+���Ãù'ü�ùe�Ãý���Vö���þbd p^��ù�Xùe�Vû ö�þ�p!ü"��ù'ý4��Xù!�������Vù,ö�ù���tü�ùe����ûX÷­ö��s���_[^] h�Ä+ùPp7÷������XþPøVù���ùe�Ãþ�prü"���tü¤ü"��÷­û/��þ�ù,û·ö�þVü����÷­öü"��ùvû�ùe�f��K÷Ãü}ýþVÿ�ü"��ùvû"����ùedÈù�h��� ù�����ù�� �����_¸ ø Ë!ó �"!�#%$���ñcÐk�z��ñ®Ð!ËkôkñcÐ&�4Ð$ó�÷���'�Ð-øqË`ø ��(�øqËk�Îø �CÏ)�®øqË�ñ�ó�÷kõ�Ð�*�ø,+ÎÐ-+.$AËfÐ"ô-�®Í&Ð-ó�Ï�ó�ø��cËÌ�ó��Öó/(�'�ÐfÍ�Ëfó�Í�ò10�ñ���ô�ó��32-��ÐfÍ�òÊ��ñ®Ð�465 ��Í óbôeõ�ÐKó���ò7�®��õ ò������Køéó�õ¢���6�4÷�ÐkÍ��8�-�Îø9�4ÐÌË;:<+����=�có�ÍÌ�Îøéô��®õ�ó�Í>$ø �+¸ øqËzó t9? »A@�v>BC�D!E# ��ñ®Ðk�F�f��Ízó�õ õ^ó/(�'�ÐfÍ�Ëfó�Í�ø©ÐÌËHG ÍI�®����ø��CÏÆø �´�Îø��4Ð ? ó���(J�+ó�K�ø��CÏáó�� �n�{Ëk�ML2���ÐkÍÌøéÐkË$�©����ñ®ÐN4�58��Í&óbôfõ�ÐN0_Ðsñ�ó�'�Ð-O

PPP �DRQ xTS ¹�º®»e¼{½ ¾ gAU�S ¹�º®»e¼{½�¶�G VIWRXZY [A\�]_^_] `A��UAa�b ¼dc�í PPP�e @gfih=j Lîí À v�ã*9k�����Ðz��ñ®ó��$ø �l@Rjnm vo0_ÐfÍ&Ð�óZ�?Ð�Ï�õöø Ïbøé÷fõ�Ð7�>�®��ô��ÎøÎ����$Nó���(pL)jnm v10_ÐkÍ"Ð�ób� �n�{ËÌ�q�®��õöò{�?���KøÎó�õ`ø �áÈ�$�ó���(

}��sr�jéÈ�vs��ñ®Ðk�t@RjéÈ�vufih=j�vxwî&ï v-øqË$�?Ð�Ï�õöø Ïbøé÷fõ�Ðd+_:¼

Page 48: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

y¤þ<z�ù�z|{�}�z¤û|~­ö��3ù7z|{�ù }���øVù�Kû|}�Kýo~bû/�.~�d=~�z�ùe�1z�þ����ö�ö�~bö���~­öoz|~qdÈù }�z^dÈþSû�z ? l�pAù$dN��û�zE{�}PøVù7L e ? h� �R����� �7ûXû���dÈù1z|{�}�z4¸�~­û�}�j ? »A@�v�Ñ��\[`] lD}Vö���}VûXû"��dÈù�p ùp{�}PøVùt}Vö�}���øVù�KûA}�Xý�G�}�z�z|}��"w�~­ö��z|{�ù�ù,ö��&Xý6��z|~ÃþSöÇû"�>{�ùedÈù }���þPøVù�~bö�z|{�ù�û�ù,ö�û�ùRþVÿ��Ãù'ÿ�z Ñ"þ<"ÑCA~��){�zq~bö���~­û�z|~bö��b��~­û|{�}���~���~�zÆý�h��+ùs�3þSö�û�z�����&z}Vö�}���øVù�KûA}�Xý��VpE{�~��A{�z�A~­ù,û�z�þÆ��~­û|z|~­ö��b��~­ûA{«¸��9Xþbd�}�A}Vö���þbd��Î��ö��&z|~ÃþSö:há[¤ùe��}����Mz|{�}�z���~bû�)~ÃøVù,öÇþ<A}��f�Ãù7}��f�3ù,ûXûMz�þ ûXþbdÈùT����ö��&z|~ÃþSö4p�{�~��A{o~­ûAù-~�z|{�ù�q}K�3þbd����Ãù�z�ùe�Ãý=A}Vö���þbd�����ö��&z|~ÃþSöþ<Aùe�bû�ù7}����ö��&z|~ÃþSöy¸ � pE{�ù�Kù���~­û/�>{�þSû�ù,ö�}�z�A}Vö���þbdzh%�����ö�ûM}VûM��þb���Ãþ�p¤ûf¶�N� \�] `� {�þ=þSû�ù�A}Vö���þbd�UPÕÖ¹�º®»e¼��[`��öpG�l�û|~�dN���.}�z|~­ö��lz|{�ù=4�5�þ<A}��f�­ù$��}����­ûMþ<��G }Vû���þb���Ãþ�p7ûf¶¡ öp~­ö�����z7j�}p¢�»&} � v"��Xþbd£Gsl���~��"w1A}Vö���þbd Ù+ÕÖ¹�º®»e¼��eä ��ù�Xýþ<A}��f�Ãù�¸éþSöp~­ö�����zÒÙNz�þH�Vù�z·þb��z�����zM¥[�¦�z"��)ö§z|{�¦�}Vö�û"p"¦��¥$ÛI}p¨©z�þHG�ª{�¦,öpG�þb��z�����zXû�~�zXûM�b��¦,ûKûEU>«¬��þ<�U�¶~��­U « ��U®z|{�¦,öp�b��¦,ûXû=¯8� û|¦e����þ<A}Vö���þbdnopj,~éh_¦�h�l�|¦�z"��KöZ¼�v~��­UA«�°��U®z|{�¦,öp�b��¦,ûXû=¯8A}Vö���þbd±j,~éh_¦�h�l�A¦�z"��Köyº�v

y¤þ<z�¦7z|{�}�zqz|{�¦T�� ö�ö�~­ö���z|~�d�¦�þ<�²�³~bû"¦,ûKû�¦,ö/z|~�}����Ãý�´Ìl�p�{�~��A{o~­û"z|{�¦T���ö ö�~­ö�� z|~qd�¦7þ<�gGsh_�~­ö��&¦!¸µ~bû}tj9´�»A@Ìv�Ñ��\[`]_l�p"¦sw�ö�þ�psz|{�}�zf¶PPP �%�¶_�®S ¹�º®»e¼�� ¾ ¶�� �<[ \�] ` �Ú¼Ia�bµ�Dd¶ ¸µS·5�¸�¹Rº À©»!à ¶)� � \�] ` �Ú¼Ia PPP e @fã j ¼�v

�¼{�¦,öZ��þ�¦,û7�Óþb��z�����z+¼d½ª¾aj�}��&z"�­ý�pE{�¦,ö¿GÙþb��z�����zXûTz|{�¦K�3þ<A|¦e�&z �b��¦,ûXûT��þ<�U�hN�P�­û�þ½ö�þ<z�¦�z|{�}�zpE{�¦,öt�1Àgû¤þ<A}��f��¦�~bû�� û�¦e����þ<A}Vö���þbdzl�z|{�¦,öt�Á��XþPø�~���¦,û�GÚpE~�z|{p}Vö�¦kj�}��&zRû|~�dN���.}�z|~­þSö½þ<�­z|{�¦�A¦-}��465�þ<>}��f��¦�h\|�{�¦,û|¦�þ<� û�¦�Xø�}�z|~ÃþSö�û�~qd����Ãý?¶

�%d¶_�TS ¹�º®»e¼�� ¾ ¶)� � [ \�] ` �Y¼IaL�)�%�¶_�®S ¹�º®»e¼�� ¾ gAU�SG¹�º®»e¼���¶/G V>W XZY [ \�]_^_] ` ��UAa©ã jéíbvÂu¦�z7ÃÅÄRÆ�Æ-Ç æÌç©è� �%d¶_�NS¹�º®»e¼�� ¾ gAU®S¹�º®»e¼��4¶�G V>W XZY [ \�]_^_] ` �µUAa���¦,ö�þ<z�¦�z|{�¦ÊË>��È>È�ÐÌË�ËK�8�+ó)(�'�ÐkÍ�Ëkó�ÍÌò=Gj,~­ö§z|{�¦7|¦-}��u¦kj6�Ŧ�A~�d=¦,ö/z�v�hD¾"ÉC��}�z|~ÃþSö�û j ¼�vM}Vö���jéíbvMû|{�þ�psz|{�}�z

Ê ÃÅÄRÆ�Æ�Ç�b ¼dc�í Ê e PPP �%R¶ ¸ËSÌ5�¸/¹�º À©»-à gAU�S ¹�º®»e¼��N¶/G VIW XCY Í \�]_^_] ` ��UAa�b ¼dc�í PPP f�@j�pE{�¦�|¦Mz|{�¦7ö�þ<z|}�z|~ÃþSöp4�5Ũ ^ � jnm�»�m v­|¦��9¦�Kû�z�þ�}Vö=~�dH}��)~­ö�}�XýN~­ö�û�z|}Vö�z|~.}�z|~ÃþSöËþ<�¬z|{�¦�¦,ö��&Ký���z|~­þSöÈû"�A{�¦ed=¦~­ö�pE{�~��>{§}n�3þbd����.¦�z�¦e�Ãý�>}Vö���þbdŸÎ~bû·û|{�}�|¦e�§~bö�û�z�¦-}��½þ<�­}Èû�¦�¦e�t�Zþ<E}4�\[^]av�hÏ®¦�Ð ö�¦®¦'ø<¦,ö/z®Ñ�Ò�ÓÔÓ<}VûD��þb���Ãþ�p¤ûf¶�p©¦7û|}Pý§ÑÅÒ�ÓÕÓVþ��f�f��Kû�j,}oÈ���õ�õöøqËÌøÎ���Ö�-ÈAÈ-�®Í�Ë�v"~��u}Vö=ýÈþ<�uz|{�¦^Ùvø�}�����¦,û��û�¦e�§z|{�Kþb���){�þb��zMz|{�¦7¦kj��Ŧ�A~�d�¦,ö�z�}�|¦ z|{�¦�û|}�d�¦�h � þSö���~�z|~ÃþSö�~bö��eþSöpz|{�~­ûq¦'ø<¦,ö�zE�)~Ãø<¦,ûf¶

PPP �%R¶ ¸ËSÖ5Ÿ�¹�º À©»-à gAUMS ¹�º®»e¼��K¶)G VIW�XCY ÍÅ\�]_^_] `¢��UAa�b ¼dc�í PPP� PPP �%d¶ ¸ S·5�¸�¹Rº Àq»!à gAU�S ¹�º®»e¼��K¶)G V>W XCY Í \�]_^_] ` �ªU Ê ÑÅÒ�ÓÕÓ a��%d¶ÕÑ�Ò�ÓÔÓ a

fá�Dd¶ ¸µS·5�¸�¹Rº Àq»!à gAU�SG¹�º®»e¼���¶/G V>W XZY Í \�]_^_] ` ��U Ê ÑÅÒ�ÓÕÓ a��%�¶ ÑÅÒ�ÓÕÓ a�b ¼dc�í PPP ãy¤þ�p$l²z|{�¦nw<¦'ý¿��þ)~­ö/z ~bûTz|{�¦=��þb���Ãþ�p�~­ö��c¶ ~��q}z�3þb���.~bû|~ÃþSö+ö�¦'ø<¦�eþ��f�f��)ûflz|{�¦,ö×j9�¬¦e��}�� û�¦np©¦H}VûXû"��d=¦z|{�¦7û�¦,ö���¦�IØR|¦e�&¦-~Ãø<¦��û|{�}�A¦�}s�3þbd=����¦�z�¦e�ÃýlA}Vö���þbdÎ�Î��ö��&z|~ÃþSö�v�z|{�¦�}���ø<¦�)û|}�Xý=G�{�}Vû©}�� û�þb����z�¦e�Ãý8ö�þí

Page 49: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

~­ö���þ<�d=}�z|~ÃþSö }���þb��z���~�zUMj,}Vö���~.zXû���Xþ<��}���~��.~�zÆý¤þ<���b��¦,ûXû|~bö��EU��3þ<||¦e�&z"�Ãý7~­ûu¦kj�}��&z"�Ãýn¼dØ�íbv�h�|�{�¦�|¦��Zþ<A¦�¶PPP �Dd¶ ¸µS·5�¸�¹Rº À©»!à gAU�S ¹�º®»e¼���¶/G VIW XZY Í \�]_^_] ` ��UAa�b ¼dc�í PPP� PPPnÙdÚ%Û ¶ ܵSÖ5Ÿ�¹�º<Ý »TÞ�ß U�Sáàdâ�ã-ä���å�G VIW XZY Í \�]_^_] `æ U Ê Ñ�Ò�ÓÔÓ a¬biädc�ç�è§m Ú%Û ¶ÕÑ�Ò�ÓÔÓ a PPPe ädc�ç7m Ú%Û ¶ÕÑ�Ò�ÓÔÓ aj9éE{�¦ Û ¦ 驦 }<ê.ë�ì�í�ë�¦7z|{�¦7�,}<î&zEz|{�}�z Ú%Û ¶ Ñ�Ò�ÓÔÓ a æ äMb Ú%Û ¶ÕÑÅÒ�ÓÕÓ a9ï>ðñ�~.ò�}<ê.ê�ó<ô�é"¦lò�¦�¦-õFz�ì1�¬ì)í�ò�õ Ú%Û ¶ÕÑÅÒ�ÓÕÓ aCðTö�{�~.ë�~.ë�z|{�¦N� Û ì<��}���~.ê.~�zxó�z|{�}�z�÷��<¦�zT}1ø=}�z|îA{¿~���÷���~.î|ùLH~�z�¦-ø=ëN}�z Û }<ò�õ�ì)øú� Û ì)ø�}�ë�¦�z�ì<�Eë|~�û�¦tàdâ�ã-ä�� Ý ðtö�{�~.ë � Û ì<��ê�¦-øü~.ë z|{�¦H驦-ê.êþýCù�ò�ìdéEò˯8��~ Û z|{�õ�}-ó� Û ì<��ê�¦-øHÿpj,ë�¦�¦ z|{�¦ ò�ì<z�¦-ëEì)òp� Û ì<��}���~.ê.~.z;ó�ï­� Û ì)ø éE{�~.îA{§~�z�~�ëqù6ò�ìRéEò§z|{�}�z Ú%Û ¶ÕÑ�Ò�ÓÔÓ a æ h=j�L � c�ç Ý ï>ðÚ ê.í��<�)~.ò���¦��<¦ Û ó6z|{�~.ò��=~�òo�)~��<¦-ë�z|{�¦7Ð�ò�}<ê Û ¦-ë|í�ê.z�åÊ ÃÅÄRÆ�Æ���biädc�ç Ê

e ädc�ç7m ÚDÛ ¶ÕÑÅÒ�ÓÕÓ a6f�@æ h=j L �ç Ý ïuf�@��

Page 50: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���� ������������������������! ���#"$ �&% $('*)�+-,/. *�0���!"�1#23�����45�0� $ ���!6��0��7�����698*�:;�����<�=���0���?>��4�����8����@A���0B

CEDGFIHKJMLKD NPO

Q R SATGU;VWSYX[Z]\_^�`ba-c-dbe�S\

. f��8g�h�3ig��jG6# ��A���;��8g�-6����� k���41��l ��=23��1�����m9�=#23���n6#�����4poq����rs�0�=2t8����2��� u ��!vw�4� ��x"g��y#��="p�<���zjG�=�0{����7��=�|���� ��=g7���8-}&~<oY8��������41g���������="_�<1#�23�����4h��oq������1�2�8-��8����|���g���k�=�����|�z���k�=���=�������k�=���=�#�t��(1��A���I��8g�41� �"&�b�G2� ��=���A�<���4j���8g�G����23�A��8����I��8g����2�8g�=jG�������� ��w��2�8#���� ��=�A�0�=2�1g�t����� ��&��8g���0�=�������z��#"����0�����741�����8����9�� ��¡��� , ������8]�4M��M�u*��1������� ������"]�4M�&�<����j��� K ��� ��= , ��8#���¢��8��p��2�8g�=j����j�jG�="��u���0�= ��5�3i!�0�=�"#��0�574�� ��_�P�0�=2�1g���¢�=�23���!6������4&��2�8��=jG�_�£���?�����#���0�t�����*{� ��=g7���8fj��=������7��=��� . �<��23��m9�4g�h23�41� �"x�������0�h��8g�I�<�4 � ���oY��g7_7��=g������ ¤��8g�������=jx�¥-¦�§n¨9©�§kª «­¬<®t¯�°3±�²´³�µ�¶<¶¸·4¹»º½¼(¾<¿�À3¯Áµ�¯ÂÀ�¯9Ã�²´·tÄ�Å�¾Æ±�¯ÈÇ3Ã�ÉkÀ3³wÀ&ÊÈ°�±�² }�Ë�Ì ¾�Å�³wÀ´ÇtÇ3µ=Í!À´ÇÏÎ|Én¾�Ã�ÉоuÇÇ�À�Ã=Ñk²0ÀÒ¾<¯�Å<ÉkÀÓÇ=À3¯gÇ=À]±s°Ï¶�À�°3Å Ë ±�² Ë ²t¾�Í�É!Å�¾<¯9Ô�¾£Ç´Å�¾<¯nÍ4Ñ!¾£Ç�Ékµ Ì ¾<¶¸¾�Å�·�Õ�ÎÖÀMÃtµ�¯×Ã�±�¯�Ç´Å�²´ÑgÃ�Åxµ�¯ÂÀ3¯9Ã=²t·tÄ�Å�¾Æ±�¯Ç3Ã3ÉgÀ3³wÀ5ʤØÖ°3±�² µ�² Ì ¾�Å�²0µ�²t· Ë ¶�À�¯nÍ4Å<ÉE³wÀ´ÇtÇ3µ=Í!À´ÇfÎ|Én¾�Ã�ÉÙ¾uÇ�Ç�À�Ã=Ñk²0ÀÚ¾£¯ÐÅ<ÉkÀ&Ç�À3¯�Ç�À[±0°�¶�À�°3Å Ë ±�² Ë ²t¾�Í�ÉnÅ_¾<¯ ËÔ�¾uÇ3Å�¾£¯!Í�Ñk¾uÇtÉ!µ Ì ¾£¾�Å�·4Û5Ü-µ�³wÀ3¶Ý·�ÕÖÊ ØÞ ~ÆßÈ� Äkµ�²�Ç=À´Ç ß µ�Ç ßáà=��â�â�â=��ßÚã ¬äÎ|ÉkÀ�²0Àxå ßÏæ å9ç } ºpµ�¯9Ô ±�ÑgŸÄ9ÑgÅ<ÇÊ Þ ~ÆßMà´�¤è Ê Þ ~ÆßÓé��¤èê�ê�ê�è Ê Þ ~Æß ãt� Ûë 8n1#��m�1#����g7��41��������74����� ¤�=#23���n6#�����4���2t8g�=jG�¢�£���?}b{ä�#���YjG�=������7��=�=mìoÖ�_j��=���=�23���!6�������9���0�������n{ ��=g7���8pjG�=������7��=�ß ���K�<�4 � ���oY�h~����Y1#��1��� Æm�í-���q��8g�h��8#�����="x�������t�:����t�0�-ß ���Yßáà=��ßÏé���â�â�â3ß ã�mgoY8g����� å ßÏæ å�ç }îg���Yï ç �I�0�Gð��ñ æóòô���k�=��� �õ æ ç � Þ ~ ñ æ �¤öMß æ÷ 1g�06#1g� ñ à3� õ à=��â�â�â�� ñ ã=� õ ã

~ . Ï�41g�A�=n�������w"��u��2�1��������4&���b�� ���m�o(�w������1�jG�G��8���� å ß å �u�A�pj¢1� �����6# ��¢���(}�ø|��8����?���=���0����23�����4[�u��=���0�5�0�����=jG�� ����¸�ù ¨�ú�§#û�ü �[g���="ý�0�M��� 2���������1� ?oY����8ý8g��oþoÖ�["g��y9g�&���=2�1g�������Ù��ý��8g�Ú�0�=��0� ���¢ ����£�s{ä����{ä����748n����"���������g741�����8����#�� ��¡���GoY8g�=x�=#23���n6#�����4p���;�����9���0�������n{� ��=g7���8pj��=������7��=�?���Y�� u ���o(�="ì�Kÿ�z1���1#�� Æm�o(��� � ���oE��8g�?��"g ��������������0�_��2�23�=���K��8g�? ����<�s{ä���0{ä����748n�Ö���t��2� ������f���(jw��*�����ujG�=�Ö���q���q ������=�=�����o(�� ����=moÖ� ³_Ñ!Ç´Å ���=���0����23�I��8g����"g ����t�������Ú���I�<�4 � ���o���IoY8g�= ��8g����"� ������������Ú��1g�9j������?���n1������]~Æß����ßáà´��0�x��8g�_������2� ��_oÖ�_����!1������h��8���� å ß� å�ç å ßáà å ��~ . f��8��¢�=#23���n6#�����4 ��2�8g�=j��_���b�� ���mó���?��8g�41� �"f�b�2� ��=���A��8����A���I���z�=���0� �0�x�����=������8g�G�=�23���n6������4&��2�8g�=jG��������8����z���=���0����23�����4Ú���z���=j��� ��="ìø|��&�<��23��m���z�u� ¾<³Äk±�ÇtÇt¾ Ì ¶�À �0�p23�4����0��1�23�?��*�f�=�23���!6������4&��2�8g�=j��GoY8���2t8f���z�0�=2�1g���¢o8g�=���8��_��"� ��������������u��� � ���o(�="f�0����1g�9j����K��o(�5"����b�����=*�s{� ��=g7���8pjG�=������7��=���¸� ü �_�0�0���=���Y��8����Yo(�-��j�6��4��� ¯9± ���=�0�0����23�����4�4-��8��( ��=�7���8������#jG�=������7��=���u Ô�¾ zÀ�²0À�¯#Å �!1g�������=�=ø���8n1��=m���8��Öy��t�0���!1g�����w~Æß����ßáà´�ó��#"h��8g�q�0�=23�4�"�n1g����� ~Æß Ø� ��ß Øà �K2���p8��= �� å ß � å��ç×å ß Ø� å �

ü 8#�� ��¢��8g����2�8��=jG�����b�� ������I�0�=2�1����G��&��8g�G�0�=��0�G���(���"���������g741�����8����#�� ��¡����m����I���Ig���?�=n�������= �����������0�<�!�ug7P�����23�¢��8g�G2���6#8g�����0�3ik�?���z��oY��23�G���I ��4�7P���?��8g�GjG�=������7��G�£���?���n6#�u2��� �� ~<���n69��2��� |�# ��k2��2���6#8g�����h8��= ���� ç }��t� ë 8g�P�����1g���� ��n1g�=�������4Ï���=�-2���ÓoÖ�P"g���b���0�0����� ë 8����I�#����g74�A1#�A�0����8��

Page 51: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

��1g���0�=23�Ö��� ³�±=Ô*À´Çh±s°hÀ�¯9Ã�²´·tÄ�Å�¾�±�¯ oY8��u2�8P�����?�0�=2�8�����n1g�=�q1����="w�0�_�=�23���!6��q ��4g7_jG�=������7��=�K1#����g7¢��# ��k2��52���6#8g������:��Y: Î�¾�Å<É����!À�Ô�¾£¯�Ä9ÑgÅ�¶�À3¯nÍ*Å<É4Ç ���l���#1��� �"#��g7h�9 ��!2��9�

. w�� � g����oY8����;�<�4 � ���oY��o(�Y������1�j��l�A:��î&�×�k���k�=��� � �G���k�=��� � �ô���k�=��� � ���"G��8��YjG�=������7��ß �0�G�b�?�=�23���!6��0�="5���q6#�������="P�u*�0���G�0��!1g�=�23�?�����5{ä�#���(�# ��k2��k�lßMà=��â�â�â���ß ã=�;ÿz �jG�4�0�q��8g�AjG�4��������1g���� ¤��1g7�7��=�0�����4x�<���Y��j��!"g�I���|�=�23���n6������4x���q��8g�A�<�4 � ���oY��g7g�

Ê Þ ~ÆßW� ç � Þ ~Æß à �óèê�ê�ê�è(� Þ ~ÆßÚãt�tâ

ë 8�����jG�k"g�A~<�kg��oYw��� À3¶�À�Ã�Å�²s±�¯#¾�ÃAÃt±=Ô*À Ì ±=± � ���"!�#%$ jG�!"�����o(���;��23��1#�� � ��¢�4g�K���9��8��K�£�41g�ÖjG�!"��=������=#23���n6#�����4x�06b�=2���y��="x�� ��4�7GoY����85��8g�'&)(q%P�# ��k2��52���6#8g������*���0�h��8#���l�£���"g�=23���!6������45�0��o(�����6�����6b���� ���m*�Wj¢1��0���b�Y�-�������=" ÄgÀ3²´³_ÑgÅäµ�Å�¾�±�¯ ���"w�<1g����8g����jG�����j-1������b�l�=���0�G�0�_��n ������z~<74�� ��=5í��t�

ë 8�����jG�k"g�f�0���=jw��7��!�k"ìml�����23�f��8g�f2���6#8g�����0�3ik�P�u����8g� ����jG�& ��=g7���8ý������8��fjG�=������7�� ��#"8g�=�23�G��8g�������u�z��p2���6#8g�����0�3ik�?�3i!69��������4,+¢�q1g���=�������z��8��u�I��2�8g�=j����0�=2�1g���-� . �AjG�4�0�A23����������� ����u�g���(���=2�1g���?��w��8��z�0�=#�0��������"���������g741�����8����#�� ��¡����m������23����(���(� Ô*À�Å�À�²t³_¾<¯�¾uÇ3Å�¾�à �=�23���!6������4w��2�8g�=jG���.  ����23��m����I"g�!�=�Ig���I�� ��=[��2t8����� �����8��GoÖ�=�������-�£����j����q�0�=2�1g�t�����]~�23���ì6����� !���41#�z ��=23��1g���=�t�oY8��������8g�h��"g ������������x���K�� � ���oÖ�="x�4# ��P�G����g74 ��.�!1g�����P�0����8g�/���&������2� �����&z�����41p�0���AoY8n�0�

ë 8����Ö8����q ��="��0�¢!1�jG�����41#������8g���K��1�7�7��=�0�����4��I~�����<��23��mg"g�=����74���g7frs7��n�k"�v¢jG�!"��=�Ö���ó�=�23���!6g{�����4��u�Ö��P��23���� ��I�����=�-���¤���=�0�=����2t89�t� ü �? ����0�Ö8g�����z��o(�-����8g���KjG�k"g�=�("g��y#g�="����(6#���������ó��8g�1&)(q%�0�����"�����"¤�2�354 ¦�§n©7698�:£¨,; <=;�¦?> 35@�3A@?BDC�2FEG2IH ªÓ¨,J�§LKÊ Þ ~ÆßW�õ �lò ���k�=��� �îg���ï ç �I�0�Gðõ æ ç � Þ ~ õ æNMìà�öáß[æä�÷ 1g�0691g� õ � � õ à ��â�â�â�� õ ã

O Þ ~ õ ����â�â�â�� õ ãt�îg���ï ç �I�0�GðßÏæ ç õ æ5MìàóöM� Mìà

Þ ~ õ æÆ�÷ 1g�0691g�lßMà=��â�â�â=��ßÚãë 8g�?�����"g�4j­�# ��!2�� õ �z���q2��� � ��="x�� ¾<¯�¾<Å�¾Æµ�¶Ý¾QP�µ�Å�¾Æ±�¯ ¿�À�Ã�Åä±�² m����"x���q2�8g�4���=5�����"g�4j� ����=��2�8x���ujG�I�g��oÏjG�=������7��Ö���b�=�23���n6��0�="¤�|%k��#23����8����ì���¤� ²sµ�¯�Ô4±�³_¾QP�À�Ô �=�23���!6������4I��2�8g�=jG��m����¤j���748*�¤6b���0�=n������ � �����_�0�=2�1g���¢�����8g�_�0�=��0�¢�������"#���0����g741#����8����#�u ��¡����� . f����23��mì��8���� ¾uÇ �P���=2�1g���_jG�!"��h���;�=�23���n6������4ìø�0���GR¡�TSó�<���Y"g�������u ����

ë 8g�?2���6#8g�����0�3i!�q8g�����?���( ��4g7����(��8���P��8g�zj��=������7��I�!�w�� µ4Ô4Ô�¾�Å�¾£¿�À ����23�0���h~����q��6�6b�4�0�="��0�¢��8��j-1� �����6# ���2������� ��¢����23�0���ho(��8#��"[�� �41g�h�u��������� |jG�k"g�G���q�=�23���!6������49�t� . Ú����23��m���8g�w2���6#8������0�3i!�A�u� ��4g7����;��8���_��8��YjG�=������7��Y�n�F� �#�����|oY8��u2�8_���|�=���0�=n������ � ��h��6�����jw�� ¤~��A��2�8��=jG�KoY����8Gg�A2���6#8g�����0�3ik��3i!6#��#�����4foÖ�41� u"p�b�-"g���0���tj������������2A���"&8g�=�23�_����0�=2�1������t�Yÿ74����ìm�oÖ�Gg���="&8g�����-�£���I�Â�0�P�b�¢��������="�6b����j¢1g���������4ìm���#"x���lj-1#�0�l��"�"��������4��� � ����b�I�VUw2����=*�l�0����* ��������2�354 ¦�§n©�W�§!§XJ�8�>0; < C�2FYZEGH ªÓ¨,J�§LKÊ Þ ~ÆßW�õ � ò ���k�=��� �îg���ï ç �I�0�Gðõ æ ç ß[æ�ö]� Þ ~ õ æ5Mìà´�÷ 1g�0691g� õ ��� õ à=��â�â�â�� õ ã

[

Page 52: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

O Þ ~ õ � ��â�â�â�� õ ãt�îg���ï ç �I�0�GðßÏæ ç õ ægöM� Þ ~ õ æNMìà´�÷ 1g�0691g�lßMà=��â�â�â=��ßÚã

ÿY74�����o(�z8��� ��z�-�����"��4j×����������� ���B=�������4� ��=23�0���q2t8g�4�0�=P�=��2�85���ujG�Y�¢g��oýj��=������7��?���;�=�23���n6��0�="¤�\ó����� $ � $ j��!"g��m|��8��u�h��2t8g�=jG�5���-���=2�1g���5���"[��8g�52���6#8������0�3i!�¢���¢ ��4g7����¢��8���Ï��8��PjG�=������7����!��4� ��/�þ�9�������;ÿzx��"g ���n����7��h������8����K��2�8g�=jG�h�� ���� $ � $ jG�!"��?�u�K��8����l�Á"g�!�=�lg���l����="x�0���b�I��������=" 6����tj-1g���������4Ó~�����<��23��mbo(�¢g�� ����?g���="f�0�P�u* ������z� ��x����"g���z�0�5"g�=23���!6��t�t� ë 8��u�Y���1��0����1� ��x�0�4jG�h23�4n�0�3ik���Y�<���l���=���0�4#�l�����VUw2����=�23���2 ¨L] @ ú�§!©fªÓ¨,J�§LK ÿ�����8g���hjG�k"g�_���IjG������ ����0�=" �n�f��8g�G�3ig��jG6# ��¢oY����8&oY8��u2�8&oÖ�G��6b�=g�="&��8��u��0�=23�����4ì���Y�=2��� � ì��8����Y�41g�Y����������� b��1g7�7��=�0�����4po(���Y�0���=�23���!6��Yß �����

ñ à�èI~Æßáà�ö]� Þ ~ ñ à´�0�¤èê�ê�ê�è ñ ã�èI~Æß ã�ö]� Þ ~ ñ ã��0�t�

�#1g�h��8#���A8����-��8g��"#������"g ���*����7��P���l8��� !���7���2���698g�����0�3ik�h��o��23�����_ ��4g7f���h��8g��6# �����n�0�3ik���p�(1����8g�����-�u�Yg�Pg���="��<���z��8g� ñ æ �0�5�� � ó���-"����b�����=*�V+h~A*��oW�����"g�4j ��� �1��=���8��41� �"x�b�h2t8g�4�0�=f�� ����������jG���xjG�=������7��w���?�=�23���!6��0�="ìø��#1g�?�4g�G����#"g�4j� ��� u1g�_6����Aj��=������7��wj���748*�I�b�_�=g�41g748ì�¸� ë 8#�����1g7�7��=�0���Y��8g�A�<�4 � ���o��g7G��jG6����� ��=jG�=n���;6#��2����t���"g�4j ñ ���"x��8g�=�23�4jG6#1g�0�A��8��A2���6#8g�����0�3ik�l�����

ñ è?~Æß à öÒ� Þ ~ ñ �0�ìèI~Æß é ö]� Þ ~ ñ_^ ���0��èê�ê�ê�è?~ÆßÚã�öM� Þ ~ ñZ^ ð"`á���0�tâ

ë 8����K��2t8g�=jG�h8#���l��"�"������� ��I�3i!6#��#�����45����23�0���=m#���"p�u�l�� ��0���0�=2�1������(%!���/R¡�TS��£���l��1g����8����l"g�������� ����

a U;X=U;`ìU�\_^�U�V

R¡��S_�Ï�|�(�= � u������m;ÿh��&z�=�����Æm�(Y�;>����k��6#���Æm����"]:����Y��74�=oq�=����ÿ $ �4�23�����0��%!�=2�1�������� ë ���=����jG�=n�G���%!�kj�jG���0���u2_(Ö�23���!6������4ì��ÿz��� ��k�����(������8��)&)(q%x�p�k"g�=�q��� ÷ 6b�����������4ì�gî ÷h$ %cbedgfn�#ÿ ����� u���# ���<���4jihkjljnmporqXqtsXslsvukwXxzy�{}|�~Xxt��{�yl�z|vq |,xtyg�Lxgq�%�-h�� �kqtmv� m0yn�vxgq0x-�t��ulyt��~�{}hXjz���g�

Page 53: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�_�0���z�V�9���Q���G�z�����t�9�X���z�v�� � � �¢¡n£z¤F¥§¦ �n�r�9�X�0¨v©T���Q�g�ª�r� � ���X«0�r�z¬z�9�t«L­n�®¯���z�5����r�z�)°z�g�0�t��­0�z��±��t�r²

³D´¶µ.·p¸º¹p´ »/¼

½ ¾ ¿�À¶Á¯ÂÿÅÄÇÆÉÈËÊ�Ì�ÍFÎFÏ�Ð9¿_È

ÑÒ� Ó¶��l���Q�g�k��Ô���z�r�G����Ó¶�Õ��­0�c©T�g�0©T�V«0���z�¶ÖI×ØnÙTÚÛ×�ÜÒÙTÝLÞßáàáâ�ãAäå×tÝçæÅ­v��©9­=�t�9� ¨0����=�r�è��0©T���X«0��t��év�Q�r�9�t�9�nêë�Q��0¬z��­/Ó¶����9�t¬z��VìX�z�v�/¬g�7�z�)���gÓ¶�Å�Tík�zÓ¶«v�Q��î¯ï.�k�Z��­0���0¬�æ��1�0���I�0�z���r�r����9�����I��­k�_���z���©V���z���%æ"�z����­k��«0�9�©V���r�¯�k�Vðv�v�Q���Q�g�)�z�0�Q�V�5��ê}�z�rê}�9�Q¬g­l�?���v�0���r�����0¬g¨0����­0�téL�������ë�"æÅ­0�V����Ó¶���k��%�z����0©T�9�l«0�����g��t���p©T�g�0©T�V�9�k��,î�ñ1��¨v��¨0�z�åì æ��p�z�����7æè��­k�p�z�k�z�V�9���t���'�r�)�z©V©T��9����­k�¢���V�N��ê}�z�rê}�á�Q¬g­n���z�9�z©V�Q�.ò�ó/�z��ÓI�z�l�����Ó¶��.�z�)�Q�.���Qôz��Vî.õ_�7æ¢�V�z�V�ì?æ¢�ªÖËölÚTãÅ�9��r�r�9��©T�1��­k�¶�z�k�z�V�9���t�9�c�z�)�5�g���Q�7æ_�V÷ZæÅ­k�� ��­k�¶�z�k�z�V�á���t�����¨kévÓI�������cøX¨k�V���=ùåúûtü9úºý�þ��r����­0� ò�óº�z�9�z©V���/æ¢�G���øX¨0�Q�9�I��­0�t�ªÿ úûnÿ�� ÿ úºý ÿ îèù ¦ ��z�n�Ç�z�p��­0���0©T���X«0���Q�g����©á­k�Ó¶��p¬g�Q�z������G��­0�.«0���V���Q�g¨0�"�Q�©T��¨k���zìk�Q�p��­0�g¨0����é��1©V�Q��t�Å��­0�t�Z�Q�p���¢��z�r���r�¶é0����tô��­k�/��0©T�9�l«0�����g� ��©á­k�Ó¶�ª�Q�"��­0���.������r�9��©T���Q�g��������ÓI�7�z��������Ç�A�z©T�Vì¯�Q�'���/äNÖ_â�× ÚáÚáä�����Ùª�r�Û©T�g�0�r�r�9¨v©T��z�n� ��0©T���X«0���Q�g�Û��©á­k�Ó¶�¶æÅ­0��©9­Û���)�r�©V¨k���ËæÅ­0��c��­0�I�z�k�z�V�9���t��� ���)�z���Q�-æ���Û�r�Õ��¨0évÓI�Q�1Ó¶������t¬z���z���0����V����l�p�Q��k¬z��­0�Vî þFÑÒ�����r������"��­0�t�pæ¢�.��Ó¶«��g�r��ÝL×Ë����r�r�á��©T���Q�g���g�ª��­k�.�Q��k¬z��­0�¢�z��Ó¶������t¬z��Å���Øtä )ÙTß�ÙTÝvã)øX¨k�V�9�Q����?��­l¨0�ì%��­k�¶ð0�9���)øX¨k�V���Éùåú ûtü9ú=ý�þ1�z�0�Û��­k�I���©T�g�0�ÇøX¨k�V���èùåú �û ü9ú �ý þ1©V�z�Ç­0��z�ÿ ú û ÿ��� ÿ ú �û ÿ î

¦ �/�z���k�z��æÅ­0�t�¯�5�g���Q�-æÅ��æ¢�Å�z����¨0ÓI�Z��®������ ÷�����ü������ �!����ü�����"$#%����ü�����" �z�0�¶��­0�ÅÓ¶������t¬z�ú �r�/é��.��v©T���l«v�r��Õ���"«v�t�9����Õ���l�r�I�I�r�øX¨k��0©T�.�z�'&ªê}év�Q�pév���X©9ôX�Zú ý ü�(�(�(7ü9ú*)Tî+-,/.103254�687'9;:�<�=><?0�@A,/B1,CB�DFEG+IHJ+LKNMO:�P32AQR�S ùåúÃþT û U ����ü�����"�k�z��V'�W�.�r�!XTZY �[� S ù TZY;\ ý1] ú Y þ

ï)¨k�r«L¨k� T û ü T ý ü�(�(�(-ü T )^ S ù T û ü�(�(�(7ü T )áþ�k�z��V'�W�.�r�!Xú Y � TZY/\ ý�] � \ ýS ù TZY þï)¨k�r«L¨k�Zú ý ü�(�(�(ü9ú*)_ ­k�)�9�z�0�k�gÓiév���X©9ô T û1���"©V�z���Q����z�Ûä5Ý0ä5ãAäa`?� äcb�`zãAäå×tÝedtÙ�Þ7ã}×tß�ì0�z�0�����"©9­k�g����ª�9�z�0�k�gÓI���I��z©9­�����Ó¶�.��k�Væ Ó¶����9�t¬z��������0©T�9�l«0�r��%î������v©T����­0���,���%��ßf`tÝ�Øg×tÖËäcb-Ù�ØF��0©T���X«0���Q�g�.�9©9­k�Ó¶�zìt�Q�%ÓI��¬g­n�%«��z�r��l�����z���Q�é��Ë�r�©V¨k�9��������­k�Ë�r��0�r���z�����0�v���r�����k¬g¨v����­0�tév�������ë�zî ¦ �c�A�z©T�Vì,��­0���Fä�ÚË�����©V¨k���ËÓ¶�X�0�'�z�¯��0©T�9�l«0�����g����r�V�����V�5�V����0©T�Jg�Gh����ª��­k��«0�9�V�X�Q�g¨v�p�Q�©T��¨k�����k�z�r��Å�N�z�_�k�V���z�����Vî

_ ­k�¯©V�Q«L­k�V���r�Tí��?­k�V���¯���?�Q�g�k¬z�V�%��­0�z����­k��Ó¶����9�t¬z�¯él�.�z�i`nØgØtäAãAä;dzÙ��5�z©T�r�z���z�5& év�Q���¯ùA�z�%�z«v«��g�����r����­k�IÓ�¨0�Q���Q«L����©V�t���Q�z�Ë�A�z©T�r�z�'æ¢�/­0�z�Ç���Û�g¨0�'���0�Q�����z��ÓI�X�k�¶�z�"��0©T�9�l«0�����g�Lþáî _ ­0���.���)����r��l�����z���Q��z«0����ÓI�z�%ùA�k�z�r�p��­v�t�¯�.��©9­k�ÓI�ZæÅ�Q��­Ë�k�.©V�Q«v­0�V���r�TíX���Tí�«v�z�0�����g�Ëæ��g¨0���Fé��p�k�V�r�V�9Ó/���0���r����©��z�0�¶­k��v©T����0�r�©V¨k�9�-þáî ¦ ����­k� �kjZ� ÓI�X�k�zì��k�©T���X«0���Q�g� �k�V��0�_�r�ª©T�gÓ¶«v¨k�r�l� \ ýS ����­l¨0�ìm� Ó�¨0�r�Åé����/ôz�V�z��âkÙTß�ÖËö�ã�`gãAäA×tÝvì��z�0���Q�ZÓF¨v�r�pé��.��n/©V�Q��l�Z�r�/���l�z�V���'ù5¬g�Q�z��c�¶«v�t������©V¨v���t�"ôz�V�po þáî

Page 54: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

+-,/.103254rqs2t2tP371@u<�=$EG+Iv HJKIMO:�P32AQR�S ùåúÃþT û U ����ü�����"�k�z��V'�W�.�r�!XTZY �çú Y ] � S ù TZY/\ ý þ

ï)¨k�r«L¨k� T û-ü T ýü�(�(�(-ü T )^ S ù T û ü�(�(�(7ü T )áþ�k�z��V'�W�.�r�!Xú Y � TZY ] � S ù TZY;\ ý þï)¨k�r«L¨k�Zú ý ü�(�(�(ü9ú*)

ñŬg�z���Gæ¢�1­0�7�z�1�F�9�z�0�0�gÓ ���0�������z���Q²�t���Q�g�I�z�©T�r�z�"©á­k�g�r�����z©9­ª����Ó¶�Å���k�VæÔÓI������t¬z�)���¯��0©T�9�l«0�r��%îw?�Qôz� �kjZ� ÓI�X�k�zì���­0���'��©á­k�Ó¶�ª���F���©V¨k���ª�z�0�Ç��­k�ª©V�Q«v­0�V���r�TíX�������Q�g�k¬z�V����­0�z� ��­0��Ó¶��9���t¬z�GéX��g�0�Q�i& éL�Q���Vî¯ñ1���z�k� �z�l���t¬z�'�z����­0���p�9©9­k�Ó¶�'�7�z�V� �xjp� Ó¶�X�0�)���p��­0�t� � �k�X��Z�k�z�Z�0�V����r�Ié��.�ôz�V�z��Û«��V�áÓF¨k���t�����g�ùA�����5�z©T�Vì�æ¢���k�V�z�V�)�k�V��c�r�����n�z�V���y� �����z�9�k�V�1�r�ª�k�©T���X«0�áþáî _ ­0���Å���_¨0�r�V�A¨0����Ò�r�gÓ¶�I©T�g�l�r�TíX���'�N�z�'����z�r�g�0�.�z�¢��n/©V�Q��v©T�c�z�0�ÇÓI�z���¶��Ó¶«��z�����z�l���Q�c�z�����7æÅ�.¨0�)�r�Õ¨v�r�I®������z�æÅ­0��©á­��t���IÝL×zã"«��V�9Ó�¨k���t���Q�g�v�Vî

{ ¾ Á¯ÂLÂm|x}ÅÁ�~��FÏ��ËÁ�È1Ï�Ð9Ê�|¢Ï�Ð9¿_È

_ ­l¨v���A�t�Z���G��­0���"©V���z���ìX��­0�1�g�v�Q�/©T���X«0�r�z¬z�9�t«v­v��©-`�âgâA� äAÞ�`zãAäA×zÝ æ¢��­0�7�z��©T�g�0�����k�V�9��G���¢��0©T�9�l«0�����g����åî �zîQìnôz�V�V«v���k¬.��­k�Å©T�g�l�r��n�����z���.�r�9�z�0��Ó/�Q�r�r��IÓ¶����9�t¬z�Fù5�z��Ó¶��9���t¬z��áþ�­v���0�k����5���gÓ �z�¶��7�z����k���z«kê«v���k¬/�z�0�z�V�9���t���zî �k�z�1��­0���_�t«0«v����©V�t���Q�g� ¥ �t�)�Q��z�r�_��­l¨v�Z�A�t� ¥ ��­k�F�z�k�z�V�á���t���Õ­0�z�Åé��V���â�` ÚáÚáä/dtÙù5�V�z�� ��­0�g¨k¬g­ �g¨0�G�k�Vðv�v�Q���Q�g�º�z���r�©V¨0�9�Q�ë�º�z�����7æ¢�� �z�Ô�z�k�z�V�9���t�9�º�r�èÓ¶�g¨v�n�/©á­k�g�r��Ô«v���z���n�r�Tí���t�r���z©�ô��_�z�0�����n�r�V�á�z©T�p���V«���t�r��0�Q�/æ_�Q��­���ò�ó��z�9�z©V���zì0��­0���"æ"�z�ZÓ¶��z�l�Z�r�IÓ¶���k���«��z�r��n�����z�����z���ê�k���z«0«v���k¬_�z�,Ó�¨0�Q���Q«L�Q�¢©V��«v­k�V���r�Tí����áþáî j ¨k�¯��­k�V���_�t���p�z��­k�V����Ó¶«��z�����z�n�¯�t«v«v����©V�t���Q�g�v��æ_­0��©9­¶���V�Vô¶�r�«0���z�r�©T�Z�t¬g�z���0���Z�z��`gÞãAä/dtÙ��z�k�z�V�9���t����æÅ­k���r�9�Q��"�r�¶���l�r�V���5�V���)æÅ�Q��­���­k�.©T�gÓIÓ�¨0�0��©V�t�����g�ªé��V��æ��V����­k�'�r��0�0�V�Z�z�0�����©T��Q�z�V�7î_ùAñŬg�z���,ìv���ª��­k�F©V�z�r���z����0©T���X«0���Q�g����­k���z�0�z�V�9���t���ªæ"�z�Å�z����¨vÓ¶��ª�g�0�Q��r�����z����k�9�z«%ì%év¨k�.�k�V�z�V�.�r�Õ���n�r�V���5�V���Ë���Û�z�n�c�z��­k�V��æ"�� æÅ�Q��­c�r�9�z�0��Ó/�������Q�g�v�Vî þ _ ­0���_é0�9���k¬g�_¨v��r�I��­k�'��ÓI«��z�9���z�n�Z�t«0«L����©V�t���Q�g���z��Ó¶������t¬z�F�z¨k��­k��l����©V�t���Q�g�,î

¦ �0�N�z�9Ó/�z���Q�zìv��Ó¶������t¬z�Ë�z¨k��­k��l����©V�t���Q�g� ��©á­k�Ó¶��ù5�z�z����� ¥ ÓI������t¬z�Ë�z¨k��­k��l����©V�t���Q�g�Û©T�X�0�-þ�z���Q�7æ_�.�ª���©T��Q�z�V���r��é��¶©T�g�n�����0©T��Û��­v�t�.��­k�IÓ¶������t¬z�nùA��þ)­k�Ë���©T��Q�z����t�9�I�z©T��¨0�z���Q� �r��n�.éX����­0�©V���z��Ó¶���r��0�k�V�î ¦ � �z��­k�V�Ëæ��z�9�v�Vì���ÓI�t¬g���k�����r��0�0�V�N� �z�0�è�����©T����z�V�I� æÅ­k�cæÅ����­Ç�r� ©T�gÓËêÓF¨0�v��©V�t�r�'���Õ�9¨0©9­c�¶æ¢�7����­0�t� � ���p¬g¨0�t�á�z�n�r�V��c��­0�t�_ÓI������t¬z��1­k�����©T��Q�z��_�t���F¨v�0�z�Q�r�V����,ì0�z�v�æ��V���F�z©T��¨0�z���Q���r��n�Åél�r�Åî¯ñ1�Z�¶����z�%æ��z�9���ª�Tí��zÓ¶«L�Q�zì0��ÓI�t¬g���0�)�ˬz��0�V�9�z�,æÅ­k�¶æ"�z�l���Z��­k���tév�����Q�ë��r�Ë����0�G©T�gÓ/ÓI�z�0�0���r�¶­k�V�¢���g���0�Q�V�9��������¨0©9­��Fæ¢�7�/��­0�t�"��­k�)�r�g���0���V�9��©V�z��é��_©T�g�l�X���0©T�����­0�t����­0��z�9�k�V�9���z�á�Q¬g���0�t�r��Ë�5���gÓ ­k�V�î¯ï1���z�g¨G©V�z�I��ÓI�t¬g���0�"�)éL�z�kô'æÅ­v��©9­¶�k�V��v���r����0��¨k�9�¢��­v�t�¯Ó¶������t¬z���Q�p���©T��Q�z��Å�N���gÓ ©V���Q��l���pæ��V�9�'�z©T��¨0�z���Q�ª�����9¨k��ªél����­k�g�r�F©V���Q��l���Vî������L�p�z���Q�-æç��­k�Ó �r�/�k�¶��­0���î

¦ �¶������Ó¶«��z�����z�l���r� �9�©T�z¬g�0�Q²V�ª��­0�t�Ë��0©T���X«0���Q�g�è����©T�gÓ¶«v�Q�V�r��Q� �z����­k�z¬z�g�0�z�"�r�ÒÓ¶����9�t¬z�ª�z¨kê��­k��n����©V�t���Q�g�,îr�¯�g¨©V�z� ¥ �z�v�Ç�z�5�r���k� ¥ ­0�7�z�/�g�k�GæÅ�Q��­k�g¨0����­k�I�z��­k�V�7î ¦ �F�9­k�g¨0���Ò�z������é��©V�Q��t�_��­v�t�_��0©T�9�l«0�����g�Õ¬z��k�V�9�z���Q�ª«0���7�����k��_�k�GÓI������t¬z�Ë�z¨k��­k��l����©V�t���Q�g�cæÅ­0�t���r�X�V�z�V�îÅñ1�_�z�c�TíXê�zÓ¶«v�Q�zì,©T�g�0�����k�V�Å��­k�Ë�g�k�Tê}����Ó¶�F«L�z��æÅ­v��©9­c«0���-�X���0��Å«��V���5�©T�1���©T���©T��æÅ­0��c�������k¬g���FÓ¶����9�t¬z�Ë�����0©T���X«0�r��,î j ¨k�.�V�z��ÇæÅ­k���� �z�0��� �9­0�t���¶��ôz�V�eonì%�z�Ò�z�k�z�V�9���t�9� ©V�z�Ò�r��v�Ò�z�n���rÓ¶������t¬z���T �r�ª��­k�����©T��Q�z�V�.�z�0�c��­0���_æÅ�����,é��F���l�r�V��«0���V�r��céX��� �z�1�G����z��Ó¶����9�t¬z� T ] o���­v�t�1æ"�z�)�r��l�él���ÅîZ�¯�z��Õæ¢�z�9�r�zì��Q�?��­0�'�r��0�k�V�Å��0©T���X«0���ZÓ¶����9�t¬z��ú �z�0�Õ�r��v�0�p�Q�FùA�åî �zîQìL�r��0�0� T � ú ] otþáì

Page 55: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�z� �z�0�z�V�9���t��� ©V�z���L�Q«���­k�Ë���z�r�1év�Q�_�z� T ¥ ¬g�Q�����k¬ T � ¥ �z�0� ��­k�����­0���9�©T��Q�z�V�)æÅ�����,�zév���z���Û�Ó¶������t¬z�Iú � æÅ­0��©á­��0����V�9�Å�N���gÓ ú ���Õ��­0�Fðv�v�z�,év�Q�Vî ¦ ÓI�t¬g���k�F��­k�F�v�zÓI�t¬z�F��­v���Å©T�g¨0���c©V�z¨0���F�Q�T æ¢�V���'�z����0©T�9�l«0�����g�ª�z��ìv�9��zìv��­0�'�zÓ¶�g¨0�l�p�z���¶év�z�kôGæÅ�Q��­0�k�á�æ"�z�a�

j �V�N�z���.©T�g�l�����X¨0���k¬kìlæ��1¬g�Q�z�.���5�z�9ÓI�z���k�Vðv�0�����Q�g�¶�z�%�g¨k�¢¬z�g�z��ùA�z�Qæ"����p�z�G��ÓI«��z�9���z�n����­0���k¬F�r��k�lþáî'ñ Ö/ÙTÚ9Ú�`��lÙr`tökã/��ÙTÝLãAäAÞ�`zãAäA×zÝÉÞá×ØnÙG©T�g�0���������_�z���ëæ¢�Õ�z�Q¬z�z�9�Q��­vÓI�V÷Å��­0�!�����Ò�z�Q¬z�z�á�Q��­0Ó æÅ­0��©á­���tôz��¯�)�r�©T���V��ôz�V�-oZ�z�0�Ë�1ÓI������t¬z�Åú �z�0�����V��¨0�9�0���������t¬����f��� �n�z�0����­k�"�z�V�9��ðv©V�t���Q�g�Ë�z�Q¬z�z�9�Q��­vÓ¡£¢c¤/¥ æÅ­0��©á­I���tôz��"��ôz�V�rogìX�'Ó¶����9�t¬z�)ú ìX�z�v�G�������t¬��I�¦���F�z�0�/���V��¨k�9�v���Å������­0�Å���t¬Ë���¯�t�z�����I�z�v����z��­k�V��æ_���r�zî _ ­k�Z��©á­k�Ó¶�1����¨0����I�z�¯�5�g���Q�-æÅ�V÷���­k�Å�r��0�0�V���z�0�¶�9�©T��Q�z�V�¢���©T���V���Q�I��­0�t�9�Z���9�z�0�0�gÓôz�V�§ogî�ÑÔ­0��Ç��­k�/����0�k�V�'æ_����­k��.�r� �r��0�ÇÓI������t¬z��ú �r����­k�/���©T����z�V�ì���­0�G�r��0�0�V�'©T�gÓ¶«v¨k�r���f��� U ����� S ùåú þ)�z�0� �r��0�0�i¨åú ü¦�¦��� ©áîIÑD­k��Ç��­k�¶�9�©T��Q�z�V�'¬z�V���F�r�9�z�0��Ó/�������Q�g�ª¨åú �Aü¦�f�����«©áì?��­0����©T��Q�z�V�)æÅ�����,�0�z�_�z©V©T�V«0�1��­k�ËÓ¶������t¬z��ùA�åî �zîQìvæ_�����%�k�z�_é��'©T�g�n�����0©T��c��­0�t�_��­k�FÓ¶��9���t¬z�Fæ"�z�)�r��l�él�I��­0�1�r��v�k�V��þ�¨0�0�Q���� ¡£¢¬¤;¥ S ùåú �Aü¦�f������þ��­�tî�ï1��©T�g¨k�9�r�zì�æ¢�_���øX¨0�Q�9�Å��­0�t�¢��­k�)��©9­k�ÓI�1é��ËÞ9×tß�ßrÙ9ÞãAì�r�I��­0�t�Å���1����� S ùåú þ¢�g¨k�r«L¨k���x�¦���kìv��­k�� ¡k¢c¤;¥ S ùåú ü¦�¦���kþ"��­k�g¨v���ª�z�Qæ¢�7�X�Å�g¨k�r«v¨k�®�tî

ÑÔ����­ ��­v���G��� ÓI���v�,ì"æ��Ò©V�z�D�k�Vðv�k� �z�D�t«0«0���z«0�á���t�r�c�k�z�����g� �z�'���©V¨k�9�Q���zî ¦ �k�5�z�9ÓI�z�����zì"�z��z�k�z�V�9���t���Õ�9­k�g¨0���ªé��'¨0�v�tév�Q�)�r�O���5�z��¬z�����I�t�z�����ª���t¬��g���z�l�ªÓI������t¬z�'��­v�t�Åæ"�z�'ÝL×gãp�r��n�_él���_î_ ­0�t�.���Vì%�z�Ò�z�k�z�V�á���t���Û��­k�g¨0���cé��¶¨0�0�tév�Q�F�r�>���N�X�g�¬�ª��­k�Ë���©T����z�V�'���l�r�ª�z©V©T�V«v�����k¬§`tÝ0à Ó¶������t¬z�ú æÅ­0��©á­ �0��� �k�z�F©T�gÓ¶���5���gÓ¯�_îcï1«0����ÓI�z���Q�zì?æ¢�/æ¢�g¨0������Qôz�I�r� «0���V�z��l�'��­k�ª�z�k�z�V�9�9�t���Ò�N�9�gÓ���N�X�g�����k¬��Z��­k�¯���©T��Q�z�V���V�z��F�t�5�r�V�?��­0���z�k�z�V�9���t�9�)���V���Ó�¨0�Q���Q«L�Q�����t¬g��©T�gÓ¶«v¨0�r����g�'Ó/�z�n�.�0����V����l�Ó¶������t¬z��Vî ¦ �Û�5�z©T�Vì?æ��¶æÅ�����?�9�øl¨0���������gÓ¶�V��­0���k¬ª�V�z��Ç���r���g�k¬z�V�÷)��­0�t�.��­k�¶�z�k�z�V�á���t��� é��¶¨0�v�tév�Q��r�����N�X�g�¬����­k�)���©T��Q�z�zìk�V�z����t�5�r�V�Z�r�V����k¬Ë���t¬g�p�5�z�)ÖËöt��ãAäeâA�QÙ'Ö/Ù�ÚáÚ�`��XÙ�Ú'×�Ü.ã/��ÙI`gØ?dzÙTßáÚ�`tß�à�° ÚFÞ��X×täåÞáÙTî

ñ ���Q�r���Q�)Ó¶�z���.�5�z�9ÓI�z�����zìkæ��.æÅ�����L�0�Vðv�k�.�z� ×tߦ`gÞ���Ù!����� S ù²± þ�æÅ­0��©á­ª���tôz��_�z�Z���k«v¨k�"�¶Ó¶������t¬z�ú �z�0�Ò���V��¨k�á�0���z���g¨0�r«v¨k����­k�i�f�����5�z����­0�t��Ó¶������t¬z�zì�©T�gÓ¶«L¨k�r��ÇæÅ����­Ò����r«��©T���r�côz�V�§o ù5�g�0�©V�z�Õ��­0���0ôI�z����­v���"�z�9�z©V�Q�'�z�Z���V«0����r��l�����k¬Ë��­k������0�k�V��þáî¯ÑÒ�'æÅ�������z���Q�-æ �z�Õ�z�k�z�V�9�9�t���G�r�G���n�r�V�á�z©T�æÅ�Q��­Û��­0���)�z�á�z©V�Q�/�z��ÓI�z�l� ����Ó¶����z���������Qôz��Vì?�9¨kévÓI�Q�r�����k¬GÓ�¨0�Q���Q«L�Q��ÓI������t¬z����z�"�Q����©á­k�g��©T�/�z�v����©T��Q�����k¬Ò���è���V��¨k�9�è��­k��©T�z���9��r«��X�0���k¬����t¬g�VîÉÑÒ�Õ���7���­0�Õ�z�k�z�V�9���t���W��é0����tô��³�Û��­k����©á­k�Ó¶�Õ�Q��Q�'©V�z�Ç�5�z��¬z�/��� �z�����e�f���Õ�5�z�'�cÓ¶������t¬z�ªú ��­v�t�'�Q�'�k�V�z�V����¨kévÓ/�Q�r�r��Û�r�c�Q�����z�9�z©V�Q�´�¯�åî �zîQì��Q����­0��z�k�z�V�9���t����©V�z�ª�g¨k�r«v¨k�"�F«v�z�Q�z¨åúçü¦�¦����©1ù5æÅ­k�V���.ú æ"�z�"�k�V�z�V�Z��¨kéLÓI�Q�r�r��G�r�!����� S ù²± þrþ¢��¨0©á­G��­v�t�¡£¢c¤/¥ S ùåú ü¦�f����þ-�µ�tî ÑÒ�ª������­k�G��©á­k�Ó¶�ª�����r�©V¨k�����Q�Z�k�c�z�0�z�V�9���t��� ©V�z�­��é0�9��tôt�c��­k�ª��©9­k�ÓI�æÅ�Q��­�­0�Q¬g­ª«v���zév�tév�����Q�ë�zî _ ­k�.�5�z�9ÓI�z�?�k�Vðv�0�Q�����g�G�5�g���Q�-æÅ�Vî¶ 2¸·�B3,;¹�,/:mB»ºF¼ Ö/Ù�ÚáÚ�`��XÙ½`tökã/��ÙVÝvãAäåÞ8`zãAäå×tÝ ÚTÞ��kÙTÖ/ÙÕùa�����,ü ¡k¢c¤/¥ þGäNÚ�ù/¾�ü8¿áþ�À�ÚVÙ9ÞVö�ßrÙ�ä Ü�Á�ÜT×zß�`?�/�x`gØ?ÀdtÙTßáÚ�`zßáä}Ù�Ú� ß�ö�Ý0Ý0ä5Ý5��ä5ÝÛãAä5Ö/Ù-`zã¢Ö¶×tÚ�ã3¾8ÁÄÃ�ÙN�t`´dtÙ�Å

®¯��gÆo U ����ü���� � ��¨åúçü¦�¦��� © U  �����uÇ8È«É Ê ÷ ¡k¢c¤;¥ S ùåú ü¦�¦���kþ��­�ZË�ú �Ì½Í h3ÎÏ¿üÃ'��ÙTß�Ù Í äNÚ�ã/��Ù'ÚVÙã¢×rÜ�Ö/Ù�Ú9Ú�`��lÙTÚ'ã/�t`zã' Ú�öu�VÖËä5ãåãëÙ9ØGã}×�äAã5Ú'×tߦ`gÞ���Ùz����� S ù²± þ�Ð

°g¨0�����r�ˬg�Q�z�.�r�gÓ¶�)«0�á�z©T����©T�_æ_�Q��­/��­0�)�0�Vðv�0�Q���Q�g�%ìgæ��)ð0�á�r���Tík�zÓI���k�)�z�����0���©V¨k���_�9©9­k�Ó¶�z÷���­0��g�k�Tê}����Ó¶�ª«v�z�è�Tík�zÓ¶«v�Q���N�9�gÓ �té��-�z�zî ¦ �è��­0���F©V�z�r�zì�æ���ÓI�7� ���Q�Væ ��­k�ª�5¨v�0©T���Q�g�Ñ����� S ùåú þ'�z����V��¨k�9�0���k¬/ú ] o¸�L�z�Q¬z�z�9����­0Ó ¡k¢c¤/¥ S ùåúçü¦�¦����þ"�9�V��¨k�9�0�®�F�Q���z�0�Õ�g�v�Q���Q�¯ú ] �f���L� onî_ï1��©T�g¨k�9���zì��­0���Å�9©9­k�Ó¶�Ë���_�k�z�_�r�©V¨0���zì��z�v���z� �z�k�z�V�9���t���Õ©V�z�c�5�z��¬z���/� �z�����½�¦���/�g���z�l��Ó¶��9���t¬z�Ëú �z�¯�����©9­k�g��©T�_�z���N�g���Q�7æÅ�÷?��­k�Z�z�k�z�V�á���t����ð0�9�r����¨kéLÓI�Q����ú$�,ù5æÅ­k�V�9�pú ���� ú þ��r�'�Q�����z�9�z©V�Q�zìl�z�0�¶���©T��Q�z�����ª���V��¨k�9���¦���kî _ ­k���z�k�z�V�á���t������­k����g¨k�r«v¨0���p�Q���p�5�z��¬z�V���*¨åúçü¦�¦��� ] ú ] ú �«©áî _ �I�r�V����­0�t�p��­0����N�X�g���p��­k�'���©T��Q�z�V�ìv�0�z�r����­0�t�

ù/�f��� ] ú ] ú � þ ] úÒ���f��� ] ú � �[onüæÅ­k�V���.��­0���p���p�r�9¨k���9���0©T�z�¦���¶æ"�z�Å�I� �z�����G���t¬G�N�z�_Ó¶������t¬z�Fú �Aî

Ó

Page 56: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ô�Õu֬״Ø�Ù8Ú³ÖcÛÝÜJÞ´ß�àpá?Ù8Ütâ¬á´Õäãå à�æ å£ç¸è´éIêìë Õ¸Û¦Ù8Þtãuíäî�Û³ÖcÞ�Õ�Û¦Þ å Ù³ÜtïuÛ¦Þ´ð´Ù8á?ïAñ¸ÜòÄÙ³Þ´ß/Ø�Ú³Ú¦Þ´Ù�ó´Þ�Õuá?Û³ñuá´Õpô®á?Û¦õ

öF÷!øzùxúªûx÷ üiý

þ ÿ ����������� ������ ���� ÿ����������������� �������� �!�#"$�

%*ØJÚ³á'& â¬á´Ú¦ÛNÛ³Ö)(!Ø!Û³ñ ØiãuØ+*äÕuÖcÛ³ÖcÞ�Õ�Þ´ßxÚ¦Ø�î�í Ù�ÖcÛsÜ�ß;Þ´Ù-á,(!Ø�Ú³Ú8á?ð´Øiá´í Û³ñ Ø�Õ5Û³Ö¬î�á?Û³ÖcÞ�ÕOÚ³î8ñ Ø-(LØ/.NÛ³ñuá?ÛNá´Õá´ã ×´Ø�Ù8Ú³á?Ù³ÜÑÚ³ñ Þ�íäâ¬ã10mØ�íuÕäá20äâcØ�Û¦Þ*ß;Þ´Ù8ð´Øpáe×�á´â«Ö¬ã Û³á?ð�Þ�Õ4325761(!Ø�Ú³Ú³á?ð´ØpÛ³ñuá?Û$&£á´ÚJÕ Þ´Û!Ø98�ïäâ¬Ö¬î�ÖcÛ³â¬Ü×�á´â¬Ö«ãuá?Û¦Ø�ã:0tÜ*Û³ñ ØJâ¬Ø�ð�ÖcÛ³Ö)(Lá?Û¦ØrÚ¦Ø�Õuã Ø�Ù-;pàpÞ´Ù³Ørß;Þ´Ù<(Lá´â¬âcÜ/=1Û³ñ ØJá´ãu×´Ø�Ù8Ú³á?Ù³Ü�ñuá´Ú-á´î�î�Ø�Ú³Ú-Û¦Þeá´Õ§Þ´Ù8á´î�â¬Ø>�?A@�BDCFEHGJI =´á´ÕuãNî�á´ÕNÙ³Ø-K5íuØ�Ú¦Û�Û³á?ð�Ú3ß;Þ´ÙL(IíäâcÛ³ÖcïäâcØ�(!Ø�Ú8Ú³á?ð´Ø�Ú�Þ´ßuñäÖ¬Ú�î8ñ Þ�Ö«î�Ø/;NM�ñ Ø�á´ã ×´Ø�Ù8Ú8á?Ù³ÜPO¦Ú³íuî�î�Ø�Ø�ãuÚ�QÖcß®ÖcÛiÞ�íuÛ¦ïäí Û³Úrá§ïäá´Ö¬ÙSRUT4VXWZY\[^]!ß/Þ´Ù_&�ñuÖ¬î8ña`cbedgf BhC E T4VXWZY\[ Ijilk á´ÕuãÏßCí Ù³Û³ñuØ�Ùm(!Þ´Ù³ØST &£á´ÚS5onqpÚ³í^0r(LÖ¬Û¦Û¦Ø�ã>Û¦Þ§Û³ñuØ >s?�@�BhCtEHGJI Þ´Ù8á´î�âcØ E ÖcßzÛ³ñ Ø�á´ãu×´Ø�Ù8Ú³á?Ù³Ü>Ú³íäî�î�Ø�Ø�ãuÚJÖ«Õ>Û³ñuÖ¬Úu&£á�Ü/=v&£Ø�Ú³á�ܪ۳ñuá?Û�ÖcÛñuá´Ú!ß/Þ´Ù³ð´Ø�ãªáe×�á´â«Ö¬ã Û³á?ð*Þ�Õw(!Ø�Ú³Ú³á?ð´ØjT I ;yx(!Ø�Ú8Ú³á?ð´Øpá´í Û³ñ Ø�Õ5Û³Ö¬î�á?Û³ÖcÞ�Õªî�Þ�ã Ø½Ö¬Ú E{z Vm| ID} Ú³Ø�î�í Ù³Ø½Öcßá´Õ¸Üeá´ã ×´Ø�Ù8Ú³á?Ù8Ü�Ù8íäÕuÕuÖ¬Õ ðrÖ¬Õ�Û³Ö)(!Ø z ñuá´Ú®Ú³íuî�î�Ø�Ú8Ú�ïuÙ³Þ/0äá20äÖ«â¬ÖcÛsÜ�âcØ�Ú³Ú�Û³ñuá´Õ:| E Û³ñuØiî�Þq(!ïäâcØ�Û¦Ø/=�ß;Þ´Ùm(iá´âã Ø+*äÕuÖcÛ³Ö¬Þ�ÕrÖ«Ú Ö¬Õ�Û³ñ ØzïuÙ8Ø�×tÖcÞ�íäÚxâcØ�î�Û³í Ù³ØNÕuÞ´Û¦Ø�Ú I ;~�޴ۦأ۳ñuá?Û3Û³ñ ØZá20�Þ�×´Ø£ã Ø+*äÕuÖ¬Û³ÖcÞ�Õ®Ú³á�ÜtÚ1Õ Þ´Û³ñuÖ¬Õuð á20mÞ�íuÛ�OfÙ³Ø�ïäâ¬á�Ü®á?Û¦Û³á´î���Ú�Q�&�ñ Ø�Ù³Ø+0tÜzá´ÕNá´ã ×´Ø�Ù�Ú³á?Ù³Ü

Ù³Ø } Ú¦Ø�ÕuãuÚ�á�(!Ø�Ú³Ú³á?ð´Ø!Û³ñäá?Ûs&ká´Ú�ïuÙ³Ø�×�ÖcÞ�íuÚ³â¬Ü�Ú³Ø�Õ¸Û�05ÜpÛ³ñ Ø!âcØ�ð�ÖcÛ³Ö)(Lá?Û¦ØlÚ¦Ø�ÕuãuØ�Ù-; ë Õ�á´â¬â�Û³ñuØ-Ø98 á/(!ïäâ¬Ø�Ú&ZØ*ñuá�×´Ø�ãuÖ¬Ú³î�íuÚ8Ú¦Ø�ã Ú³Þ ßCá?Ù-=�Û³ñ Ø�`�bed{fOá´âcð´Þ´Ù�ÖcÛ³ñ7(�&£á´Ú�Ú³Û³á?Û¦Ø�âcØ�Ú³Ú�Ú¦ÞÑÛ³ñuá?Û�ÖcßNá ïäá´Ö¬ÙSRUT4VXWZY\[^]iÖ«ÚØ�×´Ø�Ù®á´î�î�Ø�ïuÛ¦Ø�ãS05ÜpÛ³ñ ØIÙ8Ø�î�Ø�Öc×´Ø�Ù-=�Û³ñ Ø�Õ�ÖcÛs&�Ö¬â¬â�0�ØIá´î�î�Ø�ïäÛ¦Ø�ã*â¬á?Û¦Ø�Ù�Þ�Õ�Öcß�Û³ñuØ-á´ã ×´Ø�Ù�Ú³á?ٳܽٳØ�ïAâ¬á�Ü�Ú�Ö¬Û+;x�âcÛ³ñuÞ�í ð�ñ�ïuÙ³Ø�×´Ø�Õ5Û³Ö¬Õ ðiÙ³Ø�ïäâ¬á�Ü�á?Û¦Û³á´îm�tÚyÞ´ß'Û³ñuÖ¬Ú�Ú¦Þ´Ù³Û�Ö¬Ú�Ö)(!ï�Þ´Ù³Û³á´Õ5Û+= Û³ñ Ø#0�Ø�Ú¦Ûv&ká�Ü�Û¦ÞiïäÙ³Ø�×´Ø�Õ¸ÛyÚ³íuî�ñá?Û¦Û³á´î���Ú!Ö¬Ú®Þ´ß/Û¦Ø�ÕOá?ïäïäâ¬Ö¬î�á?Û³ÖcÞ�Õ } Ú¦ï�Ø�î�Öe*äî E &�ñuá?Û-Ö¬Ú®Û³ñuØJÚ¦Ø�ÕuãuØ�ÙIÚ³í ïäïmÞ�Ú³Ø�ã*Û¦Þ�ã Þ�Öcßxñ ØiâcØ�ð�Ö¬Û³Ö)(Lá?Û¦Ø�âcÜ&£á´Õ5Û³ÚiÛ¦Þ§Ù³Ø�ï�Ø�á?Ûiá*ïuÙ³Ø�×�ÖcÞ�íuÚ$(LØ�Ú³Ú³á?ð´Ø\� I á´Õuã>Ú³Þ:&ZØ�ã Þ§Õ Þ´ÛiãäÖ¬Ú³î�íuÚ³Ú!Ö¬Û�(Iíuî�ñ>ñ Ø�Ù³Ø/;!� Þ´ÙiÛ³ñuØÙ³Ø�î�Þ´Ù8ã�= Û³ñ Þ�íuð�ñ�= ÖcÛ£Ö«Ú£ÕuÞ´Ûká´â«âAÛ³ñäá?Û ãuÖe�iî�íuâ¬Û�Û¦ÞlïuÙ³Ø�×´Ø�Õ5ÛxÛ³ñ Ø�Ú¦Øzá?Û¦Û³á´î���Ú�Ö«ÕJïäÙ8á´î�Û³Ö¬î�Ø/;�� Þ´ÙxØ98 á/(!ïäâ¬Ø/=Öcß3Û³ñuØ�Ù³Ø®Ö¬Ú á!ð�â¬Þ/0äá´â�î�âcÞ�î��rÛ³ñ ØNÚ¦Ø�Õäã Ø�Ù�î�á´Õpá´íuÛ³ñ Ø�Õ¸Û³Ö«î�á?ۦخ۳ñ ØNî�Þ�Õuî�á?Û¦Ø�Õäá?Û³ÖcÞ�ÕpÞ´ß�á$(!Ø�Ú8Ú³á?ð´ØIá´ÕuãpáÛ³Ö)(!Ø�Ú¦Û³á/(!ï��mÖcß'Û³ñ Ølá´ã ×´Ø�Ù8Ú8á?Ù³Ü�Ù³Ø�ïäâ¬á�ÜtÚ�Û³ñ Ø�(!Ø�Ú³Ú³á?ð´Ølá?ÛyÚ³Þq(!Ø-â«á?Û¦Ø�Ù�Û³Ö)(LØIÖcÛs&�Ö¬â¬â�0�ØNØ�×�Ö¬ã Ø�Õ5Û�Û³ñäá?ÛÛ³ñuÖ¬ÚNÖ«Ú®á½Ù³Ø�ïAâ¬á�ÜS0�Ø�î�á´íuÚ¦ØJÛ³ñ ØLÛ³Ö)(!Ø�Ú³Û³á/(!ï�&�Ö¬â¬âL0�ØLÞ�í ÛNÞ´ß ãäá?Û¦Ø/;�M�ñuÖ¬Ú®Õ Ø�ð�â¬Ø�î�Û³ÚIá�âcÞ´ÛNÞ´ßxã Ø�Û³á´Ö¬â«ÚE &�ñuá?Û Þ´ß�Û³ñ ØIî�âcÞtîm�tÚ�Þ´ß'Û³ñ ØNÚ¦Ø�ÕuãuØ�Ù�á´Õuã�Ù³Ø�î�Ø�֬״Ø�ÙyãuÖe�mØ�Ù�0tÜ�á!ß;Ø+&WÚ¦Ø�î�Þ�ÕuãuÚ�� I 0äí Û ð�Öc×´Ø�Ú�áLñuÖ¬Õ5ÛxÞ´ßÛ³ñ Ø#&ká�Ü�Ù³Ø�ïäâ¬á�Ürá?Û¦Û³á´îm�tÚ�î�á´Õj0mØzñuá´ÕäãuâcØ�ã�;%*ØJá´î�Û³íuá´â«âcÜ�ñäá�×´Øiá´â¬â'Û³ñuØLÛ¦Þ5Þ�â«Ú®Õ Ø�î�Ø�Ú³Ú³á?Ù8Ü�Û¦Þpî�Þ�ÕäÚ¦Û¦Ù8íuî�ÛNá�Ú³Ø�î�í Ù³Øu(!Ø�Ú8Ú³á?ð´Øiá´í Û³ñ Ø�Õ5Û³Ö¬î�á?Û³ÖcÞ�Õ

î�Þtã ØJß;Þ´ÙlÚ³ñ Þ´Ù8Û�(!Ø�Ú³Ú³á?ð´Ø�Ú-;j��Ø�Û���.��'��V k�� C�� �'��V k��-��� �'��V k��-� 0�Øiá E{z Vm| ID} Ú¦Ø�î�íuÙ³Ørò�����;�%*Øî�á´Õ�á´íuÛ³ñ Ø�Õ¸Û³Ö«î�á?Û¦Ø�� } 0AÖcÛ_(!Ø�Ú8Ú³á?ð´Ø�Ú�á´Úiß/Þ�â¬âcÞ\&�Ú+.*Û³ñ ØeÚ¦Ø�ÕuãuØ�ÙJá´Õuã Ù³Ø�î�Ø�Öc×´Ø�Ù½Ú³ñäá?Ù³Ø�á Ú¦Ø�î�Ù³Ø�Û��´Ø�Ü�\�P  �'��V k�� C ;�M3Þiá´í Û³ñuØ�Õ¸Û³Ö¬î�á?Û¦Ølá�(!Ø�Ú8Ú³á?ð´Ø�T   �'��V k��-� =uÛ³ñuØIÚ¦Ø�ÕuãuØ�Ù�î�Þq(!ïäí Û¦Ø�Ú�WZY\[ i � BDC�E T I ;%�ñ Ø�Õ�Û³ñ ØIÙ³Ø�î�Ø�Öc×´Ø�ÙyÙ³Ø�î�Ø�Öc×´Ø�Ú�RUT¡VXWXY\[�]�=mñ Ø®Þ�í Û¦ïäíuÛ³Ú k Ö¢�S� BhCFE T I�i WXY\[7; ë ÛyÚ³ñ Þ�íuâ¬ã,0mØIî�âcØ�á?Ù-=äÞ�Õá´Õ�Ö¬Õ5Û³íuÖc۳֬״خâcØ�×´Ø�âU=äÛ³ñäá?Û�Û³ñuÖ¬Ú�Ö¬Ú�áJÚ¦Ø�î�í Ù³Ø >s?A@£E &�ñ Ø�Õj¤§Ö¬Ú Ú³í7�iî�ÖcØ�Õ5Û³âcÜrâ¬Þ�Õ ð I �Aß/Þ´Ù�î�Þq(LïäâcØ�Û¦Ø�Õ Ø�Ú³Ú-=&ZØ®ð�֬״ØNá!ïuÙ³ÞtÞ´ß3ñuØ�Ù³Ø/;

¥�¦¨§t©oª\§�« ¬®­�¯�° >s?A@ n2±^p³²J´{5 °mµ 3·¶�n2¸ ° ´º¹ E{z Vm|L»w¼t½ � I<¾ ¹ °�¿ ±�À °'Á

 ª\©�©oà ��Ø�Û�ÄÅ0�Ø®á´Õ½á´ã ×´Ø�Ù8Ú³á?Ù8Ürá?Û¦Û³á´îm�tÖ¬ÕuðiÛ³ñ Ø®Ú³î�ñ Ø-(!Ø/�äÖÆ;ÆØ/;e=rÄ»á?Û¦Û¦Ø-(LïuÛ³Ú Û¦ÞLß/Þ´Ù³ð´ØNáL×�á´â¬Ö«ãJÛ³á?ðÞ�Õiá®Õ Ø+&Ç(!Ø�Ú³Ú³á?ð´Ø/;���Ø�î�á´â¬âuÛ³ñuá?Û�Ä�Ö«Õ¸Û¦Ø�Ù8á´î�Û³Ú�&�ÖcÛ³ñ!Û³ñuØ Ú³î8ñ Ø-(LØ�0tÜIÙ³Ø-Ktí Ø�Ú¦Û³Ö«Õ ð®áÈ0äíuÕäî8ñ!Þ´ß�Û³á?ð�ÚZÛ¦Þ0mØ î�Þq(Lïäí Û¦Ø�ãlÞ�Õ!×?á?Ù8ÖcÞ�íuÚ�(LØ�Ú³Ú³á?ð´Ø/=�á´Õuã!á?Û�Ú¦Þq(!Øxï�Þ�Ö¬Õ5Û'Þ�í Û¦ïäí Û³Ú�á´Õ!á?Û¦Û¦Ø-(!ïuÛ¦Ø�ãLß/Þ´Ù³ð´Ø�Ù³Ü,RUT4VXWZY\[^]<;

k

Page 57: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

%*ØlÚ8ñ Þ'& ñ Þ\&WÛ¦Þ�î�Þ�Õ5×´Ø�Ù³Û®á´Õ5ÜpÚ³íuî�ñ�á´ãu×´Ø�Ù8Ú³á?Ù³ÜpÖ«Õ¸Û¦Þ�á´Õeá´ã ×´Ø�Ù8Ú³á?Ù³ÜÊÉ Û³ñuá?Û�Û¦Ù8ÖcØ�Ú�Û¦Þ�ãuÖ¬Ú¦Û³Ö¬Õuð�íuÖ¬Ú³ñ� ß/Ù³Þq( álÛ¦Ù8íuâ¬ÜiÙ8á´ÕuãuÞq( ß/íäÕuî�Û³ÖcÞ�Õ�;É Ö¬Ú�ð�֬״Ø�ÕJá´î�î�Ø�Ú³Ú£Û¦Þ-á®ßCíuÕuî�Û³ÖcÞ�ÕiÞ´Ù8á´î�âcØ/=F&�ñ Ø�Ù³Ø Û³ñäÖ¬ÚÄßCíuÕuî�Û³ÖcÞ�ÕiÖ¬ÚÄØ�ÖcÛ³ñuØ�ÙZáNî�Þq(LïäâcØ�Û¦Ø�âcÜlÙ8á´ÕuãuÞq(

ß/íuÕäî�Û³ÖcÞ�ÕpÞ´ÙyØ�â¬Ú¦Ø-á´ÕeÖ¬ÕuÚ¦Û³á´Õuî�ØIÞ´ß�� BDCFEHGJI ß/Þ´ÙyÙ8á´Õuã Þq(Lâ¬Ü } î�ñ Þ�Ú¦Ø�Õ �\� ;�É®&�Ö¬â¬â�Ú8Ö)(Iíäâ¬á?Û¦ØIá´ÕeÖ¬ÕuÚ³Û³á´Õuî�ØÞ´ß�Û³ñ Ø >�?A@ ß/Þ´ÙcÄ$Ö¬ÕJÛ³ñ Ø�ß/Þ�â¬âcÞ\&�Ö¬Õ ð�&£á�Üo.�&�ñ Ø�Õ_Ä$Ù³Ø-K5íuØ�Ú¦Û³ÚZá´íuÛ³ñ Ø�Õ¸Û³Ö«î�á?Û³ÖcÞ�ÕrÞ´ß3á�(!Ø�Ú³Ú³á?ð´Ø#T4Ëg=^ÉÚ³í^0r(LÖ¬Û³Ú�T4Ë�Û¦Þ�ÖcÛ³ÚyßCíuÕuî�Û³ÖcÞ�Õ�Þ´Ù8á´î�âcØ!á´Õuã�Ù³Ø�î�Ø�Öc×´Ø�Ú#0äá´î���Ú¦Þq(!Øl×�á´â«í Ø�WXY\[qË�&�ñuÖ¬î8ñ�ÖcÛyÛ³ñ Ø�Õ�Ù³Ø�Û³í Ù8ÕäÚÛ¦Þ�Ä�;P�1Ö«Õuá´â¬âcÜ/=LÄ Þ�í Û¦ïäí Û³Ú�RUT¡VXWXY\[�]�á´ÕuãOÖ¬ÚNãuÞ�Õ Ø/;Pɵî8ñ Ø�îm�tÚ�&�ñ Ø�Û³ñuØ�Ù�Ä Ø�×´Ø�Ù-Ù³Ø-K5íuØ�Ú¦Û¦Ø�ã§Û³ñäá?Û(!Ø�Ú³Ú³á?ð´ØsTÌ0�Øxá´í Û³ñ Ø�Õ5Û³Ö¬î�á?Û¦Ø�ã��5Ö¬ßAÚ¦Þ^=·É»ð�í Ø�Ú³Ú³Ø�Ú#OfÙ8á´Õuã Þq(­ßCíuÕuî�Û³Ö¬Þ�Õ7Q E ÖU;ÆØ/;e=¸Þ�í Û¦ïAí Û³Ú�� I á´Õuã!Ú¦Û¦Þ´ïäÚ-;Í Û³ñ Ø�Ù�&�Ö«Ú¦Ø/=LÉ Ú³í^0r(LÖ¬Û³Ú�T Û¦ÞeÖcÛ³Ú®Þ´Ù8á´î�â¬Ørá´ÕuãOÙ³Ø�î�Ø�Öc×´Ø�Ú�0äá´îm�§Ú¦Þq(!ØJ×�á´â«í ØuWXY\[tÎ�; ë ß�WZY\[ i WXY'[�Î\=Û³ñ Ø�ÕÏÉ ð�íuØ�Ú³Ú¦Ø�ÚPOfïäÚ¦Ø�íäã Þ´Ù8á´Õuã Þq( ßCíuÕuî�Û³Ö¬Þ�Õ7Q E ÖÆ;ÆØ/;e=�Þ�í Û¦ïäí Û³Ú k'I �1Þ´Û³ñ Ø�Ù�&�Ö«Ú¦Ø/=�É ð�í Ø�Ú8Ú¦Ø�Ú,OfÙ8á´ÕuãuÞq(ß/íuÕäî�Û³ÖcÞ�Õ7Q E ÖU;ÆØ/;e=uÞ�íuÛ¦ïäí Û³Úv� I ;��Ø�ÛríuÚiá´Õuá´â¬Ü5õ�Ø�òÄÙ'Ð ���ÒÑ �'��V k�� C .�É�Ó/Ô³Õ-Öº× Ø iÙk�Ú ; ë ÕªÛ³ñuÖ¬ÚJî�á´Ú¦Ø/=vÉ Ö¬Úið�Öc×´Ø�ÕÏá´î�î�Ø�Ú³ÚrÛ¦Þ á

ïäÚ¦Ø�íuã Þ´Ù�á´Õuã Þq(�ß/íäÕuî�Û³ÖcÞ�Õ�=�á´Õuã�Û³ñ Ø!Ú³Ö)(-íuâ¬á?Û³ÖcÞ�Õ�ß/Þ´Ù#Ä Ö¬ÚyØ98�á´î�Û³â¬Ü�Ø-KtíuÖc×?á´âcØ�Õ5Û�Û¦Þ�á´ÕeØ98�Ø�î�í Û³ÖcÞ�ÕeÞ´ßÄWá?Û¦Û³á´îm�tÖ«Õ ðiÛ³ñ Ø®á´î�Û³íäá´â >s?�@ ;ZætÞ^.

ò�Ù\Ð ����Ñ �'��V k�� C .·É Ó ÔUÕ Öº× Ø iÛk�Úi òÄÙ'Ð ���uÑ �'��V k�� C �'RUT4VXWZY\[^] Ñ Ä >�?A@ Ô³Õ Ög× Ø .2`cbedgf BhC E T4VXWZY\[ I�iÛk�Ü T Ý ,Þ Ú V

Û³ñ ØNÚ³íuî�î�Ø�Ú8ÚxïuÙ³Þ/0äá20AÖ¬â¬ÖcÛÝÜLÞ´ß�Ä�;Í Õ[Û³ñ Ø*Þ´Û³ñuØ�Ù�ñuá´ÕuãA=yî�Þ�ÕuÚ8Ö¬ã Ø�Ù½òÄÙ�Ð � Ñ ß Ytà\á �câ�� .�É�ÓLÖg× Ø iãk�Ú ;M ñuÖ«Ú�Ö¬Ú�Ø98�á´î�Û³â¬Ü�Û³ñuØ

ïuÙ³Þ/0äá20äÖ«â¬ÖcÛsÜÏÛ³ñuá?ÛjÄ î�á´Õ[ß/Þ´Ù³ð´Ø§áª×?á´â¬Ö¬ãFÛ³á?ð>Þ�Õ áªïuÙ³Ø�×�ÖcÞ�íuÚ³â¬Ü } íuÕrK5í Ø�Ù�ÖcØ�ãa(!Ø�Ú8Ú³á?ð´Ø/=#&�ñ Ø�ÕFÛ³ñuØ>�?A@ Ö¬Ú Ö«ÕuÚ¦Û³á´Õ5Û³Ö¬á?Û¦Ø�ã,&�Ö¬Û³ñ½á ¿ n/ä�å�² ° p ° ²æ6�ÀX325 µ n2äµß/íäÕuî�Û³ÖcÞ�Õ�;�xWâ«ÖcÛ¦Û³âcØzÛ³ñ Þ�í ð�ñ5Û�Ú³ñuÞ�íuâ¬ã½Ú³ñuÞ'& Û³ñäá?ÛÚ³Ö¬Õuî�Ø$� Ö¬Úzî�Þq(LïmØ�Û¦Ø�â¬Ü�Ù8á´Õäã Þq(P=AÄ î�á´ÕuÕ Þ´Ûzï�Þ�Ú³Ú³Öe0AâcØIïäÙ³Ø�ãuÖ¬î�ÛyÛ³ñ ØLî�޴ٳٳØ�î�Û�WZY\[�Þ�Õ�á�Õ Þ�Õ } K5í Ø�Ù�ÖcØ�ã(!Ø�Ú³Ú³á?ð´Ø�&�ÖcÛ³ñ!ïäÙ³Þ/0äá20äÖ¬â«ÖcÛsÜ�0mØ�Û¦Û¦Ø�ÙZÛ³ñuá´Õ�¼ ½ ��E Ú³Ö«Õuî�Øv¤pÖ¬Ú�Û³ñ Ø�Þ�íuÛ¦ïäí Û�âcØ�Õuð´Û³ñ!Þ´ß�Û³ñ Ø�ßCíuÕuî�Û³Ö¬Þ�Õ I ;�æ�Þ^=òÄÙ\Ð � Ñçß Y·à�á ��âÈ� .·É�ÓLÖº× Ø ièk�Ú¨é ¼t½ � ;

æ�Ö¬Õuî�ØÈ&£Ø#��Õ Þ'& Û³ñuá?Û�� Ö¬Ú�á E{z Vm| ID} ò��s��=r&ZØNñäá�×´Ø/.êêê ò�Ù\Ð �\��Ñ �'��V k��

C .·É Ó/Ô³Õ+Öº× Ø iÛk�Úoë ò�Ù\Ð � Ñìß Ytà\á ��â�� .qÉ ÓLÖg× Ø iÛk�Ú êêê é |+í

M ñuÖ¬ÚxÖ)(u(!Ø�ãuÖ¬á?Û¦Ø�âcÜrÖ)(!ïäâ«ÖcØ�Ú�Û³ñäá?Ûò�Ù\Ð ����Ñ �'��V k�� C �'RUT¡VXWXY\[^] Ñ Ä >s?A@ ÔUÕ Öº× Ø .2`�bed{f BDC E T¡VXWXY\[ I�iîk�Ü T Ý jÞ ÚLé |¨»y¼ ½ � V

&�ñuÖ¬î�ñ�ïuÙ³Þ�×´Ø�Ú�Û³ñ Ø®Û³ñ Ø�Þ´Ù8Ø-(P;

ï ð1ñ�ò��Ûó-��� ÿô�Aõ$�Z���� ���ö�÷tø,�����v���ì ���o�� ������

M ñ ØiÚ�ù�ñ Ø-(!ØJÞ´ßxÛ³ñ ØLïuÙ³Ø�×�ÖcÞ�íuÚNÚ¦Ø-ù�Û³ÖcÞ�Õ:&ká´Ú-Ú¦Ø-ù�í Ù³Ø/=L0äí Û®Þ�ÕäâcÜ�á´â«âcÞ'&£Ø�ã§á´í Û³ñuØ�Õ¸Û³Ö)ù�á?Û³Ö¬Þ�Õ§Þ´ßxÚ³ñ Þ´Ù8Û+=*^8tØ�ã } âcØ�Õ ð´Û³ñ�(!Ø�Ú³Ú³á?ð´Ø�Ú+;#xyÚ�Ö¬Õ�Û³ñ Ø�ù�á´Ú³Ø-Þ´ßZØ�Õ7ù�Ù³ÜtïuÛ³ÖcÞ�Õ�=o&ZØ�&£Þ�íuâ¬ã�â¬Öe�´ØIÛ¦ÞrØ98�Û¦Ø�Õuã�Û³ñuÖ¬Ú�Û¦Þ�á´â¬âcÞ\&á´í Û³ñ Ø�Õ5Û³Ö)ù�á?Û³ÖcÞ�ÕªÞ´ßyâcÞ�Õ ð´Ø�Ù�(!Ø�Ú³Ú8á?ð´Ø�Ú+; ë ÛLÖ¬Ú�(-í7ù8ña(!Þ´Ù³Ø�ãuÖe�uù�íäâcÛIÛ¦Þ�ãuÞeÚ³Þ�ñ Ø�Ù³Ø/;�M3Þ�Ú¦Ø�Ø�Û³ñuÖ«Ú+=Ù³Ø-ù�á´â¬âÄÛ³ñäá?ÛIÖ¬Õ*Û³ñuØ_ù�á´Ú¦ØJÞ´ßxØ�Õ7ù�Ù³ÜtïuÛ³ÖcÞ�ÕÏ&£Ø_ù�Þ�íuâ«ã§á´âe&ká�Ü�ÚIíäÚ¦ØLÛ³ñ ØiÚmù8ñ Ø-(!Ø�Ö¬ÕÏ&�ñuÖ)ù8ñ:&ZØrÚ³Ö)(!ïäâ¬Ü0uÙ³Ø�á2�-Û³ñ Øv(!Ø�Ú³Ú³á?ð´Ø�Ö¬Õ¸Û¦Þ�0äâ¬ÞFùm�tÚ�á´ÕuãlØ�Õrù�Ù³Ü5ïäÛ�Ø�á/ù8ñ$0äâ¬ÞFùm�-Ú¦Ø�ïAá?Ù8á?Û¦Ø�âcÜ E íuÚ³Ö¬Õ ð�á´ÕLÖ¬ÕäãuÖ¬Ú¦Û³Ö¬Õuð�íuÖ¬Ú³ñuá20AâcØØ�Õ7ù�Ù³ÜtïuÛ³ÖcÞ�Õ�Ú�ù8ñ Ø-(LØ I ê Û³ñuÖ«Ú�á?ïuïäÙ³Þ�á/ù8ñS(Lá�Ü�Õ Þ´Û�0�ØNÛ³ñ Ø�(!Þ�Ú¦Û�Ø+�uù�ÖcØ�Õ5Û+=�0äí Ûyá?Û�â¬Ø�á´Ú¦ÛyÖ¬Ûs&�Ö«â¬â�0�ØÚ¦Ø-ù�í Ù³Ø/úüûkí ÛrÛ³ñuÖ¬Úrã ÞtØ�ÚÊ5�n/pu&£Þ´Ù��ÏÖ«ÕÏÛ³ñ ØÏù�á´Ú¦ØeÞ´ßNá´íuÛ³ñ Ø�Õ¸Û³Öºù�á?Û³ÖcÞ�Õ�; å Þ�ÕuÚ8Ö¬ã Ø�ÙiØ98�Û¦Ø�ÕuãuÖ«Õ ðOÛ³ñuØÚ�ù8ñ Ø-(LØzÞ´ß�Û³ñ Ø�ïuÙ³Ø�×�ÖcÞ�íuÚ£Ú¦Ø-ù�Û³ÖcÞ�Õ�Ö¬ÕJÛ³ñuÖ¬Ú�&ká�Ü/;kæ�Þ�&�ñuØ�Õrá´Õ�á?Û¦Û³á/ùm�´Ø�Ù�Ù³Ø-Ktí Ø�Ú¦Û³Úxá´Õ�á´í Û³ñ Ø�Õ5Û³Ö)ù�á?Û³ÖcÞ�Õ

¼

Page 58: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Þ´ßNT i Twý�þ�T ÿ�Û³ñ Ø®Ú¦Ø�Õäã Ø�Ù�ù�Þq(!ïäí Û¦Ø�Ú á´ÕäãJÙ8Ø�Û³í Ù8ÕuÚ�WXY'[ i WXY'[ ý þ�WZY\[ ÿ =^&�ñuØ�Ù³ØsWXY\[ ý i � BDC�E Twý Iá´Õuã�WZY\[ ÿ i � BhCFE T ÿ I ;�ûkí ÛxÖcÛxÖ¬ÚkØ�á´Ú¦ÜrÛ¦Þ!ß/Þ´Ù³ð´Ø®á!Õ Ø+&[Û³á?ðiÖ¬ÕrÛ³ñuÖ«Ú£Úmù8ñ Ø-(!Ø/.Zՠ޴ۦخ۳ñuá?Û�WXY\[ ÿ þ�WXY\[ ýÖ¬Ú ál×?á´â¬Ö¬ã�Û³á?ðiÞ�ÕÊT¡Ë i T ÿ þ�T ÿ Ýi Tüú

�s(u(P;-;+;¦Û³ñuÖ¬ÚzÚmù8ñ Ø-(!Øu&£á´Ú®ÕuÞrð´ÞtÞ�ã�Ú³Ö¬Õ7ù�Ø!ÖcÛzá´â¬â¬Þ'&£Ø�ã�á´Õ�á´ã ×´Ø�Ù8Ú8á?Ù³Ü�Û¦ÞwOfÙ8Ø�Ú³ñ5í��LØ-QP(!Ø�Ú³Ú³á?ð´Ø0äâcÞ�ù���Ú+;�ætÞ�(Lá�ÜF0�Ø�&£Ø�ù�á´ÕpØ�Õ ß/Þ´Ùmù�ØlÚ¦Þq(!Ø-Þ´Ù8ã Ø�Ù8Ö«Õ ðLÞ�Õ�Û³ñ Ø�Ú¦Ø�0äâcÞ�ù���Ú ý ��� Þ´ÙzØ98�á/(LïäâcØ/=�&ZØ�(LÖcð�ñ5Ûù�Þq(!ïäí Û¦Ø >s?�@�BhCtE Twý/þrT ÿ I á´Ú'ß/Þ�â¬âcÞ'&�Ú+.�� BDC�E R k ]2þ7Twý I þ^� BhCFE RU¼q]mT ÿ I ; E %�ñuØ�Ù³Ø#R��H]3Ù³Ø�ïuÙ³Ø�Ú³Ø�Õ¸Û³Ú�Ú¦Þq(LØ*^8tØ�ã } âcØ�Õ ð´Û³ñOØ�Õ7ù�ÞtãäÖ¬Õ ð½Þ´ß Û³ñuØJÖ¬Õ5Û¦Ø�ð´Ø�Ù���=Äá´Õuã�T ý VmT ÿ á?Ù³ØrÞ´ß Û³ñ Ørá?ïuïuÙ³Þ´ïuÙ�Ö¬á?Û¦ØJâcØ�Õuð´Û³ñuÚ+; I M ñäÖ¬Úù�Ø�Ù³Û³á´Ö¬ÕuâcÜNïuÙ³Ø�×´Ø�Õ5Û³Ú'Û³ñ ØZÙ8Ø } Þ´Ù8ã Ø�Ù8Ö¬Õuðyá?Û¦Û³á/ùm�-Þ´ßäÛ³ñ ØkïuÙ³Ø�×�ÖcÞ�íuÚ1Ú�ù8ñ Ø-(LØ/;Nûkí Û�Ö¬Ú1ÖcÛ'Ú¦Ø-ù�í Ù³Ø\����Ø+*AÕuÖcÛ¦Ø�âcÜÕ Þ´Û+;�xyÕ*á´ã ×´Ø�Ù8Ú³á?Ù8ÜÊ&�ñuÞJÞ/0uÛ³á´Ö«ÕuÚ�Û³ñ ØlÛ³á?ðPWZY\[ ý þvWZY\[ ÿ ß;Þ´Ù#(LØ�Ú³Ú³á?ð´ØuT ý þsT ÿ ù�á´Õ*Ö)(u(!Ø�ãuÖ¬á?Û¦Ø�âcÜß;Þ´Ù³ð´ØLá�×?á´â¬Ö¬ã�Û³á?ð½ß/Þ´ÙÈ(!Ø�Ú³Ú³á?ð´Ø_T ý E ÖcÛ®Ú³ñ Þ�íuâ«ãj0�Ø-Þ/05×�ÖcÞ�íuÚzñ Þ\& I ; ë Û®Ú³ñ Þ�íäâ¬ã�á´â«Ú¦Þ�0mØ�ù�âcØ�á?Ù®Û³ñäá?Ûá´ÕJá´ã ×´Ø�Ù�Ú³á?Ù³Ü$&�ñ ÞIÞ/0uÛ³á´Ö¬ÕuÚ�(IíäâcÛ³ÖcïäâcØxÛ³á?ð�Ú£Þ�ÕrãuÖ¢��Ø�Ù³Ø�Õ5Û�(!Ø�Ú³Ú³á?ð´Ø�Úcù�á´Õ£OfïäÖcØ-ù�Ø�Û¦Þ´ð´Ø�Û³ñ Ø�ÙmQlãuÖ¢��Ø�Ù³Ø�Õ5Ûïäá?Ù³Û³ÚxÞ´ß'Û³ñ Ø�Ú¦Ø�(!Ø�Ú³Ú³á?ð´Ø�Ú�Û¦Þuù�Ù³Ø�á?Û¦ØIáL×�á´â¬Ö«ã�Û³á?ðLÞ�Õ½Ú¦Þq(!ØIÕ Ø+&4(!Ø�Ú³Ú8á?ð´Ø/;

ë Û'Ú¦Ø�Ø-(LÚ1Û³ñuá?Û'ás0äá´Ú³Ö)ùäá-&ªÖ¬ÕNÛ³ñ Ø�Ú¦ØkÚ�ù8ñ Ø-(LØ�Ú1Ö«Ú�Û³ñuá?Û1Û³ñ Ø£Û³á?ð�Ú�á?Ù³Øcù�Þq(!ïäí Û¦Ø�ã-á´ÕuãIÛ¦Ù8á´ÕuÚ�(LÖ¬Û¦Û¦Ø�ãÖ¬ÕiáÈ0AâcÞFùm�t&�Ö¬Ú¦Ø ßCá´Ú³ñuÖcÞ�ÕA=q&�ñuÖ)ù�ñ!ð�Öc×´Ø�ÚZÛ³ñ Ø�á´ãu×´Ø�Ù8Ú³á?Ù³Ü!Û¦ÞtÞ�(-í7ù8ñiÛ¦ÞIïAâ¬á�Ü�&�ÖcÛ³ñA;'à�á�ÜF0�Ø�á®×?á?Ù8Ö¬á?Û³Ö¬Þ�ÕÞ�Õ�Û³ñuØ-â¬á´Ú³ÛyÚ�ù�ñ Ø-(!Ø�&�Ö¬â¬âA&ZÞ´Ù���;®ætï�Ø-ù�Öe*rù�á´â¬âcÜ/=mÖ¬ÕuÚ¦Û¦Ø�á´ã�Þ´ßÄÞ�í Û¦ïAí Û¦Û³Ö¬Õ ðJÛ³ñ Ø�ù�Þ�Õrù�á?Û¦Ø�Õuá?Û³ÖcÞ�ÕeÞ´ß�WXY\[ ýá´Õuã WXY\[ ÿ =N&�ñ5Ü�Õ Þ´ÛNÞ�í Û¦ïAí ÛNÛ³ñ Ø�Ö¬Ù#8�Þ´Ù�� ë Õ§Û³ñuÖ¬Ú�ù�á´Ú¦Ø/=�&£ØJñuá�×´Ø >s?A@rBhCtE T ý þÈT ÿ I�i � BhCFE R k ]�þT ý I�� � BhCtE RU¼q]¨þ�T ÿ I#E á´Õuã�×´Ø�Ù8Öe*où�á?Û³ÖcÞ�Õ½Ö¬Úxï�Ø�Ù³ß/Þ´Ùm(!Ø�ã�Ö«Õ�Û³ñ Ø®Þ/0t×tÖcÞ�íäÚ�&£á�Ü I ; ë Ûvù�Ø�Ù³Û³á´Ö¬ÕuâcÜ�Ú¦Ø�Ø-(LÚñuá?Ù8ã Ø�Ù Û¦ÞJá?Û¦Û³á/ù��½Û³ñuÖ«ÚxÚ�ù8ñ Ø-(Lخ۳ñuá´Õ�Ø�ÖcÛ³ñuØ�Ù Þ´ß3Þ�íuÙ ïuÙ³Ø�×�ÖcÞ�íuÚxÚ�ù�ñ Ø-(!Ø�Ú+;ûkí Û®Þ´ßcù�Þ�í Ù8Ú¦Ø$&£ØLÚ³ñ Þ�íuâ«ã�ÕuØ�×´Ø�Ù#0�ØLÚ³á?Û³Ö¬ÚX*uØ�ãS&�ÖcÛ³ñ*áPù�Þ�ÕuÚ¦Û¦Ù8írù�Û³ÖcÞ�ÕeÛ³ñuá?ÛjO¦âcÞtÞ/��Ú®Ú¦Ø-ù�í Ù³Ø-Q E Ö¬Û

Ú¦Ø�Ø-(LÚxÛ³ñuá?Ûxá´ã ×´Ø�Ù�Ú³á?Ù8ÖcØ�Úxá?ٳخá´â)&£á�ÜtÚv(LÞ´Ù³ØÈù�âcØ�×´Ø�Ù Û³ñuá´Õ�ã Ø�Ú8Öcð�Õ Ø�Ù8Ú+;9;+; I ;�%*ØNïuÙ³Ø�ß/Ø�Ùká!Úmù8ñ Ø-(!ØzÛ³ñäá?ÛÖ¬ÚlïuÙ³Þ�×�á20äâ¬Ü§Ú¦Ø-ù�í Ù³Ø/ú1x�ÕäãÑß;Þ´Ù!ð´ÞtÞtãÑÙ8Ø�á´Ú¦Þ�Õ�.rÛ³ñuÖ¬Úlâ¬á´Ú¦ÛLÚ�ù�ñ Ø-(!ؽ֬ÚlÖ¬Õ ßCá/ù�ÛP5on/p-Ú¦Ø-ù�í Ù³Ø/=Z×tÖ«á�Û³ñuØß;Þ�â¬â¬Þ'&�Ö¬ÕuðJá?Û¦Û³á/ù���;-æ�á�Ü�á´Õeá´ã ×´Ø�Ù8Ú8á?ٳܽÞ/0uÛ³á´Ö¬ÕäÚ�Û³á?ðPWXY\[ ý ß/Þ´Ù�(!Ø�Ú³Ú³á?ð´ØuT ý =mÛ³á?ðPWXY\[ ÿ ß/Þ´ÙÈ(!Ø�Ú³Ú³á?ð´ØT ý þ�T ÿ =äá´Õuã�Û³á?ðuWZY\[� yß/Þ´Ù�(!Ø�Ú³Ú8á?ð´Ø�T Ëý ;�M ñ Ø�Õ�Û³ñ Ø®á´ãu×´Ø�Ù8Ú³á?Ù³Ü�ñuá´Ú â¬Ø�á?Ù8Õ Ø�ã�Û³ñ Ø®ß/Þ�â¬âcÞ\&�Ö¬Õ ð^.

WZY\[ ý i � BhCFE R k ]¨þcT ý IWZY\[ ÿ i � BhCFE R k ]¨þcT ý I�� � BhCtE RU¼q]¨þ�T ÿ IWZY\[� i � BhCFE R k ]¨þcT Ëý I í

æ�Ö¬Õ7ù�ØÈWXY\[ ý � WXY\[ ÿ i � BhCFE RU¼q]¨þ�T ÿ I =äÛ³ñ ØNá´ã ×´Ø�Ù8Ú8á?Ù³Ü,ù�á´Õ½ß/Þ´Ù³ð´ØNÛ³ñuØ®×�á´â«Ö¬ã�Û³á?ðuWXY\[ ý � WXY\[ ÿ � WXY\[� Þ�Õ½Û³ñ Ø E Õ Ø+& I (!Ø�Ú³Ú³á?ð´Ø�T¡Ëý þ�T ÿ =äá´Õuã�Û³ñuØ®Ú�ù8ñ Ø-(LØNÖ¬Ú Õ Þ´Û Ú³Ø-ù�í Ù³Ø/;M ñuÖ¬Ú�0uÙ8Ö¬Õ ð�ÚiíuÚ!Û¦ÞOáOÚ�ù8ñuØ-(!Ø E Û³ñ Ø����� ¾�������I ß/Þ´ÙJá?Ù�0äÖ¬Û¦Ù8á?Ù³Ü } âcØ�Õ ð´Û³ñw(!Ø�Ú8Ú³á?ð´Ø�Úu&�ñuÖ)ù8ñ¡´º¹

ïuÙ³Þ�×?á20äâcÜ!Ú¦Ø-ù�íuÙ³Ø/; ë Û�Ö«Ú�×´Ø�Ù³Ü�ù�âcÞ�Ú¦Ø�Û¦ÞNÛ³ñuÖ¬Ú�â¬á´Ú¦Û�Ú�ù�ñ Ø-(!Ø/=tØ98�ù�Ø�ïäÛ�Û³ñuá?Û£ÖcÛÄÖ¬ÚÄÙ�á´Õuã Þq(LÖcõ�Ø�ãA;LxyÚ�á20�Þ�×´Ø/=âcØ�ÛÈ�®.��'��V k�� C � �'��V k�� � � �'��V k�� � 0�Ølá E{z Vm| ID} ò�����;�M ñ ØlÕ Þ´Û³á?Û³ÖcÞ�ÕwR��Z]v&�Ö¬â¬â�ãuØ�Õ Þ´Û¦ØlÛ³ñ Ø����2¼ }0äÖcÛ�Ù³Ø�ïuÙ³Ø�Ú³Ø�Õ¸Û³á?Û³ÖcÞ�ÕeÞ´ßkÖ¬Õ5Û¦Ø�ð´Ø�Ù���Ö¬Õ�0äÖ¬Õäá?Ù³Ü E &ZØ$&�Ö¬â¬â¨0mØ!â¬Öº(LÖcÛ¦Ø�ã�Û¦Þ,(!Ø�Ú³Ú³á?ð´Ø�Ú®á?Û#(LÞ�Ú¦Û�¼ ��� ÿ ëÒk0äâcÞ�ù���Ú�âcÞ�Õuð^=mÕ Þ´Û�árÚ¦Ø�Ù8ÖcÞ�íuÚ�Ù³Ø�Ú¦Û¦Ù�Ö)ù�Û³ÖcÞ�Õ�Ö¬Õ½ïuÙ�á/ù�Û³Ö)ù�Ø I ;�M3Þ�á´í Û³ñ Ø�Õ5Û³Ö)ù�á?Û¦Ø!á_(!Ø�Ú8Ú¦ð�á?Ø�Tü=�ïäá?Ù8Ú³Ø�Tá´ÚsT ý þ G+G+G þcT! +=^&�ñ Ø�Ù³Ø#" T%$&" i ���2¼F; å ñ ÞtÞ�Ú¦ØNálÙ8á´Õäã Þq( ×?á´â¬í Ø�'   �'��V k��-� ½ ý á´ÕuãPù�Þq(!ïäíuÛ¦Ø/.

WXY'[ Ë i � BhCFE ��þ�' I(� � BhCtE R k ]¨þ�T ý I��!G+G+G)� � BDC�E R+*']AþcT� I íM ñ Ø�ù�Þq(LïäâcØ�ۦأ۳á?ð�Ö«ÚAWXY'[ ièE '�VXWZY\[·Ë I ;Äætï�Ø-ù�Öcß/ÜtÖ¬ÕuðxÛ³ñ ØZ×´Ø�Ù�Öe*rù�á?Û³ÖcÞ�Õ-á´âcð´Þ´Ù8ÖcÛ³ñr( Ö¬Ú1âcØ�ß/Û3á´Ú'á´ÕNØ98�Ø�Ùmù�Ö¬Ú³Øß;Þ´Ù Û³ñuØyÙ8Ø�á´ã Ø�Ù-;�� Þ´Ùyá-ãuØ�Û³á´Ö¬âcØ�ã�ã Ø�Ú�ù�Ù�ÖcïuÛ³ÖcÞ�ÕJÞ´ß3Û³ñuØ�, Í � } àÊx å á´Õuã�álïuÙ8Þ5Þ´ß�Þ´ß'Ú¦Ø-ù�í Ù�ÖcÛsÜ/=uÚ¦Ø�ØuÐ k+Ú ;-/.10)2436587:9<;)7:7:58=?>@58>BA�0)2�>)2DCEAGFH2DI�JLK&M/K?N?M/KOJ)0671587P58>6FQ=?M�RSK?TVUW;6>WA�58T)IP24N?2XAPA�=KY7:9D062<RZ2(I�0)589/0[587PK?9XA�;)K?T8T]\S7:2<9<;6M�2?^

_

Page 59: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

` ��ó-�������������

Ð k�Ú à ;�û£Ø�â¬â«á?Ù³Ø/=���;bazí Ø�Ù8Ö¬Õ�=�á´Õäãµò¨;P��Þ´ð�á-&ká�Ü/;c, Í � àÊx å Ú+.l~�Ø+& àpØ�Û³ñ Þ�ãuÚFß/Þ´Ù à½Ø�Ú }Ú³á?ð´Øjxyí Û³ñ Ø�Õ5Û³Ö)ù�á?Û³ÖcÞ�Õ Ô�Ú8Ö¬Õ ðÏ�'Ö¬ÕuÖcۦؽòZÚ¦Ø�íuã Þ´Ù8á´Õäã Þq( �uíuÕ7ù�Û³Ö¬Þ�ÕuÚ+; å Ù8Ü5ïuÛ¦Þedgf è ;�x�×?á´Ö¬â¬á20äâ¬Ø½á?Ûhji�i�kZlnmomLpop�pGqjrosutv�w1rosLxyv<t�xuwGm)w(sLt{z|s{m6}1~6hP~)zjmLkG�)k�t�zPsLm��o�{zL}1��ros�v�hji)}1� ;

ç

Page 60: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�@�����u���O�&�V�D�#�u�����L�O�o���u�G�� �c� �����u����� ���n�O�o�� G¡¢�&�V�{�£�n� � �&�o¤��n�u¥u�O�L¤|¦��§Y�&�u���W�&�n�u�©¨u�{���L�&¦��u��ª«�L�n¬

­¯®�°�±Z²e³Z® ´�´

µ ¶ ·Y¸|¸P¹Zº[·¼»¾½�¿(À�·Á¿1ÃOÄ�¹�¿1ÃOÅ@ÁÇƯūÈ�·Y¸É½�¸|·YÈÊÃ?ÁÌËÎÍ(¹ZÄ�¿(Ã&Ä�·

ÏSÐW¡��u���ZÑ��&�{ÒÓ�{ j�Ô¤��&Ð����V�{ ��������&¡� G�&�&�V�{�Õ�{�Î��Ö � ���&¦��L�#�V�����!×<ÐW�u�n��ØÙ�n�Ù¡¢�{���n�n�? �¡¢�Ô�Ù��Ö � Ñ+�u�ÚOÛoÜLÝ¢Þ Ò�ÐW�&�&�L¥uÐW�� ��O���j¥��c§Ï@ß�à � �{�G�n�n�O �¡¢�&����¥�����Ö � Ñ��u�£á ÜLâoã�ä¢Ý Ò�ÐW�&�&�L¥uÐW�����åÒ��u�&У���Væ#¡� ��V��àç��u�n�©�&��Ò�ÐuèPé�Ðå¥{�E�uÐ��{�jÐ�¡¢�{���n�n�O G¡¢�&�V�{� � �&¦jÐåê@ë©ÏSìD��Ö �í� �&¦��L�ÂéB�u���nÐW¡� j�&ÐåÑ+�u���L�&î|�V�n�O�L�O������ì�V�{�j¥#ÒïÐW�&�&�L¥uÐW��àSð!Ðå�&Ð����VÐ�éí�&¦G���[¡¢�{���&�n�O �¡¢�&�V�{�(èP�����&¡� G�&�S�V�&�@�nÐW¡� j�O�V�D�uèP�u�G�b�&¦jÐW��¥{�V�uÐå�&�{Ò�Ðñ�u�&¦jÐ����Ö � �[�&¦��L�[�L�&Ðñé[���jÐW�V�# ��nÐW�b���£¤G�O�u¡¢�&��¡¢Ðuàò£óïôïõ?öÉ÷ùøÔú©ð!ÐÕ�jÐW�&¡¢�O�Vî1Ðc�&¦������&¡?¦jÐWÒ�Ð%�û�üÒ��u�&ÐÙ¥uÐW�jÐ��?�u���V�/�ý�&¦��u�üé�ÐÙ�����ü�û�u�n���&��Ò�Ðuàþç�Ð��ÿ������������ �����������������������ïî1Ð@�����������/ì/§Ï@ß��o�&¦jЩ�nÐW���jÐ����u���#�&ÐW¡¢ÐW�V�uÐ��Bé[�����j�O¦��L�&Ð@���O�u�����{Ò� Ð��� "!#������� � àYß �%$ù�&�{Ò�Ð�¤G�L�O�uÒ�Ð��nÐ��'&)(+*,��é�Ы�nÐ�Ð�î1ÐW�V�Eéüé[¦�Ð��&Ð@�&¦G���B¤G�L�O�uÒ�Ð��nÐ��S¡¢�{ÒïÐW�S�����n�¤G���W�-�?à©ç�Ð��©�&¦jÐ��j�u�&�L�&�V�{�+.�/10[�jÐW�j�u�nÐ��&¦jÐ2�3&546���/ì�îG�V�Â�&Ð�¤��OÐW�nÐW���&�L�&���{���uÑy�����nÐ�¥uÐ��)/Z����îG�����L�O�uà874��u j�&¦jÐW���&��¡��L�nÐbÒ�ÐW�&�&�L¥uÐ:9¾è�¤G�L�O�nÐ29 �u���c�nÐ�;o jÐW��¡¢Ð£�uÑSîG�V��¡ � �<9+=��?>?>?>@��9BAÂÐW�u¡O¦C��*D4+&��/ì�îG���&��V�{�j¥jà � ¦����{�nЫ�ï�O�u���j�{Ò �L�u�� jÐ)EF!#������� �5G = �u���b¡¢�{Òï¤G j�nÐH�

IKJ@LNM ÿPO@�Q�8RSET�VU ÿWO��K��R).K��0XRY9+=��VUeÿPO@�K��R).[Z\0]RY9_^��VUa`?`?`UeÿPO@�K��R).��b0]RY9cAd��>7S¦jЫ¡¢�{Ò�¤G��Ð��nЫ�&�L¥#���e.�E� IfJ�L 0)���&¦jЫ�OÐW¡¢ÐW�V�uÐ��@�jÐ�ÐW�G�YE����£�u�O�jÐ��S�n�Ô�uÐ��O�VÑ��g�?à

ð!Ð@Òï�E�ï�j�u�nÐ[��Ñ�Ð�é �û���nÐ��&ÐW�&�&���j¥�¤1�{�����&���LîP�{ ��Y�&¦������&¡?¦jÐWÒ�ÐuàYß �V�O�&��è��V���� �?�u���j�{Òï�V¬�ÐW�]��é�Ð[�&�Wé���u�n�©�&��Ò�Ð�¦j�Eé �&¦�Ðå�jÐ��nÐ��?Òï�������&�&��¡«�uÐ��O�O�V�{�c�uÑY�&¦����©�&¡?¦jÐWÒ�Ð���� â|ÜuÞ �nÐW¡� j�&Ðuà���ÐW¡¢�{���(è1�&¦jÐ�Ò�ÐW�&�&�L¥uÐ9 ���Â�u�&�& �Ò�ÐW���n��¦��W�uÐ��VÐW��¥u�&¦�é@¦���¡O¦����Â�#Òå ��V�&�V¤|�VЫ�uÑ8��*h4B&��?ài7[¦����S�OÐW�n�n�O��¡¢�&�V�{�c���Â�j�u�Â�&ÐW�u���V��&¦��L�ï�nÐ��uÐ��&Ðuè��&����¡¢Ðù�&¦jÐ��&Ð��L�&Ð��nÐW¡� j�OÐ = éB�W�����n��¤G�u� ��Ò�ÐW�&�&�L¥uÐb�n���&¦��L�ï���&�å��ÐW�j¥u�&¦ÕîPÐW¡¢�{ÒïÐW���Ò� ����&�V¤G�VÐÂ�uÑ'��*j4k&�����¦j�Eé�Ð��uÐ��WèG�O �¡O¦�¤|�u�������j¥�Òï�E�ù�VÐW�u�£�n�#�&���V¥{¦��Z�V�{�&�B�uÑ�Ð�æÔ¡��VÐW��¡¢�l� �&����¡¢Ð�Ò��u�OС¢�{Ò�¤G j�&�L�&�V�{�G���uÑ(ÿ �L�&ÐS�&Ð�;o ��V�OÐW���n�å��Ö � �«�V�{�j¥uÐ��ÒïÐW�&�&�L¥uÐ@�?àYß �����u�����uèu�&¦jÐ[Òï�$j��Òå �Ò Ò�ÐW�&�&�L¥uТì�VÐW�j¥u�&¦��O j¤�¤1�u�&�nÐW�£îo�ù�&¦G���[�&¡?¦jÐWÒ�Ðå���m��*n4#&��P`o�[Z�pfG = 46���BîG�V�&�N� �&����¡¢Ðñ�&¦jÐñ¡¢�{ ����nÐ��<.�/K0Z����¡��û ��jÐW�é[�V�&¦£ÐW�u¡O¦�îG�V��¡ � �&¦j�{ G���£�j�u�Ô×n¡¢��¡��VÐWØT�?à

ð!ÐÂ�G���ï�j�u��Ò�ÐW���&���{�Ô���u�n�y�&��Ò�ÐS�&¦�Ð[Ðq$j�u¡¢���nÐW¡� ��O�V�/���&ÐW�O ��V�YÑ��u��&¦��û��&¡?¦jÐWÒ�Ðuè��n�ñé�Ð@�j���n��¦jÐ��OÐuàr"sutovxw@t�y z|{ ÜuÝ8}Lâ-~<}\���Lä�ÝOÚ?}LÝ�~<}uÞ:Þd}T�q�@��â�ã�Þ�Û�ä'�����i�[���F��Úq��Û�äq�Ôä@Ý��oâGâ-�+âoãm��â£ÞQ�3�Ôä"��ÝnÜ��{ãLÛ á ~��� }Lâx�£Ý&äb���Gä Ú ÞQ��â�ã2}uÞS�ïÜ)Ú¢Þ���Þd}WãuÚ��?ÝnÜ���� Þ�Úi���F� ÜLÝK}T� á ä��ZÞ�Û�ä5�|ÝnÜT�b}o�q� á � ÞQ~bÜK�«Ú��g���?ä¢ÚOÚ3��� á+á ~i�¢ÜLÝ:ãH��â�ã}Ôâ1äq�W����} á �[���Ôä ÚOÚ?}Wã�äK�|Þd}Wã<��}���Ým�+Úe}{Þ��ïÜ)Ú¢Þ Z � ^ `�ZoG-� �¡ZoG-�)� ��¢ÏS�{ j¥{¦����ï�n¤1ÐW� � ����¥jè��&¦jÐ8£��O�n�y�nÐ��OÒ �û�ï�&¦jЩ�Lî1�E�uÐÂîP�{ G���ï¡¢�{Ò�ÐW�BÑ��&�{Ò �&¦�Ð@¤��O�uîG�LîG�������/�å�uÑ���×n¡¢�{���û�Hì�&�V�{��Ø����£�&¦�Ы�O�u���j�{Ò �L�u�� jÐ)E:���&ÐW¡��u�û�(�&¦jÐ�×<îG���&�&¦����E�Ô¤��&�uî|�VÐWÒïØåÑ+�O�{Ò ¤��&Ð����V�{ ��Z�VÐW¡¢�& ��&ÐW�S�u�����&¦�Ð�j�u�nÐW�S�{��¤G�&�uîG�LîG���û�V�/�-�����&¦�Ðñ�nÐW¡¢�{�����nÐ��?Òþ¡¢�{Ò�ÐW�SÑ��&�{Ò �&¦�Ð�Ñ��u¡¢�S�&¦G�L� IKJ@L ���'¤�îG���&�S�V�{�j¥jè��u���£�&¦�Ð¥d¦P§�¨d©u¨dª�«�¨]¬«�­@­@®°¯±�²]®°¨dªT³�´d«fµ?³«�¶°¶T·K©1¸d§�´X®°´�¹Hº�»¼´[©K½K¾�¸d©�³�¿%§�¸]¨dª@©P¿%§�¶À¶°§f²]®°¯±'¸d©f«�´[§�¯TÁ´d«Kµ5Ã_ÄeÅ]Æ2Ç�È@ÉÂʼª©K¯5¨dª©

Ë ÌXÍ §�¿]ÎDÆ+ÏЫ�¯­NÎ2ÑÂÆ ÏfÒ5²¼§�¾@¶À­NÓH©'®À­@©K¯?¨d®°½f«�¶XÔÕ´[®°¯@½K©'¨dª©1µm«b¸d©YÓH§�¨dª"¬�«�­@­@©f­e§�¾�¨�¨d§"ÏKÒ�Ö[×�ØÙ³�«�¯�­e¨dª©K¯m¨dª©¸d©1½K©K®°Úq©1¸X½f«�¯¯§�¨X¾¯�«�ÛSÓ@®À±�¾§�¾´[¶ÜµÐ¨d©K¶°¶T²]ª@®À½Ýª ÛY©K´[´d«�±�©W²¼«�´]®°¯�¨d©1¯�­@©K­TÉ

Page 61: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�u�j�uÐ��O�&�L�&�Ù¡��u�Õ�u�VéB�W���ñ¥{ �ÐW�&����¡¢�u�&�&ÐW¡¢���&�L¥�é[���&¦Ù¤��&�uî|�LîG�����V�D�kZ G-� �Y�u���Ù�&¦jÐ�£G���u���nÐ��?Ò ¡¢�{Ò�ÐW�Ñ+�&�{Òþ�&¦jÐñ�nÐW¡� j�?�V�/�ù�uÑ4�&¦�Ðñ§Ï@ßyà

ð!Ð��j�Ù���u�ï¥{�V�uÐb�!¤��O���uÑ©¦jÐ��&Ðuà ßj�u�ùÒ��u�&Ðb�jÐ��&�u�û�S�Lî1�{ j���&¦jÐb�&¡?¦jÐWÒ�Ð��u���Î�!Ñ� G����¤��&�o�uÑ[�uÑ�nÐW¡� j�O�V�D�uèG�nÐ�Ð�Þ%�?ß�àø"à#ø«õ?öÉ÷ùøÔú�Ö¾é@���jÐW�V��ì/ ��nÐW����Ö � ���©îG�u�nÐW�Ù�{�!�&¦jÐ �'áS� Ò����jÐ��uÑ�ÐW��¡¢�&�o¤��&�V�{���&¦��L�ñéyÐï���û�<ì¡� ��&�nÐW��¤��&Ð����V�{ ��&�V�uàSâ[�6éyÐ��uÐ��Eè��j�u�nÐñ�&¦��L�@�&¦����S¡¢�{�G�jÐW¡¢�&�V�{�c���ZÐW���&�V�&ÐW�V�£Ñ��u�&�& ��V�n�{ �� � �&¦jÐ��OÐñ���[���u��&ÐW�u�n�{�(èZ��� ¥uÐW�jÐ��?�u�:èB�n�%�u�&�& �ÒïÐb�&¦��L�Ô�!¥u�o��� Ò����jÐb�uÑÂÐW��¡¢�&�o¤��&�V�{�eé[�û����¥{�V�uÐb�O���&Ð��n�%�Ù�nÐW¡� j�OÐ��Ö � � �u���Ù����¡¢Ð��uÐ��O�&�T�?à � �ÙÑ��u¡¢��è��&¦jÐ �'áS� ìD��Ö � ���ÕãPÐ��O���O���V¥{¦��&�V�ä� �u���%¦��u�ñ���ÕãPÐ��&ÐW�����nÐW¡� j�O���/�¤��&�u¤1Ð��&�&�VÐW���Ñ��&�{Ò �&¦jÐ �'áS� Ò����jÐ��uÑ4ÐW�G¡¢�&��¤G�&�V�{�(à

Ö@�O�& �Ò�Ðñÿå�-������� � �B������� � �æ������� � �û�[�����������/ì/§yÏ[ßç� ���u�nÐñ�&¦��L�S�&¦�Ð����j¤| j�Z�u���b�{ j�n¤| j��VÐW�j¥u�&¦��B�L�&Ð��j�6é¯�u�O�& �Ò�ÐW�#�n��îPÐÂ�&¦jÐ��&�uÒ�Ðuè�Ñ��u�Z¡¢�{���uÐW����ÐW��¡¢Ð©�{���V�-�?à�ð!ЩÒÔ�W�#��Ð?£G�jÐ �'áS� ìD��Ö ��u��Ñ+�{�û�V�Eé[���Â�&¦�Ðï�nÐW���jÐ��«�u�����OÐW¡¢ÐW�V�uÐ��ñ�&¦��L�OÐï�£�O�u���j�{Ò � Ð��B �!6������� � à�ç�Ð��)* î1Ð��&�{Ò�ÐF£g$�ÐW�¤G�L�O�uÒ�Ð��nÐ������&¦jÐÂ�u j�&¦jÐW���&��¡��L�&�V�{�£�&¡?¦jÐWÒ�Ð[é@����� ÜLâ á ~ î1Ð[�jÐ?£|�jÐW�ïÑ��u�yÒ�ÐW�O�&�L¥uÐW���uÑ��VÐW�j¥u�&¦è¤"`1*é� ��à]ÐuàVè* îG�V��¡ � ��è�ÐW�u¡O¦ä¤ îG�V�&�ñ�V�{�j¥o�?à27��c�u j�&¦jÐW���&��¡��L�nÐ#��Ò�ÐW�&�&�L¥uÐ�ê Mìë =?�?>?>?>�� ë � è �&¦jÐÔ�nÐW�G�jÐ����nÐ��&�í\î M � � �u�G�¡��Ñ��u� / M ���n��*��[�nÐ��&� íHï M ÿPO@� ë ï U íHï G =��?à)7S¦jÐå�&�L¥����Â�&��Ò�¤G��� í � àñ�o¤1ÐW¡��Õ£|¡��L�&�V�{��uÑ��&¦�Ð��uÐ��O�Õ£G¡��L�&�V�{���u�V¥u�u�O�V�&¦�Ò ���Z�VÐ�Ñ��Z�n�ï�&¦jЫ�&ÐW�u��Ð��Wà

7S¦����Â�&¡?¦jÐWÒ�Ð � ���c¡¢�{���n�O�u�&���n���&¦jÐ �'áS� Òï�o�jÐå�uÑÐW�G¡¢�&��¤G�&�V�{� � �û�@�jÐ��nÐ��?Òï�������&�&��¡��(�����&¦�С¢�{���nÐq$����uÑBÒ�ÐW�&�O�L¥uÐ#�u j�&¦jÐW���&��¡��L�&�V�{�Ù�&¦������j�oÐW�«�j�u�ñ��ÐW¡¢ÐW�&�&�L�O������¤��&ÐW�&ÐW���ñ�£¤G�&�uîG�VÐWÒ�à�ð@�u�nÐÔ�u���&��&¦��L��èG���£¡¢�{���n�O�u�n�Z�n���&¦jÐ�ê@ë©ÏSìD��Ö � è��&¦����B¡¢�{���n�n�? �¡¢�&�V�{�£é��u� � �BÑ��u��ñuò äb� Ò�ÐW�&�O�L¥uÐñ�VÐW�j¥u�&¦����{�����uà� �ýÑ �u¡¢��è@�V�£���ù¡¢�{Ò�¤G�VÐ��nÐW�V� �����nÐW¡� j�OÐ�é[¦jÐW� �L�L�O���LîG�VТì/��ÐW�j¥u�&¦ÎÒ�ÐW�O�&�L¥uÐW���L�&Ð! ��nÐW�(àÉÖÂ�ù�Õ�&��Òï¤G�VÐÐq$��uÒ�¤|�VÐ[�uÑ��u�#�L�n�&�u¡ � è��&�W�ï�&¦�Ð@�u�j�uÐ��O�O�L�&���&Ð�;o jÐW�n�&����&�L¥å�{�ÔÒ�ÐW�&�O�L¥uÐ ë =5!#��������� � �OÐW¡¢ÐW�V�o�û�j¥� � �u��� �&¦�ÐW� �&Ð�;o jÐW�n�&�#�!�&�L¥ �{� Ò�ÐW�&�&�L¥uÐ�� � �&ÐW¡¢ÐW�V�����j¥c�Ùó�àað@�u�nÐ��&¦G�L�Ô�&¦��L���ÙóZ���ï�%�)�u�û����&�L¥bÑ+�u�©�&¦jÐïÒïÐW�&�&�L¥uÐ ë =?�b���¡��éyÐï�VÐW�E�uÐ��uÐ��O�Õ£G¡��L�&���{���uÑ�&¦����ÂÑ �u¡¢���n���&¦jÐ��&ÐW�u�jÐ����?àN7S¦jÐ�Ñ��{���V�Eé@���j¥�&¦jÐ��u�&ÐWÒ �&¦jÐ��&Ð�Ñ��u�&Ð��&Ð�Ñ�Ð��O�Â�{���V���n�£�&¦jÐ�¡��u�nÐ�é[¦jÐ��OÐ"£g$�ÐW��ì/�VÐW��¥u�&¦�Ò�ÐW�O�&�L¥uÐW�è��* îG�V��¡ � �Â���{�j¥o�S�L�OÐ�u j�&¦jÐW���&��¡��L�nÐW�(àr"sutovxw@t�yõôö{ ÜuÝN}Lâ-~�}\���uä¢Ý?Úq}LÝ�~�}{Þ:Þd}\�q��+â�ã£Þ�Ûjä��]÷"�P�d�����u� Ý���â�â-��â�ã���âcÞQ���Ôä���ÝnÜ��{ãLÛ á ~�� � }Lâx�Ýnä��?�Gä Ú¢ÞQ�+â�ã�}uÞW�ïÜ)Ú¢Þ]�£Þd}WãLÚP�?ÝnÜ��ø� Þ�ÚY���F� ÜuÝK}\� á äq�Þ�Û�ä'�|Ý<Üo�b}T�?� á ��ÞQ~ùÜK�[Ú��-���?ä¢ÚOÚ���� á�á ~'��ÜLÝ ã\�+âoã�}�â1äq�W��} á �Q���Ôä¢ÚOÚq}Eã�ä1�PÞd}Wã"��}���Ým�+Úm}uÞS�ïÜ)Ú Þiùfú � ú^fû�ü ¥ �+ZoG-�)�+��¢ÏS�{ j¥{¦����å�n¤1ÐW� � ���j¥jè{�&¦�Ð5£��O�&���nÐ��OÒ¾¡¢�u�&�OÐW�n¤1�{�������n�ñ���n�u�&�Y�uÑ(¡¢�{���û���&�V�{�:��é�Ð@�j���j�u������O¡� ��&���jÐ��&�u���û�¦jÐ��&Ð@���j�&¦jЫ�nÐW¡¢�{�G�£�nÐ��OÒ ¡¢�{Ò�ÐW�ZÑ��&�{Ò �&¦jЩÑ��u¡¢�Z�&¦G�L�Z�&¦jÐ��u�j�uÐ��O�&�L�&�£¡��u���u�VéB�W����×<¥{ jÐW�&�&Ø��u��¤1ì�î|�V��&�L¥�¡¢�u�O�&ÐW¡¢�&�V�ïé[���&¦Ô¤��O�uîG�LîG�������/�FZ G-� �j�u���#�&¦jÐ@�&¦G�V�O�ï�nÐ��?Ò ¡¢�{ÒïÐW�yÑ��&�{Ò �&¦jЩ�nÐW¡� j�O���/���uÑ��&¦jЩ§Ï@ß�àÖ[¥{�u����è�éyÐ��j���j�u�å¤��&�E�����jУ�jÐ��&�u���û��¦jÐ��OÐuè�îG j�ñ�OÐ�Ñ+Ð����&¦jУ�&ÐW�u�jÐ����n�c�&¦jУéyÐW�û�Hì�éS�O�V�n�nÐW�%¤G�L¤1Ð���ÞÜZ�ßé[¦���¡?¦��jÐW�&¡¢�O�Vî1ÐW�B�&¦jÐ �'áS� ìD��Ö � �u���£¥{�V�uÐW�@��Ñ� G���P¤��&�o�uÑDàý�þ-ÿ s õ�þ�����õ?öÉ÷ùø#ú 7S¦����y�&¡O¦�ÐWÒ�ÐÂ�û��j�u�� ��nÐW�ù���ï¤��?�u¡¢�&��¡¢Ðuè�î| j��L�L�O���u���&�i� ��à]ÐuàVè��©��Ö � �Y�L�OÐuà á  ���&¦jÐÔ�&ÐW�u�Y�&ÐW�u�n�{� Ñ+�u��¤��&ÐW�nÐW���&���j¥��&¦��û�«�&¡O¦�ÐWÒ�Ð#���«�n�������n�&���� �¡¢Ð��&¦jÐÔ���u�&�V�{�Ù�uÑ �OÜ á+á �ûÚ�� Üuâ-�DÝ&ä Ú��ûÚ¢Þd}LâGÞÛ })ÚOÛi����âx�EÞQ� ÜuâjÚ è��« ��nÐ�Ñ  ���¡¢�O��¤��n�u¥u�?�L¤G¦���¡Z¤G�O��Òï�V�&���uÐ��&¦G�L�é[���û�j¡¢�{Ò�Ð@ j¤#�L¥{�u���#�û�L�nÐ��y�����&¦�Ð@¡¢�{ j�O�&Ðuà

Ö@�O�& �Ò�Ð��@¦��u�O¦ñÑ  ���¡¢�&�V�{���D�-���������8�������������&¦��L��¡¢�{Òï¤��&ÐW�&�nÐW���L�&îG���n�O�L�&��ì/�VÐW�j¥u�&¦ñ�û�j¤G j�&���n��u��¤1ì�îG�V��{ ��n¤G j��à�ð!Ы�&�E�Ô�&¦��L� ë � ë óG�OÐ�¤��&ÐW�nÐW���&�B� �?Ü á�á �+Ú��:ÜLâ<�¢ÜuÝ � �VÑ ë�Maë ó�î| j��¡� ë � M �ä� ë ó �?à� �jÑ��u�OÒï�u�û�V�uè�� ��� �OÜ á�á �+Ú��:ÜLâ-�XÝ&ä Ú��+Ú Þd}uâGÞ ��Ño�V�����S×n����Ñ+ÐW�u�&��îG�VÐWØB�n�i£G���«�@¡¢�{�������&���{�©Ñ+�u���c�{Ò��u�OÐÑ��u�OÒï�u���V�Â�� t� ������� v � z ���?â���ÜLÝ���} á � � �ûÚ ��������� �f�OÜ á�á �+Ú��:ÜLâ�Ý&ä Ú��+Ú Þd}uâGÞ�� ����ÜLÝm} á�á�� Ý���â�â-��â�ã���âcÞQ���Ôä"}{Þ �ïÜ)Ú Þ� ���yä«Û }H�Lä��

§Y�@ÞÕ� ë � ë ó �����ç� ë��M ë ó�� �ä� ë � M �¡� ë ó �Ýßu(6��>

Z

Page 62: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ð!Щ���u�nÐ@�&¦��L���&¦jÐÂ�LîP�6�uÐ��jÐ?£G���V�&���{��éy�{ G���Ô�jÐ�ÐW�#�n�åîPÐÂ�u���L¤��nÐW�#�n��¥{���uÐÂ���O�V¥u�u�&�{ G��è�¡¢�{Ò�¤|�VÐq$����/��ì�&¦jÐ��u�&Ð��&��¡Z�jÐ?£|���V�&�V�{�ñ�uÑ|¡¢�{�����û�&�V�{��ì��&ÐW�O���n�&�u��¡¢Ðuè)îG j��éyÐZ�j�©�j�u� ¥{�V�uÐZ�& �¡?¦��©�jÐ?£G���V�&���{�ñ�����&¦���� ¡¢�{ j�O�&Ðuà

� �{�����û�&�V�{��ì��&ÐW�O���n�&�u��� ¦G�u�&¦ïÑ  ���¡¢�&�V�{�G��L�&Ð@�uÐ��&�ï ��nÐ�Ñ  ��:è��u�G�Ô¦��E�uÐÂÒï�u���ï�L¤�¤G�û��¡��L�&�V�{����à � ���nÐ��&ÐW�n�<ì���j¥{�V�uè�¡¢�{�û�����&�V�{�jì��&ÐW�&���n�&�u����¦��u�&¦�Ñ  ���¡¢�&���{���L�&ÐS�&¦jÐУG�O�n��¤��O�ûÒï�V�&�V�uÐBé�Ð[¦��W�uÐÂ�nÐ�ÐW�Ô�&�«Ñ��L��&¦��L� ��}Lâ�â|ÜuÞîPÐ�¡¢�{���&�n�O �¡¢�nÐW�£Ñ��&�{Ò �u���L�&îG�V�n�O�L�O�ï�{�jТì�éB�W�£¤1Ð��OÒ� ��&�L�&�V�{�l���&¦����B�n�&�L�nÐWÒ�ÐW���[���B�&����¥{¦��&�V�ï�û�jÑ+�u�?Òï�u�[��u� � Ò�Ð���Ñ��u�{ c�L�&Ðå�����nÐ��OÐW�n�nÐW���û���&¦jÐñÐq$j�u¡¢���n�&�L�nÐWÒ�ÐW���@�uÑ��&¦G���S�&ÐW�O ��V���?àZÖ@�û�(�u�&¦jÐ��¤��O��Òï���&�V�uÐW�Zé�Ц��W�uÐù�nÐ�ÐW�!�&¦o ���Ñ �L� � ÐW��¡¢�O��¤��&���{�(è�§Ï��©�Wè4§Ï@ß ��è4§yÏ[§�Wè�Ò�ÐW�&�&�L¥uÐ#�u j�&¦jÐW���&��¡��L�&�V�{� � ¡��u�Ùî1С¢�{���n�n�O �¡¢�nÐW�cîG�u�&ÐW���{���u���b�{�jТì�éB�W��Ñ� ��G¡¢�&�V�{�(à� YÐ���¡¢�{���û���&�V�{��ì��OÐW�&���n�&�u��¡¢Ð��nÐ�ÐWÒï�Â�n�£îPÐå�#�&�n�O��¡¢�&�V��n�n�&�{�j¥uÐ��Â�u�&�& �Ò�¤G�&�V�{�£�&¦��u���{�jТì�éB�W���jÐW�&�Wà

ë©���&¦jÐñ�u�&¦jÐ��©¦��u����è|¡¢�{�����û�&�V�{��ì��&ÐW�O���n�&�u���S¦��u�&¦bÑ  ���¡¢�&���{��� ��}Lâ î1Ðñ¡¢�{���n�n�? �¡¢�nÐW��î|�u�nÐW���{� Úd��äq��?� ñ � �u�&�& �Òï¤��&�V�{���Â�& �¡?¦��u��Ï©��Öñè1¦��L�O�G�jÐW�&�S�uÑyÑ��u¡¢�n�u�?���j¥jè(�u���c¦��L�O���jÐW�O�[�uÑ¡¢�{Òï¤G j�&���j¥£�����O¡¢�&Ð��nÐ�V�u¥{�L�O�V�&¦�ÒÔ��àYßj�&�{Òþ��¤G�O�u¡¢�&��¡��u�1¤1�{�����y�uÑ��o��Ð�é«è��&¦jÐ��&Ы�L�&ЫÒï�u���ÔÐ�æ#¡��VÐW���S¡¢�{�G�n�n�O �¡¢�&�V�{�G�B�uÑ'��é[¦G�L��L�&ÐZî1ÐW���VÐ��uÐW�å�n��î1Ð@��¡¢�{�û�����&�V�{�jì��&ÐW�&���n�&�u��� ¦��u�&¦åÑ� G��¡¢�&�V�{���?���&¦jÐZÒ��{�n��é�ÐW���Hì � �j�Eé[�å�uÑP�&¦jÐW�nÐZ�L�&Ð@��âÂÖ[ì���u���!�"! � à"7[¦jÐ��&�V�& ��L�&�V�{�c¦jÐ��&Ð����©�u���u���u¥u�{ ��©�n���&¦��L�©�uÑy§yÏ[ß �?�ÂéyÐ � �j�Eé ¦j�6é �n�b¡¢�{���n�n�O G¡¢�§Ï[ß ��Ñ��&�{ÒÓ�u��� �{�jТì�éB�W� Ñ� G��¡¢�&�V�{� îG j�����%¤G�O�u¡¢�&��¡¢Ð�é�У G�nÐ#î|�V�o¡ � ¡��V¤G¦�Ð��O����� � Ð#!%$B�%�u�ïÖ&$B�é[¦�'�¡?¦£éyЫî1ÐW�('�Ð*)uЫÒï� � Ы¥u�o�o�b§yÏ[ß ��à

+ ·�,W·YÍ(·Á�Ä�·¸

Þ%��ß��%à á ÐW���û�.-&Ðuè Ïñà/�� jÐ*-0'(1(èÙ�21G� §4à�Ï43u¥{�WéB��5uàcê@ë©Ï ��Ö �76 � ð[Ð�é ��Ð*8&¦93�� 6 Ñ�32-í�bÐ 6 ì6 �L¥uÐ�Ö 98&¦jÐ�1:8;'�¡��.8;'<3�1>= 6 '(1j¥!ß?'(1�'<8nÐb§ 6 ÐW ��932-0@21G�93{Ò ß� �1�¡A8;'(3�1 6 à � -05�¤�8B3DCFE � àyÖ4).@2'(G(@Lî�G�Ð/@.8H9I:I�JLKBMNM.ONO:O�P9QNR2SUTWVXQNR.YZT[S:Y2V�M\V]R.S�^_R�M�`Xa�Hba\^9M.J�c\J�S�^bR.M�dNe�^.`XcfQNR�TWH9I\`Xg à

ÞÜZ@ßih%à á Ð�G(G(@.-&Ðuè ¨jà ªj'(G('(@21�èk@21�l §�à Ï43u¥�@Wé7@�5uà�7S¦�ÐnmoÐW¡*o9-0'(8p5q3uÑr8&¦jÐts7'(uG¦jÐ*- á G<3fv �sB¦�@2'w1�'(1j¥xh�Ð 6;6 @L¥uÐ Ö%o98&¦jÐ�1:8;'(v*@.8;'<3�1ys3Nl�Ðuà>z Ü���Ý?âÂ} á ÜK�D��Ü��8�x�jÞ/ä�Ý }Lâx�x{-~)Ú Þ/ä?� {x���3�ä¢âÂ�?ä Ú è}|��\��~\��� ~2|HZ��N~2E2E��[Z��H�H�T�?à��-OÐ�G('�Ò�'(1�@.-;5�)uÐ*- 6 '<3�1�'w1�s7-;5:u�8B3�CFE\�jà%Ö4).@2'(G(@Lî_GVÐ[email protected]:I�JLKBMNM.ONO:O�P9QNR2SUTWVXQNR.YZT[S:Y2V�M\V]R.S�^_R�M�`Xa�Hba\^9M.J�c\J�S�^bR.M9Q��bQ?TWHfI.`bg à

~

Page 63: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�&���(�2�*�0�;�<�����2���"�.�0�N�(�2���� ��� ����2�� ¢¡ ���B�0�N��£�¤A�;�<���#�B� � �;�N¥��B�2¦2�0�.¥_§��¨��;�2�����;�B�2�ª©2�����.�;§��2�"«j�.�B¬

­¯®±°³²L´DµL® ¶/·

¸ ¹»º%¼0¼¾½;¿X½;º&ÀÂÁ:ÃÅÄU¿b½0¿�ÆXÇ�À%ÆÉÈrÇL¿XÊtËi̱ÀiÍ�Æ]½;º%Ài¿

ÎÏ�#¤A���:�;�(�N£9�ÑÐ&�<�;§Ò��£9�Ó���(�;¤*£��;�;�<���Ô���;��ÕÖ�(�2�B�Â�;�wÕ±�Ñ���×�;§9�ÙØ%�2�;§fÚp�2���fÚ��"Û � ¥��.�0�2���<¦�ÕÜ���2�Ó�2£9Ú�;§9�����;�w¤*�.�;�<�����2���.�;Ý��<�B�0�.�¾�(�<�:Úp�<���9¦±Õ±���;�;�.¦2����Þ&ß���¤*�2�w�à�;§9�Ó�9�*á����(�;�<���â�2���Ù¤A���(�(�(�;�(���fÚW�;���;�(�;�;�2����§��2�0§��£���¤A�;�<���]ã±äå�*�Óæçã�è�éfê�ë\ì�íâî è�éfê�ë\ì�ï�Ýb���"�ð£���¤A�;�<���×�;�.ñN�w�9¦ò�.�;Ý_�<�B�0�.�;�:Úp�<���9¦2�;§��(�9¥�£9�;�Â�2���×�0�AÚ�;£9�0���(��¦ÙóXÚWÝ_�<�³��£9�B¥�£9�*Þ�ÛÉ¥_�2�<��ô�õàêBõ_ö<÷³�(�j�×ø¾ù.ú�úüû�ý¾û�ù.þDû�þÏæn�<�7æÿô�õ]÷�� æô�õ�öw÷%Ý_£9�³õ���Éõ�ö�ÞÑÎÏ��;���#�;§��.��æÖ�(�jô�� ê��¾÷pÚp¤A���(�(�w�;�<���Ù�;���;�(�;�;�2�����<��f���2�&�2�(�]�2�<¦2�2�0�<�;§�Õ��� �0£������(�9¦Ó�(�#�;�(Õ±�³�.�4Õ����B��� �9�;§��¥��;�2Ý��.Ý��w�(�<�p�Ñ�;§��.�� Ð4�(�(�_��£9�B¥�£�����¤A���(�(�(�0�<���#�(���<���;�7�;§��2���*Þ

ÎÏ���9�2�B�����(�2�B�³�;�(Õ��±�;§��.�¤A���(�(�w�;�<���fÚW�;���0�(�B�;�2�:�ª§��2�;§Ô��£���¤A�;�<�����³¤*�2�����2�jÝX��¤A�����B�B�0£�¤A�B���×���0��Õ�.�;Ý��<�B�0�.�0�/���9�AÚWÐ7������£���¤A�;�<�����&�2�%¥X�*�0ÕÓ£9�;�.�;�<�����*Þ³Ø4��ÐZ�*�2�*���à�;§9�*�ò¤*�2�ÔÝX�i¤A�����B�B�0£�¤A�B����Ý��2�;���ò���ý����0ø*û �Zø³�2�;�;£�Õ±¥��;�<�����Z�(��¤*�(£����(�9¦ª�;§9�ª§��.�0�������;���2�å�ð�2¤A�B�2�0�(��¦��N�;§��%ߪ�fÛ¯¥��;�2Ý��(��Õ����2�7�;§9�ª§��.�0�������;��2��¤A��Õ±¥_£9�;�(�9¦i���(�;¤A�0�*�B�ª�(�2¦��.�0�<�;§�Õ���Þ?Û%�(�B���9�;§��*�;�³�.�;�j�B��Õ±�³¥��¾�2¤A�;�(¤*�2�]¤A�����B�B�0£�¤A�;�<������2���ð£���¤A�;�(�����Ð4§��(¤¾§â�B�*��Õk�B��ÝX�j¤A���(�w�(�;�<���Ù�;���;�w�B�;�2������Ýb���;�[ÚWñN�9��Ð4�â�2Õ±���9¦Ñ�;§9���B�Â�.�;�i�fØ&Û4Ú0ëj�2���ò��� � Þ

¡ ���(�å�(�:�B�*�;���B�;�w�9¦��B�4���9�2Õ��(�����;§9�UÕ��w���(ÕÓ£�Õ¯��£9�B¥�£9�å�(���9¦2�;§jóÑ����¤A���;�;�.�;�ª���2���4§��2�;§j�ð£���¤A�;�<���j�B�Ýb�Ñô�� ê��¾÷pÚp¤A���w�(�(�;�<���#�0���;�(�B�;�2�:�*Þ���§9�³�����(�<��Ð4�(�9¦i�2���2�<�f�;�(�7Ð&�(�(�bÝX�ª�w�9���2�¾Õ��2���fÝ�£9��¤*�2�â���2�0�(�<�ÙÝb�³Õ��2���Õ±�2�;�#���f�2¤A�*Þ �9�2�"!.þ$#�§��2�;§ÿ��£���¤A�;�<���ÿæ ãè�éfê�ë\ì í î è�éfê�ë\ì ï Ð�â¤*�2�D¤A�����;�(�9�*�Ó�;§9�#�����(�<��Ð&�(�9¦�2�<¦2�2�0�<�;§�Õk�0£������w�9¦��(�#�;�wÕ±�³�;��£9¦�§��(����%N£��2�X�B�&� ã¨U�(¤0ñÙ�¾�2���9��Õ õò�2���/¤A��Õ±¥�£9�B�(')�¯æô�õX÷�9�2��*+� ë³�B�,� ã¨Z�(¤;ñÙ�0�2������Õrõ.-?���0/X�*�;���:�L���;��Õkõ¡ �1')�¯æô�õ.-W÷�;§9���/��£9�B¥�£9�Âô�õ�êBõ.-p÷�2���/�B�B�2¥

24�2�B�Ï�;§��.�#�<�³�;§��w���2�<¦2�2�¾�<�;§�Õ ��£9�B¥�£9�;�#�2�:�:�;§��w�9¦����;§9���»�<�Ù��£9�B¥�£9�;�#�ÿ¤A���(�(�(�;�(���]Þ}ί§��.�â�(�Ñ�;§��¥��;�2Ý��.Ý��w�(�<�p�±�;§��.�L�;§��(�7�2�<¦2�2�0�<�;§�ÕÉá_�����7�i¤A���(�(�(�;�(���43ÙÎÏ���(���9�<�à�;§9�³��£9�B¥�£9��2�?æx�(�7�B¥��0���2�#�;��£9¦�§��<�£����<���2�0Õ��<�#�(�Òè�éfê�ë\ì ï �_�;§9����¥��w¤;ñf�(�9¦Ñ���0�2���9��Õ õ.-Z�2���ò¤A��Õ�¥�£9�;�(�9¦5'6-7� æô�õ.-W÷��w�4�;��£�¦�§��<�â�.ñN�w��B�j¥��(¤0ñN�(��¦%�¾�2���9��Õ8'6-79Ôè�éfê�ë\ì�ïXÞ+�9�2�Z�³¥_�.�;�;�(¤*£��(�.�7*:�:�;§9���;���;§���¥��;�2Ý��.Ý_�(�(�<���³�;§��.�<'6-=�>'Ó�w���.ÝX��£9�?A@ ï Þ�Û%���i�;§9�*�;�*���2�;�6���0��£9¦�§��<�Ó�B¥X���.ñN�(��¦����;§9��¥��0�2Ý��.Ý��(�(�(�p�j�;§��.�Z�.�U�<���2�B�4ù.þB�&�2�C'ED*ê�F�F�F�êG'6Hå�w����%:£��2��B�I'Ï�(�(J�ô��:K ? ï ÷¾ÞÑλ§��.�Â�;§��(�jÕ±���2���j�(�³�;§��.���<�UÐ��Ð7�2���Âæn�B�/ÝX�"ô�� ê��¾÷pÚp¤A���(�w�(�;�<���Ô�;���0�(�B�;�2�:���àÐ�Õ�£��;�4�.�4�<���2�B��§����2�(�:K ? ïMLN���9�2� ? ï5OP�:KQ�*ÞR7£9�4�(�#�ð�2¤A�4Ð�¤*�2�â¦��<�2���2�"�2�<¦2�2�0�(�;§�Õ �0£������(�9¦i�(�#�;�(Õ±���7Ð4§��w¤0§â�(�LÕÓ£�¤0§"ÝX�*�B�B�*�4�.�4á������(�9¦¤A���(�(�(�;�(�����*ã¨U�(¤0ñ#���(�B�;�(��¤A���f�0�2���9��Õ õ=D�ê�F�F�F�êBõ4H� ��Õ±¥�£9�B��'EDS�»æô�õ=D ÷¾ê�F�F�F�êG'6HT�¯æô�õ.H�÷¡ �?�2���#�2���;§9�U'6-V ���.�;�j��%N£��2���9��£��B¥�£9�L�;§9�¤A�2�0�;���B¥X�������(��¦�õ4-WV �λ§��.�%�(���;§��j¥��;�2Ý��.Ý_�(�(�<���Ñ�2�?á������w�9¦±�ѤA���(�(�w�;�<���â�9��ÐX3 ¡ ���;§��j��£9�B¥�£9�4�2�?æ��(�&�.¦��2�(�"�2�0�;£�Õ±���"�B�Ýb�&�;��£9¦�§��(�Ó£����<���2�¾Õ �(�âè�éfê�ë\ì ï �N�;§9����¥_�(¤;ñf�(�9¦j�¾�2���9��Õ õ4-å�2���Ù¤A��Õ±¥�£9�;�(��¦U'6-;� æÿô�õ4-�÷��(�U�.ñf�(���B�¥��(¤;ñf�(�9¦Ó�0�2������ÕY'6-19Ôè�éfê�ë\ì�ïXÞ<�9�2�4á��N���M*0ê[Z"ô�Ð4�<�;§M*\��PZ:÷ ���;§��%¥��;�2Ý��.Ý��(�w�<�p�±�;§��.��'6-=�>'^]ª�(�7�;§����

ë

Page 64: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�;��£9¦�§��<� ?A@ ï ÞL�f�(��¤A�³�;§��*�;�Â�.�;�`_ Ha�b �cJ�ô�� a ÷¥��2�<�¾�³ë(de*0ê[Z5dN� ���;§9�ÂÕ±���2���4�;§��.�4�;§��j¥��;�2Ý��.Ý_�(�(�<����;§��.�\'6-7�f'^]Â���2�%�.�ª�<���2�B�&���9��¥_�2�<�&�(�4�;��£9¦�§��(�IJÑô�� a K ? ï�÷¾Þ�ôg��§��w�4�(�&�.¦��2�(�ò�2���.¥�¥��w�(¤*�.�;�<���/�2���;§��h Ý��<�;�;§������Ù¥��;�2Ý��<��Õ&ii�;§��.�4Ð�§����2�Â����¤A��£��:�B�*�;���"Ýb�*���2�;�   ÐZ���.�;�j¥��(¤0ñN�w�9¦)�7�0�2���9��Õr���(��Õ±�����;����;��Õ è�éfê�ë\ì�ïÒ�2���ÒÐ7�2�:�Â�B�"á_���Ô�;§9�Ñ¥��;�2Ý��.Ý��(�w�<�p�/�;§��.���.���(���2�B�j�pÐ���2��;§����B���.�;�Ñ��%:£��2�WÞ ÷e��§��(�Õ±���2���L�;§��.���(�åÐZ�³Ð7�2�:�4æÜ�B��ÝX�±ô��¾ê�� ÷pÚp¤A���(�(�(�0�<���#�;���;�(�;�;�2�����9Ð�jÕ�£��B�L�.�4�<���2�B�L§����2�U� a K ? ï Le���9�2�? ï OP� a KQ�*Þ

�f�(��¤A���<j ?QkGl �(�4¤*£9�;�;���:�;�<��¤A�����;�(�9�*�;���/�����2�;�(Ý��<�6�f�;§��(�7���f¥��(�2�(�#Ð&§��Ù�;§9�§��2�0§#��£���¤A�;�<�������"� �§��2��ë ?6m ÚWÝ��<�Â��£9�B¥�£9�³�2�����9Ø&Û4Ú0뱧��2��ë � é\ÚWÝ��(�j��£9�B¥�£9�*Þ&nª�Ô�;§��±�2�;§9�*��§��2���;�]Ý��<�f¤;ñ�¤*�<¥�§9�*�0�³¤*�2�Ýb�j�;��¤*£9�;�jÐ4�(�;§âÕ�£�¤¾§"�;§��2�;�B�*��ñ2�*�:Úp�;�<¬*����ô��B��¥��;�*�2���:�����9§��2£��B�;�<�2���B���.�0¤0§ò�(�#�;�(Õ±���o��Ð�Â�9�*���â�;§��ñ2�*��Úp�<����¦2�;§qpÒ�B�ò�;�.�;�(�;���`�rL ?Qs �2�tpPO �<�2¦<�   ¤A��Õ�¥��.�;�Ñ�;§��(�³�B�"�;§���¤*�2�B���2��§��2�;§Ò�ð£���¤A�;�(�����Ð4§��(¤¾§×�9�*���×�;§9�Ñ��£9�B¥_£9�Â�<���9¦2�;§�ó �B���;�.�;�(�B���`� a L ? ï��2�ÓófO ? �(�2¦��;÷¾ÞI��§��(�³���f¥��(�2�w�ÏÐ4§:���;§����£9�B¥�£9���<����¦2�;§±�2�]�j§��2�;§±�ð£���¤A�;�(���±�(���.ÝX��£9���pÐ4�w¤A���;§9�4�<����¦2�;§±�2�]�ª�p�N¥��(¤*�2�9ñ2�*���<����¦2�;§±���2�Z�³Ý��<�f¤;ñ¤*�<¥�§9�*�Âôð�.�%�<���2�B�L���2�4�;§9�j¤*�2�B���2�1�vu7�#�2���â�2�;§9�*�%¤*�<¥�§9�*�¾�¤A�����B�B�0£�¤A�B���"Ýb�*���2�;�j�±���*Ð}�2���.�0�&�.¦2�:÷¾Þ�?�(���2�(�<�6��ÐZ�4�;�B�;���;���;§��.�U�;§9���B�4�\�2�(£��������2�Zó/�.�;�&�Â�9��¤A���;�;�.�;�6�NÝ�£9�����2�U�;£�wѤ*�(�������:�;��%:£��<�;��Õ±���:�*Þ��§��.�j�(�����<�³�(�³¤A�*�;�;�2�(���(�"¥X���;�;�<Ý_�<�����2�Â�#�ð£���¤A�;�<���Ô�B�/§����2� h �(�.�0¦2��iâ��£9�B¥�£9�j�(���9¦2�;§Ôó>O ë ?6m �2����2�*�&�B�;�(�w�b���2��ÝX�ª¤A���w�(�(�;�<���Ù�;���0�(�B�;�2�:�*Þ

x ÈrÇL¿XÊÂÁ:Ç�Àry�ÁAz { ¹nª£9��Õ±�2�;�<�.�.�;�<���±���2�U�(�:�B�;�f��£�¤*�(��¦%¤A���(�w�(�;�<���fÚW�0���;�(�B�;�2�:��§��2�;§i�ð£���¤A�;�<�����?Ð7�2���B��£��B�L�;§9��ÕÅ�B���2¤0§��<�*�2�Õ±���;�;�.¦2�×�2£��;§9�����;�w¤*�.�;�<��� ���2�"�.�;Ý��<�B�¾�.�0�(�<�:Úp�<���9¦ÿÕ±���;�;�.¦2���*Þkäà�*�×ô�|�}=~àê���������÷ÙÝX�Ï�}ô��¾ê�� ÷pÚp�B��¤*£9�;�Õ±���;�;�.¦2� �2£��;§9�����;�w¤*�.�;�<��� ¤A�f�9����2�ÏóXÚWÝ��(�ÔÕ����;�;�.¦2�������2���Å�(�*�Ôæ ãòè�éfê�ë\ì í î è�éfê�ë\ì ï ÝX�D�ô��¾ê�� ö ÷pÚp¤A���w�(�(�;�<���Ô�0���;�(�B�;�2�:�j§��2�;§Ï�ð£���¤A�;�(���]ÞiÎÏ�Ѥ*�2�×�9�*á_�9���â�9�*Ð Õ±���0�;�.¦2���2£9�;§������;�(¤*�.�;�(���Ò¤A�N���ô�|�}=~�ö�ê�������� ö ÷&���2�5!Q���Aûg�g��!6� #Q� ú��*þA�����×Õ±���0�;�.¦2���j�2�³�����(�<��Ð4�*ãª�;§9���B�������*�j�2���Ô�;��¤A���<�2�*�Â�0§��.�;�±�#ñ2�*�� �2�³�(�ò�;§9���2�¾�<¦��(���2���;¤0§���Õ±�2ÞU�à�â�2£9�;§������;�(¤*�.�B���ÙÕ±���;�0�.¦2�&� 9è�éfê�ë\ì�í��X�;§9�±�B�����9�*�ª¤A��Õ±¥�£9�B���|�}=~�ö ��ô�� ÷)��|\}=~ � ôðæÿô�� ÷B÷¾ÞÒί§9����;§9�Ù�;��¤A���<�2�*�±¦2�*�;�±�ÔÕ±���;�0�.¦2�����;�.¦Ò¥��2�<�âô��}êG�G���9÷ ���;§��Ù�0�AÚ¤A���<�2�*�L�2�*�0�<á�����;§9�%�;�.¦±ÝN�ѤA��Õ±¥�£9�;�w�9¦U������� ö ��ô��}êG�G���f÷<��������� � ôðæÿô�� ÷¾êG�G�^�9÷¾Þ�ÎÏ�³�B�B�;���;��;§��.�LæÜ�w�¥��.�;�L�2�à�;§��j�9�*á����<�;�(���Ñ�2���;§9�j�;¤0§���Õ±�6���2���â�(�á��f���â�2���#ñf�9��Ð4�#�B�±�;§��ª�2���2�*�0�;�.�;�#�.�B�;�2¤;ñf�(�9¦��;§���;¤0§9��Õ��2Þ%ô�nª���(�Ñ�;§9�³ñ2�*� � �(�L£��9ñf�9��Ð4�#�B���;§9�j�2�9�2�*�¾�;�.�;�2Þ ÷���T�A�4���E� � ô�|\};~�ö�ê��S����� ö ÷Zûwý !jô�� ê��^¡r�Böw÷ ��ý��;ø�¢E�G� £&� ý0ý�!��A�\!Q¢����E�*þ.�ðûðø !6�ðû�ù.þÑý*ø��E��£&�;¤*ùQ��!Q���*û��g�G!Q�o#Q�ú��*þA������£&� ý¾ý�!��¥� ý�¦§ ���;�4¨ ÎÏ�Z�Bñ2�*�;¤¾§j�;§9��¥��;�:�2�N§9�*�;�2Þ?Û&�0�;£�Õ±�?Ð�U§����2��B��Õ±���2�9�2�*�¾�;�.�;�q©��.�B�;�2¤0ñN�w�9¦Óô�|\}=~ ö ê�������� ö ÷�2���"�¾£������(��¦i�w�/�;�(Õ±���¾Þ4ÎÏ��Ð7�2���%�B�ÙÝb��£����â�;§9�i�;£�¤*¤A���;�&¥��;�2Ý_�.Ý��(�(�<�����2���;§��(�4�2�9�2�*�¾�;�.�;��ô��;��¤*�2�w��;§��.�i�;§��(�Â�w�Â�;§9�Ù¥��;�2Ý_�.Ý��(�(�<���ò�;§��.�>© �;£�¤*¤A���;�B�ð£��(�(�ò���2�;¦2���±�ò�\�2�(�w�×Õ±���;�0�.¦2�����;�.¦Ò¥��2�<�����ÿ��þC��ªÕ±���;�;�.¦2���;§��.�ÑÐ7�2�Ñ���*�2�*�����f¥��(�(¤*�(�;�<���2£9�;§������;�(¤*�.�B���>ÝN�ÿ�;§��ò�B�������*� ÷¾Þ»äà�*�B�;�(�9¦¬«­ �9���9�2�B�"�;§��Õ±���;�;�.¦2���&�;§��.�®© �;£9Ý�ÕÑ�<�;�7�B�Ñ�<�;�7�2�0�2¤*�(��|�}=~ ö �9ÐZ�j�;§����âÐ�2�:�4�B�ÑÝb��£����]ã

¯B°²±�±B³´Iµ�¶�·� ¨���¸ �v¹ è�éfê�ë\ì s ��ô�� êG������÷ ¹ º |\}=~.»¼:½�¾ ¿ ã6������� � ô�� êG�G�^�9÷<� ëÁÀM� K98«­� Fäå�*�SÃ;ÄEÅÆÅÈÇ É�Ç ÄEÊ��9�����2�B�Z�;§9�7�*�2���:�?�;§��.��æÿô�� ÷<�¯æô��}öw÷å���2���B��Õ���� ö;9Ë«­ Þ7ô ¡ Þ �2Þ����;§��(���9���9�2�B���

�;§9�j�*�2���:�4�;§��.�Ì© Ð7�2�4�.Ý��<�³�B�Ñá����â��¤A���(�w�(�;�<���#�(�âæ�Þ ÷ÓÎÏ�j�;§9���"§����2�2ã¯;°�±�± ³´ � ¨���¸ © �;£�¤*¤A�*�����TÀ"Ã;ÄEÅÆÅÆÇ É�Ç ÄEÊ Â ¡¨��^¸ © �;£�¤*¤A�*�����1À Ã;ÄEÅÆÅÆÇ É�Ç ÄEÊ Â

?

Page 65: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ô�Ð4§9�*�;� Ã;ÄEÅÆÅÆÇ É�Ç ÄEÊ�Õ±���2�����;§��.�4��¤A���(�w�(�;�<���#�9�N�����9�2�L�f¤*¤*£9� ÷¾ÞÎÏ��Õ����â�9��Ð ���2�B�j�;§9�j�����(�<��Ð4�(�9¦9ãë.Þ4ί§9��� ©k�;£�¤*¤A�*�����i�2�����;§9�*�;�òûwý#�ò¤A���w�(�(�;�<���=���;§9����Ð��ñf�9��Ð �;§��.�iæô�� ÷t�kæô�� ö(÷³���2��B��Õ��)� ö+9¬«­ Þ���£9�;�;§9�*�0Õ��2�;�6�Í� ��Î� ö��;�(��¤A� ©É¤*�2���9�2�j�;£�¤*¤A�*���Ï�(���;§��(�ª�(�%�B�0£9�/ô��;��¤*�2�w��;§��.�q© ¤*�2�������(�/�;£�¤*¤A�*���Ï�<��� K9 «­ ÷¾Þ��N���X�;§��(�%Õ±���2���%�;§��.�q© §��2�%����£����"�#¤A���(�w�(�;�<���ò�(�æqôð�9�2�B�±�;§��.�³�*�2���:�)ÃBÄEÅÏÅÈÇ É�Ç ÄEÊÑ�2�j���*á��9���Ô�.ÝX���2��Ð7�2�j�9�2�ª���9�2¤A�;�<�Ô�;§9�±�;�2Õ±���2�³á������(�9¦Ù�¤A���(�w�(�;�<���Ù�(�âæ   ÐZ���9�*���â�B�Ñ�2�����<�;�<�����2�(�(�±�����;£9�;�³�;§��.�\� �2�����}öX�.�;�Â���0/b�*�0�����¾÷¾ÞZ�f�w��¤A�æÖ�(�jô�� ê�� ö ÷pÚp¤A���(�(�(�0�<���#�;���;�(�;�;�2�����9Ð�§����2��¨��^¸�© �;£�¤*¤A�*�����1À�Ã;ÄEÅÆÅÆÇ É�Ç ÄEÊ Â LN� ö Þ

? Þ4ί§9��� © �;£�¤*¤A�*�������2���D�;§9�*�;���(�âþ_ù6�Ñ�Ò¤A���(�(�(�0�<���;�Z�(�*� ­ µ�¶�·� è�æÿô�� ÷ ÐQ� 9Ñ«­ ìôð�WÞ �2Þ���;§��w�±�(�i�;§9�/�B�*�±�2�³§��2�;§����i�2�%�2�(�7���(��Õ±�����;���(�Ò«­ ÷¾Þ>�f�(��¤A� © �;£�¤*¤A�*���9���=�ZÐ�/ñf�9��Ðk�;§��.�������� ö � ô�� êG�G�^��÷+� ë&�2���ѧ9����¤A���S�����Ó��ôðæô�� ÷¾êG�G���9÷�� ë.Þ�nª���;§9�&�2�;§9�*�Z§��2���;�:�0�(��¤A��Ã;ÄEÅÆÅÆÇ É�Ç ÄEÊ���w�ò�9�2�³�f¤*¤*£9�³Ð�±ñN����Ð �;§��.�jæô�� ÷5K9 ­ ÞrR7£9�³�;§9���Ô© §��2�ª���;�B���:�;�(�2�(�(�/���2�0¦2���Ò�#�\�2�w�(�Õ±���0�;�.¦2�����;�.¦�¥��2�<�7�(�#�;¤¾§9��Õ±��ô�|�}=~àê���������÷U���#��þB��ª Õ±���;�;�.¦2�jæô�� ÷¾Þ�f�(��¤A�&ÐZ�³�.�;�ª¦��<�2����;§��.�Óô�|�}=~åê��S�����:÷�(�Âô�� ê��¾÷pÚp�B��¤*£��;�6���;§��w�LÕ±���2���L�;§��.�&¨���¸ © �;£�¤*¤A�*�����7À Ã;ÄEÅÆÅÈÇ É�Ç ÄEÊ Â LN�*Þ

¨U£9�B�;�w�9¦±�*�2�*�;�N�;§��(�9¦±�B�2¦2�*�;§9�*�ª�;§9��Ð4�L�;§��.� ¯B°�±�±;³´ LN�T¡e�Bö��9¥��;���f�(�9¦±�;§��j�;§9�*�2�;��ÕâÞ

Õ ÖÿÄ<×;Ø�Ä�Í�Æ8z ÄU¿_¿bÇ�Ù4ÄÒ{ Ì�ÆXÊiÄZÀ%ÆX½0Í�ÇÆX½;º%À��§:£��³���.�^�à�2�(�?��£9�Â�0¤0§9��Õ±����§����2�ÑÝb�*���×�B��¤*£9�;���.¦��2�(���B�Â�/¤A��Õ±¥_£9�;�.�;�<�����2�(�(��ÚWÝX��£��������ò�2�9�2�*�¾�;�.�;������<�6�f�2���Ñ��£���;¤¾§9��Õ±���7�����(�Ó�����;£9�;�&�;§��.��2�#�2�9�2�*�¾�;�.�;�ѧ��2��Ó�;Õ��2�(��ô�Ý�£9�Z�����fÚW¬*�*�;�:÷U¥��;�2Ý��.Ý_�(�(�<����2�X�0£�¤*¤A���;�*Þ ¡ �±�;§9�4¤*�2�;�4�2�X����¤A�;�N¥��;�<���iÐ�4Ð�*�;�4�.Ý��(�L�B���9�*á����Âôð�2�����2¤0§��<�*�2��÷��Â�9�2�;�<���±�2�X¥X�*�;����¤A��B��¤A�;��¤A�C�_¤*�2�/Ð�j�9���;§��j�;�2Õ±�j���2�4Õ±���;�0�.¦2���2£9�;§9���:�;�(¤*�.�;�<���C3

λ§��.�UÕ��<¦�§:���³�9�*á����<�;�(���Â�2�_¥b�*�0����¤A��Õ±���0�;�.¦2���2£9�;§9���:�;�(¤*�.�;�<�����(�:�2ñÓ�(�<ñ2��3ÙÎÏ���(�����B�*��Õ��(��¦��<�jÐ�ÐZ��£��w�Ï�;��%N£��<�;���;§��.���/¤A��Õ±¥_£9�;�.�;�<�����2�(�(��Úp£��:ÝX��£����9���Ô�2�<¦2�2�¾�<�;§�Õx¤*�2���9�2�Â¥X���;�;�(Ý��<�/���2�;¦2���"�\�2�w�(��;�.¦â���Ï�Ù���*Ð Õ����;�;�.¦2�6�à�;§��.�ª�w�*ã4�#Õ±���;�;�.¦2���2£��;§9�����;�w¤*�.�;�<���ϤA�N���i�w�&¥X�*�;����¤A�;�<�/�B��¤*£��;�±�<�����2�ª�2�w��2�<¦2�2�0�<�;§�ÕÑ�  Ð�§����2�2ã

¨��^¸ �v¹ è�éfê�ë\ì s ��ô�� êG������÷ ¹ |\}=~ ¼ ½�¾ ¿ ã6������� � ô�� êG�G�^��÷+� ëÁÀI� K9 ­� �¯éEFØ4��ÐZ�*�2�*���9�;§��(�U�9�*á_���<�;�<���±�(��¢fþ4!�ø��:û���Ú²!Ó�*ú��*Þ<�9�2�v!2þ$#�Õ±���;�;�.¦2�ª�2£9�;§9���:�;�(¤*�.�;�<���Ù�;¤¾§9��Õ±�6�f�2���2�9�2�*�;Ú�;�.�;�³¤*�2�Â�2�<Ð7���f�à�����;§9�������(�<��Ð&�(�9¦9ã]¦�£9���;���L�0�2������Õ¯ñ2�*� � ö 9Ôè�éfê�ë\ì s �2���Â��£9�B¥�£9�Uô�� ê�|�}=~ � » ô�� ÷B÷ôð�2���Ô�2�Bñ"�9�M%:£��*�0�<���%�B���;§��)|\}=~��2�0�2¤*�<��÷¾ÞU2&�2�B�i�;§��.�³�<���;§9�i�2�9�2�*�0�;�.�0�"¦�£9���;�;���%¤A�2�0�;��¤A�;�<��ôð�2���� ö=� � ÷��;§9�����;§9�i�2�9�2�*�0�;�.�0�"�;£�¤*¤A�*�������]�;§:£����b�;§��Ó�2�9�2�*�¾�;�.�;�CV �&¥��;�2Ý��.Ý��(�(�p�â�2�U�;£�¤*¤A���;�ª�(�&�.�³�<���2�;�ë^K ? s ôð�2���/§9����¤A�j���2�4é�÷¾Þ

Û&�����;�(����¥b���w������ÐZ�U���2�B���;§��.�å�;§��w�å�B�2�;�å�2�f�.�B�;�2¤;ñ#ôð��Þ �2Þ��\¦�£9���;�;�(��¦L�;§9��ñ2�*��÷X�9�N���à�9�2�à¤A���:�B�0�2���w¤A��;§9���ð�2¤A�Ó�;§��.���;§��Ñ���9�AÚW�;�(Õ���¥��2�×����¤A�;�N¥��;�<�����;¤0§9��Õ��Ù�(�Â¥X�*�;����¤A�;�(�Ò�B��¤*£9�;�2Þ��4§��(�j�(��¥��.�;�;�(�Ô��£9��B�Ï��£9�±�9�*á����<�;�(���Ò�2��¥X�*�;����¤A�±�B��¤A�;��¤A���w�×�;§9�â¤*�2�B�â�2�4����¤A�;�N¥��;�<���;��Ý�£��Ó�(�Ó�2�(�B�Ï��£��Ñ�B�Ï�;§9�Ù���2¤A��;§��.�7�;§9�ª¦2���2�X�2���B��¤A�;��¤A�Ù�(���£������2Õ±���:�;�2�(�<������/b�*�;���:����;��Õ �;§9�%¦2���2�X�2��Õ����;�;�.¦2�j�2£9�;§9���:�;�(¤*�.�;�<���åÞ¡ �×�;§9�Ѥ*�2�B���2�7Õ����;�;�.¦2���2£��;§9�����;�w¤*�.�;�<���;��¦�£9���;�;�(��¦â�;§9��ñ2�*�Ò�(��¤A�2�0�;��¤A�;�<�PÛ�ù^� ý�þ_ù6�vÛ.û�£iû�þ�û�ý��Ü���E�!�ÛQÚQ���0ý�!Q�o#;Ý ý��Wù6��!.úA�4�BùÓ�:!Ó�*û�úüûg�g#Ñù�¤4ýo¢�ø0ø �Aý0ý6�9��Þ �2Þ����;§��&�2�9�2�*�0�0�.�;���;£�¤*¤A�*�����UÐ4�<�;§Ñ¥��;�2Ý��.Ý��w�(�<�p�Ó�.�Z�<���2�;�ë^K ? s �;�*¦��.�0���<���0�7�2��Ð&§��.��§��.¥�¥X�����7Ð4§����9�*�2�*�4�<�L¦�£9���0�B�����;§9�j�w��¤A�2�;�;��¤A�Lñ2�*�2ÞL�N�9ã

¨���¸ ¯;°�±�±  � ¨��^¸ ¯B°²±�± À#¤A�2�;�;��¤A�4¦�£9���;�  ¡D¨��^¸ ¯B°²±�± À#�(��¤A�2�0�;��¤A�L¦�£9���;� ÂÞ ¨��^¸ ¯B°²±�± À#¤A�2�;�;��¤A�4¦�£9���;�  � ? @.s F

ß

Page 66: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

nª�Â�;§��U�2�;§9�*��§��2���=�\�(�j�;§9�¤*�2�;�Z�2������¤A�;�:¥��;�<���j¦�£9���0�;�(�9¦��;§��ZÐ��0���9¦�ñ2�*�¤*�2�i§N£9�;�å�;§9�7�2�9�2�*�0�0�.�;�4V ��B�2�;�2��¥��;�2Ý_�.Ý��(�(�<���j�2�]�;£�¤*¤A���;���:��Þ �2Þ��N�<���;§9�&�2�9�2�*�0�;�.�;�i�(�?�B�;�f�(�9¦³�B���9�*�B�*�0Õ��(���Ð&§9�*�;§9�*�U�j¤*�(¥�§9�*�;�B���f�à �;�*¥��0���B�����;�U�2�±����¤A�;�N¥��;�<���i�2�XÕ±���;�;�.¦2�v� l �2�Á�eDU�(�i�;§9�L�����AÚW�;�(Õ±��¥��2�±�0¤0§9��Õ±�6�N�;§9���2�9�2�*�¾�;�.�;����;�B���:�;�(�2�(�<�§��2���B��¦�£9���;�ÑÐ4§9�*�;§9�*���;§��"ñ2�*�Ð�2� � l � àcá � l �2� � D�� àËá �eD*Þ>ί§9���>�<�¦�£9���;�B����¤A�2�0�;��¤A�;�<�6���<�?¤A�2�;�;��¤A�;�(�Ó�9�*�B�*�¾Õ��(�9����Ð4§��(¤¾§iÕ±���;�;�.¦2�LÐ7�2�?����¤A�;�N¥��B���]Þ+R7£9�?Ð4§9���i�<��¦�£9���;�B����(��¤A�2�;�;��¤A�;�(�6���<�7Ð��;���9¦��(�Ù�9�*�B�*�¾Õ��(�9���7Ð4§��(¤0§âÕ±���0�;�.¦2�ÂÐ7�2�L����¤A�;�N¥��B���]Þ7�N�9ã

¨U�^¸ ¯B°�±�± Â.â ë^K ? � ¨��^¸ ¯B°²±�± À#¤A�2�;�;��¤A�4¦�£9���;�  ¡D¨��^¸ ¯B°²±�± À#�(��¤A�2�0�;��¤A�L¦�£9���;� Â.â ë^K ?� ë^K ? ¡é â ë^K ? �¯éEF

ôðÛ&���#�;§��j�2�9�2�*�0�;�.�;�ÙÐ��£��(�#§����2�ÂÝb�*��� h ÝX�*�B�B�*���Q/�ii¦�£����;�;�(�9¦i�0�2���9��Õ��<��Ð&§9���â�<�L���(�Ù�9�2�L¦�£9���;��;§9�¤A�2�;�;��¤A�4ñ2�*�   �2��¤A��£��0�B�6���<��§��2���9�±Ð7���â�2��ñN�9��Ð4�(�9¦±Ð&§9���#�;§��(��§��2�L�N¤*¤*£��;���;ã ÷

ÎÏ�4�;§9��ÐZ���Ñ�.Ýb���2�L�;§��.���9�³Õ±���;�0�.¦2�4�2£9�;§9���:�;�(¤*�.�;�<���±�;¤¾§9��Õ±��¤*�2�±¥��;�*�2�������2�±�2���2�*�0�;�.�;�Ó���0��Õ�;£�¤*¤A�*�����(��¦Dô�Ð&�<�;§9��£9�i�*�2��� Õ��.ñf�(�9¦Ô�2�:�q%N£9�*�0�<���Ó�B�Ï�;§9�I|\}=~�2�0�2¤*�<�6ã ÷ Ð4�<�;§ÿ¥��0�2Ý��.Ý��(�(�(�p� [email protected] �Ð4§9�*�;�MpÒ�(�Â�;§9�Ññ2�*�:Úp�<���9¦2�;§]Þ ¡ ���w�Â�2�(�B�ò�B�0£9���;§��.���9���;¤0§���Õ±��¤*�2��¥��;�*�2���:�Ó�2�ÿ�2�9�2�*�0�0�.�;�Ï���0��Õ�;£�¤*¤A�*�����(��¦/Ð4�<�;§Ï¥��;�2Ý��.Ý��(�w�<�p� ?A@ ï �àÐ&§9�*�;��ó �(�Â�;§9�Ñ�<����¦2�;§Ï�2�L�;§9���;�.¦9ãi¤A�����;�(���*�Â�2�×�2�9�2�*�¾�;�.�;�Ð4§9��¦�£9���0�B���&�Ñ�0�2���9��Õ �B�B�0�(��¦)ä&9Òè�éfê�ë\ì�ïò�2���â��£9�B¥�£��;�4�2�%�<�;�����2�;¦2�*�;�×ô�� êGä�÷¾Þ&�f�(��¤A�³Ð�Âñf�9��Ð�;§��.�\� §��2���.�4�<���2�;�L���9�j�.�2�(�(�Ù�;�.¦Ñ�(��è�éfê�ë\ì ï ���;§9�³¥��;�2Ý��.Ý��(�w�<�p�±�;§��.��äi�w�L�i�\�2�(�w�#�;�.¦Ñ�(�L�.�4�<���2�;�? @ ï_Þ��4§:£����Ó¤*ùQ�,!.þ$#M£&� ý0ý�!��A�&!Q¢����E�Aþ4�ðûðø !6�ðûðù2þÒýAø��E��£&�)ª�û����Iå���##ú��AþA�Ó���æpç!.þ4ÛI��!^�#ú��*þA������ó7è�!.þ!�ÛQÚQ���0ý�!Q�o#�ø�!2þq!.úéª�!Q#\ý ¤Aù6�g�¥�&!IÚ²!.úüûgÛI£&� ý0ý�!��A��ê4��!��,�E!.û��Ñù.þç!âþC��ªf£&� ý0ý�!��A�,ª�ûg���&�4�BùÓ��!Ó�Aû�ú ûg�g#ë!��ú���!\ýo�?Õ��²�bè [email protected] ê ?A@ ï ì�Þ

�N����¤*�2�ÒÐZ�ѤA�����B�B�0£�¤A�j�/Õ����;�;�.¦2�Ñ�2£9�;§������;�(¤*�.�;�(���Ò�;¤0§���Õ±�Ñ�(��Ð4§��w¤0§Ï�2�Ò£��:ÝX��£����9���Ô�2�9�2�*�;Ú�;�.�;�CV ���;£�¤*¤A���;���w�Ñ�(�(ÕÑ�<�B����B�ÿÕÑ�²�_è [email protected] ê ?A@ ï ì²3 ¡ � �ð�2¤A����ÐZ��¤*�2���9�2�Ù�2¤0§��<�*�2���;§��(���<�ªÐZ���2�(�<��Ð�;§9�±�2�9�2�*�0�;�.�;�"�B�#�w���B�*�0�2¤A�ªÐ4�(�;§"�;§9�,|\}=~��2�0�2¤*�<�/ôð¤*�2�Ô�2��£Ô¤A�����B�B�0£�¤A�ª�®¸ �B�0�(�N�(�2�  �;¤¾§9��Õ±�iÐ4§��(¤¾§�2¤0§��<�*�2���Ó�;§��w�j�<�*�2�����2�L�B��¤*£��0�<�p�Ï�.¦��2�(���B���2���2�9�2�*�0�;�.�;�ÔÐ4§��"�9�*�2�*���w���B�*�0�2¤A�;��Ð4�<�;§Ò�;§9�5|\};~��2�;Ú�2¤*�<��3�÷¾Þ"nª�7¤A��£��0�B�6��¥��;��§��<Ý_�<�;�(�9¦Ù�;§9�Ñ�2�9�2�*�¾�;�.�;�Ï���;��Õx�(�:�B�*�0�2¤A�;�(��¦/Ð4�<�;§Ï�;§��æ|\};~��2�¾�2¤*�<�ÑÕ��.ñ2����;§9�±¥��;�2Ý��(��Õ ���/�<���9¦2�*�³�2�*�;�Ï�(�:�B�*�;���B�;�(��¦�ã�ί§��.�³Ð�±Ð4�(�(�?�9�â�(���;�B���2�Ï�(�ªÝX��£������;§9���N£�ÕÂÝX�*�ª�2��;�(Õ±�����;§9���2�9�2�*�¾�;�.�;�פ*�2�ÿ�(�:�B�*�0�2¤A��Ð&�<�;§Ò�;§9���2�0�2¤*�<�#�2���ÿ�B�*�ÑÐ4§��.��Ð��¤*�2��2¤0§��<�*�2���;§����»ô��;§��w��(�Â�2���2�(�2¦2��£��Â�B�ò�;§9��¤*�2�;���2�L¥X�*�;����¤A���B��¤A�0��¤A�6��Ð&§��(¤0§×ÐZ�Ù¤*�2���2¤¾§��<�*�2�Ù�����<�ÔÐ4§9���ÒÐZ�#�(�(Õ��(�%�;§���2�9�2�*�0�;�.�;�#�B�Ñ�w���B�*�0�2¤A�;�w�9¦����;�(��¦��<�ª�;�(Õ±�³Ð4�<�;§#�;§��&ìEíò�2�¾�2¤*�<��÷¾Þ

Page 67: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

î\ï$ð�ñ6ò�ó�ô:ð�õWö5÷6ø+ù�úQó�ö¥û�ú6ï.üý ùëþ ýSÿ�������� ïÓõGó�÷¥ü���õ:ð�÷�ïMõG÷ ý ó:ö �$õG÷��6ó�ú����Óö�<ó:÷6ø�ò�ô:ôG÷6ó��6÷�ï$úQõ��$ú6ï��UúQõ��

����� �"!$#"� %'&

( )+*-,/.0*-132/4�5�678*-1-9:,/*<; *>=�=@?"AB*DCE9�2GF:*IHJ2GKL1M?N2GK�OJH

P®ò8�÷�ï$ô:ð�ü$ò�óS÷Q�$óN�6÷�ú6û �SR ó�ð�òUT$ö,ð ïÓõGó:÷Eü��ò�üMû�ú6ô:õSõ:ðWV)ò � V)÷6ó:ò �úQó:ò�øX�$û�û�ö&ï$÷ZY\[-]�ò0�ú6û�ûBõ��$úQõ"YSòYSú6ïAõÍõG÷J�÷�ï$ôGõGóL��õ3^B_a`4ô/YB�$ðWL�UúQó:òÁôGò0U��ó:òSú���ú6ð�ï$ôGõÍú6ï��$ï R ÷Q�.ï$ü�ò�üvú6ü�ñ6ò�ó�ô:úQó�ö�[3bc�.ð�ô/Y�ð�û ûd�.ú�ñ6ò�õ��$òú6ü�ñ²ú6ïAõ:ú��6ò&÷6øSú�L�$ð�ò�ñ¥ð�ï�Mô:ò0U��ó:ò�V)ò�ô:ô:ú��6ò5ú���õ���ò�ïAõ:ðe�úQõ:ð�÷�ïgfMhXikj l�mniJo�pqrmsp�t�u�l�v�wUpgo�xLxymsz{t@iXh|l�pnx�}ï�÷6õGò®õ��$úQõMõ���ò~�$ôGò��>]{�1ô��gú6ô�YSò®ü$ð�ü>ð�ï��.ó:ò�ñ¥ð�÷Q�$ô'�÷�ï.ôGõGóL��õ:ð�÷�ï.ôy��YÁò®ï�ò�ò�ü>õG÷Pú6ô:ô��V,òëõ��.úQõôG÷QV)ò(÷�ï$ò���YSú^öIøk�.ï�õ:ð�÷�ï�ò���ð�ôGõ:ôU[�P®òrï�÷6õGò�ü�û ú6ôGõ õ:ðeV)ò�õ��$úQõ�øX�$û�û;ô:ò0U��ó�ð�õWöMú���ú6ð ï$ôGõ\ú6ï��$ï R ÷Q�.ï$ü�ò�üú6ü�ñ6ò�ó�ô:úQó:ö5ð�ô�ï�÷6õÁó:ò�ú6û�û�ö:�B÷�ô:ô:ð R û�ò��Íø�÷6ó�÷�ï$ò\õ��$ð ïn�n�Að�õÁð�ô�ðeV��C÷�ô�ô:ð R û�ò�õG÷��$ó:ò�ñ6ò�ïAõÁú6ïæú6ü�ñ6ò�ó�ô:úQó�ö,ø�ó�÷QVô��U�ò�ò�ü$ð�ï��Y�ð�õ����$ó:÷ R ú R ð û�ð�õ[ö`úQõrû�ò�ú6ôGõ�V&úd�����������L������Q��YB�$ò�ó:ò��Üð�ôUõ���ò'�6ò�ö��[û�ò�ïn�6õ�� ú6ï.ü��eõ��$òõ:ú����[û�ò�ïn�6õ�� ÷6øSõ��$ò5ô�y��ò0V)ò�[����ó:õ���ò�óyV)÷6ó:ò��7ð�õ R ò0�÷QV)ò�ô(ðeV��$ó�ú��õ:ðe�ú6û+õG÷ëú�L�$ð�ò�ñ6ò5ôGò0U��ó ð�õ[ö��$ï$û�ò�ô:ôYÁò R ÷Q�$ï$ü"õ���ò,ï �V R ò�óv÷6øN ���ò�ó ð�ò�ôXõ���ò,ú6ü�ñ6ò�ó�ô:úQó�ö��ú6ï�V,ú��6ò)õG÷�õ��$ò�^{_a`®÷6ó ú��û�ò�[)þE÷n�;õ���ò R ò�ôGõYÁò8�ú6ï���÷��BòUð�ô�õG÷&ú�y�$ð�ò�ñ6ò(õ��$ò�ø�÷�û�û�÷ZY�ð�ï�rû�ò�ñ6ò�û;÷6ø+ôGò0U��ó�ð�õWö��¡£¢�¤M¥§¦©¨�¦kª@¥¬«�­®z�w¯xyx�o0° w£o�mnikjswUp�iXhX±yo�iXh|l�p²xU±�jswUz�w£hWx�x�o�h|³�i�l�´Lw�µ¶�[ô:ò0U��ó:òMúQø�õGò�ó�·��$ôGò�ô�h ¸L¹M¸Ul�uo�pq$oQ³�v�wUuLxUo�u¯qºx¯m�´�z:hki|iXhkp�°$o�i�z�ldx¯i{·�z�w�xLx�oZ°�w¯x�i�l+hXikx'^{_/`»l�u¶o�±U¼ew¯¹�ikjsw�t�u�l�´Lo�´�hk¼½hXiXq$ikj o�i:ikjswoQ³�v�wUuLxUo�u¯q£±yo�p�ikjsw�p�l�mni½t�mni"o'vdo�¼¾h|³'i�oZ°£l�p�o�pGw�f¿z�w�xLxUo0°�w�h©x\¼Àw�xLx8ikj o�p£µ�Á

 ï>ú6û�õGò�ó�ï$úQõGòÃY�ú�öe÷6ø\����ó�ú6ô:ð�ï�®õ��$ð�ô&ü$òUÄ.ï$ð�õ:ð�÷�ïNð�ô,õG÷qó:ò0  �$ð�ó:òIõ��$ò�ø�÷�û�û�÷ZY\ð�ïn�n�`û�ò�õ�Åëó:òU�.ó:ò�ôGò�ïAõúr�6ò�öÌø�÷6ó(õ���ò�V)ò�ô:ô:ú��6òMú���õ���ò�ïAõ:ðe�úQõ:ð�÷�ïqô�y��ò0V)ò��+ú6ï$üÜú6ô�ô��V)ò�Y\ð�õ���÷Q��õ(û�÷�ô:ôU÷6ø"�6ò�ï�ò�ó ú6û�ð�õ[ö®õ��.úQõõ���ò�ô�y��ò0V)ò�ð�ô&ü�ò�õGò�óLV,ð�ï.ð�ôGõ:ðe�[$Æ=ò�õ'Ç ÈÉ�0Ê~Ë��UÌUÌUÌ���Ê�ͯ� R òIúÜôGò�õ,÷6ø V)ò�ô:ô:ú��6ò�ô5ú6ï$üeû�ò�õ'ÎÏÈ�0жÑ�Ò Ë �UÌUÌUÌ���жÑ�Ò Í � R ò�õ���ò �÷6ó:ó:ò�ô��C÷�ï.ü$ð�ïn�tôGò�õ�÷6ø=õ:ú���ô�ø�÷6óSõ��$ò�ôGò V)ò�ô:ô:ú��6ò�ô8�gð�[ ò�[À� Ð�Ñ�Ò Ë ÈÓ^{_a`�Ô��kÊ~˯�y�ò�õ��[½�y[Nbc�$ò�ïMø�÷6ó�ú6û ûaÇ ú6ï$ürÎ��4ú6ï$ü�ø�÷6ó�ú6ïÓö�ÊÖÕ× Ç ú6ï$ü�ú6ïÓö�жÑ�Òn�YSò(ô��$÷Q�$û�ü��$ú�ñ6ò��

�<ódØÙ^B_a`@Ô0�kÊ��-È¿Ð�Ñ�Ò�ÚdÇÛ��Î�ÜÞÝ�µUÌ

ß\÷5ö6÷Q�IôGò�ò\Y{�Óöæõ��$ð�ô�ð�ô ò0 ��$ð�ñ²ú6û�ò�ïAõ�õG÷�ßXòUÄ.ï$ð�õ:ð�÷�ï�àZáP®ò�Y�ð�û�ûÞ��ð�ñ6ò:�÷�ï$ôGõGóL��õ:ð�÷�ï$ôX÷6øÁô�y��ò0V)ò�ô�ú�L�.ð�ò�ñ¥ð ïn�5õ��.ð�ô\û�ò�ñ6ò�ûT÷6øÁôGò0U��ó�ð�õWö�[\âSò�ø�÷6ó:ò)ü�÷�ð ïn�æô:÷n�

��÷ZYSò�ñ6ò�ó0��YSòUõ:ú��6òtú R ó�ð�ò�ø1ü$ðÀ�6ó:ò�ô�ô:ð�÷�ïMú6ï$üIü$ð ô�U�$ô:ôNÄ.ï$ð�õGò Ä$ò�û�ü$ôU[ã ä�KLH�K�2@*<ä�K�*>4Lå�=

bc��òvï�÷6õ:ð�÷�ï5÷6øÍúBæ�w�¼À³,ò��EõGò�ï$ü$ôSõ���òvï�÷6õ:ð�÷�ï5÷6øÍú��6ó:÷Q�n�a� R ö��÷�ï.ô:ð�ü�ò�ó�ð ïn�(ôGò�õ:ôSø�÷6óNY{�$ðeL�&õçYS÷rõ[ö �Cò�ô÷6ø1÷��Bò�ó�úQõ:ð�÷�ï$ô úQó:òUü�òUÄ4ï�ò�ü/[¡£¢�¤M¥§¦©¨�¦kª@¥�謭éæ�w�¼À³��Xê"�yë��QìI��hWx£oÃxUw0i"êÉfMhkikjºx�tnwL±Uh|o�¼{w�¼ew�z�wUp�ikx£í�o�p�³�îd¹�o�¼ïl�p °�fMhXikj+iXf>llytnw�u�o�iXh|l�pnx�ð�ë:�QìI� ³�w|æ-p@w�³�l�p�tso�hkuLx8l¶¸8w�¼ew�z�wUp�ikx l�¸�êñxym±�j�ikj o�iIikjswM¸Ul�¼k¼ïl�fMh©p�°�±Ll�p�³�hXiXhXl�px j l�¼À³QòîQÁ:ðXê"�yë8� h©x8o�±yl�z:z:msi�oQiXh©v�w\°�u�l�m�t�¹>fMhXikjrh|³�w�p�iXhXiXq�íQÁ

à

Page 68: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ó Á£ôÞjsw'ìélyõnwUu¶oQiXhXl�p�hWx�odxyx�l0±0hXoQiXh©v�w¯ö�¸Ul�u�o�¼©¼�÷��Lø���ù × ê:¹§÷�ì�ð|øJìrù0�-ÈéðX÷�ì�øU�Nìrù�Áú Á£ôÞjsw'ìélyõnwUu¶oQiXhXl�p�hWx�±yl�z:z�mni�o�iXhkv�wZò�¸Ul�u�o�¼k¼�÷��Lø × ê:¹Þ÷�ìrøcÈÓøJìr÷GÁû Á£ôÞjsw:wU¼ÀwUz�wUp�i�î�h©x�o�p~h|³�w�p�iXhXiXqB¸Ul�u�ì"ö�¸�l�u�o�¼k¼�÷ × ê:¹>à\ìr÷�È�÷�ì�à{È�÷GÁü Á£ôÞjsw)ü$ð�ôGõGó�ð R ��õ:ð�ñ6òvû�ú0Yýh©x x�o�iXh©x©æNwL³dö�¸Ul�u�o�¼©¼�÷��Lø���ù × ê:¹§÷�ì�ð|ø-ëgù0�-È»ðX÷'ì�øU�aëþðX÷'ìrù0��Áÿ Á ­J¼k¼Gp�l�p���w�u�l�w�¼ew�z�wUp�ikx\h©p�ê j o�v�w:o�pÃhkpv�w�uyxUw�msp�³�wUu�ì�ö¸Ul�u�o�¼k¼�÷ × ê"��÷��È�� ¹>ikjswUu�w�w���h©x¯ikxo�p�wU¼ÀwUz�wUp�i3÷ � Ë × ê<xym±Uj�ikj oQi3÷�ì£÷ � Ë È àQÁ

��÷6ó�ú6ïÓö:Ä$ò�û�ü�ê���YSò\ü�ò�ï�÷6õGò R ö:ê��õ���ò\ôGò�õ<÷6ø;ð�ïAñ6ò�ó:õ:ð R û�ò�ò�û�ò0V,ò�ïÓõ:ô�÷6øGêýðgï�÷6õGò õ��$úQõÁõ���òB�$ô�ú��6ò�÷6ó:ó:ò�ô��B÷�ï$ü$ôtõG÷®÷Q��ó:�$ó ð�÷6ó��$ô:ú��6òMø�÷6ó� �� �y[� �ïçõ���ò£�ú6ôGò�÷6øXú~Ä$ò�û�ü/�-ê � ð�ôrô:ðeV��.û�öÌú6û�û�ï�÷�ïn��ò�ó�÷ò�û�ò0V)ò�ïAõ:ôU[

b�ö �.ðe�ú6û�ò��Eú�V��.û�ò�÷6ø Ä.ò�û�ü$ô&úQó�ò�õ���òëó:ò�ú6û�ï �V R ò�ó�ô,÷6óæõ���ò��÷QV��.û�ò��Nï �V R ò�ó�ô0[�bB��ò�ôGò"úQó�òR ÷6õ��5ò���ú�V��.û�ò�ôS÷6øÍð�ïÄ.ï$ð�õGò"Ä$ò�û ü$ô{ðgð|[ ò�[À�sÄ$ò�û ü$ô-Y�ð�õ��5ú6ïMð�ïnÄ4ï$ð�õGò ï �V R ò�óÁ÷6ø=ò�û�ò0V)ò�ïAõ:ôy�y[��Xø=ð ïÓõGò�ó:ò�ô:õð�ï$�ó:ö �$õG÷��6ó�ú����AöqúQó�ò£Ä.ï$ð�õGò�Ä$ò�û�ü.ôU[  ô,ú6ïçò���ú�V��.û�ò��Sï�÷6õGòIõ��$úQõ���)ð�ôrú�Ä.ò�û�üg�$ï$ü�ò�ó)ú6ü$ü$ð�õ:ð�÷�ïú6ï$ü�V��$û�õ:ðÀ�.û ðe�úQõ:ð�÷�ï$ðXV)÷Eü�$û�÷ � �y[�P®ò��$ú�ñ6òæú6û�ó:ò�ú6ü�ö`ôGò�ò�ïÜõ��$úQõ � ð�ô(ú£�6ó:÷Q�n���.ï$ü�ò�óUú6ü$ü$ð�õ:ð�÷�ï/[���ó:õ���ò�óyV)÷6ó:ò��sYÁò �ú6ï�L�$ò0��æõ��$úQõÁó�ò0 ��$ð�ó:ò0V)ò�ïAõ:ô���� � úQó:òvô:úQõ:ð�ô�Ä$ò�üÃðgô�ð�ï�òXõ���ò�ö���÷�û�ü5÷^ñ6ò�ó�õ��$òvô:ò�õ��à��UÌUÌUÌ�� � ���÷�ï.ô:ð�ü�ò�ó:ò�üMú6ô ú)ôL� R ôGò�õ�÷6ø1õ���òUð�ïAõGòU�6ò�ó�ôy�y[>bc�$òvï$÷�ïs��õGó�ð�ñEð�ú6û4÷�ï�ò�õG÷�L�$ò0���ð�ô"�÷�ï$ü$ð�õ:ð�÷�ï� � R �$õUõ��$ð�ô\�ú6ï R ò,ñ6ò�ó�ðÀÄ$ò�üÌ÷�ïÜúr�ú6ôGò�� R ö��ç�ú6ôGò R ú6ô:ð�ô�ðgð�[ ò�[À�Tõ���òæð�ïAñ6ò�ó�ôGò,÷6ø���ð�ô�� } ÿ ð�ôUð�õ:ô�÷ZY�ïð�ïÓñ6ò�ó ôGò��y[��vï(õ��$ò�÷6õ���ò�óM�$ú6ï$üa�Qï$÷6õGòÁõ��$úQõ���1ð�ô>p@l�iÍú{Ä$ò�û�ü/[3��÷6ó7ò��Eú�V��.û�ò�� ÿ �.ú6ô1ï$÷{V��$û�õ:ðÀ�.û ðe�úQõ:ð�ñ6òð�ïÓñ6ò�ó ôGò'ð�õGó:öæõG÷�Ä.ï$üM÷�ï$ò��½�y[

ïIø�ú��õU��YÁò\�ú6ï"ôGõ:úQõGò(õ���ò�ø�÷�û�û�÷�Y�ð�ïn�)û�ò0V�V,ú)ô���÷�Y�ð�ïn�)õ���ò�ò���ð�ôGõGò�ï��òU÷6øMV,ú6ïÓö�Ä.ï$ð�õGòJÄ.ò�û�ü$ôU��I¢ �!�!"g«�#§wUi%$ ´Lw:o8õ�u¯h©z�w�p�m z�´yw�uUÁ'ôÞjswUp&('�h©x�o{æNw�¼À³�msp�³�w�u:oQ³Q³�hXiXh|l�pºo�p�³�z:ms¼ïiXh õ�¼¾h|±Lo�iXh|l�pz�lZ³�ms¼ïl)$�Ábc��ò\�.ó:÷A÷6øT÷6ø7õ��$ð�ô�ð�ô ô�ðeV��.û�ò�[  ô�ú R ÷^ñ6ò��Cð�õ ð�ôc�û�ò�úQó õ��$úQõ* ' ð ô ú��÷QV�V���õ:úQõ:ð�ñ6ò��6ó:÷Q�n���$ï.ü�ò�óú6ü$ü$ð�õ:ð�÷�ïa��ú6ï$ü(õ��$úQõ§�÷�ï$ü.ð�õ:ð�÷�ï$ô+��� � ��÷�û�üa[Þbc�$úQõ3�÷�ï$ü$ð�õ:ð�÷�ï � ��÷�û�ü$ôÍð�ô=ò�ú6ôGö�õG÷Xô���÷�Y\���$ô:ð�ïn��õ��$ò<ø�ú��õ

ðgü$ð�ô�U�$ô�ôGò�ü��$ó:ò�ñEð�÷Q�$ô:û�ö&ð�ï£�û�ú6ô:ô¯�Áõ��$úQõ*��' ð�ô�ú�±UqQ±U¼¾hX±-,Qu¶l�m�õ��$ï$ü�ò�ócV��$û�õ:ðe�.û�ðe�úQõ:ð�÷�ï~ð�ø�÷6óc�$ó�ðeV,ò.$/�y[/ ò0�ú6û�û4õ��$úQõSõ��$ð�ô>V,ò�ú6ï$ôSõ���ò�ó:ò�ò���ð�ôGõ:ôSôG÷QV)òJ�6ò�ï�ò�ó�úQõG÷6ó�0 × �' ø�÷6óNYB�$ðey�1 �' Ȭ�2043Q�50 Ë �UÌUÌUÌZ�50 ' �%6d�ðkYÁò�V&ú�öPôGõG÷�� �$ò�ó:ò��Sô�ð�ï�ò70 ' � Ë ÈÉà�È8043Iú6ï.üçõ���ò��ös�û�òIó:òU�Bò�úQõ:ôy�y[ ý ÷�ï$ô:ð�ü�ò�ó)ú6ïAöçò�û�ò0V)ò�ïAõ÷ × �' [EP®ò��¥ï�÷�Y õ��$úQõ£YÁò��ú6ï�Y ó ð�õGòÃ÷Nú6ô70:9çø�÷6ó�ôG÷QV)ò�; × ���s�UÌUÌUÌZ�<$�=>���Q[ýâN�$õæõ��$ò�ï0 9 0 ' �?9�� Ë È@0 ' � Ë È à�ú6ï$ür��ò�ï��ò8÷ � Ë È@0 ' �?9�� Ë }.ô:ð�ï�ò ÷�Y�ú6ô úQó R ð�õGó úQó:ö&õ��$ð ô"V)ò�ú6ï$ô õ��.úQõ�ò�ñ6ò�ó:öï�÷�ïn��ò�ó:÷,ò�û�ò0V)ò�ïÓõ{�$ú6ô ú6ïIð�ïAñ6ò�ó�ôGòUú6ï.ü£YÁò�úQó:ò(ü�÷�ï�ò�[

��÷6óc�÷QV��.û�ò�õGò�ï�ò�ô:ôNYÁò�ï�÷6õGòvõ��$úQõS÷6õ���ò�ó"Ä.ï.ð�õGòBÄ$ò�û ü$ô<ò���ð�ôGõU}n��÷�YÁò�ñ6ò�ó0�nYSò�Y�ð�û�û.ï�÷6õN�.ôGò\õ��$ò�ôGò�ð�ïõ��$ð�ôc�û�ú6ô�ôU[A B¿OJH:=2G,a9:132GKyH:A ? )+*-,/.0*-132DCFEHG

 óLV,ò�ügY�ð�õ�� õ���ò£V,ú�IL�$ð ï�ò�ó:öÌ÷6øJÄ.ï$ð�õGò'Ä$ò�û�ü$ô0�MYÁòJI�ú6ïeï�÷�Y ��ð�ñ6ò�ú�I�÷�ï$ôGõGóL�?I�õ:ð�÷�ïç÷6ø\ú~V)ò�ô:ô:ú��6òú���õ���ò�ïAõ:ðKI�úQõ:ð�÷�ï)ôLIy��ò0V)ò�õ��$úQõ+ð�ô�µ¶�[ôGòMIU��ó�ò�úQø�õGò�ócà��$ôGò�[3P®ò"Y�ð�û�û�ò��¥õGò�ï$ürõ��.ð�ô R ò�û�÷�YçõG÷J��ð�ñ6ò�ô�IL��ò0V)ò�ôôGòMIU��ó:ò(úQø�õGò�óc· �$ôGò�ô0��ø�÷6ó�úQó R ð�õGó�úQó:ö�·d[ ô�ô��V)òU÷Q�$ó{V)ò�ô:ô�ú��6ò�ô��.ú�I�òtð�ô�ú�Ä.ï$ð�õGò\Ä.ò�û�ü£êDð�÷6øNI�÷Q�$ó�ôGò��4ú6û û/YÁò(ó:ò�ú6û�û�ö�ï�ò�ò�ü"ð�ô�õG÷ R ò(ú R û�òõG÷�ò2O1I�ð�ò�ïAõ:û�ö�V,ú��+V)ò�ô:ô:ú��6ò�ô � ð�ï®ú�÷�ï�ò���õG÷���÷�ï�ò�V,ú6ï$ï�ò�ó � õG÷�ò�û�ò0V)ò�ïÓõ:ô(÷6ø"ê��y[�bB��ò&ôGò�ï$ü$ò�óú6ï$üÌó:òMI�ò�ð�ñ6ò�ó�ô��$úQó�ò&úr�6ò�ö®÷6ø�õ���ò5ø�÷6óLV ÷��Lø��3YB��ò�ó:ò�÷`ú6ï$ü+ørúQó:ò&ó ú6ï$ü�÷QV,û�öPIL��÷�ô:ò�ï ð�ï�ê�[�bB��ò

Page 69: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ú���õ���ò�ïAõ:ðKI�úQõ:ð�÷�ïPõ:ú��®÷�ïeú~V)ò�ô�ô:ú��6òrÊ Y�ð�û�û R ò'Ð�Ñ�ÒgÈ®÷�ÊDë øZ[Óð<Q\÷6õGòIõ��$úQõ,ð�õ)ð�ôtò�ú6ôGöqø�÷6ó)õ��$òó:òMI�ò�ð�ñ6ò�ó�õG÷&ñ6ò�ó�ð�ø�ö7I�÷6ó:ó�òMI�õ:ï�ò�ô:ô ÷6ø+ú���ð�ñ6ò�ï�õ:ú��,÷�ï�ú:��ð�ñ6ò�ï�V)ò�ô:ô�ú��6ò�[½��I¢ �!�!"�èSR+Tnw�z�wVU�UUoM,�w:o�mniWTswUp�iXh|±Lo�iXh|l�p�U�±2Tsw�z�w:o�´Ll�v�w:hXU\àZÕnÚ ê'ÚZY[UUwL±Umsu�w�o¶¸¯içwUu�î'm�UUw0Á

\^]�ª/ª`_ î\ô:ð ïn��õ���ò)ó:ò�ø�÷6óLV��$û�úQõ:ð�÷�ï`÷6ø"ßXòUÄ.ï$ð�õ:ð�÷�ï+à��§YÁò5ô���÷ZY õ��$úQõUø�÷6ó(ú6ïÓö~Ê ú6ï$ü®ú6ïAö`õ:ú��ba¯�ú6ï$ü�ø�÷6óvú6ïÓörÊJc�dÈ Ê ú6ï$üëú6ïÓö�õ:ú��&aec©�@YSò��$ú^ñ6ò���ógfÙ^{_/`%h2i j0ðkÊJce��Èka[caÚ�^{_/`%h2i jUðkÊ���Èla[m�ÝéàZÕnÚ ê'Ú½[Q�÷6õGò�õ��$úQõU�

�<ó�fÙ^{_a``h2i jUðkÊ c �MÈna c Ú�^{_a`?hMi j�ðkÊ��>Ènaom@È �<ógfÙ^{_a`?hMi j0ðkÊJcÀ�-È�a[cqp£^{_/`%h2i j0ðkÊ��>Ènaom�<ó�fÙ^{_/``hMi j�ðkÊ��>Ènaom Ì

Æ=ò�õ:�$ô\Ä$ó ôGõI�÷Q�$ïAõ(õ���òMï �V R ò�óU÷6ø"�B÷�ô:ô�ð R û�ò��6ò�öEôU["þEð�ï%I�ò�÷��Lø × ê��1õ���ò&õG÷6õ:ú6û�ï �V R ò�ó�÷6ø"�6ò�ö¥ôð�ô�Ú ê'Ú r�[s�Xø\õ��$ò�ôGò��6ò�öEôU��YÁòtI�û�ú6ðeV õ��$úQõ&ò��Eú�I�õ:û�ö²ê ô:úQõ:ð�ôGø�ö$^B_a`%h2i j0ðkÊ���Èua¯[v �ï$ü�ò�ò�ü/�Sõ��$ð�ô)ð ôò0 ��$ð�ñ²ú6û�ò�ïAõUõG÷ëó:ò0  �$ð�ó�ð�ï�æõ��.úQõ�÷�Ê<ë�ø:Èway�1÷6ó�ø�Èxa.=$÷�Ê�["þ¥÷n�3�.ûe�n����ð�ï�Ið�ïÜú6ïAö®ñ²ú6ûW��ò,ø�÷6ó�÷ö¥ð�ò�û ü$ô�ú'�B÷�ô:ô:ð R û�òUôG÷�ûe��õ:ð�÷�ï�ø�÷6ó ø�};ô:ð�ï?I�ò�õ���ò�ó:ò�úQó�ò£Ú ê'Ú%IL�$÷�ðKI�ò�ôXø�÷6ó�ús�Cõ��$ð�ôJV)ò�ú6ï$ô\õ��$úQõ\õ��$ò�ó:òtúQó�òÚ ê'ÚQôG÷�ûe�$õ:ð�÷�ï$ô+ð�ïtõG÷6õ:ú6û|[MP®òJ�$ú�ñ6ò�y��$ôGõ<ôL��÷ZY�ïrõ��.úQõ>�<ó�fÙ^{_a`%hMi jUðkÊ��-Èna[m@È»Ú ê'ÚÙÕnÚ ê'Ú r È àZÕnÚ ê'Únðgï�÷6õGòõ��$úQõ õ��$ð�ôB��÷�û�ü$ôSø�÷6ó\úQó R ð�õGó�úQó:ö�Ê��za��y[P®ò{I�û�ú6ðWV8øk�$ó:õ���ò�ó<õ��$úQõ�ø�÷6ó�ú6ïAö�ÊJc�dÈ¿Ê ú6ï$ü,ú6ïAö|a[ck�Óò���ú�I�õ:û�ör÷�ï�òc�6ò�ö,ô�úQõ:ð�ô�Ä$ò�ô>^{_a`?hMi j0ðkÊJcÀ�>È

a[c[p{^B_a`%h2i jUðkÊ��-Èna¯[�Q�÷6õGò õ��.úQõ1õ��.ð�ôTð ôTò0  �$ð�ñQú6û�ò�ïAõTõG÷�ó�ò0 ��$ð�ó�ð�ïn� õ��.úQõ�÷�Êgë'ø"ÈnaÍú6ï$ü:÷�ÊJc ë�ø"Ènaeck[þ¥÷�û�ñEð�ïn��õ���ò�ôGò�ò0 ��$úQõ:ð�÷�ï$ô/��ð�ñ6ò�ô3÷:È»ðWaV=*a c ��ðkÊ�=BÊ c �L� Ë ú6ï.ü8øcÈna2=cÊr÷:ðgï�÷6õGòÁð�õ=ð�ô}I�óL�?I�ð ú6ûQ��ò�ó:ò<õ��.úQõÊJc�dÈ�Ê � ÷6õ���ò�ó�Y�ð ôGò��ðkÊ~= ÊJcW�=ü�÷¥ò�ô1ï�÷6õ3�.ú�ñ6ò ú6ïtð�ïÓñ6ò�ó ôGò��½�y[Mbc���.ôÍõ���ò�ó�òSð�ôTò���ú�I�õ:û�öU÷�ï�ò�ôG÷�ûe�$õ:ð�÷�ïõG÷,õ��$ð�ô�ò0  �$úQõ:ð�÷�ïa�$ú6ï$ü£YSò8�$ú�ñ6ò�ô���÷ZY\ï�õ��$úQõB��ógfÙ^B_a`%h2i j0ðkÊ c �-Èna c pr^B_a`%h2i j0ðkÊ��MÈna[mGÈ àZÕnÚ ê'Ú rQ[

��ûe�����ð�ïn��ð�ïÜõG÷"õ��$ò&ò0  �$úQõ:ð�÷�ïqú R ÷�ñ6òæôL��÷ZY�ô�õ��$úQõ:�<ó�fÙ^{_a`%hMi jUðkÊJce��È�aec/Ú�^{_a`?hMi j0ðkÊ��8ÈwaomcÈàZÕnÚ ê'Ú½�4ú6ï$ürYÁò(úQó�ò(ü�÷�ï�ò�[þ¥÷n�d��ð�ñ6ò�ï(úB�.úQó:õ:ðKIU�$û úQóaV)ò�ô:ô:ú��6òSô��.ú�I�ò>Y\ð�õ��^� V)ò�ô:ô:ú��6ò�ô�ð�ó:òU�$ó:ò�ô:ò�ïÓõ:ð�ï��ú6û�û��B÷�ô:ô:ð R û�ò�V)ò�ô:ô:ú��6ò�ô

YÁò�V,ðÀ�Q�AõvôGò�ï$ü��y�@YÁò�I�ú6ï`ò2O1I�ð�ò�ïAõ:û�öIú���õ��$ò�ïÓõ:ðKI�úQõGò)õ��$ò�ôGòtú6ôXø�÷�û�û�÷�Y�ôU�B�ÁðKI��Iú��$ó�ðWV)ò�$Ìô��?Iy�"õ��.úQõ$t�v�E[-bc�$ð�ôIY�ð�û�û4ü�òUÄ.ï�òvú�Ä$ò�û�ü7('\ð ï'Y{�$ðKIL�'YSò-I�ú6ïMò0V R ò�üæ÷Q��ó"V)ò�ô:ô:ú��6ò�ô��.ú�I�ò�[ Ë bB��ò�ôLIL��ò0V,òú R ÷�ñ6ò(õ���ò�ï�ú�IL�$ð�ò�ñ6ò�ô�àZÕ�$�[ôGòMIU��ó ð�õ[öIúQø�õGò�ó ÷�ï�ò8�.ôGò�[

bc�$ð�ô�ð�ô�ú6û�û@YSò�û�ûBú6ï$ü��6÷A÷Eü/[À[À[ R �$õ"YB�$úQõ�ð�ø=÷Q��ó"V)ò�ô�ô:ú��6òUô��.ú�I�òUð�ô�ô�V&ú6û�û©á£ßX÷¥ò�ô�õ��$ð�ô�V)ò�ú6ï�õ��.úQõYÁò(úQó�ò�õ���ò�ïMó:ò�ôGõGó�ðXI�õGò�ü�õG÷,ú�IL�$ð�ò�ñ¥ð�ï�rô:òMIU��ó�ð�õWöÃàZÕ�$b� àZÕ���� ðkYB�$ðKIy�£V,ðÀ�Q�Aõ�ï$÷6õ R ò�ú�I2I�òU�$õ:ú R û�ò(ð�ø� ð�ôSõG÷¥÷,ô�V,ú6û�ûk��á�ádáJQ\÷6õ�úQõ ú6û�û|[�P®òI�ú6ïJIy��÷¥÷�ôGò�$�ú6ô û�úQóL�6òvú6ôcYÁòUû�ðe�6ò:ðgô�� R y�òMI�õ�õG÷-$P�v� �Sú6ï.üõ���ò�ï�w�z�´LwL³ë÷Q��ó�ðgô�V,ú6û û©�cV,ò�ô:ô:ú��6ò,ô��.ú�I�ò:Y�ð�õ��$ð�ï"õ���òrðgû�úQó��6ò�ó¯�BÄ$ò�û�ü�('�[^ �ï`õ��$ð�ô{Y�ú�öÃYÁò)÷ R õ:ú6ð ïôGòMIU��ó�ð�õWö�àZÕ�$Mú6ï$ü/� R ötô:ò�õGõ:ð�ïn��$&û�úQóL�6ò�ò�ï�÷Q�n�Q�/�:I�ú6ï5ú�IL�$ð�ò�ñ6òcYB�$úQõGò�ñ6ò�óSû�ò�ñ6ò�û�÷6øBôGòMIU��ó�ð�õWö�YSò ü�ò�ô:ð�ó:ò�[���o� ���:�.�����N�o���v���%���N�4���M�k���e���?���b�.���%�bc��ò�ôLIL�$ò0V)òUú R ÷^ñ6ò\�$ôGò�üMú)ü�òU�6ó�ò�ò��Là�ðgð|[ ò�[À�$û�ð�ï�ò�úQó��M�B÷�û�öEï�÷QV,ð�ú6û@÷�Ê ë+ø õG÷)ú�Iy�$ð�ò�ñ6ò(ôGòMIU�$ó�ð�õ[öæúQø�õGò�ó÷�ï�ò��.ôGò�[JbB�$ð�ô�ôL�n���6ò�ôGõ:ôXõ���òtø�÷�û�û�÷ZY�ð�ï�,ò��¥õGò�ï.ô:ð�÷�ïë÷6ø<õ���òtôLIL��ò0V,òtø�÷6óvôGòMIU��ó�ð�õWö�úQø�õGò�ó{·��$ôGò�ôU���Íò�õê R òXú�Ä.ï$ð�õGò{Ä$ò�û�ü/[�bc��òvôGò�ï$ü�ò�ó�ú6ï$üæó:òMI�ò�ð�ñ6ò�ó ôL�$úQó:òc·§ë²à�ó�ú6ï.ü�÷QV ò�û�ò0V)ò�ïAõ:ôN÷ 3 �UÌUÌUÌ0��÷QÍ × ê�[-bT÷ú���õ���ò�ïAõ:ðKI�úQõGò�ú�V)ò�ô�ô:ú��6ò{Ê × ê��Eõ���ò\ôGò�ï.ü�ò�ó�I�÷QV�����õGò�ôIжÑ�Ò:ÈÓ÷ 3 ëÃ÷�Ë¶Ê ëÃ÷ r ÊbrGë! 2 2 ¯÷�Í�Ê Í [MP®òû�ò�ú�ñ6òæð�õvõG÷Iõ���ò)ó:ò�ú6ü�ò�ó(õG÷�ñ6ò�ó�ð�ø�ö"õ��$úQõ�õ��.ð�ô���ð�ñ6ò�ôUú�ôLIy��ò0V)ò�Y{�$ðKIL�®ð ô�àZÕnÚ ê'Ú �[ôGòMIU��ó:òæúQø�õGò�ó\·:�$ô:ò�ôð�õ���ò��$ó:÷¥÷6ø>�.ó:÷�I�ò�ò�ü.ôUú6ôUú R ÷^ñ6ò��7ú6ï$ü��$ôGò�ôUõ���ò)ø�ú�I�õ(õ��$úQõ(ú�ü�òU�6ó:ò�ò��|·��B÷�û�öEï�÷QV,ð�ú6û1÷�ñ6ò�ó�ú£Ä$ò�û üëð ô�$ï$ðe  ��ò�û�ö&ü$òUÄ.ï�ò�ü R öæð�õ:ô ñ²ú6ûe�$ò�ô�÷�ïIú6ïAö'·>ë�àUü.ð�ôGõ:ð�ï?I�õN�B÷�ð�ïAõ:ôy�y[

¡�¢¤£¦¥�§�¨�©«ª�£�¥�¬�¨�¥+­�§N­}®«©¯©�°g£¦¥±¬�¨z²V§�¥�£�³�®¯´oµ-¨�¶·¥�£2£�©Z¨L¸�¹º§(»¼¨½¥�¬�§5£�¸�§5¾À¿Á¸�£¦¾@°MÂ�¾.Ã�§e¸}¥�¬�§5£�¸�Ä)ª<¥o¨L¥�§5ª±¥�¬�¨�¥+¿Å£�¸¨�©¯©�ÆÇ¥�¬g§e¸�§�§eÈ�®¯ª<¥�ª+¨�³g¸�®¯¾�§�®¯°)¥�¬g§�®¯°2¥�§e¸�²º¨�©`É ÆJÊo˺Æ&ÌKÍ

Page 70: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

University of MarylandCMSC456 — Introduction to CryptographyProfessor Jonathan Katz

Lecture 25

1 Algorithmic Number Theory

Before turning to public-key cryptography, it will be useful to re-examine some elements ofnumber theory. Until now, we have mainly presented number-theoretic results from a very“theoretical” point of view, rather than from a “practical” point of view. To give a specificexample, we discussed in class that if p is an odd prime then exactly half the elements inÎ∗

p are quadratic residues; but we did not really discuss how to tell when a given elementis a quadratic residue nor did we discuss how to find a square root of a quadratic residue.Of course, from a purely abstract point of view the problem is (typically) easy: given aquadratic residue y ∈

Î∗

p we can always exhaustively search through all elements ofÎ

p

until we find a correct square root. But such an answer is unsatisfying in our case, sincewe will only be interested in algorithms running in polynomial time; thus, the questionbecomes (e.g., in the case of the above example): how can we compute square roots in

Î∗

p

in polynomial time.Before continuing, a word is in order regarding “polynomial time”. Note that this

always means polynomial time in the length of the input (i.e., |p| in the case above) and not

polynomial time in the magnitude of the input (i.e., p). This distinction is crucial in thecase of designing efficient algorithms (note that |p| = dlog2 pe = O(log p)). The numbersthat are used for cryptography are typically huge (say, 512 bits long) and the differencebetween an algorithm running in time linear in p = 2512 and one running in time linear in|p| = 512 is enormous!

1.1 Simple Arithmetic Operations

Since we will be dealing with such large numbers, it is worth verifying that the “simple”arithmetic operations we are used to can actually be evaluated in polynomial time. We statethe following facts without proof (in the following, a, b are positive integers with a ≥ b):

1. Addition/subtraction of a and b can be done in time O(|a|).

2. Multiplication of a and b can be done in time O(|a| · |b|). In fact, this can be improvedto O(|a| · log |a| · log log |a|) using the discrete Fourier transform (DFT); we do notgive details here.

3. Division of a by b (returning both the quotient and the remainder) can be done intime O(|a| · |b|). Note this means that we can compute a mod b in time O(|a| · |b|).This can be also improved using the DFT.

1

Page 71: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

4. Modular exponentiation (i.e., computing ab mod N) can be done using O(|b|) mul-tiplications modulo N . Since each multiplication modulo N (including the modularreduction) can be done in time O(|N |2), this gives an O(|b| · |N |2) algorithm for expo-nentiation. (Of course, if the complexity of multiplication is improved using the DFTthen the total complexity of exponentiation also improves.) We discuss an efficientalgorithm for exponentiation below.

Addition, subtraction, and multiplication can all be done using the simple, “grade-school” algorithms for these operations. But we elaborate for the case of exponentiation.A naive way of computing ab mod N is to compute a2 = a · a; a3 = a · a2, . . . , ab = a · ab−1

and then finally reduce ab modulo N (what is the time required for this approach?). Asomewhat less naive method is to reduce the value at each step; i.e., compute a2 mod N =a · a mod N ; a3 mod N = a · a2 mod N, . . .. Note, however, that this algorithm has b stepsso the running time will be exponential in |b|!

A more careful algorithm — repeated squaring — is needed. Pseudocode for this algo-rithm follows (we assume b > 0 for simplicity).

Input: a, b,Nif (b = 0) return 1ans = a, tmp = 1// we maintain the invariant that our solution is tmp ∗ ansb mod Nwhile (b > 1) {if (b is odd) {tmp = tmp ∗ ans mod Nb = b − 1 }

ans = ans ∗ ans mod Nb = b/2 }

return (tmp ∗ ans mod N)

Note that this algorithm performs at most 2|b| multiplications and each multiplicationtakes time O(|N |2) (we assume here that a < N — if this is not the case we reduce a beforestarting). So the total running time of the algorithm is O(|b| · |N |2), as claimed above.

1.2 The Euclidean and Extended Euclidean Algorithms

Very frequently, it is necessary to compute gcd(a, b) for two integers a and b. We nowdiscuss an algorithm — the Euclidean algorithm — that can compute this in polynomialtime.

Assume that a > b. The first thing to notice is that gcd(a, b) = gcd(b, a− b) (verify thisyourself). This suggests the following algorithm:

gcd(a, b)if b > a then return gcd(b, a)if b divides a then return breturn gcd(b, a − b)

This recursive algorithm will certainly return the correct answer (the arguments to the gcdfunction always decrease but remain greater than 0, so the algorithm will terminate; when

2

Page 72: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

the algorithm terminates it always returns the correct answer). But what is the running timeof this algorithm? Well, we can’t really say much other than the fact that the largest inputto gcd drops by O(1) each time a recursive call is made. Thus, O(a) recursive calls might benecessary to evaluate gcd(a, b). (In fact, the algorithm may take this long. Let a be odd andconsider the computation gcd(a, 2). This will recursively evaluate gcd(a−2, 2), . . . , gcd(3, 2)which is roughly a/2 recursive evaluations.) But this is not polynomial time!

In fact, a better algorithm is possible. Instead of recursively calling gcd(a − b, b), whynot subtract off as many multiples of b as we can? In other words, we may recursivelycall gcd(a mod b, b). The algorithm then becomes the following (known as the Euclidean

algorithm):

gcd(a, b)if b divides a then return breturn gcd(b, a mod b) (note that a mod b < b)

We can bound the number of recursive calls via the following claim.

Claim The value of the smallest argument to the gcd algorithm drops by at least a factor

of 2 in every two recursive calls.

Proof Let the initial call be gcd(a, b). Assuming this is not the final call (i.e., assumingthat b does not divide a) the next recursive call will be gcd(b, a′ = a mod b). If a′ ≤ b/2 thenwe are done. Otherwise, the next recursive call will be gcd(a′, b − a′). But since a′ > b/2then b − a′ < b/2 and we have proved the claim.

The claim indicates that the algorithm makes at most 2 log b = 2|b| recursive calls, andsince a mod b can be computed in time |a| · |b| the total running time is O(|a| · |b|2) (and,in particular, the algorithm runs in polynomial time).

It is also true (we do not prove it here) that if gcd(a, b) = r then there exist two(possibly negative) integers X,Y with |X| ≤ b/2 and |Y | ≤ a/2 such that Xa + Y b = r.In the particular case when gcd(a, b) = 1 we get Xa + Y b = 1. Computing such X andY will be very useful (we show why below), and in fact can be done using the Extended

Euclidean algorithm. While we do not give pseudocode for this algorithm, we hope thefollowing example and discussion will allow the reader to develop the algorithm on theirown.

Note that every time the Euclidean algorithm recurses, we evaluate a mod b. Insteadof just keeping track of the remainder, we may also keep track of the number of times bdivided a. Thus, besides recursively calling gcd(b, a mod b) we can imagine also recordingthe fact that a − nb = a mod b (where n = b a

bc).

Now, If gcd(a, b) = 1 then the final recursive call of the algorithm will be something likegcd(a′, 1). Let the second to last call to the algorithm be gcd(b′, a′). Then we know thatb′ mod a′ = 1 and moreover that

b′ − n′a′ = 1

for some integer n′. Let the third-to-last call be gcd(a′′, b′). We again know that a′′ mod b′ =a′ and furthermore that

a′′ − n′′b′ = a′

3

Page 73: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

for some integers n′′. Here is the key to the algorithm: substituting for a′ gives:

b′ − n′(a′′ − n′′b′) = 1

⇒ (1 + n′n′′)b′ − n′a′′ = 1.

Continuing in this manner (substituting each time) yields, for each recursive call gcd(ai, bi),integers ni, n

i for which niai + n′

ibi = 1. Ultimately, working backward to the first (initial)function call, we get integers X and Y for which Xa + Y b = 1.

We conclude with an example. Consider gcd(55, 19). The Euclidean algorithm will runas follows:

gcd(55, 19) 55 − 2 · (19) = 17

gcd(19, 17) 19 − 17 = 2

gcd(17, 2) 17 − 8 · (2) = 1

gcd(2, 1) output 1.

Working backwards, we see that 17 − 8 · (2) = 1. Substituting for 2 gives:

17 − 8 · (19 − 17) = 1

⇒ 9 · (17) − 8 · (19) = 1

Substituting for 17 gives: 9 · (55 − 2 · (19)) − 8 · (19) = 1 or:

9 · (55) − 26 · (19) = 1,

the desired answer.

4

Page 74: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ï{Ð?ÑKÒ�Ó2Ô�ÕLÑ×Ö[ØÚÙ�Û(Ü�Ý�Ô�Ø�ÞKÝ�Ð%ßà ÜPá à�â:ã�äHåçæ Ð:ÖzÔ�Ù�ß?è%éVÖLÑ×ÙqÐ&ÖzÙ à ÔLØ�ê?ÖzÙ�ë�Ô�Ý�ê`ì:ØíNÔLÙ�ÛWÓMÕLÕzÙ�Ô-î�ÙqÐ?Ý�ÖLì?Ý�Ð�ï^Ý�Özð

ñ�ò|ó·ô�õÀö�ò ÷bø

ù úüû�ý�û�þ ÿ ��� ý����vþ �� û� ÿ�ý��� tû� ��� ý����vþ ��� ÿ�� �����

��Ó�ì?Ý�Ò�Ó.ÕzÓ2ÓMÐHÝ�Þ×ÔLÓMÝ�ß�Ø·Ý{Ð4è�����Ó2Ô}Ù�Û Ó�� Ý���ê%Þ×ÓMÕ+ÑXÐ �*ì?ÑKé¦ì^Ñ×Ö!�½Ý�Õ+Ð?ÓMéVÓMÕLÕLÝ�ÔLØ·ÖzÙ{ë�ÓMÐ�Ó2Ô�Ý�ÖzÓ.ÞXÝ�ÔLë�Ó�ê?Ô¦Ñ"�|ÓÐ4è�����Ó2Ô�Õ$#&%'��Ý��|ê%Þ×ÓMÕ�ÑKÐ?é2ÞXè?ß�Ó�ÖLì?Ó)(-á+*~éVÔLØ�ê?ÖzÙqÕzØ ÕzÖzÓ,�.-<ÑKÐ/�*ì%ÑKé�ì�Ñ×Ö�ÑKÕ�Ð�ÓMéVÓMÕLÕLÝ�ÔLØHÖzÙ·ë�ÓMÐ�Ó2Ô�Ý�ÖzÓ10325476Ô�Ý�Ð?ß�Ù8� ê?Ô¦Ñ"�|ÓMÕ¤Ý�Ð?ß9�Hè%Þ×ÖLÑ×ê%Þ×Ø-ÖLì�Ó,� ÖzÙ·Ù��%ÖLÝ�ÑKÐ|Ý:�|Ù�ß%è?ÞKè?Õ;}Ý�Ð?ß�ê�Ó2ÔLÛWÓMéVÖ<��ÓMÕLÕLÝ�ë�Ó�Ý�è�ÖLì�ÓMÐ4ÖLÑKé2Ý�ÖLÑ×ÙqÐ-<ÑKÐ=�{ì?ÑKé�ìë�ÓMÐ�Ó2Ô�Ý�ÖLÑKÐ?ë{ÙqÐ�Ó�ê?Ô�Ñ>�|Ó�? å Ý�Ð%ßHÐ�Ù�Ö�Ð�ÓMéVÓMÕLÕ�Ý�Ô�ÑKÞ×Ø·Ô�Ý�Ð%ß�Ù8� å �½Ý�Õ¤Õ�èA@1é2Ñ×ÓMÐ4Ö;#&B{ÙC��Ó2Ò�Ó2Ô,6�.Ó1ì%ÝMÒ�ÓÚÐ?Ù�Ö^Ø�Ó2Ö=�|ÓMÐ4ÖLÑ×ÙqÐ�ÓMßD�{ì�Ó2ÖLì�Ó2ÔÑ×Ö^ÑXÕ-ê ÙqÕ�ÕLÑE�%Þ×Ó�ÖzÙbë�ÓMÐ�Ó2Ô¦Ý�ÖzÓÚÕLè?é¦ì�ê?Ô�Ñ>�|ÓMÕGF3HJI,KLF$MN03OQP7#R��ÓÓ���ê%ÞKÙ�ÔLÓJÖLì?ÑKÕ|ÑKÕ�ÕLè�Óbì�Ó2ÔLÓ�#S-T��ÓtÙqÐ?Þ×ØFëqÑ×Ò�Ó�Ý�ì?ÑKëqì+UoÞ×Ó2Ò�ÓMÞ½Ù�Ò�Ó2Ô�Ò�Ñ×Ó$�SÙ�Û-ÖLì?ÑKÕ�ÖzÙ�ê%ÑKé76½Ý�Ð?ßÀÔ�Ó2Û Ó2Ô�ÖLì?ÓÑKÐ:ÖzÓ2Ô�ÓMÕzÖzÓMß7ÔLÓMÝ�ß�Ó2Ô�ÖzÙWVYX[Z�ÛWÙ�Ô�ÝHÒ�Ó2ÔLØ�ÔLÓMÝ�ß%Ý7�%Þ×Ó{Ý�Ð%ß\�|Ù�Ô�Ó{ÑKÐ+Uoß?Ó2ê?ÖLì�ÖzÔLÓMÝ�Ö]��ÓMÐ:Ö½Ù�Û}ÖLì�Ó-ÑKÕLÕ�è�ÓMÕ�ì�Ó2Ô�Ó�#Q;

*{Õ�ÕLè��|Ó^�.ÓJì?Ý�Ò�ÓbÕzÙ8�|ÓJÝ�Þ×ë�Ù�Ô�ÑKÖLì��`_baLcedgfihjc k3lnm�o�p�k1�*ì?ÑXé�ìFÖLÝ7q�ÓMÕ|Ý�Õ|ÑKÐ�ê%è?ÖHÝ�Ð ÑKÐ4ÖzÓ2ë�Ó2Ô:?ÀÝ�Ð%ßÔLÓ2ÖLè�Ô�Ð?ÕGr5ê?Ô�Ñ"�|Ó,s1Ñ×ÛN?�ÑKÕ�ê%Ô�Ñ"�|Ó·Ý�Ð?ßtrzéVÙ8�|ê�ÙqÕLÑ×ÖzÓ,s�Ù�ÖLì?Ó2Ô]�*ÑKÕzÓ^-T�.Ó��*ÑKÞKÞ ß?ÑXÕLé2è?ÕLÕ½Ó���Ý��|ê%Þ×ÓMÕ*Ù�ÛNÕLè?é¦ìÝ�Þ×ë�Ù�Ô�Ñ×ÖLì��1Õu��ÓMÞ×Ùv�g;#uw�ì?ÑXÕ*ß�Ù�ÓMÕ{Ð?Ù�Ö)Ð�ÓMéVÓMÕ�ÕLÝ�Ô�ÑKÞ×Ø&ÓMÐ?Ý7�`Þ×ÓHè?Õ{ÖzÙ7ë�ÓMÐ�Ó2Ô�Ý�ÖzÓ|ÞKÝ�ÔLë�ÓW-WÔ�Ý�Ð?ß�Ù8�^;½ê?Ô�Ñ>�|ÓMÕè?Ð?Þ×ÓMÕLÕx��Ó-é2Ý�Ð&ÕLÝ�ØÚÕzÙ8��Ó2ÖLì?ÑKÐ�ë�Ý7��Ùqè�Ö.ÖLì?Ó-ß%ÑKÕzÖzÔ�ÑE�`è�ÖLÑ×ÙqÐ�Ù�Û+ê%Ô�Ñ"�|ÓMÕ$#<y}Ó2Ö{z�-T|�;�ß?ÓMÐ�Ù�ÖzÓ-ÖLì�Ó·Ð4è�����Ó2ÔÙ�Û}ê?Ô�Ñ"��ÓMÕNÞ×ÓMÕLÕ.ÖLì?Ý�ÐÚÙ�Ô.Ó,}�è?Ý�Þ`ÖzÙ�|b# æ Ö.ÑXÕ5q Ð�ÙC�{Ð~-<ÑKÐ�ÛWÙ�Ô���Ý�ÞXÞ×ØA;�ÖLì?Ý�Öxz�-T|�;�� ���� �\� ������[� � #'��Ó)é2Ý�Ðë�ÓMÐ�Ó2Ô�Ý�ÖzÓHÝ�Ô�Ý�Ð?ß�Ù8���iU��`Ñ×Ö½ê?Ô�Ñ"�|Ó·è?Õ�ÑKÐ�ë�ÖLì�Ó^ÛWÙqÞKÞ×Ùv�*ÑKÐ�ë|Ý�Þ×ë�Ù�Ô�ÑKÖLì��W�

� o7�no7a�fvk�o�_baLced�o�-�X$��;(*Ó2ê�ÓMÝ�Ö*è?Ð4ÖLÑKÞ ÕLè?é2éVÓMÕLÕLÛWè?Þ��à ì�Ù�ÙqÕzÓ·Ô�Ý�Ð?ß?Ù8��?��`�C�+�,Xn�C�æ Û)_baLcQd:f�h�c k3l7m�o�p�k$-Q?�;.Ùqè�Özê`è�ÖLÕRr5ê?Ô�Ñ>�|Ó,s�ÖLì�ÓMÐJÙqè�Özê`è�Ö<?tÝ�Ð?ßJÓ���Ñ×Ö

B*Ùv���1Ý�Ð:Ø�ß?ÑY��Ó2ÔLÓMÐ4Ö{?��*ÑKÞKÞ<��ÓÚÐ�Ó2ÓMß�ÖzÙtÖzÓMÕLÖHè?Ð4ÖLÑKÞ'��ÓG %Ð%ß�ÙqÐ�ÓG�{ì?ÑKé�ì�ÑKÕ^ê?Ô¦Ñ"�|Óv¡¢��ÓMÞKÞ�6¤ÕLÑXÐ?éVÓz�-T|�; � ������� � 6+ÖLì?Ó|Û Ô�Ý�éVÖLÑKÙqÐ�Ù�Û.ê%Ô�Ñ"�|ÓMÕ·ÑKÕR-T|�£¤Þ×Ù�ëi¤�|�;�£v|¥�¦XC£¤Þ×Ù�ëi¤�|FÝ�Ð?ß ì�ÓMÐ%éVÓ|ÖLì�Ó|Û Ô¦Ý�éVÖLÑ×ÙqÐ�Ù�Û�iU��%Ñ×Ö-ÑKÐ:ÖzÓ2ë�Ó2Ô¦Õ:�*ì?ÑKé¦ìPÝ�Ô�Ó�ê%Ô�Ñ"�|Ó�ÑKÕ)ÔLÙqè�ëqì?Þ×اXC£7��#�á�Ù��.Ó��*ÑKÞXÞ+Ð�Ó2ÓMßPÖzÙJÖzÓMÕzÖ å ÙqÐ�ÝMÒ�Ó2Ô�Ý�ë�Ó å �Ð4è�����Ó2Ô�Õ�è%Ð:ÖLÑKÞ��.Ó� %Ð?ß�ÙqÐ?Óu�*ì?ÑXé�ì�ÑKÕ�ê?Ô�Ñ"��Ó�#&w�ì�è?Õ$6�Ý�ÕLÕLè��1ÑKÐ�ëR_baLcQd:f�h�c k3l7m�o�p�k(Ô¦è?Ð?ÕNÑKÐ1ê�ÙqÞ×Ø�Ð?Ù8��ÑKÝ�ÞÖLÑ"�|Ó�6?ÖLì�Ó·ÓMÐ4ÖLÑ×ÔLÓ � o7�vo7a�fvk�o�_¨a�cQd�oÝ�Þ×ë�Ù�Ô�Ñ×ÖLì���Ô¦è?Ð?Õ�ÑKÐ&ê�ÙqÞ×Ø Ð�Ù8��ÑKÝ�Þ ÖLÑ"��Ó�#

©�ª«© ¬G­�®°¯²±�³°®L´,µ·¶g¸N¹v´�®°º�»w+ÙJéVÙ8�|ê%ÞKÓ2ÖzÓ�Ùqè?Ô-Ý�ÞKë�Ù�Ô�Ñ×ÖLì��8ÛWÙ�Ô)ë�ÓMÐ?Ó2Ô�Ý�ÖLÑKÐ�ë&ê?Ô�Ñ>�|ÓMÕ$6��.Ó|Ð�Ó2ÓMß�ÖzÙJÕzê�ÓMé2Ñ×ÛWØbݼ_¨aLcedgfihjc k3lnm�o�p�k*Ý�Þ×ë�Ù7UÔ�Ñ×ÖLì��W#�%.Ý�Ô�Þ×Ø7ÑKÐ&ÖLì�ÓÕzÓ,��ÓMÕzÖzÓ2Ô,6%í�ÔLÙ�Û°#¾½·Ý�ÕLÝ�Ô�é�ìtëqÝMÒ�Ó�Ý|ëqè�ÓMÕzÖ{Þ×ÓMéVÖLè�ÔLÓ^ß?ÑXÕLé2è?ÕLÕLÑXÐ�ë�Ý~-WÒ�Ó2ÔLØ&ÔLÓMéVÓMÐ:Ö;¿ F$0«F$ÀÁ�KTM�K>Â�03K3I)Ý�Þ×ë�Ù�Ô¦Ñ×ÖLì�� ÛWÙ�ÔNÖzÓMÕzÖLÑKÐ�ëê?Ô�Ñ"��Ý�ÞXÑ×ÖoØ�#¤Ï{Ð�ÛWÙ�ÔLÖLè?Ð%Ý�ÖzÓMÞ×Ø�6qÖLì?ÑKÕ(Ý�ÞKë�Ù�Ô�Ñ×ÖLì�� ÑXÕ(ÕLÞ×ÙC�=Ã4ÑKÐ�ê?Ô¦Ý�é[UÖLÑKéVÓ�6.Û<Ý�ÕzÖzÓ2Ô~À�Ä7M ¿ 47Á�KEÅvF ¿ Ý�Þ×ë�Ù�Ô�Ñ×ÖLìN��Õ|Ý�ÔLÓbè?ÕzÓMß!#�*)Õ9��ÓÆ�*ÑKÞXÞ.ÕzÓ2Ó�6½ì�Ùv�.Ó2Ò�Ó2Ô,6½ÖLì�ÓMÕLÓJÔ�Ý�Ð?ß�Ù8��ÑKð2ÓMßÝ�Þ×ë�Ù�Ô�Ñ×ÖLì��1Õ�ì?ÝMÒ�ÓHÝ�Õ]��Ý�ÞKÞ�ê%ÔLÙ��%Ý7�%ÑKÞXÑ×ÖoØ�Ù�Û¤ÔLÓ2ÖLè�Ô¦Ð?ÑKÐ�ë�ÖLì�Ó ��Ô�ÙqÐ�ë�Ý�Ð?Õ���Ó2Ô,#ÇuÈTÉvÊ$Ë�ËCÉvÌ�Í w�ì?ÓÆ ?Ô�ÕzÖÚÝ�Þ×ë�Ù�Ô�Ñ×ÖLì��Î�.Ó�éVÙqÐ?ÕLÑXß�Ó2ÔG�*ÑXÞKÞ�Ð�Ù�ÖÚéVÙ8��ê%Þ×Ó2ÖzÓMÞ×Ø~ÕzÙqÞKÒ�Ó�Ùqè�ÔÚê?ÔLÙ��%Þ×Ó,�Æ6Ï�`è�Ö�*ÑKÞKÞ¨� Ó�Ý1ë�Ù�Ù ßPÑXÐ?Ñ×ÖLÑKÝ�Þ}ÕLÖzÓ2ê!# Ð.Ó2ÛWÙ�ÔLÓ���ÓHê?Ô�ÓMÕzÓMÐ:Ö)ÖLì?ÑKÕ)Ý�Þ×ë�Ù�Ô�ÑKÖLì��W6�Þ×Ó2Ö$ѯÕ-ÕzÖzÓ2ê¼�%Ý�é]q�Ý�Ð%ßPÕLÓ2Ó��*ì%Ý�Öê?ÔLÙ�ê�Ó2ÔLÖLÑ×ÓMÕ'��Óg�*ÑKÞKÞ+�½Ý�Ð:Ö.Ùqè�Ô�_baLcedgfihjc k3lnm�o�p�kNÝ�Þ×ë�Ù�Ô�Ñ×ÖLìN�DÖzÙ�ì%ÝMÒ�Ó�#ÏÒ)ê%ÖLÑ"��Ý�ÞKÞ×Ø�-WÙqÐ?éVÓu��Ó-Ý�ÔLÓu�*ÑXÞKÞKÑKÐ�ëÖzÙÝ�é2éVÓ2ê?Ö.ÝÕ]��Ý�ÞKÞ ê?Ô�Ù��%Ý7�%ÑKÞKÑKÖoØ·Ù�Û�Ó2ÔLÔLÙ�Ô[;b��Ó���Ùqè?ÞKß|ÞKÑ"q�Ó�ÖLì%Ý�ÖNÛ Ù�ÔgÄ7O�O�ê?Ô¦Ñ"�|ÓMÕ$6�ÖLì�Ó�ê%ÔLÙ��%Ý7�%ÑKÞXÑ×ÖoØ^ÖLì%Ý�Ö

X

Page 75: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

_¨aLcedgfihjc k3lnm�o�p�k^ÔLÓ2ÖLè?Ô�Ð?ÕÓr5ê%Ô�Ñ"�|Ó,stÕLì�Ùqè?ÞKßÔ��ÓÚì?Ñ×ëqì!ÃNéVÙqÐ:Ò�Ó2Ô�ÕLÓMÞ×Ø�6NÛWÙ�ÔWÄ�O�O�éVÙ8�|ê�ÙqÕLÑKÖzÓ7Ð�è���� Ó2Ô¦Õ$6±ÖLì?Óê?ÔLÙ��%Ý7�%ÑXÞKÑ×ÖoØ&ÖLì?Ý�Ö�_baLcQd:f�h�c k3l7m�o�p�k*ÔLÓ2ÖLè�Ô�Ð?Õ^rzéVÙ8�|ê�ÙqÕLÑ×ÖzÓ,s&ÕLì?Ùqè?ÞKßÆ��Ó�ì?Ñ×ëqì�# à ÙqÐ?ÕLÑKß?Ó2Ô{ÖLì�Ó�ÛWÙqÞKÞ×ÙC�{ÑKÐ�ëÝ�Þ×ë�Ù�Ô�Ñ×ÖLì��Æ�

_baLcQd:f�h�c k3l7m�o�p�k�-Q?�;ÛWÙ�Ô�Õ<�ÖX·ÖzÙ9× ê%ÑKé�qÚÔ�Ý�Ð%ß�Ù8�ÙØ�ÚÛ��X7�$Ü$Ü$Üv�3?\ÝJXn�Ñ×Û+ëqé2ߨ-3ؾ�3?�;:Þ�ÖX·ÔLÓ2ÖLè�Ô�ÐßrzéVÙ8��ê ÙqÕ�Ñ×ÖzÓ,s1Ý�Ð?ß&Ó���Ñ×ÖÑ×ÛbØnà7á�â1Þ��X��|Ù ßg?¨6?Ô�Ó2ÖLè�Ô�ÐßrzéVÙ8�|ê�ÙqÕLÑ×ÖzÓ,s1Ý�Ð?ßJÓ���Ñ×ÖÑ×ÛN?Pì?Ý�Õ�ê%Ý�ÕLÕzÓMßbÝ�ÞXÞ�ÖLì�Ó^Ý7��Ù�Ò�ÓÖzÓMÕLÖLÕ$6?ÔLÓ2ÖLè�Ô¦Ðãr5ê?Ô�Ñ"��Ó,s

ä*Ù�ÖzÓ�ÖLì?Ý�Ö*Ñ×Ûb×lÑKÕ*Ý1éVÙqÐ?ÕzÖLÝ�Ð4Ö/-WÙ�ÔC6%ÑKÐJë�ÓMÐ�Ó2Ô¦Ý�ÞL6?ê�ÙqÞ×Ø Ð�Ù8��ÑKÝ�Þ�ÑKÐæå ?'å�;�ÖLì�ÓÝ7��Ù�Ò�Ó�Ý�Þ×ë�Ù�Ô�Ñ×ÖLìN�wÔ�è?Ð?Õ*ÑKÐê ÙqÞKØ�Ð�Ù8�1ÑKÝ�Þ ÖLÑ"�|Ó�#'��Ó��1ÝMØJÐ�Ù�ÖzÓ^ÖLì�ÓÛ ÙqÞKÞKÙC�*ÑKÐ?ëHÛWÓMÝ�ÖLè�ÔLÓMÕ*Ù�Û¤ÖLì%ÑKÕ�Ý�Þ×ë�Ù�Ô�Ñ×ÖLìN�W�ç/èTé È3ê ë�ì ?¥K�Â)í¾À[K�ÁGF�îx0Tï+F�Ä�ð�47ñ7F�Ä7O�ò�4�ÀK30Tï�ÁóÄ7Oe25Ä7PnÂ=À�F,03ô+ÀMAÂ¥õYí¾À[K�ÁGF+ö�÷ø Évù�ù¾ú æ Û�? ÑKÕ)ê?Ô�Ñ>�|Ó�6�ÖLì�ÓMÐPÛ Ù�Ô^Ý�ÞKÞ&ؼÚÔû�üà �.Ó�ì%ÝMÒ�ÓRØnà7á�â��ÙX��|Ù ßg?¨#=w*ì4è?Õ,6�? ê%Ý�Õ�ÕzÓMÕ)Ó2Ò�Ó2ÔLØÑ×ÖzÓ2Ô�Ý�ÖLÑ×ÙqÐJÙ�Û�ÖLì�Ó^Ý7��ÙgÒ�ÓHÝ�Þ×ë�Ù�Ô�ÑKÖLì��wÝ�Ð?ßJÖLì�Ó2Ô�Ó2Û Ù�ÔLÓÖLì�Ó^Ý�Þ×ë�Ù�Ô�Ñ×ÖLìN�ÇÝ�ÞE�½ÝMØ Õ�Ùqè�Özê%è?ÖLÕRr5ê?Ô�Ñ"��Ó,s+#�nì?Ý�Ö�é2Ý�ÐG��Ó{Õ�ÝMØ��*ì�ÓMÐ1?7ÑKÕ(ê%Ô�Ñ"�|Óv¡7Ï)Ð�ÛWÙ�ÔLÖLè?Ð?Ý�ÖzÓMÞ×Ø�64ÖLì�Ó2ÔLÓ)Ý�ÔLÓ{éVÙ8�|ê�ÙqÕLÑKÖzÓ*Ð4èN�=��Ó2Ô�Õ�?ÚÛ Ù�ÔÏ�*ì?ÑKé¦ìØ à7á�â �óXu�|Ù�ßg?ÀÛWÙ�Ô�Ý�ÞXÞxØãÚ²û üà Ã�ÖLì?ÓMÕzÓ7Ð�è�����Ó2Ô�ÕÝ�ÔLÓ&é2Ý�ÞKÞ×ÓMß�ý'Ä7À[Á�K3I$ï�Ä�F�O)M�ô+ÁGð�F$À�Â[#Dþ�Ù�Ô�ÕLè?é¦ìÐ4è�����Ó2Ô�Õ$6�_¨aLcedgfihjc k3lnm�o�p�k<�*ÑKÞKÞ Ùqè?Özê%è�Ö�ÖLì?Ó/-<ÑKÐ?éVÙ�Ô�ÔLÓMéVÖ;(Ý�Ð?Õ]�.Ó2Ô�r5ê?Ô�Ñ"��Ó,s^è?Ð?Þ×ÓMÕ�ÕNÑ×Ö�ì?Ý�ê%ê ÓMÐ%Õ(ÖzÙHê`ÑKé]qÝ�ÐæØPÛWÙ�Ô��*ì%ÑKé�ì�ëqé2ß&-3ØN�3?�;�Þ�ÿX7Ã(ÖLì?ÑXÕ1�*ÑKÞXÞ(ì?Ý�ê?ê�ÓMÐD�*Ñ×ÖLì�Þ×ÙC�üê?ÔLÙ��%Ý7�%ÑXÞKÑ×ÖoØ�#�þ�Ù�Ô�Ð�ÙqÐ+U à Ý�Ô��ÑKé�ì%Ý�ÓMÞÐ4è�����Ó2Ô�Õ$6+��Óé2Ý�Ð�é2ÞKÝ�Ñ>��ÖLì�Ó^ÛWÙqÞKÞ×ÙC�{ÑKÐ�ë��*Ñ×ÖLì�Ùqè?Ö�ê%ÔLÙ4Ù�Û��ç/èTé È3ê ë�ì ? K�Â^I47Á�í+4nÂK30«FÛÄ7M ¿ 0TïAF�À]FÓF � K>Â�0TÂ�Ä�0gO"F]ÄnÂ�0=47M�F9ØãÚtû�üà ì 47À^2<ï�K3I$ï~Ønà7á�â~Þ� X��|Ù ß?���KL÷3F,÷"î�? K>¼M¾4�0^Ä ý'Ä7À[Á�K3I$ï�Ä�F$O1M�ô+ÁGð�F�À��CîÆ0Tï+F$MÖF�Ä8I�ï·K30«F�À�Ä�03K34�MÖ4 ì _¨aLcedgfihjc k3lnm�o�p�k[-Q?�;§47ôA0Qí¾ôA0TÂõ°I�4�ÁuíA4nÂK30«F�ö/2'KT0Tï/í¾À�4�ð]Äið�KTOQK303PÆÄ�05O"F]Ä7Â[0)XC£��A÷w�ì4è%Õ$6�ÑKÖzÓ2Ô�Ý�ÖLÑKÐ�ë=×�ÖLÑ>�|ÓMÕg-<Ý�Õ�ß?ÙqÐ�Ó{Ý7��Ù�Ò�Óv;.ÓMÐ%ÕLè�ÔLÓMÕ.ÖLì?Ý�Ö�ÖLì�Ó{ê%ÔLÙ��%Ý7�%ÑKÞXÑ×ÖoØ�ÖLì?Ý�Ö:_¨a�cQdgfih�c k3lnm�o�p�k��{ÑKÞKÞÔLÓ2ÖLè�Ô�ÐFÖLì�Ó^��ÔLÙqÐ�ë�Ý�Ð?Õ]�.Ó2Ô|ÑKÐ!ÖLì?ÑKÕ�é2Ý�ÕzÓbÑKÕ�Ý�Ö9�|ÙqÕzÖWXC£���5#§w�ì%ÑKÕê%ÔLÙ��%Ý7�%ÑKÞXÑ×ÖoØ�é2Ý�Ð���Ó\��Ý�ß?Ó&Ý�ÕÕ]��Ý�ÞKÞ Ý�Õ{ß�ÓMÕLÑ×ÔLÓMß���Ø&ÑKÐ?éVÔLÓMÝ�ÕLÑXÐ�ë/×=#

æ Ð�ÕLè��R��Ý�Ô�Ø�6�ÖLì�ÓÝ7��Ù�Ò�ÓÝ�ÞKë�Ù�Ô�Ñ×ÖLì��Çì%Ý�Õ�ÖLì�Ó·ÛWÙqÞKÞ×ÙC�{ÑKÐ�ë�ê?ÔLÙ�ê�Ó2ÔLÖLÑ×ÓMÕ,�X7# æ ÛN?PÑKÕ½ê?Ô�Ñ"��Ó�6�ÖLì?ÓMЧ_¨a�cQdgfih�c k3lnm�o�p�k½Ý�ÞE�½ÝMØ Õ�ÔLÓ2ÖLè�Ô¦Ð?Õ9r5ê?Ô�Ñ"��Ó,s+#��# æ ÛA?�ÑKÕ�Ý à Ý�Ô��ÑKé�ì%Ý�ÓMÞ Ð�è�����Ó2Ô,6 ÖLì�ÓMÐD_¨a�cQdgfih�c k3lnm�o�p�k:-<Ý�Þ"�|ÙqÕzÖ;�Ý�ÞE�½ÝMØ Õ�ÔLÓ2ÖLè?Ô�Ð?Õ�ÖLì�Ó·ÑKÐ?éVÙ�ÔLÔ�ÓMéVÖ

Ý�Ð?Õ]�.Ó2ÔGr5ê%Ô�Ñ"�|Ó,s+# # æ Û�?7ÑKÕ.éVÙ8�|ê�ÙqÕLÑ×ÖzÓ{Ý�Ð%ß1Ð�Ù�Ö�Ý à Ý�Ô��1ÑKé�ì?Ý�ÓMÞ`Ð4èN�=��Ó2Ô,6qÖLì?ÓMÐÓ_baLcQd:f�h�c k3l7m�o�p�k(Ô�Ó2ÖLè�Ô�Ð?Õ(ÖLì?Ó{éVÙ�ÔLÔLÓMéVÖ

Ý�Ð?Õ]�.Ó2ÔGrzéVÙ8��ê ÙqÕ�Ñ×ÖzÓ,s|Ó���éVÓ2ê?Ö��*Ñ×ÖLì&ê%ÔLÙ��%Ý7�%ÑKÞXÑ×ÖoØ1Ý�Ö��|ÙqÕzÖ=XC£���Ï#*{Õ�Ýê%Ô�Ý�éVÖLÑKé2Ý�Þ���Ý�ÖzÖzÓ2Ô,6 à Ý�Ô���ÑXé�ì?Ý�ÓMÞ`Ð4è�����Ó2Ô�Õ�Ý�Ô�Ó{ÔLÓMÞXÝ�ÖLÑ×Ò�ÓMÞ×Ø�Ô�Ý�ÔLÓ�-<Ý�Þ×ÖLì�Ùqè�ëqìÚÖLì�Ó2ÔLÓ-Ý�ÔLÓ{ÑXÐA %Ð?Ñ×ÖzÓMÞKØ��Ý�Ð:Ø�;#�á�ÙA6�Ñ×Û ÖzÔ�Ø�ÑKÐ?ëHÖzÙ�ë�ÓMÐ�Ó2Ô�Ý�ÖzÓ^ÝHÔ�Ý�Ð?ß?Ù8�üê%Ô�Ñ"�|Ó�-<è?ÕLÑKÐ�ë � o7�vo7aTfCk�o�_baLcQdgo)Ý�Õ�ß%ÑKÕLé2è?ÕLÕLÓMß1Ý7��Ù�Ò�Óv;6ÖLì�ÓNê?ÔLÙ��`Ý7�%ÑKÞKÑ×Ö[Ø*Ù�Û ê%ÑKé�q�ÑXÐ�ë�Ý à Ý�Ô��1ÑKé�ì?Ý�ÓMÞ:Ð4è�����Ó2Ô+ÑKзÖLì?Ó� ?Ô�ÕzÖ ê`ÞKÝ�éVÓ�ÑKÕ}ÞKÙC�1#(á�ÖLÑKÞKÞL6C�.Ó���Ùqè?ÞKß^ê?ÔLÓ2ÛWÓ2ÔÝ�Ð Ý�Þ×ë�Ù�Ô�Ñ×ÖLìN� �*Ñ×ÖLì�Þ×ÙC� ÛWÝ�ÑKÞXè�ÔLÓ�ê?ÔLÙ��%Ý7�%ÑXÞKÑ×ÖoØJÛWÙ�ÔGÄ7OTO�éVÙ8��ê ÙqÕ�Ñ×ÖzÓ�Ð4èN�=��Ó2Ô�Õ$#/w�ì�Ó|Ð�Ó�� Ö^Ý�Þ×ë�Ù�Ô�Ñ×ÖLìN��*ÑKÞKÞ�Ý�é�ì%Ñ×Ó2Ò�ÓÖLì?ÑXÕ$#� ê �&Évù������ é¾è�� ùNÉ7È�Ë�bêJÍ w�ì?ÓÚÐ�Ó�� ÖHÝ�Þ×ë�Ù�Ô¦Ñ×ÖLì�� å q Ð�ÙC�{ÐFÝ�ÕHÖLì�ÓJÜ�ÑXÞKÞ×Ó2Ô�U«({Ý7�%ÑKÐ Ý�Þ×ë�Ù�Ô¦Ñ×ÖLì�� å�*ÑKÞKÞ�ì?Ý�Ò�ÓÖLì�Ó·ê?Ô�Ù�ê Ó2Ô�ÖLÑ×ÓMÕ)�.ÓHß�ÓMÕLÑ×ÔLÓ�#

Page 76: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

� cQhQh o7a���f��8ce��-Q?�;��Ô�ÑKÖzÓ)?\ÝãX^Ý�Õ�������Û Ù�Ô*Ù�ß?ß��ÛWÙ�Ô�Õ&��X^ÖzÙ9×ê`ÑKé]q7Ô�Ý�Ð?ß�Ù8��Ø\Ú¼��X7�$Ü$Ü$Üv�3?^ÝãXn�ÑKÛ+ëqé2ßb-3ؾ�3?�;:Þ�ÖX·ÔLÓ2ÖLè�Ô�ÐßrzéVÙ8��ê ÙqÕ�Ñ×ÖzÓ,s�Ý�Ð?ßJÓ���Ñ×ÖÑKÛbØ� �Þ�ÖX{�|Ù�ß ?tÝ�Ð?ßãÞ !A��ÚæV �+�#"uÝãX[Z¤ÕLè?é�ìbÖLì?Ý�ÖuØ ¤�$ ���Ý1X{��Ù�ß:?ÔLÓ2ÖLè�Ô�ÐßrzéVÙ8��ê ÙqÕ�Ñ×ÖzÓ,s�Ý�Ð?ßJÓ���Ñ×Ö

Ùqè�Özê`è�ÖRr5ê?Ô�Ñ"��Ó,s�Ý�Ð%ßJÓ�� ÑKÖþ¤ÑKÔ�ÕzÖ$6?Ñ×Ö*ÑXÕ�Ñ"�|ê�Ù�ÔLÖLÝ�Ð4Ö�ÖzÙ1Ð�Ù�ÖzÓ^ÖLì?Ý�Ö*ÖLì?ÑKÕ�Ý�Þ×ë�Ù�Ô¦Ñ×ÖLì���Ô�è?Ð%Õ�ÑKÐ&ê�ÙqÞ×Ø�Ð?Ù8��ÑKÝ�Þ ÖLÑ"�|Ó�#�á ÖzÓ2êÔX^ß�Ù�ÓMÕ1M¾480ÑKÐ:Ò�ÙqÞKÒ�Ó*Û<Ý�éVÖzÙ�Ô�ÑKÐ�ë\-Q? ÝÛXC;Ã�Ý�ÞKÞ��.Ó*Ý�Ô�Ó{ß�ÙqÑKÐ?ë^ÑKÕ(ê%è?ÞXÞKÑKÐ�ë)Ùqè�ÖNÛ<Ý�éVÖzÙ�Ô�Õ.Ù�Û%��6��*ì?ÑXé�ì�ÑKÕNÓMÝ�ÕzØ�#�á�ÓMéVÙqÐ?ß%Þ×Ø�6é�ì�ÓMé�q�ÑKÐ?ë\�*ì?Ó2ÖLì�Ó2Ô*ÖLì�Ó2Ô�Ó^Ó�� ÑXÕzÖLÕ{Ý^�ÆÚ�V �+�#"gÝJX[Z¤Õ�è?é�ì�ÖLì?Ý�ÖgØ ¤ $ � Ý1Xu�|Ù�ßg?PÔLÓ,}�è?Ñ×ÔLÓMÕ�é¦ì�ÓMé]q ÑKÐ�ëÝ�Ö���ÙqÕzÖ&"7ÕLè%é�ìFÒ�Ý�ÞXè�ÓMÕ$Ã'�%è�Ö&"&ÑKÕ�Ý�Ö/�|ÙqÕLÖ�Þ×Ù�ëi¤�? ÕzÙPÖLì?ÑKÕHÑKÕ�ÕzÖLÑKÞKÞ�ê ÙqÞKØ�Ð�Ù8�1ÑKÝ�Þ�ÖLÑ"�|ÓD-WÙ�Û*éVÙqè�Ô�ÕLÓ�6éVÙ8�|ê%è�ÖLÑKÐ?ë1ÖLì�Ó�Ó���ê�ÙqÐ�ÓMÐ4ÖLÑKÝ�ÖLÑ×ÙqÐPé2Ý�Ð�Ý�ÞKÕzÙ^��ÓHß�ÙqÐ�Ó�ÑKÐtê ÙqÞKØ�Ð�Ù8�1ÑKÝ�Þ ÖLÑ"�|Ó�Ý�Õ)ÖLì�ÓHÓ�� ê�ÙqÐ�ÓMÐ4Ö)Ð�Ó2Ò�Ó2Ôë�Ó2ÖLÕu�%ÑKë�ë�Ó2Ô�ÖLì?Ý�Ð�?\ÝJXC;#ç/èTé È3ê ë�ì ?¼í+ÄnÂ�Â,F[ÂGÄ7M§K30«F�À�Ä�03KL47M¥4 ì 0Tï+F('¼KTO�O"F�À*),+{Ä�ð�KTMÔ0«F�Â[0 ì 47ÀRÄ�í+Ä7À�03K3I,ô+OYÄ7À9ñ7Ä7Oeô�FG4 ì Ø�î{0Tï+F�MØvà7á�âx��Xu�|Ù�ß ?ßÄ7M ¿ ïAF�M¾I[F5?�254�ô�O ¿ í+ÄnÂÂ/Ä7M¼K30«F�À�Ä�03K34�MÔ4 ì _baLcQd:f�h�c k3l7m�o�p�k ì 47À=0Tï+Ä�0xÂ�Ä7ÁGF:Ø�÷ø Évù�ù¾ú ä{Ù�ÖzÓ&ÖLì?Ý�Ö9Øvà�á�âR� Ø ¤.- �|Ù ßg?¨# æ Û'?Àê%Ý�ÕLÕzÓMÕ�Ý�ÐÀÑ×ÖzÓ2Ô¦Ý�ÖLÑ×ÙqÐFÙ�Û�ÖLì�ÓbÜ�ÑXÞKÞ×Ó2Ô�U«({Ý7�%ÑKÐ�ÖzÓMÕzÖÖLì�ÓMÐ ÓMÑ×ÖLì?Ó2Ô=Ø �¦X��|Ù ß:?�Ù�ÔÓMÞXÕzÓ|ÖLì�Ó2ÔLÓ�Ó���ÑKÕzÖLÕÝÓ�¥Ú V �+�#"1Ý X[Z.ÕLè%é�ì�ÖLì?Ý�Ö�Ø ¤ $ �¦Ý1Xu�|Ù�ßg?b#æ ÐÀÖLì?ÓW ?Ô�ÕzÖ|é2Ý�ÕzÓ�6)��Ó�ì?ÝMÒ�Ó~Ø ¤ - Ó�Î-3Ø� ,; ¤ - � X ¤ - � Xu�|Ù�ßg?b# æ Ð ÖLì�ÓbÕLÓMéVÙqÐ?ßvé2Ý�ÕzÓ�6Ï��Ó�ì?ÝMÒ�ÓØ ¤ - � -3Ø ¤ $ ; ¤ -�/ $ � -�Ý=XC; ¤ -�/ $ �|Ù ßg?¨#Fá ÑKÐ%éVÓ�"10¦��6��.ÓJé2Ý�Ð���Ô¦Ñ×ÖzÓÚÖLì%ÑKÕ�Ý�ÕÆ-�-�Ý1XC; ¤ ; ¤ -�/ $ /�2 �X ¤.-�/ $ /�2 �ÖX��|Ù ßg?¨#æ Ð�Û<Ý�éVÖ$6�Ý�Õ ��Ý�ÕÖLì�Ó1é2Ý�ÕzÓG�*Ñ×ÖLìß_baLcQd:f�h�c k3l7m�o�p�k[6+ÖLì?Ó1Ü�ÑKÞXÞ×Ó2Ô�U«(*Ý7�`ÑKÐPÖzÓMÕzÖÝ�ÞE�½ÝMØ Õ^Ùqè�Özê`è�ÖLÕÆr5ê?Ô�Ñ>�|Ó,s�*ì�ÓMÐJëqÑ×Ò�ÓMÐ�Ý|ê?Ô�Ñ"��Ó-Ý�Õ*ÑKÐ�ê%è�Ö$#ç/èTé È3ê ë�ì ?¥K�Â)í¾À[K�ÁGF�îx0Tï+F3'¼KTOTOEF$À4),+{Ä�ð$K�MDÄ7O�ò�47À[KT0TïiÁÿÄ�OQ25Ä7PnÂ/À�F,03ô�À[MAÂÔõEí¾ÀKTÁGF�ö$÷ø Évù�ù¾ú ��ÓJÕLì�Ùv��ÖLì?Ý�ÖHÛWÙ�Ô|Ý�ÞKÞÏØßÚS��X7�$Ü$Ü$Üv�3?ÛÝ Xn�Ó�¦û üà 6NÝ�Ð Ñ×ÖzÓ2Ô�Ý�ÖLÑ×ÙqÐ!Ù�Û�ÖLì�ÓJÜ�ÑXÞKÞ×Ó2Ô�U«({Ý7�%ÑKÐÖzÓMÕzÖ·è?ÕLÑKÐ�ëÚÖLì?ÑKÕ{Ò�Ý�ÞXè�ÓÙ�ÛÏØbÕLè?é2éVÓ2ÓMß?Õ,#:þ�ÑY�Óؾ# à ÙqÐ?ÕLÑXß�Ó2Ô*ÖLì�Ó�ÕzÓ,}�è�ÓMÐ?éVÓÆ-T|651��Ø ��| â ��Ø

¤ ��| ¤ �Ø ¤ � �$Ü$Ü$ÜC��| � �·Ø

¤ - ;�-T�*ì�Ó2ÔLÓ-Ý�ÞKÞ`Ò�Ý�ÞXè�ÓMÕ½Ý�ÔLÓ{ÖLÝ7q�ÓMÐW��Ù�ß?è%Þ×Ù{?�;#.á ÑXÐ?éVÓ����4�=�æ?�ݧX76+��Ógq Ð�Ùv�nÖLì%Ý�ÖÖLì�Ó1ÞKÝ�ÕzÖÖzÓ2Ô�� ÑKÐ�ÖLì%ÑKÕ·ÕzÓ,}�è�ÓMÐ?éVÓ1ÑKÕGX7#\ä{ÙC�16�Ñ×Û)Ø7 9� X{�|Ù�ß ?Ô�.ÓÚÝ�ÔLÓ1ß�ÙqÐ�Ó�6¤ÕLÑKÐ?éVÓ|ÖLì%ÑKÕ^Ñ"�|ê`ÞKÑ×ÓMÕÖLì?Ý�Ö*ÖLì�ÓHÑ×ÖzÓ2Ô�Ý�ÖLÑ×ÙqÐtÕLè?é2éVÓ2ÓMß?Õ$#�Ò)ÖLì�Ó2Ô��*ÑKÕzÓ�6�ÖLì�Ó2Ô�Ó=�Hè%ÕzÖ���Ó^Ý�Ô�ÑKëqì:Ö]�|ÙqÕLÖ�ÓMÞ×Ó,�|ÓMÐ4Ö�Ù�Û(ÖLì�ÓÕLÓ,}4è�ÓMÐ%éVÓ-<ÕLÝMØ�6!|68L;u�*ì%ÑKé�ìPÑKÕ�M¾4�0=X7#1нè�Ö-ÕLÑKÐ%éVÓ�|68:9 â �¢|

¤8 �|Ù ßg?¨6 Ý�Ð?ß�ÕLÑKÐ?éVÓ�-T�4ØPÝ�ÕLÕLè��|ê%ÖLÑ×Ùqо;)|68;9 â �ÙX76�.Ó9��è?ÕzÖ·ì?ÝMÒ�Ó�|68��=<�X^-<Õ�ÑKÐ?éVÓg? ÑKÕ*ê%Ô�Ñ"�|Óv;# w�ì�è?Õ$6�|>85��Ý1X{��Ù�ß:? Ý�Ð?ß!6�Õ�ÑKÐ?éVÓ/|>8���Ø ¤.? 6�ÖLì?ÑXÕÑ"�|ê%ÞKÑKÓMÕ.ÖLì%Ý�Ö�ÖLì�Ó^Ñ×ÖzÓ2Ô�Ý�ÖLÑKÙqÐbÕLè?é2éVÓ2ÓMß?Õ,#��ÓHÕzÖLÝ�ÖzÓHÖLì�Ó·Û ÙqÞXÞ×ÙC�*ÑXÐ�ë��*Ñ×ÖLì�Ùqè�Ö½ê?Ô�Ù4Ù�Û�#ç/èTé È3ê ë�ì ?ßK>Â=I�4�ÁuíA4nÂK30«FA@ F$ñ7F$M~K ì ?�K>Â\ý'Ä7À[Á�K3I$ï�Ä�F$OB@ 0Tï+F$MæF]Ä8I$ïWKT0«F$À�Ä�03KL47Mæ4 ì 0Tï+F3'¼KTOTOEF$À4)+{Ä�ð�KTM§0«F[Â�0Ï4�ô+0Qí�ô+0TÂÔõ�I�47Á�í+47ÂK30«F�ö�2'K30Tï/í¾À�4�ð]Ä�ð$K�OeK303PWÄ805O"F]ÄnÂ�0 £ â ÷w�ì4è%Õ$6)ÑXÐnÝ�ß%ß?Ñ×ÖLÑ×ÙqÐnÖzÙß A��ÑKÐ�ë ÖLì�Ó�ê?ÔLÙ��`Þ×Ó,� Ù�ÛÛ<Ý�ÞKÕzÓMÞ×ØnÑKß�ÓMÐ4ÖLÑ×ÛWØ�ÑKÐ?ë à Ý�Ô��1ÑKé�ì?Ý�ÓMÞ·Ð�è�����Ó2Ô�Õ$6*ÖLì?ÓÜ�ÑKÞXÞ×Ó2Ô�U«(*Ý7�`ÑKÐ7ÖzÓMÕzÖ*Ý�ÞKÕzÙ�Ý�é¦ì?Ñ×Ó2Ò�ÓMÕ)Ý�Þ×Ùv�.Ó2Ô�Û<Ý�ÑKÞKè?ÔLÓ-ê?ÔLÙ��%Ý7�%ÑXÞKÑ×ÖoØ1ê Ó2Ô�ÑKÖzÓ2Ô�Ý�ÖLÑ×ÙqÐDC

*{Õ{Ý7��Ù�Ò�Ó�6N��Ø&Ý�ßFEzè%ÕzÖLÑKÐ�ë/×S��Óé2Ý�ÐbÝ�é¦ì?Ñ×Ó2Ò�ÓHÝ�Õ{Þ×ÙC�lÝ�ÐJÓ2ÔLÔLÙ�Ô�ê%ÔLÙ��%Ý7�%ÑKÞXÑ×ÖoØ1Ý�Õu�.Ó^ß?ÓMÕLÑ×ÔLÓ�#

G ûIHMûNþ û�ý&J(û5VYX[Z/y�#jä�# à ì%ÑKÞKß?Õ$#LK ý'47M¾I,À�F,0«F ë MN03À�4 ¿ ô�I,03KL47MÔ0�4NM1KYò�ï+F�ÀOKgO�òiFvð�À�ÄP�,Q8M ¿SRu¿ K303K347MT�v# á�ê%Ô�ÑKÐ�ë�Ó2Ô,6�XVUWU ã #

Page 77: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

XOY[Z;\W]_^#`�Z:a�bdcWeBfhg�^#b�i;gWY>jk fPl knm7oWp(qsr Y7a.^#c�j[t>uva�Z:cTY�a.c k ^�b�w[a.cWxW^#g�w6y7bzI^�cWe{]V`�`.cW^}|WcTY[g�a�y[gWYh~�g�a.�

���N�3������� ���

� �����&���D�n���#�&�������4���I� �¡�&¢¤£��I�[�D�#�&�¥�����#�¦�¡§#�.��¨l�]_]©YLcWa.]V`ªe�cW^ª«%]Vuva�tL^�](¬ p�­

® �°¯N±N§4��²´³�µ¶�B¨¸·¹��¨(º(��»ª�ª�D�¡ºN¼3¨½ y�t>`}e{g�^©Z¾Y�a�y[]Su_i;gW`�`_¿�Àn]Ay[gÁ\W]ScTY[i:bPj[Z;`�u_t>`�`.]VjPa�yL]Nw[^#Z;\Fg�a.]vÂ,ÃW]_bÅÄÆgWi;`.cÇÃÈYLcÀªY1gW`ÊÉ.`.]Vuv^�]_aËÂ,ÃW]_bÈÌcW^ÊÉ.`.b�ÍAÍN]_a.^#Z;u*Â,ÃW]_bÈÌ7Î}`.]_a.a�Z;Y[x ­(Ï ]_^�]W¿�a�ÀncÇw>g�^�a�Z;]V`}ÀªyLcÐÀngWY�a.]Vj1a.cÊuvcTÍSÍÑt[Y[Z;u_g�a.]N`.]Vu_t[^�]Vi:b�y>gWja.cÐ`.cTÍS]VyLcÁÀ `�y[g�^�](Z;YÊgWjL\�gWY[uv]&gS^#gWY[j[cTͶÃW]_b6Ò�eÆtL^�a�yL]_^4ÍNcW^�]W¿[a�yL]ÑuvcTÍNw>i:]_a.]Ñ`.]Vuv^�]VuvbÇcWeBa�y[Z¾`�ÃW]_bYL]_]VjL]Vj¹a.cÔÓ�]ÕÍSgWZ;Y�a�gWZ;YL]Vj°g�a�gWi¾i¡a�Z;ÍN] ­ ½ y[Z¾`Ag�w[w[^#cTgWu#y¹a.c�uv^�b�w>a.cWxW^#g�w>y�b�y[gW`ÐgÖY�t[Í(Ó�]_^AcWej[Z;`�gWjL\�gWY�a�g�xW]V`¡Àªy[Z¾u#y�Àn]©`.ÃW]_a�u#yPyL]_^�]W×Ø ­ ½ y[](w>g�^#a�Z:]V`3Í(t>`.a}Ó�]Ng�Ó>i;]Ña.cÊ`�cTÍN]VyLcÁÀ�`�y[g�^�]Ng�^#gWY[j[cTÍÙÃW]_bÕZ;YÕgÇ`.]Vu_tL^#]&ÍAgWY[YL]_^ ­ÑÚ t[aZ:eÛa�yL]_bÜu_gWYÝjLc�a�y[Z;`_¿Þa�yL]_bÜÍSZ;xTy7a¡ß.t[`�a©gW`ÑÀà]Vi;ià`�y[g�^�]da�yL]VZ:^ÑÍN]V`�`�g�xW]�Z;YÜa�y[]Ð`�gWÍS]d`.]Vu_tL^#]ÍSgWY>YL]_^Vá�Ä,â�e�uvcTtL^#`.]W¿Èa�y[Z¾`�Z;`àYLcWaàgWi:ÀÛgVbÈ`nw�cT`�`�Z:Ó>i;] ­´ã cW^Û]väLgWÍNw>i:]W¿�a�yL]ªw6g�^�a�Z:]V`�ÍAgVbSÓ�]Og�Ó6i:]a.cÜ`.]Vu_t[^�]Vi:bÔ`#y[g�^�]ÊgÕÃW]_b¹å6æ�çèÓ>t[aÑÀOZ;i;iàÀngWY�aNa.c uvcTÍSÍÑt[Y[Z;u_g�a.]Ça.cTÍScW^�^�cÁÀéÀªyL]VY�`.]Vu_tL^#]a.^#gWY>`�ÍSZ;`�`#Z:cTY�Z;`�YLcSi:cTYLxW]_^ªw�cT`#`�Z:Ó>i:] ­ Î

¬ ­Oê i:cTYLxSa�y[]©`�gWÍN]Ñi;Z;YL]V`_¿[Y[cWa.]©a�y[g�a�Z;YÊ`.cTÍN]Ñu_gW`.]V`O]_\W]VY�ÍN]_]_a�Z¾YLxdgd`�Z;YLxTi:]3a�Z¾ÍN]�a.cd`�]Vu_tL^�]Vi:bg�xW^�]_]dtLw�cTYPg�ÃW]_b1Z;`3Y[cWa3w�cT`#`�Z:Ó>i:] ­(ë y>g�a�Z:e�cTtL^(`.]VY[jL]_^�gWY[jÕ^#]Vuv]VZ:\W]_^©g�^�]SZ;Y1gÇuv^#cÁÀªjL]Vj^�c�cTͦÄ{cW^ªg�^#]©gWi:ÀÛgVbÈ`ªÍNcTY[Z;a.cW^�]Vj6În`.cSa�y>g�a¡a�yL]_b�u_gWY[YLcWaª`�y>g�^�]�gS`.]Vuv^�]_aªÃW]_bLì

í ­ªr Y�a�yL]nw[^#Z;\Fg�a.]vÂ,ÃW]_b©ÍSc�jL]Vi,¿bWcTt&gWi:ÀÛgVbÈ`îYL]_]Vj©a.c}ÃÈYLcÁÀïÄÆZ;YÑgWjL\�gWY[uv]ÎDa�y[]àw>g�^#a�Z:]V`%ÀOZ:a�y�Àªy[cTÍbWcTt¹ÀÛgWY7aÐa.c�uvcTÍSÍ(t>Y[Z;u_g�a.] ­ ê u_uv]_w[a�Z¾YLxÖuvcTÍAÍ(t[Y>Z;u_g�a�Z:cTY¹e�^#cTÍðgWY�b�Ó�cÈjLb�Z;`d]V`�`.]VY�a�Z;gWi;i:bZ;ÍSw�cT`#`�Z:Ó>i:]ÑZ;Y1a�yL]d`.]Vuv^�]_aËÂ,ÃW]_b `.]_a.a�Z¾YLx ­ ÄÆñIäÈgWÍNw6i:]V`�cWeÛÀªyL]_^#]NbWcTtÔÍSZ;xTy7a3ÀÛgWY7a(a.c�gWu_uv]_w[auvcTÍSÍÑt[Y[Z¾u_g�a�Z:cTYÜe{^�cTͦgWY�bWcTYL]ÐÍSZ:xTy�a(Z;Y[u_i¾t[jL]W×�ghuvcTÍSw>gWY7b1a�y>g�a©ÀÛgWY7a�`Ña.c�gWu_uv]_w>aÑuv^�]Vj>Z:au_g�^#j�Y�t>Í©Ó�]_^#`Ûe{^�cTÍégWY�bWcTYL]W¿�cW^OgAxWcÁ\W]_^#Y[ÍS]VY7a}g�xW]VY[uvbÇa�y[g�aOÀngWY�a�`OgWY�bWcTYL](a.cAÓ�]©g�Ó>i:]©a.c`�t[Ó>ÍSZ:aÛ`�tLxWxW]V`.a�Z;cTY[`4Î

mL­Oã Z;Y[gWi¾i:bW¿BZ:e�Àà]�ÀÛgWY7aNa.c1`.]Vu_t[^�]Çg�YL]_a�ÀàcW^#ÃÔÀªZ;a�y ò¹w>g�^�a�Z:]V`ÇÄ{^�]_w>^�]V`.]VY�a�Z;YLxL¿I`�gÁbW¿�ò¹w�]_cWw6i:]ÀncW^�ÃÈZ;YLxSe{cW^ªa�yL]Ñ`�gWÍN]ÑuvcTÍNw>gWY�bLΡa�yL]VYÊÀn]&YL]_]VjÊ]_\W]_^�bWcTY[](a.c�`�y[g�^�](gAÃW]_bÇÀªZ;a�yÊ]_\W]_^�bWcTYL]]Vi;`�]W¿Le�cW^ªgSa.cWa�gWi�cWe�óÆô õ_öO÷ùøAÄ{ò õ ÎÛÃW]_b�`ªcÁ\W]_^#gWi;i ­

½ y[Z;`ÑÍNcWa�Z:\�g�a.]V`&a�yL]ÇZ;Y�a.^�cÈj[t[uva�Z:cTY cWe�ú6û>üvýÿþ��������������4ú� ,æ������4ú���� ­ lLZ;Y[uv]AÀn]�ÀªZ;i;i�Ó�]_xTZ;Y ÀOZ:a�yg j[Z;`#u_t[`�`�Z:cTYÖcWe}w>tLÓ6i;Z;u*Â,ÃW]_bÖ]VY[uv^�b�w[a�Z:cTYD¿àÀà]Pj[Z;`�u_t[`#`&a�y[Z;`N]väLgWÍNw>i:]ÊY[cÁÀ ­�Ï ]_^#]W¿ÛgÜw>g�^�a�b�ÀOyLcÀªZ;`�yL]V`Aa.cÝ^�]Vuv]VZ:\W]ÕuvcTÍSÍ(t>Y[Z;u_g�a�Z:cTY°xW]VYL]_^#g�a.]V`�a�ÀàcÝÃW]_bÈ`_× gÔw>tLÓ>i;Z¾u*Â,ÃW]_bÝgWY>j°gÖuvcW^�^�]V`.w�cTY[j>Z;YLx`.]Vuv^�]_aÛÃW]_bPÄ{cW^Ûw[^#Z;\Fg�a.]OÃW]_bLÎ ­Ië ]}ÀªZ¾i;i[jL]VYLcWa.]�a�yL]V`.]�Ó�b�����gWY[j����Ö¿�^�]V`.w�]Vuva�Z:\W]Vi:b ­ ½ y[]Ow>t[Ó>i;Z;u*ÂÃW]_bdZ;`àÀªZ;jL]Vi;b&j>Z;`.a.^#Z:Ó6tLa.]VjD×�]_\W]_^�bWcTY[]AÄÆZ;Y[u_i;t[j>Z;YLx�gWY�gWjL\W]_^#`�g�^#bLÎ�Z;`ngW`�`�t[ÍN]Vjda.c&Ã�Y[cÁÀïg�Ó�cTtLa�a�y[Z¾`

Ø

Page 78: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

w�]_^4`.cTY�� `Ow>tLÓ>i;Z¾u3ÃW]_b ­� cÀ�¿�gWY�bWcTYL]Nu_gWYPuvcTÍAÍ(t[Y>Z;u_g�a.]&ÀªZ:a�y�a�y[Z;`Ow>g�^�a�b�t[`�Z;YLxdcTY[i:bÇa�y[Z¾`ªw>t[Ó>i;Z;uÃW]_b6Ò6a�yL]3^�]Vuv]VZ:\W]_^�u_gWYhjL]Vuv^�b�w[a¡t[`#Z;YLx&cTY[i:bÐy[Z;`�`.]Vuv^�]_aªÃW]_b ­

cWa.]ha�y[g�aSÀn]Êu_gWY�\�Z;]_À¶a�y[Z;`&]VY[uv^�b�w[a�Z:cTYÝZ;YÝa�ÀàcÔj[Z"!�]_^�]VY7a_¿�Ó>tLa&]$#�t>Z:\FgWi;]VY7a_¿àÀngÁb�` ­Åã Z:^4`.a_¿Z;ÍSg�xTZ;YL]}a�y[g�a&% gWY[j('�ÀngWY�a�a.cAuvcTÍSÍÑt[Y[Z¾u_g�a.]W¿LÓ>tLa�y[gV\W](YLcWa¡`.]_a¡tLwÊgN`�y>g�^�]Vj�ÃW]_bÐZ¾YÐgWj[\FgWY[uv] ­½ yL]_b�u_gWYhjLcNa�yL]©e�cTi;i;cÁÀªZ;Y[xL×Ø ­ '�xW]VYL]_^#g�a.]V`�gNw>tLÓ6i;Z;u*Â,ÃW]_b*)w[^#Z:\�g�a.]vÂ,ÃW]_bdw6gWZ:^©Ä+���-,.���ÜÎ ­¬ ­ ' `.]VY[j[`/����a.c�% ­ ½ yL]�`.]Vuv^�]_aªÃW]_bÐ^#]VÍSgWZ;Y[`�`.]Vuv^#]_a ­í ­ %�cWÓ[a�gWZ;Y>`0��� gWY[j�]VY>uv^�b�w>a�`AÍN]V`�`#g�xW]21 gW`_×4365 798;:(Ä�1 ÎÊÄÆYLcWa.]�a�y[g�aS]VY[uv^#b�w[a�Z;cTYÍSZ;xTy7a�Ó�]}^#gWY[jLcTÍAZ:�_]Vj6Î ­ % `.]VY[j[`/3 a.c�' ­

mL­ '�cWÓ[a�gWZ;Y>`<3¶gWY[jÜj[]Vuv^�b�w>a�`�Z:a�gW`_×=1 ÷?><@�:ÑÄA3©ÎNÄ{a�b�w6Z;u_gWi;i:bW¿ÞjL]Vuv^�b�w[a�Z:cTYÕÀªZ;i¾i´Ó�]SjL]_a.]_^�ÂÍSZ¾Y[Z;`.a�Z;uÁÎ ­

cÀ�ÄÆZ;YLe{cW^#ÍSgWi¾i:bLÎ4¿�a�yL]�`.]Vu_t[^#Z:a�bdÀà]3jL]V`#Z:^�]}e�^#cTÍ `#t[u#yÇgN`�u4yL]VÍN]�Z;`na�yL]3e{cTi;i:cÀªZ;YLxL×îgWYÊgWjL\W]_^�Â`�g�^�b�ÀªyLcNw>gW`#`�Z:\W]Vi:bd]VgV\W]V`�jL^#cWw>`ªcTYB��ý�ýîuvcTÍSÍÑt[Y[Z;u_g�a�Z;cTYÇÓ�]_a�Àn]_]VY-% gWY[j�' ÄÆZ;YÇw6g�^�a�Z;u_t[i;g�^Á¿�ÀOyLc`.]_]V`C��� gWY[j3©Î¡`�yLcTt[i;jhÓ�](t[Y[g�Ó6i:]�a.c�jL]_a.]_^#ÍSZ;Y[](gWY�b�a�y[Z¾YLxAg�Ó�cTtLa�a�yL](ÍN]V`#`�g�xW]01 ­ Ä ë ]ÑÀOZ;i;i^#Z:xWcW^�cTt[`#i:bÊe{cW^#ÍSgWi;Z:�_]&g�YLcWa�Z:cTYPcWeà`�]Vu_tL^#Z:a�bhe{cW^�a�y>Z;`}`.]_a.a�Z;YLx�cÁ\W]_^©a�yL]NYL]vä�a}e{]_À i:]Vuva�tL^�]V` ­ Î cWa.]u_g�^�]_e{t>i;i:bPÀªy[g�a©a�y>Z;`�ÍN]VgWY[`ÑZ:enÀà]Ðu_gWYÔgWu4y[Z:]_\W]Ða�y[Z;`_×ÑZ:a©ÍN]VgWY>`©a�y[g�a(a�Ànc�w�]_cWw>i;]SZ;YÜghuv^#cÁÀªjL]Vj^�c�cTÍ q ÀªyLcÊy>gV\W]AYLcWa©]_\W]VY `�y[g�^#]VjÜgWY7bÕZ;YLe{cW^#ÍSg�a�Z;cTY1Z;Y1gWjL\�gWY[uv] q u_gWYÔ`�yLcTtLa�Z;Y[e�cW^#ÍAg�a�Z:cTYa.cA]VgWu#y�cWa�yL]_^}gWY[jÇÃW]_]_wPa�yL](uvcTY�a.]VY�a�`ªcWeBa�yL]VZ;^ªuvcTY7\W]_^4`�g�a�Z:cTYPy[Z;j[jL]VY�e{^�cTÍé]_\W]_^�bWcTYL]©]Vi¾`.]©Z;YÇa�y[]^�c�cTÍÊá

ë ](ÍSgÁbÇgWi;`.cN\ÈZ:]_Àùa�yL]3]VY[uv^#b�w[a�Z;cTYÊgW`¡y[g�w[w�]VY[Y>Z;YLx&Z;Y�a�ÀncL¿6j[Z¾`.a�Z;Y[uva�`.a�g�xW]V` ­àr YÇ`�a�g�xW]©cTYL]W×Ø ­ '�xW]VYL]_^#g�a.]V`�gNw>tLÓ6i;Z;u*Â,ÃW]_b*)w[^#Z:\�g�a.]vÂ,ÃW]_bdw6gWZ:^©Ä+���-,.���ÜÎ ­¬ ­ 'éÀªZ;jL]Vi:bhw>tLÓ>i;Z¾u_Z:�_]V`D�<� a.cÊgWY�bWcTYL]NÀªyLcÇÍSZ;xTy7a3]_\W]_^�ÀÛgWY�a3a.cÊuvcTÍSÍ(t>Y[Z;u_g�a.]NÀªZ:a�y1y>Z;Í ­Ï ]�ÍSg�ÃW]V`�`�tL^�]}a.cAÃW]_]_w��� `.]Vuv^�]_a ­

ê aO`.cTÍN]�i¾g�a.]_^ªw�cTZ;Y�aÛZ¾Y�a�Z;ÍN]W¿E% ÀngWY�a�`¡a.cduvcTÍSÍ(t>Y[Z;u_g�a.]�ÀªZ;a�y('dÒ[`#yL]�jLc�]V`ª`.cSgW`ªe�cTi;i;cÁÀª`_×Ø ­ %GF>Y[j[`Ûa�y>g�aª`�yL]�Z;Y>jL]_]Vj�Ã�Y[cÁÀª`&�<�Ô¿La�y[]�w>tLÓ>i¾Z;u¡ÃW]_bÇcWe�' ­¬ ­ % ]VY[uv^�b�w[a�`¡ÍN]V`#`�g�xW]H1 gW`ªg�Ó�cÁ\W]W×I3J5K7 8;: Ä�1 Î ­ % `.]VY[j>`D3 a.c�' ­í ­ '�cWÓ[a�gWZ;Y[`L3 gWY[jÊj[]Vuv^�b�w>a�`¡Z:a�gW`_×M1 ÷N> @O: ÄA3(Î ­

Ï ]_^�]1ÄÆg�xTgWZ;YD¿BZ;Y[e�cW^#ÍAgWi;i:b[Î4¿�a�yL]�`.]Vu_tL^#Z:a�bÕÀà]�jL]V`�Z:^#]Se�^#cTͦ`�t[u4yÔg�`�u#y[]VÍN]ÐZ¾`�a�y[g�aÑgWYÔgWjL\W]_^4`�g�^�b]VgV\W]V`�jL^#cWw[w>Z;YLxÐcTYPgWi;i´uvcTÍAÍ(t[Y>Z;u_g�a�Z:cTY�Ó�]_a�Àà]_]VY% gWY>j-'Ù`�yLcTt[i;jÊÓ�]Ñt[Y[g�Ó>i;]©a.c�jL]_a.]_^#ÍSZ;Y[]P1 ­Ú t[aOZ;Yha�y[Z;`ª`�]_a.a�Z;YLxL¿�a�yL]©cTY[i;b�uvcTÍAÍ(t[Y>Z;u_g�a�Z:cTY�Ó�]_a�Àà]_]VY2% gWY[j2'¶Z;`¡a�y[](u_Z:w>y[]_^�a.]vä�aQ3&¿�`.cAa�y[Z¾``.]_]VÍS`da.cÔÓ�]hgÔÀà]Vg�ÃW]_^�^�]$#�t[Z:^#]VÍN]VY7aAa�y[gWY�Ó�]_e{cW^�]Wá r YÅeÆgWuva_¿¡Àà]P`�yLcTt[i¾jÅy[gV\W]1ÍNcW^#]�u_g�^�]_eÆt[i;i:b`.a�g�a.]VjÇcTtL^Û`.]Vu_tL^#Z;a�bAxWcTgWi,×IgWYÐgWj[\W]_^#`�g�^�b�ç���æSþSRC�TRUR4û9V���WX ,æY�å6æ�çB���¦gWY[jÐÀªy[c(]VgÁ\W]V`�jL^�cWw>`ÛcTYgWi;i>uvcTÍSÍÑt[Y[Z¾u_g�a�Z:cTYSÓ�]_a�Àn]_]VYX%ïgWY[j�' `�y[cTt[i;jZR Æþ�ý{ý�Ó�]Ot[Y[g�Ó>i:]ªa.c(jL]_a.]_^#ÍAZ;YL]D1 ­îë ]�`.a.^#]V`�`�a�y>g�aÀà]NgW`�`�t>ÍN](a�y[]&gWj[\W]_^#`�g�^�bÊÃÈYLcÀª`C��� Ó�]Vu_gWt[`.]='éÍAg�ÃW]V`3YLcd]!�cW^�a3a.c�w[^�]_\W]VY�a}Z:a�`�ÀªZ;jL]V`.w>^�]VgWjj[Z;`�`.]VÍAZ;Y[g�a�Z:cTYDÒ�Z;a}Z¾`OcTY>i:bheÆgWZ:^3a.cÊgW`�`�t>ÍN]Na�yL]NgWjL\W]_^#`�g�^#b�ÃÈYLcÁÀO`���� ­(r a3`�yLcTt[i¾j�Ó�]Nu_i:]Vg�^�a�y>g�aa�y[Z;`¡jL]�F6Y[Z:a�Z:cTYÐcWeî`.]Vu_tL^#Z:a�bÐZ¾`¡YLcÁÀ ]$#�t[Z;\FgWi:]VY�a�a.cSa�yL]©w[^�]_\ÈZ:cTt[`ncTYL] ­

«�]_a�� `&Ó[^4Z:]�[[b1`.]_]ÇyLcÁÀ w>t[Ó>i;Z;u*Â,ÃW]_bÜuv^�b�w[a.cWxW^#g�w6y7b u_gWYÝgWj[j[^�]V`�`Ñ`.cTÍN]�cWe¡cTtL^&uvcTY[uv]_^#Y[`&ÀOZ:a�yw[^#Z:\�g�a.]vÂ�i:]_b�uv^�b�w>a.cWxW^#g�w>y�b ­

¬

Page 79: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ø ­ zîg�^�a�Z:]V`¡ÀªZ;`�y[Z¾YLx(a.cduvcTÍSÍ(t>Y[Z;u_g�a.](j[cÇå6æ� �y[gÁ\W]�a.cd`�y>g�^�]3ÃW]_b�`OZ;YÇgS`.]Vuv^#]_aªÍSgWY[YL]_^ ­\ cWa.]W¿yLcÀà]_\W]_^Á¿�a�y>g�ana�y[]_bSjLcÑY[]_]VjAa.cÑ]väÈu4y[gWYLxW]�Z;Y[e�cW^#ÍAg�a�Z:cTYSZ;YÐ`.cTÍN]�É.gWtLa�yL]VY�a�Z;u_g�a.]Vj[Ì&ÍSgWY[Y[]_^Ä{Àn]©gW`�`�t>ÍN]VjÇgNw>gW`�`#Z:\W]�gWjL\W]_^#`�g�^#b�g�Ó�c\W]�Ó>tLa¡Z:a¡`#yLcTt[i;jÐÓ�]�u_i:]Vg�^¡a�y[g�aOgWYÊgWjL\W]_^#`�g�^#bÐÀOyLcu_gWYSZ¾Y7a.]_^�e{]_^�]�ÀªZ;a�y&a�yL]¡uvcTÍSÍÑt[Y[Z;u_g�a�Z:cTYNu_gWYduvcTÍNw>i:]_a.]Vi;b©^#t[Z¾Y(a�y[]ª`.]Vu_tL^#Z;a�b©cWe�a�yL]¡`#u#yL]VÍN]Î ­ë ]dÀªZ;i;iÞ^�]_a�tL^4Y1a.cha�y[Z¾`�Z;`�`�t[]hÄ{Ó[^#Z:]�[[b[ÎOÓ�]Vi:cÁÀ©¿´gWY[jÔÍSgVbÜj[Z¾`�u_t[`�`}a�y[Z¾`3w�cTZ;Y7a�i¾g�a.]_^(Z;Y1a�y[]`.]VÍS]V`.a.]_^OgW`�Àn]Vi;i ­

¬ ­ ½ y[]3w>g�^�a�b�w>t[Ó>i;Z;`�y>Z;YLx(y>Z;`Ûw>tLÓ>i¾Z;u�ÃW]_b�j[c�]V`©å6æ� �YL]_]Vjha.cdy[gÁ\W](gWY�b�Z¾jL]VgNÀªyLcNÀOZ;i;i�t[`.]�a�y[]ÃW]_bÖa.cÜuvcTÍSÍÑt[Y[Z;u_g�a.]ÇÀªZ;a�yÖy>Z;Í ­ ê Y7bWcTY[]ÇÀªyLc1i:]Vg�^#Y[`Ña�yL]Çw6tLÓ>i;Z;uAÃW]_b�ÄÆgWY>j]'¦a.^#Z;]V`&a.cw>t[Ó>i;Z;u_Z:�_]�Z:a¡`.cSa�y>g�aªgW`¡ÍSgWY�b�w�]_cWw6i:]3gW`¡w�cT`�`�Z:Ó6i:]�i;]VgWYL^¡Z:a4ÎÛu_gWYÊuvcTÍAÍ(t[Y>Z;u_g�a.]�ÀªZ:a�y(' ­

í ­ªr YhgdYL]_a�ÀncW^�ÃÇcWeBò w�]_cWw>i;]W¿[]VgWu#yPw>g�^�a�bÊYLcÁÀ YL]_]Vj[`ªcTY[i:bÇj[Z;`�a.^#Z:Ó>tLa.]�y>Z;`�w>tLÓ6i;Z;u�ÃW]_bÊa.cAa�y[]cWa�yL]_^3ò2^ Ø w�]_cWw>i;]ÑZ;YPa�yL]SY[]_a�ÀncW^�à ­ ½ yL]Ña.cWa�gWiIY�t[Í(Ó�]_^}cWeàÃW]_b�`�Z¾`�Y[cÁÀ òùÄ{^#g�a�yL]_^3a�y[gWYøAÄ{ò õ ÎngW`ªÓ�]_e�cW^�]Î ­

Ï gÁ\�Z¾YLxªxTZ:\W]VY©a�yL]V`.]ng�^�xTt[ÍN]VY�a�`´Z¾Y3e{gÁ\WcW^´cWeLw6tLÓ>i;Z;u*Â,ÃW]_bO]VY[uv^�b�w[a�Z:cTYD¿FÀªy[g�a´g�^�]�a�yL]àjL^#gVÀªÓ>gWu�ÃÈ`4ìÄ r e�a�yL]_^�]¡g�^�]¡Y[c}j[^#gVÀ¡Ó6gWu�ÃÈ`Ba�yL]VYNÀªy�b©Ó�cWa�yL]_^Ii;]Vg�^#Y[Z;YLx}g�Ó�cTtLaîw[^#Z:\�g�a.]vÂ,ÃW]_b&uv^�b�w>a.cWxW^#g�w>y�bÑg�a�gWi¾i á Îë ](ÍN]VY�a�Z:cTYÊgNe{]_À yL]_^#]W×Ø ­Oã ^#cTÍ�gPw[^4gWuva�Z;u_gWi�w�cTZ;Y�a©cWeª\�Z:]_À©¿Bw>tLÓ>i;Z¾u*Â,ÃW]_bÕuv^#b�w[a.cWxW^4g�w>y7bÖZ;`ÑÍ(t[u4yÝi:]V`�`(]�_Au_Z:]VY�aÑa�y>gWYw[^4Z:\Fg�a.]vÂ,ÃW]_b°uv^�b�w>a.cWxW^#g�w>y�b ­ ½ cÔxTZ;\W]PgÔ^�cTtLxTy�]väÈgWÍNw6i:]W¿¡]VY[uv^�b�w[a�Z;Y[x `�yLcW^�a�ÍN]V`�`#g�xW]V`ÐZ¾`g�Ó�cTtLa Ø�`�`�` a�Z;ÍN]V`}eÆgW`.a.]_^(t[`#Z;YLx�w[^#Z:\�g�a.]vÂ,ÃW]_b1uv^�b�w[a.cWxW^#g�w>y�b�a�y>gWYÜt[`�Z;Y[xÐw6tLÓ>i;Z;u*Â,ÃW]_bPuv^�b�wLÂa.cWxW^#g�w6y7b ­ â�e�uvcTtL^#`�]W¿�g�i:cWaBcWe�^�]V`�]Vg�^#u#ydy[gW`IÓ�]_]VYSgWY[jAuvcTY7a�Z;Y�tL]V`Ba.c(Ó�]ne{cÈu_t[`.]VjNcTYdÍSg�Ã�Z¾YLxw>t[Ó>i;Z;u*Â,ÃW]_bdcWw�]_^4g�a�Z:cTY[`¡gW`ª]�_Au_Z:]VY�aªgW`ªw�cT`�`#Z:Ó>i:]NÄ{Àà]Ñj[Z;`�u_t>`�`�`.cTÍN]�cWeîa�yL]V`�]©ÍN]_a�yLcÈj[`¡i;g�a.]_^Z;Y�a�y[]�`.]VÍN]V`.a.]_^*Î ­

¬ ­Oã ^#cTÍ g�a�yL]_cW^#]_a�Z;u_gWiÈw�cTZ;Y7a´cWe[\ÈZ:]_À�¿Ww>tLÓ6i;Z;u*Â,ÃW]_b3uv^�b�w[a.cWxW^#g�w>y�b(`.cTÍS]_a�Z;ÍN]V`Þ^�]$#�t[Z:^�]V`Þ`.a.^�cTYLxW]_^gW`�`#t[ÍNw[a�Z:cTY>`¡a�y[gWYPw[^#Z;\Fg�a.]vÂ,ÃW]_bPuv^�b�w>a.cWxW^#g�w>y�b ­}ê `}gWYP]väÈgWÍNw6i:]W¿�Àà]NYLcWa.]VjPa�y[g�a3Z;Y[j>Z;`.a�Z;YÈÂxTt[Z¾`�y[g�Ó>i:]Aw[^#Z:\�g�a.]vÂ,ÃW]_bÜ]VY[uv^�b�w[a�Z:cTYÖu_gWYÖÓ�]dgWu#y>Z:]_\W]Vj�t[Y[jL]_^(a�yL]�gW`�`�t[ÍNw>a�Z:cTY cWeªcTYL]vÂ,ÀngÁbeÆt[Y[uva�Z:cTY>` ­ z�tLÓ>i¾Z;u*Â,ÃW]_bA]VY[uv^#b�w[a�Z;cTYÇu_gWY[YLcWa¡Ó�]�gWu#y>Z:]_\W]Vjht[Y>jL]_^�a�y[Z;`�`�gWÍS]©gW`�`�t[ÍSw[a�Z:cTY ­

í ­ zàtLÓ>i;Z;u*Â,ÃW]_b3uv^�b�w[a.cWxW^#g�w6y7b�^#]Vi;Z:]V`%`�a.^�cTYLxTi:b3cTY&`.cTÍN]Û`.bÈ`.a.]VÍ e�cW^î^�]Vi;Z¾g�Ó>i:b}j[Z;`.a.^#Z;Ó>tLa�Z;YLxÛw>t[Ó>i;Z;uÃW]_bÈ` ­a cWa.]Oa�y[g�aÛZ:anZ;`�YLcWaÛgWi:ÀÛgVbÈ`à`�c(]VgW`.bAa.c©w>tLÓ6i;Z;u_Z:�_]¡g(w>tLÓ>i¾Z;unÃW]_bAZ;YAg(^�]Vi;Z¾g�Ó>i:]¡ÍSgWY[Y[]_^ ­ê `NgWY�]väLgWÍNw>i:]W¿àcTYL]ÇÀngÁbÝcWeOw>t[Ó>i;Z;u_Z:�VZ¾YLxhgÕw>tLÓ6i;Z;uSÃW]_b�Z;`&a.cÜi¾Z;`.aNZ:aNZ;YÝa�yL]ÊYL]_ÀO`.w>g�w�]_^ ­ë ]Vi;i,¿LÀªy[g�a¡Z:e r w>t[aÛÍ(b�w>tLÓ>i¾Z;u¡ÃW]_bÜÄ{e{cW^ªÀªy[Z¾u#y r ÃÈYLcÀ a�yL]�gW`�`�c�u_Z;g�a.]Vj�`.]Vuv^�]_a¡ÃW]_b[ÎÛZ;Y�a�y[]YL]_ÀO`.w>g�w�]_^�gWY>jPu_i¾gWZ;Í�a�y[g�a}a�y[Z;`}Z;`�^�]VgWi;i:bÊ`�cTÍN]_cTYL]N]Vi;`.]�� `SÄÆ`#gVbW¿ ê ÍSg��_cTY�� `4ÎOÃW]_b[ì r YPa�y[Z¾`u_gW`.]W¿ r ÀªZ;i;i%Ó�]&g�Ó>i:]Ña.c�^�]VgWj1gWi;iÞÍN]V`�`�g�xW]V`SÄÆZ¾Y[u_i;t[j[Z¾YLxAuv^�]Vj[Z;a�u_g�^#jÕY�t[Í(Ó�]_^4`_¿6]_a�u ­ Î1Àªy[Z;u4yÀn]_^�]NZ;Y�a.]VY[jL]VjPe�cW^ ê ÍSg��_cTY ­=b ]V`�cTi:\�Z¾YLx�a�y[Z;`}Z;`�`�t[](a�tL^4Y[`�cTtLa}a�yL]NÓ�] q w�]_^#y[g�w6` q a�y[]Ó>Z;xWxW]V`.a¡jL^#gÁÀ¡Ó>gWu�Ã�a.cAt>`�Z;YLx&w>tLÓ6i;Z;u*Â,ÃW]_bduv^#b�w[a.cWxW^4g�w>y7b�Z;YÇw>^#gWuva�Z;uv] ­

mL­Oã Z;Y[gWi¾i:bW¿7Àn]3YLcWa.]}a�y[g�anZ¾YAa�yL]3u_gW`.]}cWe´w[^#Z:\�g�a.]vÂ,ÃW]_bAuv^�b�w[a.cWxW^#g�w>y�bAÓ�cWa�ydw>g�^�a�Z:]V`&ÉËÃÈYLcÁÀOÌ�ÄÆZ¾Y`.cTÍS]A`.]VY[`.]Î3]väÈgWuva�i;b1a.chÀªyLcTͦa�yL]_bÜg�^�]A`�w�]Vg�ÃÈZ;YLxÖÄÆZ:eàa�ÀàcPw>g�^�a�Z;]V`©`�y[g�^#]AgWYÜ]VY[uv^#b�w[a�Z;cTYÃW]_bW¿Èa�yL]VY4c Z;YLe{cW^#ÍSgWi;i;bÑ`.w�]Vg�ÃÈZ;YLxTd6ÍS]V`�`�g�xW]V`Û`.]VY�aàÓ�bSgWY7bAcWa�yL]_^àw6g�^�a�bSÀOZ;i;iÈÓ�]Ot[Y>Z;Y7a.]Vi¾i;Z:xTZ:Ó>i;]gWY�b�ÀÛgVb[Î ­ â3Y a�yL]�cWa�yL]_^Sy>gWY[jD¿IZ:e¡`.cTÍS]_cTYL]Çw>tLÓ>i¾Z;`�yL]V`3a�y[]VZ:^(w6tLÓ>i;Z;uSÃW]_bÖgWY[jÖ^�]Vuv]VZ;\W]V`NgÍN]V`#`�g�xW]W¿>a�yL]_bÇy[gÁ\W](YLcAÀngÁbÇcWeÞ^#]Vi;Z;g�Ó>i:bAa.]Vi;i;Z;YLx&Àªy[cA`.]VY�a¡a�y[g�aOÍS]V`�`�g�xW] ­ Ä,â�eBuvcTtL^#`.]W¿[a�y[]`.]VY>jL]_^3u_gWY1u_i;gWZ;Í `.cTÍN]NZ;j[]VY7a�Z:a�b�Z;YPa�yL]NÍN]V`#`�g�xW]SZ:a�`.]Vi:e ¿�Ó>tLa3yLcÀ jLc�Àà]NÃÈYLcÁÀ y[]&Z¾`�Y[cWai:bÈZ;YLx�ìFÎ

í

Page 80: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

eDf*gih�j�kUl�gnm�oqp�rtsvuTkUoxwiu�fEyz s2{ z}|�~��H��� f�m�kUpxy*�E�m�gnp�fZm�p z k�ox�*m�p���kUuT����o�ak�p�r�j$l�l�p�kC��p�f*uTm��*u�fv��uTm��

���Y�Q���]��� ���

� �<�a�a�=�����N�Y Y¡U¢U�¤£O¥¦�t§©¨«ª=�t�¬§H­H®¬¢U¯Dª

°j2±¬j���gif²p��³kZl�m��*y³o]p�rH�E�³±Ew´gi��µA¶�j�o«j$f*�k�ox�*m�gnp�fNu�lq·}j2±�j���u�fNp��³kZl�m��*y*o¸p�r��*kUgih¹uTm�jµA¶�j�o¸j$f³µ�k�oO�Em�gnp�f�ºv»&�x�*l�¼�·}jXl�m�uTk�mH±xo½�p�f*l�g´y³j�kUgif³�-u-f³p�m�gip�f¾p�r}¿³À�ÁSÂ$À�ÃÅÄCÆ�ÀUÃ�Á�À�Ã$ǽu�f*yBl�j�jq·/�*j�m��³j�kPu�fOo�E�³±Ewig´��µA¶�j�o-j$f*�kUoO�*m�gip�fl��U�³j$È0jq��u�f¾lUuTm�gil�r�o2gnm�º(ÉC�³k<y*j�ÊEf*gnm�gnp�f4�³j�kUj0·/giwiw˱¬j0u�f*u�wnp���p��El�m�pvm��*jy³j�ÊEf*gnm�gip�fvp�rM��j�kUrSj$�mQl�j$�k�j$�o-rSp�kQ�EkUgnhTuTm�jµA¶�j�o-j$f*�kUoO�*m�gip�f�̬m��³j=p�f*wnovy*gnÍ�j�k�j$fE�j=f*pÅ·ÎgilLm��*uTmQ·}ju�l�l��*ÈYjQm��*j<u�y³h�j�kUl�uTkUoX¶9f³pÏ·/l � gif(u�yEy*gnm�gnp�fZm�p0m��*j���gn�E�³j�k�m�jÐ9m � m��*jQk�j$��gn�Egnj$fOm�ÑÒl��E�*±Ewigi�D¶�j�o�º

Ó(Ô�ÕtÖ¤×SØÏ×�Ù�ÖGÚ�ÛN¿�Ü*Ý�Þàß�Ã�á�â�À�Ç�Àã�Ã$Á.Ç.¿�Ä+ß�äTã-ÆÃå³Àæ�ÀQßSÆ��¬j�k�r�j$�m�wno0l�j$�k�j�m�ß Â�Âä�ÁQç�ÞSÞO¿�ÜEÝÞèß�ÃCâ�À�ǹÆIé�êBëçTÞSÞìæ0ÀÆUÆ�ç$íOÀÆ/î�ï¹ð�î4ñ�ëMçTÞSÞËÃ�ßò¿Eå9ÀÁÄóÀ�ô�Ä�ÆCõPëMç�ã�öZçTÞSÞ�çTÞ÷í�äTÁ�ß�Ä�åOæHÆLø¦ùMÀ<åxçTú�À$û

�\kÏü øPý+é<êvð.õHþ\ÿ�����õ������Pý�î�ï$þ �¬ÿ��\kÏü øPý+é<êvð.õHþ\ÿ�����õ������Pý�î4ñ�þ ���

�&j$��u�wiw¤m��*uTmDg´f(m��³jH��u�l�jHp�r��*kUgih¹uTm�jµA¶�j�o�j$fE�k�oO�Em�gnp�f�¼*�¬j�k�r�j$�m/l�j$�k�j$�ov·}u�lLu��U�*gij�h¹uT±Ewijqý+u�wnm��³p��³���f³p�m<h�j�k�oj��q��gnj$f�m�wio³þC�El�gif³�(m��³jqp�f³jµAm�giÈYjq�Eu�y¾lU�U�³j$ÈYj�º��DpÅ·}j�h�j�k$¼a·}j�f³pÏ· l��*pÅ·?m��EuTm<�¬j�k�r�j$�ml�j$�k�j$�o�gilCß�æD¿9ä¹ÆUÆ�ß�Ý�ÞnÀYgifZm��³jQ�E�³±�wigi��µA¶�j�oq��u�l�j�º�MÔ������«Ú��HäQ¿�ÜEÝÞèß+Ã$áóâ�ÀÇXÀ�ã�Ã�Á�Ç.¿�Ä+ß+ä�ã4ÆÃå³Àæ�ÀHß´ÆI¿*ÀÁ�Â�ÀUÃ$Ä+ÞèÇ0Æ�ÀUÃ�Á�À�Ä��

���ÏÙ�Ù � !Dl�lU�*ÈYjQî ï#"ÿNî ñ u�f*y��p�fEl�giy³j�k�m��³j<rSp�wiwipÅ·/gif*�=u�wi��p�kUgnm��*Ȧø%$ÉCf(gif*�E�³mQý+é�êvð.õHþ&$'³p�kLj$u��U�v�¬p�l�l�gn±EwijL�.�³p�gi�j�p�r�m��³jQkUu�fEy³p�Ȧ±Egnm�l(${xj�mLõ}ï/ÿ)����<ý�î�ïÅþgnraõ ÿ õ}ïCp��³m��E�*m*�0u�f*y(j$f*y

p��³m����³m�+�u�f*y(j$fEy»&�*gil�u�wi��p�kUgnm��*È �j�k�m�u�gif*wnom�j�k.È0gif*uTm�j$l�Ìaf*p�m�jqm��*uTmHm��³j�k�jXuTk�jqp�fEwnou�ÊEf*gim�j�fO�EÈ<±¬j�k<p�r��¬p�l�lUgn±Ewnj�U�³p�gi�j$l¤r�p�kìm��³jak.u�f*y³p�È ±�gnm�l;�*l�j$y�±OoCm��³jaj$fE�k�oO�Em�gnp�f�u�wn��p�kUgim��*È?ý+l�gifE�jam��³jaj$f*�kUoO�*m�gip�f�u�wn��p�kUgnm��EÈÈH�*l�mtm�j�kUÈ0gif*uTm�j � g´fHr+u��m�¼�·}j&·/g´wiwOu�wi·}uÅoxlau�lUl��*ÈYjIm��*uTmaj$fE�k�oO�Em�gnp�f=��u�fY±¬jIy*p�f³j&gifP�¬p�wnoxf*p�È0giu�wm�giÈYjÏþ.º}ÉLr��p��³kUl�j�¼³m��³j<uT±¬pÅh�j<u�wi��p�kUgnm��*È ·/g´wiw¬f³p�m�±¬jqÀ-, Ã�ßAÀã�ÄLý+g�º÷j�ºn¼E·/giwiw¬f³p�m�kU�Ef(gifZ�¬p�wnoxf*p�È0giu�wm�giÈYjÏþ}±E�*m}m��EuTm/gil�p�¶Z±¬j$��u��*l�jQ·}j<uTk�j<y*j$u�wigif³�=·/gnm��Z�¬j�k�r�j$�m&l�j$�k�j$�o�º

°j�f³p�m�jYm��³j=r�p�wiwnpÏ·/gif³�q±�j$�Eu$h9gnp�kQp�r\m��³jYu�wn��p�k.gnm��*È�$Yý.+Åþ � rIõ gilCu�f4j$f*�k�ox�*m�gnp�f2p�r}î�ïϼ�m��*ju�wn��p�kUgnm��*È u�wn·Iu$o9lMp��*m��E�³m�l/�qý+l�gifE�j/m��³jLu�wn��p�kUgnm��*È?j�h�j$f�m��Eu�wiwnoYm�kUgnj$lMm��³jLl�u�ÈYj/k.u�f*y³p�È ±Egnm�lam��EuTm·Mj�k�j4�*l�j$y ·/�³j$fN�p�ÈY�E�³m�g´f³� õ<þ.º ý10�þ � rHõ gil�u�f²j$f*�k�ox�*m�gnp�f p�rQî4ñ$¼�m��³j-u�wn��p�k.gnm��*È u�wn·}uÅoxlp��³m��E�³m�l2+Tºa»&�*g´lMgil}l�p=u�l�wnp�f³�Pu�lI·MjCu�l�l��*È0jDm��*uTmIm��³jCl��.�³j$ÈYjQgilCÃUäTÁ�Á�ÀUÃ$Ä�ÌEg�º÷j�ºn¼9m��*uTm}m��³jCk�j$�j$gnh�j�ku�wn·}uÅoxlCy³j$�kUoO�*m�lL�p�k�k�j$�m�wioBý+f³p�m�jPm��*uTmLgiraõ �p��*w´y�±¬jHu�f-j$f*�k�ox�*m�gnp�fvp�rt±�p�m��-î�ï�u�f*y-î4ñDm��*j$fm��*gil0·}p��*wiy²��u��*l�j2uBy³j$�kUoO�*m�gip�f¸j�k�k�p�kþ.ºG{xp³¼&�akÅü ø=ý+é�ê-ð.õ<þvÿ�����õ3� ���Pý�î�ï$þ �Pÿ4+vu�fEy�akÏü ø=ý+é�ê-ð.õ<þ\ÿ5����õ6������Hý�î4ñ�þ ��ÿ5�0u�f*y(�¬j�k�r�j$�m&l�j$�k�j$�o�gilIh9gnp�wiuTm�j$y;º

+

Page 81: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

{9gif*�jI�¬j�k�r�j$�mal�j$�k�j$�oYgiltf³p�mau��.�*gnj�hTuT±Ewnj�r�p�ka�E�³±�wigi��µA¶�j�o�j$f*�kUoO�*m�gip�fYl��U�³j$È0j$l�¼�·}j�m�k�oPm�p�k�j$wiu¹Ðm��³jXy*j�ÊEf*gnm�gnp�f½m�p2l�j�jq·/�*uTm=��u�f¾±¬jXu��U�*gnj�h�j$y;º7!LlHgif¾m��³jq��u�l�jXp�r��EkUgnhTuTm�jµA¶�j�o½j$f*�k�ox�*m�gnp�f�¼Ë·}j�p�f*l�giy³j�k�u=k�j$l�m�kUgi�m�j$yZ��wiu�l�lIp�rËu�y³h�j�kUl�uTkUgij$l�Ì*f*u�ÈYj$wno�¼9·}jQ�p�f*l�giy³j�k�u�y³h�j�k.l�uTkUgnj$lI·/gim��Xl�p�ÈYjC±�p��Ef*yp�f¾m��³j$gik<m�p�m�u�w}kU�*f*fEgif³�(m�giÈYj�º98}j�r�p�k�jZgif�m�kUpxy*�E��gif³�(p��³kHÊ*kUl�mPy³j�ÊEf*gnm�gip�f4p�r/l�j$���³kUgimóo�¼�·Mjq��gnh�jZu�*k�j$��gil�j�y*j�ÊEf*gnm�gnp�fXp�rtu=�E�³±�wigi��µA¶�j�o�j$fE�k�oO�Em�gnp�f�l��.�³j$ÈYj�º

Ó(Ô�ÕtÖ¤×SØÏ×�Ù�Ö�:GÛ �E�*±Ewigi��µA¶�j�o]j$f*�kUoO�*m�gip�fNlU�U�³j$ÈYj]ßSÆ4ç¸Ä+Á.ßò¿�ÞiÀ½ä�Â�¿9äTÞèÇTã�äTæ=ß+ç�Þàá�Ä+ßSæ�À«çTÞ÷í�ä�Á.ß+Ä�å�æPÆý-;=ð<�að<=Yþ<Æ�Ü*Ãå(Ä�åxç�ÄAû

> ; ß´ÆHçXÁ�çTã�ö�äTæ=ß@?ÏÀ�ö¶�j�o(��j$f*j�kUuTm�gnp�fvu�wi��p�kUgnm��*ÈA�*B�ÄIÄAç¹â�À�ÆHç¹ÆHßSãT¿�Ü9Ä�ç�Æ�ÀUÃ�Ü9Á�ß�Ä+Ç<¿9çTÁ�çTæ�À�ÄóÀ�Á ñ+�C çTã�ö½äTܳÄà¿�ܳÄ�Æ(çq¿9çTß�Á�ä�Â0â�ÀǹÆXý+é�ê-ð&D�ê½þE�GF-ÀZÃ.çTÞ�ÞËé�ê Ä�å9Àv�E�³±�wigi�=¶�j�o çTã�öAD�ê Ä�å9Àl�j$�kUj�m&¶�j�oH�

> �¸ßSÆ�ç9I"¿³ä¹ÆUÆ�ß�Ý�ÞàÇqÁ�çTã�ö�äTæ=ß@?ÏÀ�öKJYj$f*�kUoO�*m�gip�fqu�wn��p�k.gnm��*ÈA�LB�ÄaÄAç¹â�À�Æ�ç¹ÆLß�ãT¿�ܳÄ}çL¿�ÜEÝÞèß�ÃCâ�À�Ç�é�êçTã�öXçqæ�À�Æ.Æç$íxÀDî ç�ã�öZäTܳÄà¿�ܳÄ�ÆPçZÃ�ßò¿Eå9ÀÁÄóÀ�ô�Ä�õ��MF-À<ùtÁ.ß+ÄóÀPÄ�å�ßSÆHç¹ÆûDõN������Hý�î-þE�

> =¦ßSÆ/çPöOÀ�ÄóÀÁ�æ=ßSãEß´ÆÄ+ß+ÃCy³j$�kUoO�*m�gip�fYu�wn��p�kUgnm��*È7�OB�ÄËÄAç¹â�ÀÆDç¹Æ&ßSã�¿�ܳĤç<Æ�ÀUÃ�Á�À�Ĥâ�À�ÇPD�ê©çTã�öPç=Ã$ßSá¿Eå³ÀÁÄóÀ�ô�Ä�õ çTã�ö=äTܳÄà¿�ܳÄ�ÆDçHæ�À�ÆUÆ�ç$íOÀ\îQ�%IRB�ĤߴÆ&Ä�å9ÀUäTÁ�À�Ä+ß�ÃUçTÞ�ÞàÇD¿9äTÆUÆ.ßAÝÞiÀLÄAä<åxçTú�ÀCç�Á�çTã�ö�ä�æPßS?ÅÀUöö�ÀUÃ�Á�Ç.¿�Ä+ß+ä�ã¾çTÞ íOäTÁ�ß�Ä�å�æUT&çTÞ�ÞìÄ�å9ÀYÀ�ôOçTæD¿�ÞiÀ�Æ<ùMÀHùtß�ÞSÞ;Æ�ÀUÀëtåxäTùMÀ�úTÀÁ.ëMùtß�Þ�ÞËÝUÀPö�À$ÄóÀÁ�æPß�ã*ßSÆ�Ä+ß�ÃV�WJ

F-ÀCç¹ÆUÆ�Ü9æ0À�Ä�å9ÀDÆ�Ãå9À�æ0ÀCß´ÆQ�p�k�k�j$�mETtÄ�åxç�Ä�ßSÆUë9Âä�ÁLçTÞ�Þ�ý+é�ê-ð&D�ê½þCäTܳÄà¿�ܳÄaÝÇP;0ý.+ C þUëOÂ�äTÁLçTÞ�Þ�î4ëtçTã�öÂäTÁ=çTÞ�Þ�õ äTܳÄà¿�ܳÄIÝ�ÇX� �� ý�î-þ�ùMÀ<åxçTúTÀÅû/=XY ýAõ<þtÿ�îQ�

�DuÅhxgif*��y³j�ÊEf³j$y(u�f�j$f*�k�ox�*m�gnp�f�l��.�³j$ÈYj�¼*·}j<��u�f�f*pÅ·J�Ek�p��¬p�l�jQu0y³j�ÊEf*gim�gnp�fXp�r�l�j$���³kUgnm�o�ºÓ(Ô�ÕtÖ¤×SØÏ×�Ù�Ö)Z\[\ß"ô4çZÆ$À�Ã$ÜxÁ�ß+Ä+Ç=¿9çTÁ�çTæ�À�ÄóÀ�Á]+ C I�Æ�ä�ùMÀ0ã�ävÞ"äTãxíOÀÁYùtÁ�ß+ÄóÀ0ß+ÄDÀUô�¿�Þèß+Ã$ß�Ä+ÞèÇKJ^��Ûο�ÜEÝ�Þàß�Ã�áâ�ÀǽÀã�Ã�Á�ÇU¿¬Ä+ß+äTã¸ÆÃå³Àæ�ÀXßSÆýR_�ða`.þóµól�j$���*k�j(uT��u�gif*l�mP��gn�E�³j�k�m�jÐ9m�µAp�f*wno½uTm�m�u��U¶xlvß Â.ë�ÂäTÁZçTÞ�ÞMæ�À�Æ.Æç$íxÀ�Æî�ïTð�î4ñ=çTã�öZçTÞ�Þ¤çTÞ÷í�ä�Á.ß+Ä�å�æPÆLø¦Á�Ü9ã*ã*ß�ãOíXß�ã2Ä+ß�æ�À=ç�Ä\æ0ä¹ÆÄb_QùMÀ<åxçTúTÀÅû

`dc � �\kÅüný+é�êvð&DtêþO�e;=Ì.õ\�����Pý�î�ïÅþO$�ø=ý+é�ê-ð.õ<þ\ÿ5�f�g �\kÅüný+é�ê-ð&D�ê½þh�e;=Ì.õ\������Hý�î4ñ�þO$�ø=ý+é�ê-ð.õ<þaÿ5�f�i�K�

� f m��³j�*kUgnhTuTm�jµA¶�j�o ��u�l�j�¼Qm��³j�k�jjгgil�m�y*j�m�j�kUÈ0gif*g´l�m�gi�-j$f*�k�ox�*m�gnp�f l��.�³j$ÈYj$lvl�j$���³k�j½uT��u�gif*l�m��gn�E�³j�k�m�jÐ9m�µAp�f*wnoZuTm�m�u���¶9l�º}°j<l��³pÏ·Jm��*uTm/m��*gil�gil�giÈ0��p�lUl�gn±EwnjLgifZm��³j����³±Ewigi��µA¶�j�oX��u�l�j�º

�MÔ������j:kI-[;äTÁ�çTã*ÇCÁ�À�ç¹Æ�äTã�ç�Ý�ÞnÀ/úTçTÞàÜEÀÆ}ä�Âb_.ða`<J/ã�ä�ö�À$ÄóÀÁ�æ=ßSã*ßSÆ�Ä+ß�ÃË¿�Ü*Ý�Þàß�Ã�á�â�À�ÇCÀ�ã�Ã�Á�Ç.¿�Ä+ß+ä�ã0ÆÃ�å9Àæ�ÀÃUçTãBÝ.ÀHýR_�ða`.þ.á�Æ�ÀUÃ�Ü9Á�À=ç$í�çTß�ã³ÆÄIÃ$ßÒ¿�å9ÀÁÄóÀ�ô�Ä+á�äTãEÞàÇ(ç�Ä�ÄAç�Ã�âÅÆE�

���ÏÙ�Ù � lìj�m4ý-;=ð<�að<=0þq±¬j2u y³j�m�j�kUÈ�gif*gil�m�g´�(j$f*�k�ox�*m�gnp�fNl��U�*j$ÈYj�Ì�g�º÷j�ºn¼&p�f³j2r�p�kX·D�*gi�U�m�ÎgilXuy³j�m�j�kUÈ0gifEgil�m�gi�Cu�wn��p�kUgim��*È(º z p�f*l�g´y³j�k�m��³jQr�p�wiwnpÅ·Dgif³�Yu�wn��p�kUgnm��EÈ�øX$

ÉQf�gif³�E�*mCý+é�ê-ð.õ<þ&$�p�ÈY�E�³m�j=õ}ï/ÿm���Pý�î�ïÅþgnr�õGÿ õ}ïCp��³m��E�*m*�Yj$wil�jQp��³m����³m�+

n oHpfq�r1q.s.tVu�vWwyxLzf{au {&|}q~w�qiu��/s.{&�*��q�w�pf�Kt��Kp�w �&��{&r���qiw�qiu�|}vW��v�����w�pfq��EqixO�Wq.�f�aw�p*�&��w�pfq�r1s pfq.|}qK�EvS� qK�W��� ���#�E�����

0

Page 82: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�/p�m�j(m��*uTmY·D�³j$fQ� g´lPy*j�m�j�kUÈ0gif*g´l�m�gi�T¼�ø u�wn·Iu$o9l=p��³m��E�³m�l#�4gnrDõ g´lPu�f«j$fE�k�oO�Em�gnp�f¾p�rDî ï u�fEyu�wn·}uÅoxl0p��³m��E�³m�l�+ZgnrQõ gilYu�f«j$f*�k�ox�*m�gnp�f p�r/î4ñ�º«»&�³jXkU�*f*fEgif³�vm�g´ÈYjZp�rDø gilPj$l�l�j$fOm�giu�wiwno½m��*jkU�*f*f*g´f³�Zm�giÈYj0p�r�m��³j�j$f*�k�ox�*m�gnp�f½u�wn��p�k.gnm��*È��\º�{9p³¼¤r�p�kHu�fOo4k�j$u�l�p�f*uT±Ewij�hTu�wi�³jYp�r}_0ý+�j�k�m�u�gif*wio�¼m��³jHu�wiwipÅ·}j$y�kU�*fEf*gif³�Pm�giÈYj�p�rtøÎl��*p��*wiyZ±¬j�wnp�f³��j�kDm��*u�f�m��³j<m�giÈYj<f³j�j$y*j$y�m�p�j$fE�k�oO�Em.þIu�f*yvu�fOo`*�N+T¼*m��*j<l��U�*j$ÈYj<gil&f*p�m<ýR_.ða`�þóµól�j$���³k�jPuT��u�gif*l�m&��gn���³j�k�m�jÐ9m�µAp�f*wnoZuTm�m�u���¶9l�º8Mj$��u��El�j<m��*g´lIk�j$l��*wnm&g´l�l�p�giÈY�¬p�k�m�u�fOm�¼9·MjHl�m�uTm�j<m��³j<rSp�wiwipÅ·/gif*�Zý�j$u�l�o*þ��p�kUp�wiwiuTk�o�º� Ù��¹Ù �-�R����� Ú¡ E¢�ÀUô�ÄóÝ�äÅä¹â�£2¤2¥xÛ¦Àã�Ã$Á.Ç.¿�Ä+ß�äTãßSÆPf³p�mHÆ�ÀUÃ�Ü9Á�À(�ý-8Mo§¦�m�jÐ9m�±¬pOp�¶�¨©�L{ª! j$f*�k�ox�*m�gnp�fZ·}j�ÈYj$u�f(m��*j�l��U�³j$È0j<gifX·/�*gi�.�Zm��³jQ�E�³±�wigi�&¶�j�o(�p�f*lUgil�m�lIp�r�uÈYpxyE�*wi�*l¬«©u�f*y4uX�E�³±Ew´gi�QjÐ9��p�f*j$f�m�­�¼;u�fEyuXÈ0j$l�l�uT��j$lCî3®�¯*°± gilLj$f*�k�ox�*m�j$y2±Oo2�p�ÈY�E�³m�g´f³�õJÿNî9²MÈYpxy#«Bº}{9g´f*�jCm��*gil&l��.�³j$ÈYj<g´l�ö�À�ÄóÀ�Á.æ=ß�ã*ß´ÆÄ+ß+ÃPgnm&��u�fEf³p�m�±¬j�l�j$���³k�j�ºàþ

� f r+u��m�¼\gnmHm�uT¶�j$lYl�p�ÈYjZ·Mp�kU¶½m�p2È�uT¶�jM�C{�!¦l�j$���³kUj�¼\u�f*yB·MjZy³j�r�j�kPm��Egil<�*fOm�giw}wiuTm�j�k=gif¾m��*jl�j$ÈYj$l�m�j�k$º � fEl�m�j$u�y�¼³·}j��*k�j$l�j$fOm&u0l�g´ÈY�Ewnj�k�ý�o�j�mDl�j$���³k�jÏþ}j$f*�kUoO�*m�gip�f(l��U�³j$È0j�Ê*kUl�m�º

³ ´ �<�a�a�=��� �²�Y Y¡U¢U�¤£O¥¦�t§©¨«ª=�t�¬§H­H®¬¢U¯Dª��<�¶µ=�O· �

8Mj�r�p�k�jP�*k�j$l�j$fOm�gif³�Yp��*k/Ê*kUl�m/l��U�*j$ÈYj�¼E·}j<k�j�h9gnj�· u0fO�EÈ<±¬j�k&p�rtr+u��m�l/r�k�p�È fO�*ÈH±¬j�k&m��³j�p�k�o½ý�m��*uTm·Mj��*uÅh�j�l�j�j$f�u�wnkUj$u�y³o�gifXm��*gilI��wiu�lUl.þ.º\»&�³kUp��³���³p��³m}m��*gilIl�j$�m�gnp�fZ·}j�wnj�m}« ÿ�¸ ¹H±�jQuP�*k�p9y*�*�m}p�rmó·}pqy*gil�m�gif*�m�¼9pxyEyZ�*kUgiÈYj$l$º

>»º Ð9u��m�wno0�Eu�wnr�m��³j/j$wij$ÈYj$f�m�l}p�r¶¯*°±«�Eu$h�j���u��p�±Eg�l�o9È<±¬p�wH¼½+T¼xu�fEy�jгu��m�wno��*u�wir;�*u$h�j���u��p�±�gl�o9ÈH±�p�w g +TºP»&�³j�k�jYgilCuZ�¬p�wno9f³p�È0giu�w"µAm�g´ÈYj<u�wn��p�kUgim��*È rSp�k��p�ÈY���³m�gif³�Xm��³j���u��p�±�gtl�oxÈH±¬p�wp�r}u�f4j$wnj$ÈYj$fOmP¾»®Q¯ °± Àú�Àã½ùtß+Ä�åxäTܳÄ/âÏã�äTù�ÞiÀ�öÅíOÀqä�Â0Ä�å9À&Âç�ÃÅÄAäTÁ�ß@?Åç�Ä+ß+ä�ã]ä�Â2« ý�·Mj0yEgiy-f*p�m�pÏh�j�kDm��*g´l�u�wn��p�kUgnm��*Ȧg´f(��wiu�l�l.þ.º

> ÉLr/m��³jZj$wnj$ÈYj$fOm�lYgif»¯ °± ·/gnm��]��u��p�±�g}l�oxÈH±¬p�w}¼#+T¼ajÐ9u��m�wio �*u�wir�p�r/m��³j$l�j�uTkUjM¿x�*u�y³k.uTm�gi�k�j$lUgiy*�³j$l¤u�f*y=�*u�wnrEuTk�j}¿x�*u�y³kUuTm�g´�Mf³p�f³µAk�j$l�giy*�*j$l�º º h�j�kUo%¿x�*u�y³kUuTm�gi�Mk�j$l�g´y*�³jM�Eu�l¤r�p��³kty*gil�m�gif*�mlÀ¿x�*uTk�jQkUpOp�m�l&g´f�¯ °± º

>ÂÁ gih�j$f2u�f4j$wij$ÈYj$f�mþ»®Q¯*°± ·/gnm��4��u��p�±Egtl�oxÈH±¬p�w�¼#+T¼;gnmLg´lL�p�f^Ä�j$�m��³kUj$ym�pZ±�jŦ��*uTk.y�¨Xm�py³j$��g´y³jH·D�³j�m��³j�kCp�kQf³p�m2¾½gilCuÆ¿x�*u�y³k.uTm�gi�<kUj$l�giy*�³jPgnram��³jPr+u��m�p�kUgn�$uTm�gnp�fp�rO«©gilL�*f³¶9f³pÅ·Df�º°jqk�j�rSj�kPm�pvm��Egil�u�lHm��³j�åxç�Á�öTã¬À�ÆUÆ(ä�ÂXöOÀ�Ã�ß�öTß�ãOíÅÇ�Ü*ç�ö�Á�ç�Ä+ß�Ã(Á�À�Æ�ß+ö�ܳäTÆ.ß+Ä+ǹº7lìuTm�j�k$¼�·}j���gnh�jXu�*kUj$��gil�jQy³j�ÊEf*gim�gnp�fXp�rË·D�*uTm�m��*gil�ÈYj$u�fEl�º

> °��³j$f2m��³jPr�u��m�p�kUgi�$uTm�gnp�f4p�rO« ßSÆ=¶xf³pÏ·/f�¼�m��³j�k�jYgilLuX��p�wioxf³p�È�giu�w"µAm�giÈYjHu�wn��p�kUgnm��*È r�p�kCy*jµm�j�kUÈ�gif*gif³�P·/�³j�m��*j�k&u�f(j$wnj$ÈYj$fOm}¾7®9¯*°±²gil�u#¿x�*u�y³kUuTm�g´�DkUj$l�giy*�³jCp�k/f*p�m�º/ý-!DlDy*gil����*lUl�j$yZgif��wiu�lUl�¼�m��*giltu�wi��p�kUgnm��*ÈÎ�*l�j$ltm��³j©¦ z �*g´f³j$l�jIk�j$È0u�gifEy³j�kUgif³��¨Lk�j��Ek�j$l�j$fOm�uTm�gnp�fYp�r�¾�·/gnm��=kUj$l��¬j$�mm�pȸqu�f*y�¹xÌ9y³j$��giy*g´f³�ÿO�*u�y*kUuTm�gi��k�j$lUgiy*�³p�l�gimóoHÈ0pxy*�EwnpQ�*kUgiÈ0jIfx�*ÈH±¬j�kUlagiltj$u�l�ovü u�l\o�p��ql��³pÅ·p�f��*p�ÈYj�·Mp�kU¶ | �Aºàþ

»&�³j&�*uTk.y*f³j$l�lËp�r¬y³j$��giy*gif*�P¿x�*u�y³kUuTm�g´�MkUj$l�giy*�³p�lUgnmóo<lU�³����j$l�m�lam��*j�rSp�wiwipÅ·/gif*�Dj$f*�kUoO�*m�gip�fYl��U�³j$È0jrSp�k&j$fE�k�oO�Em�gnp�f(p�rL+�µA±Egnm&ÈYj$l�lUuT��j$l�$+^É�;0ý.+ C þ=��j$f³j�kUuTm�j$l4ý+�*l�gif³�4m��³jv��p�wioxf³p�È�giu�w"µAm�giÈYjq�*kUgiÈ0jX��j$f*j�kUuTm�gnp�f u�wi��p�kUgnm��*È0lYyEgil����*l�l�j$y]gif

��wiu�lUl.þ�m�·Mp(k.u�f*y³p�È �*kUgiÈ0j$lʸìðÀ¹q·/gnm��§� ¸��*ÿË� ¹ª�EÿÍÌ2u�f*y4�p�ÈY�E�³m�j$lë ÿm¸ ¹xº�»&�*jH�E�*±Ewigi�¶�j�o(gil}«Ku�f*y(m��³jHl�j$�k�j�m&¶�j�o(gil<ýθ¤ðÀ¹�þ.º

Ï

Page 83: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

0É�»ìpqj$f*�k�ox�*m]¦<��¨9¼�m��³jHl�j$f*y³j�kD�.�³pxp�l�j$lCõ?u�lLu0kUu�f*y*p�ÈË¿x�*u�y³kUuTm�gi��k�j$l�giy*�*j�gif9¯ °± ÌEm�pXj$f*�k�ox�*m¦�+�¨9¼�m��³j<l�j$fEy³j�k/�U�*pOp�l�j$l�õ u�lDuYk.u�f*y³p�ÈË¿x�*u�y³k.uTm�gi�Qf³p�f9µAk�j$lUgiy*�³j�gifA¯*°±&ºI»&�³j<��gi�E�³j�k�m�jÐ9mgil/õYº/ý-!DlDf³p�m�j$y�uT±¬pÅh�j�¼�gnm&gil&�k.�*��giu�w¬m��*uTm&j$fE�k�oO�Em�gnp�f(gilCÁ�çTã�ö�äTæ=ßS?ÅÀUöTºàþ

Ï É�e/�¬p�fQk�j$�j$gihxgif*�&��gn�E�³j�k�m�jÐ9mËõY¼Åm��³jakUj$�j$gnh�j�kIý+�El�gif³�b¸ìðÀ¹�þ¬y³j�m�j�k.È0gif³j$l�·D�³j�m��³j�kËõ gil;u*¿x�*u�y³k.uTm�gi�k�j$lUgiy*�³jLp�kDf³p�m�¼�u�f*y(y³j$�k�ox�*m�l&u����p�kUyEgif³��wno�º

� f³r�p�kUÈ0u�w´wno�¼tu�fBj$uÅh�j$l�y³k�p��E��j�kYl�j�j$l=p�fEwnoj« u�f*y«u�f j$wnj$ÈYj$fOm�õЮ�¯*°±/º¾{9gif*�jXgnmPg´l7¦��*uTk.y�¨2m�py³j�m�j�kUÈ0gif*jQ·/�³j�m��³j�kLõÎgil&u©¿x�*u�y³kUuTm�g´�DkUj$l�giy*�³jCp�kLf³p�m�¼³m��³j<l��.�³j$ÈYj<l�j�j$È0l/l�j$���*k�j�º

°jDf³j�j$y0m�p�±¬j&��uTk�j�r��Ew�¼�m��³p��³���;ºb'�gnkUl�m�¼�f³p�m�j&m��*uTmMgnmagilt�*uTkUy=m�pHy³j$��giy³j*¿x�*u�y³kUuTm�g´�Ik�j$l�giy*�*p�l�gnmóoäTã*ÞèÇ-r�p�k�j$wnj$ÈYj$fOm�lPõ¦·/gnm��½��u��p�±Egtl�o9È<±¬p�wh¼½+Tº�8I�³m�gir\m��³j�l�j$f*y³j�kQ�Egi�U¶xl<uZkUu�f*y³p�È�¿x�*u�y³k.uTm�gi�f³p�f9µAk�j$l�g´y*�³jXõ ·/�³j$f¾j$f*�k�ox�*m�gif³�2uѦ�+�¨9¼am��*j$f¾·/gim��½�*k�p�±�uT±Egiwignm�oÒ0�Ó Ï gnmH·/giwiwa±�jqm��³jq��u�l�jXm��EuTmõ �*u�lD��u��p�±Eg¤l�oxÈH±¬p�w g +�ý�·D��o�ÔÏþ.º�{9gif*�jQm��*jP��u��p�±�g;l�o9È<±¬p�w;��u�fv±�jH�p�ÈY�E�³m�j$y�g´f(�¬p�wnoxf*p�È0giu�wm�giÈYj�¼�u�f¸j$uÅh�j$l�y³k�p��*��gif³�½u�y³h�j�kUlUuTk�o]��u�f¸m�j$wiw�·D�³j$f]m��*gil0px�����³k.l�u�f*y]m��*jvl��U�*j$ÈYjv·/g´wiwIf³p�m0±¬jl�j$���³k�j�º}{9p³¼*m��³j�l��.�³j$ÈYj<f³j�j$yEl�m�p0±¬j�ÈYp9y*gnÊ*j$yZu�l&r�p�wiwnpÏ·/l�$0fÕRÉ�»¤pYj$f*�k�ox�*m©¦<��¨9¼�m��³j�l�j$f*y³j�k&�.�³pxp�l�j$lLõ u�l&u=kUu�f*y³p�ÈË¿O�*u�y*kUuTm�gi�Lk�j$l�giyE�³jQgif�¯ °± Ì*m�p�j$f*�k�ox�*m

¦�+�¨9¼�m��³jIl�j$f*y*j�k��.�³pxp�l�j$laõ u�ltuDkUu�f*y*p�È�¿x�*u�y³kUuTm�g´�\f³p�f9µAkUj$l�giy*�³jMgif%¯ °± ùtß+Ä�å½Ö9ç�Ã.ä�Ýß�Æ�ǹæ�ÝUäTÞ¼½+Tºa»/�³j���gn�E�³j�kUm�jÐxm&gilLõ=º

�/pÏ·�¼�m��³j���gn���³j�k�m�jÐ9maõNu�wi·}uÅoxla�Eu�la��u��p�±Eg³l�o9ÈH±�p�w�¼½+Iu�f*yY�*j$f*�j � uT�*�¬j$u�wigif³�Dm�pQm��³j&�*uTkUy*f*j$l�lp�r�y*j$��giy*gif³�½¿x�*u�y³kUuTm�gi�CkUj$l�giy*�³p�lUgnmóoqrSp�k&j$wij$ÈYj$f�m�l&·/gnm�����u��p�±�g;l�o9È<±¬p�w¶¼½+ � m��³j�l��U�*j$ÈYj<l�j�j$È0ll�j$���³k�j�º-ýAÉCrI�p��*kUl�j�¼ì·}j0·/g´wiw¤l�m�giwiwtf³j�j$ym�pv�*k�pÅh9giy³j0u�kUgn��p�k�p��*lC�Ek�pOp�r}p�rIl�j$���³kUgnm�o4m�p�h�j�kUgnr�o-p��*kgif�m��Egnm�gnp�f�º�°j<y³p�m��*gil�gifZm��³jHf³jÐxm&��w´u�l�l�ºàþ

�/jÐ9m�¼³·}jLf³j�j$yqm�p=h�j�kUgnr�oYm��*uTmIu�wiwEu�wn��p�k.gnm��*È0l}gif0m��*jDj$f*�kUoO�*m�gip�fql��.�³j$ÈYjC��u�fX±�jDjÐxj$���*m�j$y(gif��p�wioxf³p�È�giu�w¬m�giÈYj�º}»&�³jQ¶�j�o(��j$f³j�kUuTm�gip�fvu�wn��p�kUgim��*Èk; �j�k�m�u�g´f*wnoXkU�*f*l&g´fZ��p�wioxf³p�È�giu�w¬m�giÈYjQ��gnh�j$fm��³j<u�wn��p�kUgim��*È0lIr�p�k&��j$f³j�kUuTm�gif*�Y�*kUgiÈYj$lIm��EuTm&·Mj��Eu$h�jHu�wik�j$u�y³oZl�j�j$f�ºa»&�³j�y*j$�k�oO�Em�gnp�f(u�wn��p�kUgnm��EÈkU�*f*l&g´f(�¬p�wnoxf*p�È0giu�w¬m�giÈYj�¼�l�gif*�j�·}jH�Eu$h�jHl�m�uTm�j$y2uT±¬pÏh�j<m��*uTmLy³j$��giy*gif*�©¿O�*u�y*kUuTm�gi�Ck�j$l�g´y*�³p�l�gnm�oXg´lj$u�l�oH·D�³j$f=m��³j�r+u��m�p�kUgn�$uTm�gnp�fYp�r�« g´l¤¶9f³pÏ·/f�º�8I�³m�·/�*uTmauT±¬p��³m�m��³j�j$f*�k�ox�*m�gnp�f=u�wn��p�k.gnm��*È]ÔM�DpÅ·��u�fm��³jYl�j$f*y³j�k � ·/�*pZy*pOj$lCf³p�mQ¶9f³pÅ·Îm��*jPr+u��m�p�kUgn�$uTm�gip�fp�rO« � �U�³pxp�l�jYkUu�f*y*p�È׿x�*u�y³k.uTm�gi�k�j$l�giy*�*j$l�u�f*y�f³p�f9µAkUj$l�giy*�³j$lKÔ

°jHf³p�m�j<m��³j�r�p�wiwnpÅ·Dgif³�(ý+gif�u�wiw���u�l�j$l$¼E·}j�u�l�l��*ÈYj<m��³jQr�u��m�p�k.gn�$uTm�gnp�f�p�r�«Kgil�f³p�m&¶9f³pÏ·/f�þ&$

> z �*pOp�l�g´f³�YuYkUu�f*y³p�ÈؾҮ9¯ °± ·/gnm�����u��p�±Eg;l�o9È<±¬p�wÙ¼#+Qgil©¦�j$u�l�o�¨9ºI{9g´ÈY�Ewno0��gi��¶ZuYkUu�f*y*p�Ⱦڮ�¯*°± u�f*y«m�j$l�m�·/�³j�m��³j�k�gim0�*u�l���u��p�±�g&l�o9È<±¬p�wȼ½+Tº � rLgim0y³pOj$l$¼Mm��³j$f¸p��³m��E�³m½¾ìº � rf³p�m�¼¬k�j��¬j$uTm�ºPÉQf2uÅh�j�kUuT��j-ý+l�g´f*�jP�Eu�wnram��³j=j$wnj$ÈYj$fOm�lLp�rO¯ °± �*u$h�j0��u��p�±�gtl�oxÈH±¬p�w�¼#+Åþ&m��*ju�wn��p�k.gnm��*È m�j$l�m�lP0=j$wnj$ÈYj$fOm�l&±¬j�r�p�k�jQÊEf*y*g´f³�Hp�f*jQ·/gnm�����u��p�±�g;l�o9È<±¬p�wÙ¼½+Tº

> {9g´È0giwiuTkUwio�¼Å�U�³pxp�l�gif*�DuDkUu�f*y³p�È6¾7®9¯ °± ·/gim��P��u��p�±EgOl�oxÈH±¬p�w g +Mgil¬¦�j$u�l�o¨Yý�h9giu&m��³j}p�±xhxgnp��ElÈYp9y*gnÊ���uTm�gnp�fXp�rËm��*jQ�*k�j�h9gnp��*l�u�wn��p�kUgim��*Èqþ.º

> z �*pOp�l�g´f³�YuYkUu�f*y³p�ÈÛ¿x�*u�y³k.uTm�gi�Ck�j$l�giyE�³j2¾2gilIj�h�j$fvj$u�l�gnj�k($Ml�giÈY�Ewio0�Egi�U¶XuYk.u�f*y³p�ÈÛÜM®9¯ °±u�f*y l�j�m�¾ ÿ¡Ü�ÝIÈYp9y©«¾º z wij$uTkUwno�¼/¾NgilquQ¿x�*u�y³kUuTm�g´�(k�j$l�giyE�³j�ºm'*�*k�m��³j�kUÈYp�kUj�¼Igim�gil�f*p�m�*uTk.yvm�pZl��³pÏ·¦ý+lUgif*�jPj�h�j�k�o7¿x�*u�y³kUuTm�g´��k�j$l�g´y*�³jH�Eu�lDr�p��³kCy*gil�m�g´f*�m/lÀ¿x�*uTk�jPk�pxp�m�l.þ/m��*uTm¬¾½g´luZÁ�çTã�ö�ä�æÞ¿x�*u�y³k.uTm�gi�Ck�j$l�giyE�³j�º

8}�*mL�³pÏ·Î��u�f4m��³jPl�j$f*y³j�kC�U�³pxp�l�jYu�k.u�f*y³p�È׿O�*u�y*kUuTm�gi�qã�äTã�µAk�j$l�giy*�*j�Ô »ìk�ovm�pqm��*g´f³¶�uT±¬p��³m/m��*g´l�*f�m�g´w�f³jÐxm/m�giÈYj�º�º�º

|

Page 84: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

߬à�áSâ�ã�äaåÀá@æ~ç]è�é�êAë^äaçìSë�à�íî êðï î/ñóò�ô%õ÷ö àóæ<äaèí�ø�ùEæÀá@è�àMæ<è î äÀçú�æ<è�û�äaë^ú üóçýÊäÀè�éRã(åÀå<è�ä2þ�è�à�ë^æÀü�ë�àAÿ�ë^æ��

���������� ��

� �����������������! #"%$&�!'( *)*+,��-.$0/2134�657-8$:9;�&15�',1<+,����=>�?3(��5@�&-83A��+B

C¶ë�å<æUæÀáED#ãGF!H/ãÆåaëIHØæÀü�ë^æ½ë�àQã(à�ùEäaç�ú�æÀáSè�à�åÀù&üªãID#ãMáSàJH¬ü�áSùaü ëLK�MGNOHLë�åUäÀã�ú�äÀã(åÀã(àóæ<ã(í ë�åQP-åÀë(çSR%ëT ø�ë�í�äaë^æÀáSù#äÀã(åaáSí�øªã©ë�à�íQëUKWVWN�HLë�å%äÀã�ú�äÀã(å<ã(à�æ<ã(íQë�å%ë T ø�ë�íªäaë^æÀá ù#àªè�àYX�äÀã(åÀá í�øªã@D©á@û�ü�æ�ZHã�å<ã(ù�øªäaãG[߬àªéRè�äÀæÀø�à�ë^æ<ã(ì@çGFOæÀü�ã\K<à�ë�á@â�ãINÅåÀù&üªãID#ã�H/ã�û�ë(â�ãðìSë�å<æ#æÀáED#ã9í�áSí»àªè�æ©å<ã�ãID æ<èJZHã�áED#ú ì@ãID#ã(à�æÀë]Z�ì@ãP-åÀáSà�ùEãÈáSæ!H/ë�å/àªè�æhù�ì@ã(ë^ä/üªè^HÑæÀüªã¬å<ã(à�íªã�ä?H/è�ø�ìSí½û�ã(àªã�äaë^æ<ã2ëÃäaë�à�íªè_D T ø�ë�íªäaë^æÀáSù}à�è�àYX�äÀã(åÀáSí�øªã`R�a�bÅãàªè^H\í�áSåÀù�ø�åaå/åÀè_D#ãcHLë(ç�å*æ<è©û�ã�æPë^äÀè�ø�à�íMæÀü�á å}å<ã�â�ã�äÀã%í�äaëIHdZ ë�ù�eA[

bÅãgf�ä&å<æ}àªè�æ<ã�ë&fihjHÈü�áSù&üÆåÀè�ì@â�ã(å}è�øªä*ú�äÀèGZ�ì@ãIDkP-å<è_D#ãWH¬ü�ë^æ�R<Z�øªæ}ü�ë�åÈíªäaëIHlZ�ë�ù�e�å}è�é�á@æÀåLè^HÈàm[V]nco@PpVWqGR*û�ã(àªã�äaë^æ<ã(å2ærHOèÆäaë�à�í�è_DËú�ä&áED#ã(åtsvu�w@HÈá@æÀü%x s!xByzx wSxYy;{9ë�à�íðùEè_D#ú�øªæ<ã(å.|}y\s4wB[l~Èüªã

ú�øSZ�ìSáSùle�ã�çMá åd|¡ë�à�í�æÀüªã�å<ã(ùEäÀã�æ8e�ã�ç�áSåcP�svu�w_R�[� nc~¶è%ã(à�ùEäÀçú�ægK�MGNYFæÀüªãÈå<ã(à�í�ã�äOùaü�è�è�å<ã(å6�*áSà�íªã�úHã(à�í�ã(àóæÀì@çXäaë�à�íªè_D T ø�ë�íªä&ë^æÀáSù}äÀã(åÀá í�øªã(å�áSà@�d��l�æ<è

ã(à�ùEäaç�ú�æcKWVWNYFæÀüªãPå<ã(à�íªã�ähù&üªèè�å<ã(å!�*áSà�í�ã�ú�ã(à�íªã(àóæÀìSç�äaë�à�íªè_DGã(ì@ãID#ã(à�æÀå?HÈá@æÀü]þ�ë�ùEèGZ�á å<çBD*ZHè�ì� V][?~Èüªã�ù�á@ú�üªã�äaæ<ã�hæ*ùEè�à�åaáSå<æÀå}è�é�ë�ìSìA�2ã(ì@ãID#ã(à�æÀåW[

� n�ßÈúHè�à©äaã(ùEã(á@âá àªûXù�á@ú�ü�ã�äÀæ<ã�hæ��l��uW�W�W�`u��?��FæÀüªã¬äÀã(ùEã(á@â�ã�ä�P-ø�åÀáSàªû�s�u�wGRÊíªã�æ<ã�ä�D©á àªã(å?HÈü�ã�æÀüªã�ähã(á@æÀü�ã�äPpV^RUã(ë�ù&üL���ÃáSå#ë T ø�ë�íªä&ë^æÀáSùÆäÀã(åÀáSí�ø�ãÆè�ä�P � R½ë^æ�ì@ã(ë�åÀæ#è�àªã9è�éPæÀüªã����2áSå©àªè�æ©ë T ø�ë�íªä&ë^æÀáSùäÀã(åaáSí�øªãGFªë�à�í�íªã(ùEäaç�ú�æÀåÈë�ù�ùEè�äaí�áSàªû�ìSçG[

�báSäaå<æXàªè�æ<ã]æÀü�ë^æ�æÀü�á å�åÀùaü�ãID#ã]áSå%áED#ú�ìSãID#ã(àóæÀë]Z ì@ã�P�HOãÆí�áSåÀù�ø�åaå<ã(íÅìSë�å<æ�æÀá�D#ã�üªè`H æÀüªãÆå<ã(à�íªã�äXù�ë�àú�áSù�eÆäaë�à�í�è_D T ø�ë�í�äaë^æÀáSùPäÀã(åÀáSí�øªã(åLë�à�í�üªè^H6æÀüªã�å<ã(à�íªã�ä*ù�ë�à9ú�áSù�e]äaë�à�íªè_DØã(ìSãID#ã(àóæÀådHÈá@æÀü9þ�ë�ùEèGZ áå<çBD*ZHè�ì � V^R�[Q��øªäaæÀüªã�ä�D#è�äÀã�PRæÀüªã�ú�äaè�è�éLáSåXåÀè_D#ãWHÈü�ë^æXá àóâ�è�ì@â�ã(í�ë�à�íJH/ã@HÈá ìSì�àªè�æ�û�á@â�ãÆá@æXüªã�äaã`R�FæÀüªã%åÀù&üªãID#ãUáSåÈå<ã(ù�ø�äÀãG[�/ø�ælH/ã%ù�ë�àðàªè�æ<ãXærHOèMíªäaë^HdZ�ë�ù�e�å¬è�é�æÀüªãXåÀù&üªãID#ãG[d��á@äaå<æWF æÀüªã%ù�áSú�üªã�äÀæ<ã�h�æì@ã(àªû�æÀüÂü�ë�å&ZHã�ã(à§áSà�ùEäÀã(ë�å<ã(í�Zç ëÒé-ë�ùEæ<è�ä�è�é8�Æë�à�í\HOã9ë^äÀã9åÀæÀáSìSì/ã(à�ùEäÀçú�æÀáSàªûðè�à�ì@ç ëjåÀáSà�û�ì@ã�Z á@æWaïã(ùEè�à�í,FÊæÀü�áSå%åaùaüªãID#ã9ü�ë�åUë������G���� 4¡�¢¤£���£�¥���¥§¦§¨%¢p©Qª¤«�¬W¡�¨� (¦§¥§¢]­2«�¡�¡�¢]¡�[ ö à ú�ë^äaæÀáSù�ø�ìSë^äIFtHÈü�ã(à�æÀü�ãå<ã(à�íªã�ä%ã(à�ùEäÀçú�æÀå%ë�KWVWNAæÀü�ã(à®HÈáSæÀüjú�äÀèGZ ë]Z�áSìSá@æ~ç%P �¯ R

� æÀü�ã]å<ã(à�í�ã�ä%ùaü�è�è�å<ã(å*� T ø�ë�íªäaë^æÀá ù#äÀã(åÀáSí�øªã(åë�à�í�å<ã(à�í�åÊæÀüªãID�[?�/ø�æÊæÀüªã(à]æÀüªãÈäÀã(ùEã(áSâ�ã�ä�HÈá ìSìªáSà�æ<ã�äÀú�äÀã�æÊæÀü�áSåhë�åOëOK�MGNYa�°Pé�ùEè�øªä&å<ãGF�æÀü�ãÈã�±Hã(ùEæOæÀü�á åü�ë�å*áSàMú�ä&ë�ùEæÀáSùEã�ù�ë�àQZHãcD©áSà�á�D©á²��ã(í�Z�çÆå<ã�æ<æÀáSàªû&�ÃìSë^äÀû�ãÃã(àªè�ø�û�ü,[6�Løªæ}æÀüªã(à�æÀü�áSåLå<ã�â�ã�äÀã(ì@çMäÀã(í�ø�ùEã(åæÀüªã�ãW³�ù�á@ã(à�ùEç�P-áSà9æ<ã�ä�D©å}è�étZHè�æÀü�ùEè_D#ú�øªæÀë^æÀáSè�à9ë�à�í9ùEè_D@DUø�à�á ù�ë^æÀá@è�à4ROè�é�æÀüªãXåÀù&üªãID#ãG[

ïèðæÀüªã T øªã(å<æÀáSè�àjäÀãID©ë�á à�åW´#ù�ë�à%HOãMùEè�à�å<æ<äaø�ùEæUëðåÀùaü�ãID#ãÆá àJHÈü�á ùaü ëJ��¥�­�µ_�²« T ø�ë�í�äaë^æÀáSù©ã(ìSã�XD#ã(àóæ&P-á¶[ ãG[²F�äÀã(åaáSí�øªã`·fàªè�àiX�äÀã(åÀáSí�ø�ã`Rhá åLæ<äaë�à�å�D©áSæ<æ<ã(í,F�ë�à�íQH¬ü�áSùaüAáSå*áED#ú ì@ãID#ã(à�æÀë]Z�ì@ã@P-ë�à�í�è�é�ùEè�øªäaåÀãå<ã(ù�øªäÀã`R�¸ ö à]é-ë�ùEæWFSH/ã�ù�ë�àjZ�çÆø�åÀáSà�ûUë#àªã(ë^æ@K.æ<ä&áSù�eYN&HÈüªã�äÀãWZç�æÀü�ãPäaã(ù�á@ú�á@ã(à�æ/ü�ã(ì@ú�å/è�øªæ}æÀüªãÃå<ã(à�í�ã�äáSà½æÀüªãLéyè�ì ì@è^HÈá àªû¹H/ëVç4´tºÈè�æ<ãÈæÀü�ë^æhá@æ�áSå�ã(ë�å<ç.©W¢]¡l¦�»Y«l¡�«�¬�«W¥�¼G«�¡2æ<è�ùaü�è�è�å<ã¬ë2äaë�à�íªè_D T ø�ë�íªäaë^æÀáSù}à�è�àYXäÀã(åÀáSí�ø�ã&P-åÀá à�ùEãÃæÀüªãÃäÀã(ùEã(á@â�ã�äleàªè`HÈåLæÀüªãÃé-ë�ùEæ<è�äaá²�(ë^æÀá@è�àAè�é�|½R�[6~*üªã2äÀã(ùEã(áSâ�ã�ä¬ù�ë�à�æÀüø�å�K<üªã(ì@ú�è�ø�æ�NæÀüªãÈå<ã(à�í�ã�ä6ZçUáSà�ù�ìSø�í�áSà�ûPë T ø�ë�íªä&ë^æÀáSù}àªè�àYX�äÀã(åaáSí�øªã¾½ë�åÊú�ë^äaæÊè�é�ü�áSå�ú�øSZ�ìSáSù�e�ã�çG[?ºÈè`HcFæÀüªãÈå<ã(à�í�ã�äù�ë�àÑùaü�è�è�å<ã7ëÅäaë�à�íªè_D T ø�ë�í�äaë^æÀáSùMäÀã(åÀá í�øªã�ë�å�ZHã�éyè�äaã®PRæÀü�ë^æ]áSåIF6Zç ù&üªè�è�åaáSàªûÅäaë�à�í�è_DÀ¿>ÁÂ�d��ë�à�íÒå T ø�ë^äaáSàªûÆá@æ�R�[c�Løªæ2æÀüªãUåÀã(à�íªã�äÃù�ë�àÅàªè^HÍë�ìSåÀèMù&üªèè�å<ã#ë]ä&ë�à�íªè_D T ø�ë�íªäaë^æÀáSùUàªè�àYX�äÀã(åaáSí�øªãcZçùaüªèè�åÀáSàªû]ë½äaë�à�íªè_D T ø�ë�íªä&ë^æÀáSù2äÀã(åÀáSí�øªã�éë�à�í9ùEè_D#ú øªæÀáSàªû�ÃÅÄ�Æ�Ç`ÈɾA[?bÅãXû�á@â�ãÃæÀü�ãXíªã�æÀë�áSìSå}à�è^Hc[

V

Page 85: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

V]nco@PpVWqGRÊû�ã(àªã�äaë^æ<ã(å/ærHOè½äaë�à�í�è_D ú�ä&áED#ã(å(svu�wcHÈá@æÀü�x s!x_yÊx wSxGyË{]ë�à�í�ùEè_D#ú øªæ<ã(å�|Ìy�sAw�[!~*ü�ã(à,Fù&üªè�è�åÀã]ë�àQë^ä�Z�á@æ<äaë^äÀç T ø�ë�íªäaë^æÀáSù�àªè�àYX�äÀã(åÀá í�øªã&¾�HÈá@æÀüQþóë�ùEèGZ�áhå<çYD*Z�è�ì � VÍP-àªè�æ<ãÆæÀü�ë^æ�æÀü�á åù�ë�à7àªè^HÎZ�ã�í�è�àªãÃãW³�ù�á@ã(à�æÀì@ç�åÀáSà�ùEãÃæÀü�ã�éRë�ùEæ<è�ä&á²�(ë^æÀá@è�à7è�ét|�áSåe�àªè`HÈà4R�[�~ÈüªãÃú�øiZ�ì áSù.e�ã�çMá åP§|Ou�¾iR}ë�à�í�æÀüªã�å<ã(ùEäaã�æde�ã�ç9áSåcP�svu�w_R�[

� nc~*üªã#å<ã(à�íªã�äÃù&üªè�è�åÀã(å�ëMäaë�à�íªè_DÏ¿�Á®� �� ë�à�íÒùEè_D#ú�øªæ<ã(å�ÃQyп ¯ [*º¬è�æ<ã#æÀü�ë^æcÃ�áSåÃëÆäaë�à�í�è_DT ø�ë�íªäaë^æÀá ù½äÀã(åÀáSí�ø�ãG[Å~¶è7ã(à�ùEäÀçú�æ�K�MGNYF�æÀüªã]åÀã(à�íªã�äXå<ã(à�í�å*�ÑyzÃS[�~ÙèAã(à�ùEäÀçú�æXëUKWVWNYFbæÀü�ãå<ã(à�íªã�ä¬å<ã(à�í�å8�>yÂÃcÒ^¾A[dº¬è�æ<ã%æÀü�ë^æPáSà9æÀü�ãgf�äaåÀæÈù�ë�å<ãGFm�GáSå¬ë©äaë�à�íªè_D T ø�ë�íªäaë^æÀá ù�äÀã(åÀá í�øªãG[ö àAæÀü�ãUå<ã(ùEè�à�íðù�ë�åÀã�� áSå¬ëÆäaë�à�íªè_D T ø�ë�íªäaë^æÀáSù%àªè�àYX�äÀã(åÀá í�øªãcHÈáSæÀü7þ�ë�ùEèGZ ábåÀçBD*ZHè�ì � VÓP�HOãì@ã(ëVâ�ã�â�ã�äaá²f ù�ë^æÀá@è�à�è�é�æÀü�áSåLé-ë�ùEæ*æ<è�æÀü�ãÃäÀã(ë�íªã�ä�R�[

� n�ßÈúHè�àÃäÀã(ùEã(áSâáSà�û*ù�á@ú�üªã�äÀæ<ã�h�æÔ��FVæÀüªãÊäaã(ùEã(á@â�ã�ä�P-ø�åÀáSàªûÔsvu�w_RHíªã�æ<ã�ä�D©áSàªã(å,H¬üªã�æÀüªã�äÔ�ÑáSå�ë T ø�ë�íªä&ë^æÀáSùäÀã(åaáSí�øªãPè�ä¬àªè�æWF ë�à�í�íªã(ùEäÀçú�æÀå*ë�ù�ùEè�äaí�áSàªû�ì@çG[

ö æÈåaüªè�ø�ìSíÓZHã�ù�ì@ã(ë^ä*æÀü�ë^æÈæÀüªã�äÀã(ùEã(á@â�ã�äPë�ì²HLë(ç�åÈíªã(ùEäÀçú�æÀå*ùEè�äaäÀã(ùEæÀì@çG[bÅã�àªè`HÚí�áSåÀù�ø�åÀå¬æÀüªã©å<ã(ù�ø�äaá@æ çAè�éhæÀüªã#åÀù&üªãID#ãG[*�/ã�éRè�äÀã�H/ã½ù�ë�àjéyè�ä�D�ë�ìSì@ç�ú�äÀè�â�ã©å<ã(ù�øªäaáSæ çGF,H/ã

D%ø�åÀæ*éyè�ä�D©ë�ìSì@ç�åÀæÀë^æ<ãXè�øªäÈü�ë^äaí�àªã(åÀå}ë�åaåÀøSD#ú�æÀáSè�àMäÀã�û�ë^äaí�áSà�û½í�ã(ù�áSí�áSàªû T ø�ë�íªäaë^æÀá ù2äÀã(åÀáSí�ø�è�åÀá@æ çG[

ÕQÖ¤×!Ø�Ù�Ú`Ù�ÛAØÎÜÞÝO«Ó�W�]¨�ª¤«�¬W¥¶ª]¥�­�µ�ßIàS�_ª]¡��G¦§¥§¬O¡�«���¥§ª]àS¢á��¥�¦§¨®¥���P�È�u�â�R�ãä»Y�]¡pªLåä¡�«W�æ�G¦§¥�¼]«Í¦ç¢��W¢]�Å«��]� µ�¢]ã¡�¥§¦�»¤�zo©W¢]¡8µB«�­A«W¡p�G¦§¥�­�µj�@¢IªGàB�è¥�|Íé�¥ ©�êS©W¢]¡ë�]�����]�쵤¢G¡�¥§¦�»¤�ë�.íz¡�àY­S­S¥�­�µÓ¥�­½¦§¥��Å«dÈgî�«�»Y�]¼]«^ïðð ýÊä^ñ |óòôo&��¾Åò:õö � ´¤í&P§|Ou�¾iR?y>V�÷4øjýÊä^ñ |óòôo&��¾Åò:õöëùLú �� ´_í&P§|Íu�¾SR6y>V�÷ ððSû âWu

ît»Y«�¡�«&õö � ª¤«W­4¢G¦ä«���¦�»Y«¹�W«W¦!¢p©�ßWàS�_ªG¡p�G¦§¥¶¬g¡�«���¥¶ª]àü«��¹��¢^ª]àY�æ¢g|ý�]­Aªjõö*ù�ú �� ª¤«�­4¢_¦ä«��.¦�»i«.�W«I¦?¢p©ßWàS�_ª]¡��G¦§¥¶¬&­4¢]­SãÉ¡�«���¥¶ª]àü«��*¥�­j�d��Ðî!¥§¦�»jþi�_¬�¢¤£�¥!��¨]�@£�¢]� � V_ÿ

bÅã*D©ëVçMà�è^H6å<æÀë^æ<ãUë�à�í�ú�äaèVâ�ã�æÀüªã%å<ã(ù�øªäaá@æ~ç]è�ébæÀüªãXåaùaüªãID#ãG´��� Ö�Û��`Ö�� Ü�������àY�&¥�­�µ@¦�»B�_¦�ª¤«�¬W¥¶ª]¥�­BµÓßIàS�_ª]¡��G¦§¥§¬*¡�«���¥§ª]àS¢á��¥�¦§¨Oåç¡�«��²�G¦§¥�¼G«*¦ç¢co.é�¥���P�È�u�â�R�ãä»Y�]¡pª]ê<¦�»Y««�­4¬I¡�¨� A¦§¥¶¢]­���¬W»Y«��Å«&¢]ài¦§�è¥�­(«�ªQ�¤£�¢]¼]«*¥��*P�È�u � â�R�ãä�I«�¬IàB¡�«��Iµ��]¥�­i��¦�¬W¥  ü»Y«W¡�¦ä«_¦§ã�¢]­S�è¨j�G¦¶¦ç�¤¬��^�Wÿ ��`Û,Û�� �*ã(ù�ë�ìSì áSàªûÒæÀüªãAíªãWf à�á@æÀá@è�à è�éÃå<ã(ù�øªäaáSæ ç§ë^û�ë�áSà�å<æ©ù�áSú�üªã�äÀæ<ã�h�æpX�è�à�ì@ç»ë^æ<æÀë�ù�e�åWFdH/ãAàªã�ã(íÑæ<èåÀüªè^H�æÀü�ë^æ*éRè�äÈë�à�çÆë�ì@û�è�ä&á@æÀüSDÑí�äaø�à�à�áSàªû½áSàMæÀáED#ã�È<HOã%ü�ë(â�ãG´

� â�� xiýÊä^ñ²P§|Ou�¾iR�òôo&���Îò�� ��� � P§M¤R<´�íëP§|Ou�¾4u��*R?y�Má÷øjýÊä^ñ²P§|Ou�¾iR�òôo&���Âò�� ��� � PpV^R�´¤íëP§|Ou�¾Au��gR?y�Má÷Éx��

�ªè�ä¬åÀá�D#ú�ìSáSù�áSæ çGFYHOã%ø�å<ãÃæÀüªãÃéRè�ìSì@è`HÈáSàªû�PRã T ø�á@â^ë�ì@ã(àóæ � R/éRè�ä�DUø�ìSë^æÀá@è�àm´

�������! #"%$yÀx � Ò(ýhä`ñ²P§|Ou�¾iR?òÀo&�&dò('^MYuIV*)_���>ò+� ��� � P,&�R�´¤íëP§|Ou�¾Au��gR?y�&�÷üø\V¤x û � âW�b.-Xë�åÀåÀøSD/-Pæ<è`H/ë^ä&í9ë#ùEè�0óæ<ä&ë�í21Sù�34165�073Àü�ë�383Àü91;:<1;:.­4¢G¦=3?>aø@-G�21¶[A-G[²F@3Àü9B�383Àü@-C>4-D-�h�1E:?34:8BF0GBFH@ûF5F>163ÀüüDíI>aø202021E0ªûJ1E0G341EDK-cÈ<LM5F>lHÈü91Sùaü ���F� �ON � âW[�b.-ëHP1;H;HQ3Àü@-R07ø2:?-�3Àü21E:SBFH@ûF5F>163ÀüüDT3?5]ù�5�09:?3?>aø�ù�3PBF0BFH@ûF5F>163ÀüSD íVU9W@-(ùC1;W21E0ªû T ø2BFW@>B�341SùP>4-R:41;W�ø@5�:4163YX�HP163Àü©ú9>45GZ9B]Z91;HE163ZXëZ[-C3?3?-C>=3Àü9BF0jâIF\BUù�5�0]3?>BFW21Sù�34165�0m[

î 5�02:41EW@-C>V3Àü2-�LM5�H;H65`HP1;0ªû^BFHSûF5F>163ÀüSDÏí U HÈü21Sù&üAûF-C34:_BF:D1;0�ú�ø@3PBÓDK5�W�ø2HSø2:8|�BF02W`BF0a-RH6-IDK-R0]3¾ÓÁcb ú �� P�HÈü21 ùaüd1;:<-R1;3Àü@-C>PB T ø2BFW@>B�341 ùD>4-R:41;W�ø2-V5F>e0@5F3�R�´fZg�hCi*jlknmnolprqtsrpvuwmxzy|{%}4k�p?~Sj�qesro�mnx,xr�Fmq�~ejlx�x��,{Zmnjl�n�R�,��}n{,u[m{Z~��

Page 86: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

íPUçP§|Ou�¾iR&ò('^MYuIV*)ù&ü@5]5�:4-c¿jòô�d��l�9:?-C38Ã*yL¿ ¯16L�&y�M/:?-C3.�ÎyLÃ16L�&yÐV�:?-C3.�ÎyLÃG¾>aø90Óí&P§|Ou�¾4u��*R<3?5©ûF-C3S5�ø23<ú�ø@3P& U16L�&y�& U 5�ø23<ú�ø@3cVD-RH;:?-�5�ø@3<ú�ø@3dM

b.-/B�>4-K1;0�3?-C>-R:?3?-RW.1;0c3Àü@-KW21æ±[-C>4-R0�ù�-ëZQ-C3rH�-C-R0`3Àü2-Uú9>45GZ9B]Z91;HE163ZX73Àü2B�3�í U 5�ø@3<ú�ø234:ëVëHÈü2-R0O¾c1;:DBT ø2BFW2>B�341Sù_>4-R:1;W�ø@-�BF02W!3Àü@-�ú2>5GZ9B]Z91;H;1;3ZX^3Àü9B�38í U 5�ø23<ú�ø@34:gVgHÈü@-R0Q¾G1;:VB T ø2BFW@>B�341Sù�0@5�0YX%>4-R:41EW�ø@-P-ù�Lr[9�e-Wf9091634165�0½V^R�[C�-C39ø2:jf9>:?3�-�h�BGD/1E0@-�HÈü2B�3�ü2B^ú�ú[-R02:ÓHÈü@-R0L¾�1;:7B T ø2BFW2>B�341Sù`0@5�0YX%>4-R:1;W�ø@-�P�HP163ÀüNþ�B�ù�5GZ�1

:?XBD*Z[5�H � V^R�[ ö 0`3Àü21E:PùCBF:4-GFz0@5F3?-K3Àü2B�3cíVUv-�h�B�ù�34H6X`:41EDUø2H;B�3?-R:D3Àü2-K-R0�ù�>4Xú234165�0`:Àùaü2-IDK-KL�5F>*íg�v1;05F3Àü@-C>cH=5F>W2:WF�3Àü2-J��16-WH�5FL�í�1E:V3Àü2-/:4BGDK-/BF:�HÈü9B�3D1;3�H�5�ø2HEWa:4-C-/16L�B7>BF09W@5_D×ú�øiZ9H;1 ùceF-CX�H�-C>-ûF-R0@-C>B�3?-RW.BF02W!3Àü@-R0cB�>BF02W25_DÑZ9163P-R0�ù�>4Xú23?-RW,[Pï�1E0�ù�-gí U 5�ø@3<ú�ø@34:ëV�-�h@B�ù�34H6X�H¬ü@-R0OíÍù�5F>>4-(ù�34H6Xû�ø@-R:4:?-R:P3Àü@-���BFHSø@-D5FL�3Àü@-_-R0�ù�>4Xú23?-RWQZ91;3WFiH�-Xü9BR�F-G´

ý�>áñ |Ìòôo&��¾�ò õ�öëùLú �� ´_í U P§|Íu�¾SR6yÐV�÷y ý�>`ñ²P§|Ou�¾iR?òÀo&�&dò('^MYuIV*)_���>ò+� ��� � P,&�R�´¤íëP§|Ou�¾Au��gR?y�&�÷ç�

°D0d3Àü2-�5F3Àü@-C>2ü2BF02W,FSH¬ü@-R0Q¾G1;:PB T ø2BFW2>B�341Sù_>4-R:1;W�ø@-_3Àü@-R0�¦�»Y«ë¼`¥¶«Wî�¢p©.í�¥��g¥�­Aª¤«r i«W­4ª¤«W­ü¦d¢p©¦�»Y«�£�¥§¦Q&*£�«�¥�­�µ�«�­A¬W¡�¨� (¦ä«�ª][6~Èü21;:�1;:�3Àü@-*ùCBF:?-lZQ-(ùCB�ø9:?-8>4-�û�B�>W2H;-R:4:�5FL�3Àü2-8�*BFHSø2-=5FLw&`F�3Àü@-ÈùC1Sú�ü@-C>43?-�h�3��HP1EH;H�BFH²H=B|X\:*Z[-/B�P-ø20916L�5F>�D/H6X!W21E:?3?>1²Z�ø23?-RW4R T ø2BFW@>nB�341SùK>4-R:41EW�ø@-G[Å~*üø2:WFz-C�F-R0�BF0�BFH;HæX�ú[5^H=-C>4L-ø2HíËùCBF0O5�02H6Xjû�ø2-R:4:J3Àü@-7�*BFHSø2-�5FLV&@HP163Àü�ú2>45GZ9B]Z�1;H;163YX�V|� � F=BF02W»ü@-R0�ù�-�3Àü@-Mú2>45GZ9B]Z�1;H;163YXa3Àü9B�3&íVU5�ø@3<ú�ø@34:cV�1;073Àü21;:}ùCBF:?-J1;:<-�h@B�ù�34H6X½V|� � [

ï�1;0�ù�-KW@-(ùC1EW21;0ªû T ø2BFW2>B�341SùJ>4-R:41;W�ø@5�:4163YX!1;:DBF:4:ÀøSD/-RWc3?5ÍZQ-QP�È�u�â�RäX ü2B�>W�P�BF02W.:41;0�ù�-&íVUz>aø902:D1;0>45�øªû�ü2H6X�341EDK-�È�R�F9163dDUø2:?3Z[-_3Àü@-XùCBF:4-�3Àü2B�3W´

ððððý�>`ñ²P§|Íu�¾SR?òôo&�&8ò('^MYuIV*)_���Ðò+� ��� � P,&WR<´�íëP§|Ou�¾4u��*R?y�&�÷4ø V�

ðððð ûâW�

�/ø23S3Àü21;:81;:S- T ø21;�*BFH6-R0]3<3?5 ����� � û � âWFSH¬ü21SùaüG1E:HÈü2B�3H=-�0@-C-RW@-RWd3?5©ú2>45t�F-G[

Page 87: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�V�2�;�F�C��4�6�Y �¡F¢¤£!¥�� \¦;¥F�9§¨ £a© ¨=ª�«F¬�­¯® ���?�¡\§2°9±��4�6¡��7�?¡ ¨ �4 \²2�?¡F³F�¥�²�´� µ��4¡F¢M�R�4�?¡F�D¶F¡��2¥��4´2¥F�!·�¥��?¸

¹»ºK¼_½8¾À¿8º ÁdÂ

à ĻÅÇÆTÈÊÉ?ËÌÆ�Í]ÎÏÆ�ÐÐnÑÓÒSÔÖÕØ×wÙVÚKÐÆ�ÑÓÛÜÝ¥F�?�Ç�4�;ÞK��ß=�7�´@¡|ß=�R§à¥F���R�9±��4 ]²9�4�6¡��à�4±´@�RÞ/��á�¥F�?�R§�¡��à�4´2��´2¥��n§2�@�R�4��¡F¢P§@�R±C�;§9�;�@³câ\°2¥F§@�n¥��4�;±�4�R�4�;§2°2¡��4�6�Z Fã��e�@¢M¡F�4�4°2�2¥��?�R¦6 Fä8�4´9�;�^�4±n´@�RÞK�Àå�¥F��±C°2�4�4�R�]�4¦6 À§@�Cæ9�@�R§�çÇ¡��2¦6 À¥F¦;¦6¡tßP�/�R�2±�� ]²2�4�;¡��è¡F¢é#ê á9�6��ÞK�R��4¥�³F�R�Cë�ì��F�R�.ß=¡F��?�Fäv�4´@�/�4±´2�RÞK���;�_�E�@�Cí^±C�6�R�]�Cî_�R�2±��4 \²2�4�;�@³d¥!�4�;�@³�¦6�Çá9�;�e��R�4°2¦6�4���E�.¥±C�6²9´@�C�4�?��ï��¤¡F¢w¦6�R�@³F�4´/ðGåMßP´2�C�4�S�4´2�;�¤Þ^�6³�´��¤á[�<¡��K�4´@�P¡F�§@�C�¤¡F¢ éCñ�ò ª ¡F���4¡]çnã�óe�K��ï�²9¥F�2�4�;¡��Ç¢M¥F±��?¡F�¡F¢¤ð7�E�8�Z \²9�;±C¥F¦;¦; ^�?¡\¡^�;�@�Cí^±C�;�R���8�?¡^°9�?�Fã

ô ¡�§2¥| Fävß=�^ßV�;¦;¦¤�?�C��¥F�õ�R�2±��4 \²2�4�6¡����4±´@�RÞ/�aåMá9¥F�?�R§�¡���¥!�4�R¦;¥��4�6�F�R¦6 .�?�?�¡��@³ ­ á�°@���4�R¥F�?¡�� ê¥�á9¦6� ­ ±��4 \²2�?¡F³F�¥�²�´2�;±�¥F��4°2ÞK²2�4�;¡���ç=ßP�6�4´c��ï�²9¥F�2�4�6¡��!¢M¥F±��?¡F� ò ãeö2°@�4�4´2�C�ÞK¡F�4�Fä[�4´@�Ç�4±´2�RÞK��ßV�;¦;¦�;Þ/ÞK�R§2�E¥��?�R¦6 K¥F¦;¦6¡|ß»�R�2±�� ]²2�4�;¡��/¡F¢Ý¦6¡��2³JÞK�R��4¥�³F�R�Cã�óe�=°9�4°2¥F¦,ä]ß��eßP�;¦;¦2áQ�C³��E�^á\ /�;�]�?�4¡\§9°2±C�;�@³�¥F�9§¢�¡F�Þ^¥F¦;�6¸R�;�@³J¡�°@�S´2¥��n§2�@�R�4�<¥F�4�4°9ÞK²2�4�6¡��2�Rä]�4´@�R�G±�¡��9�?�?�°2±��S¥Ç�4±n´@�RÞK�Fä9¥F�2§7�4´2�R�7²2�4¡|�F���4�R±C°@��6�Y ^¡F¢�4´@���4±n´@�RÞK�Fã

ö@¡F�D�4´@����RÞ/¥F�;�2§@�C�P¡F¢¤�4´2�;�8�?¡F²9�E±�ä9ß=��ßP�;¦E¦[áQ��ß�¡F�4÷��;�@³^¡|�F�C�_¥/æ9�9�6�?�Fä�±� �±C¦;�;±_³F�¡�°@²cøDã�©��E�2±��øù�;��±� \±C¦;�E±�äv�4´9�;��ÞK�R¥F�2���4´@�C�4���;��¥aú]ûCüQû�ý?þFÿ���ý���� ø ¢�¡F�JßP´2�;±n´õø������ ���������� ��� �����.åM�4�R±C¥F¦E¦�4´2¥������ ����� é ������?¡^�4´2��±� �±C¦6���4�C²[�R¥��4�e¥��V�4´9¥��P²[¡��;�]�nçnã��P¡F�?�J�4´2¥��Pá[�R±C¥F°2�4���d�;�P¥^³F�R�@�C�¥��?¡F�Rä¦6¡F³! #"c�;�<ß=�R¦;¦ ê §@�Cæ9�2�R§7¢�¡F�P¥F�] �"$�dø ¥F�S�4´@��°2�9�;â]°2�D�E���?�C³F�C��%`�;��& � �!� ¢M¡F�SßP´9�;±´'�!()�*"wã+ �6�F�R��øDä�ß=�cÞ/¥| è§@�Cæ9�@�!¥O�\°2Þ�á[�C�/¡F¢_²9�4¡Fá9¦6�RÞ/��,P�6¢D�4´@�R�?�c²2�4¡Fá9¦6�RÞ^�/¥��4�`´2¥��§Àß=�aÞ^¥R ¢�¡F�Þ^¥F¦;�6¸C�_�4´@�R�?��¥F�S±��4 \²2�?¡F³F�¥�²9´2�E±_¥F�4�4°2ÞK²9�4�6¡��2�Cã.-.��§9�;�4±C°2�4�<�?¡�Þ/��¡F¢��4´@�R�?�_²2�4¡Fá�¦6�RÞ/�8�@¡tß�ã/1032547698�:58�;=<�>�?�6@03:9ACB DC69<�EC;=8FBHG ô ´@�7§2�;�4±��4�C�?��¦6¡F³�¥���6�4´9Þ̲9�4¡Fá9¦6�RÞÌ�;���4´2��¢M¡�¦;¦6¡tßP�;�@³@îdz��6�F�R�åM�¥F�2§@¡�Þ�ç^³F�R�@�C�n¥��?¡F��� ¥F�2§��¥F�2§@¡�Þ �R¦6�RÞ/�R���$"I� øDäJ±�¡�Þ/²9°@�?�.¦6¡F³ "�ã ® ¢��4´2�;��²2�4¡Fá�¦6�RÞ(�E�´2¥��§wä\ß��D�4¥| cÿ=J�û)K�L3MONCý4ûCÿZû�PQ�4ú�þ�ýRL�ÿ=J!S þ@MTMTUFSWVQÿXLY��ü$JF��PZK�M[LMü^øDã��V¡F�?�D�4´2¥��<�6�=�E��¥F¦6ß<¥R ��]\r�R¥F�? F^Ç�?¡±�¡�ÞK²9°@�?�S§2�E�4±��4�C�?�S¦6¡F³�¥���;�4´2Þ/��ßP�6�4´J²2�¡Fá9¥�á9�;¦;�;�Z  é5_a` ø ` î<�4�;Þ/²9¦6 �³�°@�R���¥D�¥F�2§@¡�ÞI�R¦;�RÞK�R�����E�b& � ��� ä¥F�2§.�4´2�;�DßV�;¦;¦vá[�J�4´2�^±�¡F�4�4�R±���¥F�9�?ß��C��ßP�6�4´`²2�4¡Fá�¥�á9�;¦;�6�Y  é5_a` ø ` ã ô ´]°2�RäÝ�6¢�ß=�/ß<¥F�����4´@�^§2�;�±��4�C�?�¦6¡F³�¥���6�4´2ÞT�?¡/á[�1\?´2¥��§c^�ä2�6�8ßV�;¦;¦[´2¥R�F���?¡/á[�_�4´@��±C¥F�?���4´9¥�� ` ø ` �;�8¦E¥��4³F�Fãd <�BeDCf#:5?a:@03<�g#?�;�/103he8!ikj$8F;=;XBe?�gml d /�jonpDC69<�EC;=8FBHG ô ´2�E�ݲ2�¡Fá9¦6�RÞI�;�v�4´@�8¢M¡�¦;¦6¡|ßV�;�@³@î�³��6�F�R��¥F�2§@¡�Þ ³F�R�@�C�n¥��?¡F���Ø¥F�2§»�¥F�9§@¡�Þ �R¦6�RÞK�R�]�4�1" � �k"�qr�ÊøDäK±�¡�ÞK²9°2�?�s��tvuxwzyk{F|�}�~Ytvuxwzyk{F|5��~då��,ãA�Fã6äD�6¢" � ��� ( ¥F�2§�" q ���!�K�4´2��¡�°@�?²�°@�e�´@¡�°2¦;§!áQ�)� ( �*çnã ® ¢¤�4´2�;�V²2�4¡Fá�¦6�RÞù�E�V´2¥��n§wä�ß=�K�4¥R Oÿ=J�û1���[�þ@MkMRU�SWVQÿXLY��ü�JF��PZK�M�L�üGøeã

�=�C¢�¡F�4�d±�¡��]�4�;�\°2�;�@³@ä¤ß=�7³��6�F�G¥F�à��ï�¥FÞ/²9¦6�Fã ¨ ¡��2��;§@�C�J�4´@�7³F�4¡�°@²m&���z� ��� é ����5� éCñ �c°2�9§@�C�Þ�°2¦;�4�6²9¦;�;±C¥��4�;¡��wãa©��;�9±�� éFé �;��²2��;ÞK�Fä��6�Ç�;�J¥!¢�¥F±��Ç�4´2¥���& ��z� �;�J±� �±C¦;�;±`åM¢�¡F�Ç�4´@¡��?��ßP´@¡`´2¥|�F�7´9¥F§�?¡�ÞK�KÞK¡F�4�Ç¥F§@��¥F�2±��R§.¥F¦6³F�Cá2�¥�ä[�4´2�E�V�;�Vá[�R±C¥F°2�?��&���z� �;�V¥�æ2�R¦E§c¥F�2§a�4´@�Ç�@¡�� ê ¸C�C�¡��R¦;�RÞK�R���4�e¡F¢=¥æ9�2�6�?�Çæ2�R¦;§a¢M¡F�Þ ¥G±� �±C¦;�;±J³F�4¡�°@²Qçnã ® �`�4´2�;�D±C¥F�?�Fäz�;�D±C¥F�õáQ�Ç�F�C��;æ2�R§a�4´2¥�� ò �;�_¥7³F�R�@�C�¥��?¡F�|ã/©�¡@ä¥F�d�;�2�?�4¥F�9±��e¡F¢v�4´@� ¨��[� ²2�4¡Fá9¦6�RÞÊÞ/�;³�´��8á[�Fî�³��6�F�R� ò ä��]ä9¥F�9§1�\ä2±�¡�ÞK²9°@�?� ò t�uxwzy �7� ~Ytvuxwzy � � ~ ã�©��E�2±��ò � �*�Ç¥F�2§ ò�� �*�\ä2�4´@��±�¡F�4��R±��P¥F�2�?ß=�C�P�;� ò �T� ��� wT� ���� ò � ���\ã

-.�JÞ^¥R !�;Þ^ÞK�R§2�;¥��?�R¦6 ��@¡F�?�J�4´2¥��P�4´@� ¨��[� ¥F�4�4°2Þ/²2�4�6¡��d�;��M#ÿ�ýz��ü]ú]ûCý/�4´2¥F�c�4´@��§9�;�4±��4�C�?��¦;¡F³ ê¥���6�4´2Þ ¥F�4�°2ÞK²2�4�6¡���ã ® �»²�¥��4�4�;±C°2¦;¥��|äS�4´@� ¨��[� ¥F�4�4°2ÞK²9�4�6¡��»�;ÞK²9¦;�;�R�/�4´@�õ§2�E�4±��4�C�?�.¦6¡F³�¥���6�4´9Þ¥F�4�4°2ÞK²9�4�6¡��wä@¥F�S�4´@��¢�¡�¦;¦;¡|ßP�;�2³J±C¦;¥F�EÞT�4´@¡tßP�Cî

é

Page 88: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

d ;=?�0XB  3¡�ÿ=J�û]K�L3MONCý4ûCÿZû)PQ�4ú]þ�ýTL�ÿ=J!S¢V�ýz�!£OP;ûOS�L3M�¤rû4þ@MR¥�¦�L�ü7øùÿ=J@û�ü`ÿ=J�û§���[�*V�ý���£OP;ûOS�L¨Mo¤ ûþ@MT¥�¦L�üGø�©ª�69<�<�« óe�4�4°2Þ/���4´@�ǧ2�;�±��4�C�?�J¦;¡F³�¥���6�4´2Þù²2�4¡Fá9¦;�RÞ �;�V�R¥F�? c�E�cøDä¤�?¡7�4´2¥��e�4´@�C�4�K�;�e¥F�a�Cí^±C�6�R�]�¥F¦6³F¡F��6�4´2Þ­¬Ê�4´2¥��DZ�¡�ÞK²9°@�?�R�J§2�;�4±���C�?�^¦6¡F³�¥���6�4´9Þ/��ßP�;�4´õ�?¡�ÞK��²2�4¡Fá9¥�á��;¦;�6�Y ¯®7åM¡|�F�C�K±´2¡��;±���¡F¢��¥F�2§G�R¦6�RÞK�R�]�W"Qçnã°-.��Þ/¥| 7�4´@�R�d±�¡��2�?�?�n°2±��S�4´@�_¢M¡�¦;¦6¡tßP�;�@³K¥F¦6³F¡F��;�4´2Þ±¬�²[¢M¡F�S�4´@� ¨³�´� ²2�¡Fá9¦6�RÞGî

¬ ² å3���k" � �k"cq|çµP°9��¬Jå3���k" � ç=�?¡/³F�C�V¡�°@�?²9°@�³%�6¢#� ( ��" � ¡�°2�?²9°@��" (q¡F�4´@�C�4ßV�;�?�Fä2�4�;Þ/²9¦6 ^¥�á[¡F�4��P¡F�?���4´2¥���¬ ² ±C¥F�c¥F¦6ß<¥R ��S�?�R¦;¦[ßP´2�C�4´@�C��¬I´2¥F�8¡�°@�?²�°@�8�4´@��±�¡F�4��R±��P¥F�2�?ß=�C�Pá\ �±�¡�Þ/²9°@�4�;�@³]��(K�?¡�?�C��ßP´2�C�4´@�C�P�4´2�E�SÞ/¥��4±´2�R�´" � ã=óe¦;�?¡@ä�¬W²w¡�°@�?²9°2�4�8�4´@��±�¡F��4�R±��V¥F�2�4ß��C�VßP´@�R�@�C�F�C�´¬ ¡�°2�?²9°@�4�S�4´2�±�¡F�4�4�R±��_¥F�2�?ß=�C�Çå���;�2±��b"�q[�*�!uxwzy { | � äQ�4´@�J¡�°@�?²9°@�S¡F¢¶¬ ² �¥��4�;�?æ2�R�´"a(q ����tvuxwzy { | } ~Xt�uxwzy { | � ~Zçnã_©\¡@ä�¬ ²¡�°@�?²9°@�4�<�4´@��±�¡F��4�R±��S¥F�2�?ß=�C�8ßP�6�4´�²2�4¡Fá�¥�á9�;¦;�6�Y ·®8¥F�2§G�6¢z�4´@�_§2�;�4±���C�?�D¦;¡F³�¥���6�4´2Þ ²2�4¡Fá�¦6�RÞÊ�;�<�R¥F�? å��,ãA�Fã6ä�®V�;�8¦;¥��4³F�tç=�4´2�R�d�?¡^�;�<�4´2� ¨��[� ²9�4¡Fá9¦6�RÞGã/�8�4!0=250=<�g#?�;´/103he8!ikj$8F;X;XB¸?�g¹lº/�/�j�n·DC69<�EC;=8FBHGK£d¡F�4�;�*¥��?�R§.á\ d�4´@� ¨��[� ²2�4¡Fá9¦6�RÞdäQ§@�Cæ��@�».¼�½ åY" � �k"cq|çe¥F���ctvuxwzyT{�| } ~Ytvuxwzyk{F| � ~?ã ô ´@� �[�´� ²2�¡Fá9¦6�RÞ �;�D�4´@�/¢M¡�¦;¦6¡tßP�;�@³@îe³��6�F�R��¢�¡�°@�_�R¦;�RÞK�R���4�å3���k" � �k"cq��k" � ç�åMßP´@�C�4���9äw¥F�V°9�4°2¥F¦,äQ�;�P¥�³F�R�@�C�¥��?¡F�#çnäw§@�C�?�C�nÞ/�;�@��ßV´@�C�4´@�C��" � � »¾¼�½ åY" � �k"cq|ç<¡F��@¡F�Cã ® ��²9¥��4�4�;±C°2¦E¥��=åMß=�<§2�Cæ9�@���4´2�E�ݲ2�¡Fá9¦6�RÞ Þ/¡F�4��¢M¡F�Þ/¥F¦;¦; D��;�2±���ß=�=ßP�;¦;¦�°9�?���6�z�?¡V²9�4¡|�F�<�?�R±C°@��;�Z ¡F¢�¥F���R�2±��4 \²2�4�6¡����±´@�RÞK�Fä�á[�R¦6¡tßeçnä*§2�Cæ9�@��¥W��L ¿ØûOÀ���ûP3P�S/þ�ü/ÿXU9V�P6û��?¡Vá[�På3���k" � �k"�q � »¾¼�½ åY" � �k"cq*ç?ç¥F�2§^§@�Cæ��@�S¥Ký?þ�ü�K7��STÿXU9V�P;ûV�?¡�á[��å3���k" � �k"cq��k" � ç8åMßP´@�C��W" � ßV�;¦;¦\á[�8�¥F�2§@¡�Þ ¥F�9§/�;�2§@�C²[�R�2§2�R����¡F¢" � �k"cq|çnã ô ´2�R���4´2� �[�[� ²2�4¡Fá9¦6�RÞ �;�v�?¡_§2�;�?�4�E�@³�°2�;�4´�áQ�C�Yß��C�R�/�¥F�2§@¡�Þ �4°@²9¦;�R�z¥F�2§ � �6í/� ê � �R¦;¦;Þ^¥F��4°@²9¦6�R�Cã

£d¡F��J¢M¡F�Þ/¥F¦E¦6 Fä9ß=�JÞ^¥R !�4¥| d�4´2¥��e�4´@� �´�[� ²9�4¡Fá9¦6�RÞù�;�Jå=ÁR�k®nç ê ´9¥��§`�6¢�¢�¡F�D¥F¦;¦v¥F¦6³F¡F��6�4´2Þ^��¬�°2�2�2�E�@³J�E�7�4�;ÞK��Á=ß���´9¥R�F�Fî

 µ��9à %Ä�zÅ�ÆÇ& � �7� î!¬Çå3����� ( ��� � ��� ( � ç¾� éRÈ�É µ��5à %#�zÅ��ºÊ]ÆË& � ��� î7¬Jå3����� ( ��� � ����Ìtç¾� éRÈ Â ��P¡F�?�c�4´2¥��/�4´@�dæ9��?�K�4°@²9¦;�d�;�à�4´@�!¥�á[¡|�F�c��ï\²9�4�R�4�4�6¡��À±�¡F��4�R�?²[¡��2§2�Ç�?¡O¥ � �6í/� ê � �R¦;¦;Þ/¥F�à�4°@²�¦6�FäßP´2�;¦6�D�4´@���4�R±�¡��2§!±�¡F�4��R�?²[¡��2§2�<�?¡^¥K�n¥F�2§@¡�ÞÊ�4°@²9¦;�Fã

-.�eÞ^¥R K�;Þ/ÞK�R§9�;¥��?�R¦6 ��2¡F�?�P�4´2¥����4´2� �[�´� ¥F��4°2ÞK²2�4�;¡��K�;���?�?�4¡��2³F�C���4´2¥F�^�4´@� ¨��[� ¥F�4�4°2Þ/² ê�4�6¡��wã ® ��²�¥��4�4�;±C°2¦;¥��|äv�4´2� �[�[� ¥F��4°2ÞK²2�4�;¡����;Þ/²9¦;�6�R���4´@� ¨³�´� ¥F�4�4°2Þ/²2�4�6¡��w䤥F�Ç�4´@�7¢M¡�¦;¦6¡|ßV�;�@³±C¦;¥F�;ÞT�4´2¡|ßP�Cîd ;=?�0XB  3¡�ÿ=J�û§���[�*V�ýz�!£OP;ûOS�L¨Mo¤rû4þ@MR¥�¦�LMüGøùÿ=J@û�ü.ÿ=J�û��[�[�*V�ý���£OP;ûOS�L¨M¸¤rû4þ@MR¥�¦�LMü7ø�©ª�69<�<�« óe�4�4°2ÞK���4´@� ¨��[� ²2�4¡Fá�¦6�RÞT�;�8�R¥F�4 d�;�døDä��?¡^�4´@�C����;�S¥F�!�Cí^±C�6�R�]�V¥F¦6³F¡F�n�6�4´2Þͬ �4´2¥��Cä³��6�F�R�Î���k" � �k"cq�ä�¡�°@�?²9°2�4� »¾¼�½ åY" � �k"cq|ç¤ßP�6�4´K²9�4¡Fá9¥�á9�;¦E�6�Z ]®Cã.-.�eÞ^¥R Ç�4´@�R�^±�¡��9�?�?�°2±����4´@�P¢M¡�¦;¦6¡tß ê�;�@³/¥F¦6³F¡F��;�4´2Þ±¬ ² ¢M¡F�S�4´@� �[�[� ²2�4¡Fá9¦;�RÞGî

¬W²%å3���k" � �k"cq��k" � çµP°2��¬Çå3���k" � �k" q ç=�?¡^³F�C�P¡�°@�?²9°2��"a²��6¢C" ²� ��" � ¡�°2�?²9°@� é¡F�4´@�C�4ßP�E�?�Fä@¡�°@�?²9°@� ñ�P¡F�?���4´9¥��S�@¡|ß �4´2�C�4���;�<�@¡Kß<¥R �¢�¡F��¬W²��?¡^�F�C��6¢M /ßP´@�C�4´@�C��¬ ´2¥F�<¡�°@�?²�°@�<�4´@��±�¡F�4�4�R±��P¥F�9�?ß��C�8¡F��@¡F��åMßP´] aÏ*çnã

ò

Page 89: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ü��C�Ðl��¥F�2¥F¦6 \¸C�8�4´@�8¡�°@�?²9°@��¡F¢�¬�²�ã.-»´@�R�då3���k" � �k" q �k" � çv�;�¤¥ � �6í/� ê � �R¦;¦;Þ/¥F�J�4°@²9¦;�Fä!¬�²\¡�°@�?²9°2�4�é �ZÑ�¬ ¡�°@�?²�°@�4�8�4´@��±�¡F��4�R±��P¥F�2�?ß=�C�Rã=©@�;�2±��D�4´2�;�8´2¥�²9²Q�R�9�<ßP�6�4´7²2�4¡Fá�¥�á9�;¦;�6�Y Ò®Cä@ß=��´2¥|�F�Fîµ��5à %Ä�zÅ·ÆË& � �!� î7¬ ² å3����� ( ��� � ��� ( � ç°� éRÈ ��®�

Ó �.�4´@�/¡F�4´2�C��´2¥F�2§�ä�ßP´@�R�èå3���k" � �k"cq7�k" � çD�;�_¥d�¥F�2§@¡�Þ �4°2²9¦6�Fä#¬ ² ¡�°@�?²9°@�4� é �ZÑ�¬Ê´2¥�²9²Q�R�9�D�?¡¡�°@�?²9°@��"c²� �Ô" � ã^©��;�9±���" � �;�e�¥F�2§@¡�Þ ¥F�2§.�;�2§@�C²[�R�2§2�R���e¡F¢�" � ¥F�2§¸"cq*äz�4´2�;�D´2¥�²2²[�R�2�e��ï@¥F±��4¦6 ßP�6�4´7²2�4¡Fá�¥�á9�;¦;�6�Y  é5_a` ø ` ã_©\¡@î

µ��9à %Ä�zÅ��ºÊ]ÆË& � �!� î!¬ ² å3����� ( ��� � ��� Ì ç°� éRÈ � é5_a` ø ` �ô ´@�ǧ2�ZÑ[�C�4�R�2±��Ç�;�!²9�4¡Fá9¥�á9�;¦E�6�4�6�R�8�;�e�4´@�R��® ÉØé5_a` ø ` ã�©\¡@äw�6¢Õ®D�;�DÞ�°9±´a³F�4�R¥��?�C�_�4´2¥F� é5_a` ø ` ä��4´2�R�¬ ² ±C¥F��§2�;�4�4�;�@³�°2�;�´�á[�C�Yß��C�R� � �6í/� ê � �R¦;¦;Þ/¥F� �4°@²9¦6�R��¥F�2§è�n¥F�2§@¡�Þ �4°2²9¦6�R�/�R¥F��;¦6 Fã åX�V¡F�?�a�4´9¥��®1Ö é5_a` ø ` §2¡]�R�Ç�@¡F�/±�¡��2�4�4�6�4°@�?�m\rá2�4�R¥�÷��;�@³7^ ¨��[� ,��6�J�E�Ç�R¥F�? ��?¡õ¥F±n´2�6�C�F�d�4°2±C±��R��K²2�4¡Fá9¥�á��;¦;�6�Y é5_a` ø ` ¢M¡F�S�4´@� ¨��[� ²2�4¡Fá�¦6�RÞÊá] 7�4�;Þ/²9¦6 ¸\r³�°@�R��4�;�@³7^/¥Ç�n¥F�2§@¡�ÞϳF�4¡�°2²d�R¦6�RÞK�R�]�Cë ç

ô ´\°2�Cä@ß=��´2¥R�F�J�4´@¡|ßV�7�4´2¥��S�4´@� �[�[� ¥F�4�4°2Þ/²2�4�6¡��7�;ÞK²9¦E�6�R���4´2� ¨��[� ¥F��4°2ÞK²2�4�;¡��7ßP´2�;±n´G�;��4°@��K�;ÞK²�¦;�6�R���4´2�S§2�;�4±��4�C�?�8¦6¡F³�¥��n�6�4´2ÞI¥F�4�4°2Þ/²2�4�6¡��wã ® �dzF�R�@�C�n¥F¦,ä��4´@�8±�¡��]�F�C��?�S¡F¢Q�4´@�R�?�S�?�4¥��?�RÞ/�R���4��;�S�@¡F�S�?�n°@�Fã ô ´@�C���¥��4��³F�4¡�°@²��8�;�GßP´2�E±´7�4´@��§9�;�4±��4�C�?��¦6¡F³�¥��n�6�4´2Þʲ2�4¡Fá�¦6�RÞÏ�;�S±�¡��@×?�R±��4°@�4�R§!�?¡/á[�´2¥��§õá�°@���4´@� �´�[� ²9�4¡Fá9¦6�RÞÌ�;�_÷��@¡|ßP���?¡!á[���R¥F�? F䤥F�2§��4´2�C�4��¥��4��¥F¦;�?¡c³F�4¡�°@²9���;�.ßP´2�E±´õ�4´2�¨��[� ²9�4¡Fá9¦6�RÞÏ�;�8±�¡���×r�R±��4°@��R§d�?¡/á[��´2¥��§7á�°@�8�4´@� �[�[� ²2�4¡Fá9¦6�RÞÊ�;�S÷\�@¡tßP�G�?¡/á[�D�R¥F�? Fã

Ø ÙÛÚ�ÚKÐnɺܤÒpÝwÉ4ÙVÔÞÝ[ÙÇß�ÆÕÜ°àÇ×wÆâáOÔ]ܤ×�ãbÚ)Ý[É4ÙeÔ-��6�4´d�4´2�;�8Þ/¥F±n´2�;�@�C�4 ��;�7²9¦;¥F±��Fä9ß���±C¥F�c�@¡|ß ±�¡��9�?�?�°2±��S¥/Þ/¡F�4�_�Cí^±C�6�R�]�S²9°@á�¦;�;± ê ÷F�C /�R�2±�� ]²2�4�;¡���4±´@�RÞ/�Fã + �6�F�R�!¥Kæ@ï��R§7³F�4¡�°@²!øDäzß���§@�Cæ��@�_�4´@�bä³PÕå8þ�S/þ�P�ûCü�NCýR¥TVQÿXLX�FüæMONOJ@ûOS^û§Ã éOÈ ¥F�S¢M¡�¦;¦6¡tßP�Cîé ãWçر´@¡\¡��?�R�v¥P�¥F�9§@¡�Þ ³F�R�@�C�n¥��?¡F�#�§�døØ¥F�9§�¥S�n¥F�2§@¡�Þ �\°2Þ�á[�C��%¯��& � ��� ã ® �Ý�4´@�R��±�¡�ÞK²9°@�?�R�"'�è� ( ã ô ´@�_²�°@á9¦;�;±V÷F�C G�;��å3���k"QçS¥F�2§d�4´@���?�R±��4�C�V÷F�C 7�E��%Ýãò ã ô ´2��ÞK�R�4�4¥�³F�J�?²9¥F±���ßV�;¦;¦Qá[�_�4´@��³F�4¡�°@²a�6�4�?�R¦6¢ ã ô ¡/�R�2±��4 \²2�P¥^ÞK�R�4�4¥�³F�)é��!øDäz�4´@���?�R�2§2�C�²9�E±4÷��8¥K�¥F�2§@¡�Þâê]��& � ��� ¥F�2§d�?�R�2§9�8±C�6²9´@�C�4�?��ï���å3�!ë@�k"aëOécçnã

ì ã ô ¡c§@�R±�� ]²2��¥!±C�6²9´2�C�4�?��ï\�7åX¬��ºí/çnä��4´@���4�R±��R�6�F�C�K±�¡�ÞK²9°@�?�R�)éÞ�±í _ ¬ ( ã�å�ö@¡F�K¥F�� .³F�4¡�°2²�R¦6�RÞ/�R���4����²=�k"c²�ä2�4´@���@¡F�4¥��4�6¡��1�!² _ "a²w�4�;Þ/²9¦6 ^ÞK�R¥F�9�³�!²%åY"a²Eçk���Cã ç

-.���F�C��6¢M ��4´2¥��P§@�R±��4 \²2�4�6¡��G�E�8§@¡��@��±�¡F�4�4�R±��4¦; Fã<©��;�9±��KåX¬)�ºí/ç°��å3� ë �k" ë écç<¢�¡F�P�4¡�ÞK��ê�ä2ß���´9¥R�F�Fîí¬ ( �

" ë éå3� ë ç ( �

" ë éå3� ( ç ë �

" ë é" ë ��é��

-.�eÞ/¥| K�;Þ/ÞK�R§2�E¥��?�R¦6 J�2¡F�?�P�4´2¥���¢�¡F���4´@�P�4±n´@�RÞK�P�?¡�á[�S�?�R±C°@�4�Fä�¥����4´@�VÞ/�;�2�;ÞJ°2Þ ß=�PßP�;¦E¦��4�Râ]°9�6�4��4´@��§2�;�±��4�C�?��¦6¡F³�¥���;�4´2Þʲ2�4¡Fá9¦;�RÞÊ�?¡^á[��´2¥��§G�;�!ø å��6¢��@¡F�Cä2�4´@�R�c¥F�� F¡��@��ßP´@¡/÷��@¡|ßV�8�4´@��²9°2á9¦;�;±÷F�C Øå3���k"�ç�±C¥F�O±�¡�ÞK²9°@�?�Ò%m�T¦6¡F³ "O¥F�2§�¦6�R¥�����4´@���?�R±���C��÷F�C 2çnã ® ��¢�¥F±��Cä��4´@���?�R±C°2��6�Z .¡F¢8�4´2��4±´@�RÞ/�e��R¦;�6�R��¡��^�4´@�_�?�?�¡��@³F�C� �´�[� ¥F�4�4°2ÞK²9�4�6¡��^�;��øeã�-.�DßP�;¦;¦9³��6�F�D¥�²2�¡]¡F¢�¡F¢z�?�R±C°@��6�Y K¢�¡F�8ì�¦+ ¥FÞ/¥F¦��R�9±��4 ]²9�4�6¡��G�@��ï��S�4�;ÞK�Fã

î Æ°ïRÆ�×wÆ�Ô]ܤÆ�Ûà éRÈJô ãwì�¦ + ¥FÞ^¥F¦,ãwó µ�°2á9¦;�;± ê ·_�C  ¨ �4 ]²9�?¡��? \�4�?�RÞ ¥F�2§`¥!©��6³��2¥��4°@��K©�±n´@�RÞK���<¥F�?�R§.¡�� � �;�4±���C�?�

Ü�¡F³�¥��n�6�4´2Þ/�Cã� zä¶äÕäÔð@ý?þ�üaMCþ7NRÿXLY��üaMÎ��ü$ nü�¡��ýRSKþ�ÿXLX��üèð#J@ûº��ýR¥ ì�é å ª çnî ª�¬ �5ñ ª � ò å é � � « çnã

ì

Page 90: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

òWócôöõ ÷økùºôQú�ûÒü ý¾þ$ÿ�økû��öÿ ó��� þ�� ������ �� ó!úzøkü������OúºôQü7ó§úzü � øºû��cúzü�� økÿ����!û�°øºü ý=÷�ùºùzü ø�� ü7ócÿ�ú��cÿ ó���ÿ�ú �

!#"%$'&)(+*)" ,.-

/ 0214365879581:02;=<?>A@�B�CADFEG;

H �7ÿ ôöóJILKp÷GKWôM�M�N�Q÷ú�OP�a÷�ócü úz÷´ÿRQ�ócôQúz÷�I��Oû��S�öôT�8� øºüU�N�WVYX�÷Z�ÿ��M��ú��a÷\[�[�]*ÿ ùºùF��^%�cúºôQü7ó·ôöó_O�`)Kp÷´ùkÿ�ûú��a÷R[�[�]P��øºü�a��Q÷Z^±ôöóbOÛôöùRcedgfFhjilkl�cÿ�øF�$ôQý#ý=ü øWÿ��M��ÿ��M� ü økôQú���^Îù)m øF��ócócôöóN�bôöó§úºôM^�÷'d�Kp÷R�cÿ�õ ÷�`

nn �°øpo qrf s%tvu'w xUwA`m=czy�f{y|�f{y}Lf{y|S}Ui?~��g�����Õø�o qWf s�f��=t�u'w xLwA`Um�czy�f{yL|�f{y}�f{yL�pi?~��g� nnS���÷)ÿ��öùzü�øk÷Z�ÿ��M��ú��a÷��a÷�ùF�Oøkô��cúºôQü7ó§ü ýCú��a÷R���W�[ÿ�^Îÿ��#÷�ó��Oøºû��cúºôQü7ó�ù��j�a÷Z^�÷�Kp÷��7ÿ�õ ÷=�öÿ ùzú³úºôT^�÷�`

��VG���F�aü�ü7ùz÷�ùCÿ�økÿ ó��aüU^�� ÷�óa÷øTÿ�úzü øWy��:Oèÿ ó���ÿ�øTÿ ó��aüU^ ó���^�a�÷øJq��bu'w xLw�V � ú#ú��a÷�ó��OüU^%���aúz÷�ù� ~�y | V��8�a÷'���Na��öôM�G� ÷û�ôöù�czy�f � i�ÿ ó��1ú��a÷�ùz÷Z�Oøº÷úG� ÷û§ô¨ù�qWV

� VG�8�c÷�^�÷�ùkùºÿ�� ÷�ù ��ÿ��O÷)K�ôT�M�Ua�÷pú��a÷)� øºüU�N�%O�ôQúºùz÷Z�öý�V��#ü�÷�ó��OøkûL�cú.ÿ'^�÷�ùºùkÿ�� ÷)� �:O\I�ú��a÷³ùz÷�ó��c÷ø��ôT���Fù³ÿ�økÿ ó��aüU^¢¡=�bu'w xLw�ÿ ó��1ùz÷�ó���ù)�ô����a÷øºúz÷¤£Fú�czy¥¦f � ¥¤�§ijV

¨ VG�Äü§�a÷Z�OøkûL�cú)ÿ��ô����c÷øºúz÷¤£�ú�c�mRf�©ªijI.ú��a÷Òøº÷Z�O÷�ôQõ ÷ø%�OüU^%���aúz÷�ù��«~¬©ª­gm | V.c�®aü ø�ÿ ó!û4� øºüU���÷Z�Q÷Z^Î÷�ó!úºù�yL¯ef � ¯�Icú��a÷�óaü úºÿ�úºôQü7ó:y¯T­ � ¯�ùºôM^ª���Qû_^�÷�ÿ ó�ù)y¯°c � ¯TiF±³²SV´i

H ù�K¶÷�óaü úz÷Z�%�¨ÿ ùzú.úºôM^�÷�I�S�Q÷�ÿ�øj�Qû�ú��a÷³÷�ó��OøkûL�cúºôöü7óbùF�F�a÷Z^�÷�ô¨ù�µz¶N·Z¸�¹Zº�»�¸�ôQý�ú��a÷)�cô¨ù��Oøº÷úz÷)�Qü��7ÿ�økôQú���^�cøºü�a��Q÷Z^�ôöù´÷�ÿ ùzûæôöó�O�V_]�üpK¶÷õ ÷ø�IÄôQú[ôöù[óaü ú[÷�óaüU�N�U�æý=ü ø[ú��a÷%�côöù��Oøº÷úz÷%�Qü��7ÿ�øTôQú���^¼�cøºü�a��Q÷Z^ úzüba�÷�cÿ�øF�¸ô¨óæü øF�a÷ø�ý=ü ø�ú��c÷·ù��F�c÷Z^�÷·úzü§a�÷Òùz÷Z�S�aøº÷�½¾ôöó�ýXÿ��OúSIYK¶÷'óa÷÷Z�oúzü¯ÿ ùºù���^Î÷�ú��a÷Òùzúzøkü7óN� ÷ø�[�[�]ÿ ùºù���^%��úºôQü7ósô¨ó�O�V%]�üpK¶÷õ ÷ø�I�ú��a÷�[�[�]Ûÿ ùkù���^%�cúºôöü7ó:K�ôM�T�Ja�÷)ù���¾_�ôQ÷�ó�ú�úzü��cøºü9õ ÷)ú��a÷=�¿�?�[ÿ�^Îÿ��÷�ó��Oøºû��cúºôQü7ó1ù��j�a÷Z^�÷�ùº÷Z�S�aøº÷�Icÿ ù8Kp÷�óaüpK ù��aüpK�`

À�ÁrÂLÃ�ÄpÂÆÅ ÇÉÈzÊ'ËeÌ ¸\Í\Í�ÎÐÏ�»{ÑLÒ¤ÓM¸¤ÔÕµz·�cedgfFhjijÖ Ì�× » ØÙµz¶ÚO\Û ËeÌ ¸¤¶ ËeÌ ¸\Ü�Ó?Ý × Ô × Ó�¸¤¶�¹S»gÞFÏ Ë µ�Ñ�¶§·¤¹ Ì ¸¤Ô_¸µT·�cedgf � hjijÖ�·S¸F¹SºÆ» ¸ ×Zß× µe¶N· Ë ¹ZµàÏ Ì ¸¤» Ë ¸�á Ë Ö Ñ�¶�ÓâÞ ×�ËãË°× ¹¤äp·¤å

æ ÄpóÃ�ç H ùºù���^�÷�K¶÷��cÿ�õ ÷æÿ óèÿ��cõ ÷økùºÿ�øºûèméKê��ôM�F�è�ÿ óìë{acøº÷�ÿ��Æíeú��a÷sùz÷Z�S�cøkôQú�û�ü ý����\�[ÿ�^Îÿ��÷�ó��Oøºû��cúºôQü7ó³½cócÿ�^Î÷Z�Qû�Iaú��a÷øº÷�÷¤£aôöùzú³ú�K¶üÚ^�÷�ùºùºÿ�� ÷�ùê�:î¦f � ² ý3ü ø8KG�côM�F��cXô¨óaý3ü øj^Îÿ��M�Qû�ij`

ï �°ø�oàðñt�ò�óJô�ce�:î�i�`m=c�õ�ö�fjð�i�~#÷¦���.�°øpoàðPtøò�óJô�ce� ² i�`Um=c�õ�ö�fjð�i¿~#÷¦� ï ~Éù � c{��i

��÷�K�ôM�T�¶ù��cü�Kâú��cÿ�ú_ù�^��cùzú%a�÷1ù�^Îÿ��T�ãV � ó2��ÿ�øºúºôT�S���öÿ�øZI�Kp÷$ù��cü�KÕ�aüpKâúzü��cùz÷1ÿ��cõ ÷økùºÿ�øºû2m�úzü�Oü7ócùzúzøF���Oú�ÿ ó�ÿ��M� ü økôQú���^úmG¯rKG�côM�F�û�ÿ óoùºüU�Qõ ÷�ú��a÷%[�[\]ì�cøkü�a��Q÷Z^¼K�ôQú��û�cøºü�a�ÿ�a�ôM�öôQú�û�ù¦­ � V_�Fô¨ó��O÷K¶÷��FóaüpK ú��cÿ�ú�ú��c÷�[\[�]ü�cøºü�a��Q÷Z^±ôöùRcedgfFhjilkl�cÿ�øj�³INK¶÷�^��cùºúê�cÿ�õ ÷�ù�ý � h�ÿ ó��:K¶÷�ÿ�øº÷��aü7óa÷�V

��÷�Qcøkùºú³øº÷¤k°K�økôQúz÷[÷Zþ��cÿ�úºôQü7ó.c{��i³ôöó§ÿ ó�÷ZþL��ôQõ@ÿ��ö÷�ó!ú�ý=ü øF^ ceK¶÷��cÿ�õ ÷R�cü7óa÷�ú��côöù�a�÷ý=ü øº÷�ùzü%K¶÷��cüóaü ú�øº÷S��÷�ÿ�ú³ú��a÷R�c÷økôQõ�ÿ�úºôQü7ób�a÷øº÷pij`

ï �\ÿ �°ø�o��8t���÷ÆfZ���U½jðüt ò�óJô�ce���Fi�`Um=c�õ�ö�fjð�i¿~��F���è� ï ~Éù ��üpK�I��a÷SQ�óa÷�ÿ���� ü økôQú���^Õm ¯ ý3ü øWú��a÷�[�[�]��cøºü�a��Q÷Z^Iÿ ù�ý3üU�M�öü�K�ùS`

Page 91: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

m ¯ czy�f � ² f �� f ��� i��÷úGõ�ö ~ czy�f � ² i�8t ��÷ÆfZ���ùz÷ú�ðñ~ c �� f ��� ÿ ����iøF��óbm�c�õ�ö§fjðRi³úzüª� ÷ú�üU�cú ���aú��j¯ôQý��)~��j¯zI��U�a÷�ùºù��_c=ú���ôöù)K�ôM�T��øº÷S��øº÷�ùz÷�ó�ú�ú��a÷'�U�a÷�ùºùÙë [[ô�¾Î÷¤kl]W÷Z�M�M^Îÿ ó�íiü ú��a÷øFK�ôöùz÷�IN�U�a÷�ùkùê÷�c=ú��cô¨ù�K�ôM�M��øº÷S�cøk÷�ùz÷�ó!ú³ú��c÷��U�a÷�ùºùªë�øTÿ ó��aüU^ªíi

��÷ú8q������~¼�Qü���� � ² f ¡������~Õ�öü���� �� Icÿ ó��:�������~¼�Qü���� ��� V��ü úz÷�ú��a÷�ý=üU�M�Qü�KWôöóN�N`� õ�ö�ôöùÕÿ���K�ÿ�ûFù�ÿ�õ�ÿ��öôM�_���Na��¨ôM��� ÷ûªK�ôQú��Ù�Oü øºøº÷�ù ��ü7ó���ôöóN��ùz÷Z�Oøk÷ú�� ÷ûÙq4c=ü ýr�OüU�aøkùz÷�Iaóa÷�ôQú��c÷ø�móaü ø8m ¯ ��ócü�KñqrINa��aú³ú��cô¨ù³ôöù³ôQøºøº÷Z�ö÷õ@ÿ ó�újijV

� � ýR��~ ¡¦qìcXôãVx÷�V�I=czy�f � ² f �� f ��� i'ôöù§ÿ2[[ô�¾Î÷¤kl]W÷Z�M�M^Îÿ órú��N���ö÷pijI�ú��a÷�ó�ú��a÷4�ô����a÷øºúz÷¤£Fú:ð ô¨ùÿ �Q÷S�7ôQúºôM^Îÿ�úz÷s÷�ó��Oøºû��cúºôQü7ómü ý�^Î÷�ùºùºÿ�� ÷����¤V��8�côöùÎô¨ù·ùzü a�÷Z�ÿ���ùz÷ûð�ôöùÎú��c÷�ó ü ý�ú��a÷sý3ü øj^czy¥�f{y¥ | ÿ ����i�~ czyL¥¦f � ¥ ² ÿ ���FijV¿�8���cùS`

�Õø�o qWf ¡Rt u w xw `Um ¯ czy�f{y | f{y ¥ f{y ¥ | i�~ �g�A~#�°ø�o��8t���÷ÆfZ���U½jðüt�ò�óJô�ce���Fi�`m=c�õ�ö�fjð�i�~��F� �

�! ó¸ú��a÷·ü ú��c÷ø=��ÿ ó��³I.ôQý)�¯ô¨ùR�cócôöý3ü øF^_�Qû§�cô¨ùzúzøkô�a��cúz÷Z�³I#ôöó��c÷S��÷�ó��a÷�ó!ú�ü ýêq�ÿ ó���¡ÆI.ú��a÷�ó¸ú��c÷�ô����a÷øºúz÷¤£Fú\ð¢ôöù�ceK�ôöú����cô��U���cøºü�a�ÿ�a�ôM�öôQú�ûNipóa÷�ôöú��a÷øWÿ ós÷�ó��Oøºû��cúºôQü7ó$ü ý�� î óaü øG� ² V � ó$ý=ÿ��OúSIôöóÒú��côöù��ÿ ùz÷�ú��c÷[ùº÷Z�Oü7ó��b�OüU^%��ü7óa÷�ó�úpü ýÄú��a÷'�ô����a÷økúz÷¤£�ú�c ��� ÿ ����i¶ôöù��cócôöý3ü øF^_�Qû%�côöùzúzøkôMa��aúz÷Z�ôöóbO�I.ôöó��c÷S��÷�ó��a÷�ó!úpü ý?�:î¦f � ² Icü ø"�pV��8���cùSI�¸$#�¸¤¶�µ Ê m¢µz· × ÓeÓNÏÆÑ&%�¸S» Ê ºÆÓ°ôQú8�ÿ ócóaü úG��ü7ùºùºô�a��öûüU�aú ���aú�� ¯ ~��GK�ôöú����cøºü�a�ÿ�a�ôT�öôQú�ûÒÿ ó!ûb�cô('�÷øº÷�ó�ú³ý3øküU^ ��­ � V��8��ôöù�ôM^%���öôö÷�ùS`

�Õøpo qWf ¡¦f��%t�u'w xwA`Um ¯ czy�f{y|�f{y ¥ f{yL��i�~ �g�A~���­ � �

�8�L��ùS`nn �°ø¦o qWf ¡Rt u'w xUwA`m ¯ czy�f{y | f{y ¥ f{y ¥ | i�~ �g�����°øpo qrf ¡pf��%t�u'w xwA`Um ¯ czy�f{y | f{y ¥ f{y � i?~��g� nn

~ ï �Õøpo��)t���÷ÆfZ���U½jð�tøòÆóJô�ce���Fi�`Lm�c�õ�ö§fjðRi¿~��F���è��­ � ï~ ù�­ �ý h �

��÷��Oü7ó��S�M���a÷�ú��cÿ�úGù�ý � hZI��cøºü9õ�ôöó��bú��c÷�ú��a÷ü øº÷Z^bV

) 0214365879581:02;=<?>A@�B�CADFEG; Dj;+* >³5)<�CADF<�,

�8�a÷)ÿ�a�ü9õ ÷���øºü�ü ý.ü ýÕùº÷Z�S�aøkôQú�û:�côM�$ócü ú�øº÷Z�Qû�ü7ó¯ÿ ó�û:�cøºü���÷øºúºôQ÷�ù�ü ý?� øºüU�N�ûOÛü ú��c÷ø´ú���ÿ ó$ú��a÷)ý=ÿ��Oúú��cÿ�ú³ôQú�Kpÿ ù³ÿ�Q�ócôöúz÷�I��OûÆ�S�öôM�G� øºüU�N��ôöóÙKê�côM�j�'ú��a÷\[�[�]P�cøºü�a��Q÷Z^6K�ÿ ù%ë �cÿ�øF��íÆV��8���cùSIFÿ��M��K¶÷�óa÷÷Z�úzüª�aübô¨ù�Q�ó��§ÿ�Q�ócôQúz÷�IÆ�Oû��S�öôT�ê� øºüU���1ôöó_KG�côM�F�'ú��a÷'[�[�] ÿ ùºùF��^%�cúºôQü7ó'ôöù�a�÷Z�öôQ÷õ ÷Z�'úzü%�aüU�M�§ÿ ó��ÚKp÷�cÿ�õ ÷bÿÎùz÷Z�S�aøº÷�÷�ó��OøºûL��úºôQü7ó�ù��F�c÷Z^�÷.-¿��÷��7ôöõ ÷R�a÷øº÷�ü7óc÷�÷¤£Fÿ�^%���Q÷�ü ý¾ù����F�$ÿ%� øºüU�N�JV

��÷_^�÷�ó�úºôQü7óa÷Z�æôöóæÿ óæ÷�ÿ�øF�öôö÷ø��ö÷Z�Oú��aøº÷]ú��cÿ�ú�ú��a÷%^����Qúºô����¨ôM�ÿ�úºôQõ ÷R� øküU�N��u0/1 c=ý3ü ø32��cøTôM^�÷pi�ôöù[ÿ�Oû��S�öôT�%� øºüU�N�JVÙ�°õ ÷�ó4a�÷úzúz÷øZICú��a÷_�cô¨ù��Oøº÷úz÷ª�Qü��7ÿ�økôöú���^ú�cøºü�a��Q÷Z^ ôöóæú��côöù�� øküU�N�eôöù�a�÷Z�öôö÷õ ÷Z��úzü:a�÷�cÿ�øF�ûc=ý3ü ø��öÿ�ø�� ÷42AijV°ò´óaý3ü økú��cócÿ�úz÷Z�Qû�cXÿ ùpû üU�ÚK�ôM�M��ù��aü�K�ijI�ú��a÷�[�[�]*ÿ ùºù���^ª�cúºôQü7ó§ØÑ�¸g·'¶�Ñ Ë �aüU�M�'ôöóú��côöù)� øºüU���JI�^Îÿ��FôöóN��ôQú8�cócùF�côQúºÿ�a��Q÷´ý=ü ø�ú��a÷R���W�[ÿ�^Îÿ���÷�ó��OøºûL��úºôQü7ó�ù��F�c÷Z^�÷�V

Page 92: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

��÷�óa÷÷Z�sÿ'ù��öôM�U�!ú��Qû�^�ü øº÷=�OüU^%���¨ôM�ÿ�úz÷Z�$÷¤£aÿ�^%���ö÷�V���÷ú42§~ �&5"6 ��Kê�a÷øº÷�2æÿ ó�� 5 ÿ�øº÷�a�ü ú���cøkôM^�÷�V���÷ú'O �a÷�óaü úz÷bú��a÷�ùz÷ú[ü ý¿þ��cÿ��aøkÿ�úºôT��øº÷�ùkôM���a÷�ùWôöó§u /1 VR�Fô¨ó��O÷�Kp÷���óaüpK¹ú��cÿ�ú´÷¤£Fÿ��Oú��Qû��cÿ��Qýú��a÷�÷Z�Q÷Z^�÷�ó�úºù°ü ýAu0/1 ÿ�øº÷GþL�cÿ��cøkÿ�úºôM��øº÷�ùºôT���a÷�ùSI�Kp÷ê�cÿ�õ ÷ ï O ï ~ c72G����i ­ � ~ 5 V � ú°ôöù°óaü ú¾úzü�üR�cô�¾Ù�S���Qúúzüoù��aü�K±ú���ÿ�úªO ôöù]ÿû� øºüU�N��c��có��a÷ø%^����Qúºô����öôM�ÿ�úºôQü7ó�ijI¾ÿ ó���ü7ó��Qû¸ÿ4�öôQúzú��ö÷Ú�cÿ�øj�a÷ø]úzüoù��aü�K±ú���ÿ�úôQú�ôöù�ôöóæýXÿ��Oú)ÿ��Oû��S�öôT�=� øºüU�N�JV � ú�ôöù�ÿ��¨ùzü1÷�ÿ ùzûæúzü$ôM�c÷�ó!úºôQý=û¯÷Z�Q÷Z^�÷�ó�úºù�ü ý�ú��côöù'� øºüU����c=ü øZI#ý=ü ø�ú���ÿ�ú^Îÿ�úzúz÷øZI�úzüR�F�aü�ü7ùz÷Wøkÿ ó��aüU^¹÷Z�ö÷Z^�÷�ó!úºù¶ôöó]ú��côöùY� øküU�N��i.ùºô¨ó��O÷�ôQú°ôöùY��ü7ùºùºôMa��Q÷�úzü�÷S¾_�ôQ÷�ó�ú��Qû=�a÷úz÷øF^Îô¨óa÷Kê�a÷ú��a÷ø)÷Z�Q÷Z^�÷�ó�úºùRq#� u /1 ÿ�øº÷Ùþ��cÿ��aøkÿ�úºôT��øº÷�ùºôM���a÷�ù�ü ø]óaü úSV:®.ôöócÿ��M�Qû�I#ú��a÷Ù[�[�]âÿ ùºù���^%�cúºôQü7óoô¨ùa�÷Z�¨ôQ÷õ ÷Z�¸úzü4�aüU�T�¸ôöóeú��côöùR� øºüU�N�JVû�8�côöùR� øºüU�N�môöù�^�ü7ùzú=�OüU^ª^�ü7ó��öû��cùz÷Z�eúzüæô¨ócùzúºÿ ó�úºôöÿ�úz÷Òú��c÷Ú����[ÿ�^Îÿ���÷�ó��OøºûL��úºôQü7ó�ù��F�c÷Z^�÷�V

�8�a÷øº÷�ÿ�øk÷�ü ú��a÷ø�÷¤£aÿ�^%���ö÷�ù³ü ý��OûÆ�S�öôM��� øküU�N��ùRc�^�ü7ùºú8K¶÷Z�T� k°��ócü�K�ó1ÿ�øk÷�ú��aü7ùz÷�a�ÿ ùº÷Z��ü7ó1÷Z�M�¨ô��cúºôM��S�aøºõ ÷�ùji�ôöó4Kê�côT�F��ú��a÷Ù[\[�]±ÿ ùºùF��^%�cúºôQü7ó¸ôöù'a�÷Z�öôQ÷õ ÷Z��úzü��aüU�M�³VÙ��÷Òùzúzøk÷�ùºù�ú���ÿ�ú�ÿ ù��öü7óN�$ÿ ù�ú��c÷[\[�]�ÿ ùºù���^%�cúºôQü7ó�ôöùYa�÷Z�öôö÷õ ÷Z�)úzü��aüU�M�bcXÿ ó��Îÿ ù��öü7óN��ÿ ù�^����Qúºô����¨ôM�ÿ�úºôQü7óbôöóbú��a÷8� øºüU�N�ª�ÿ óªa�÷)�aü7óc÷÷S¾_�ôQ÷�ó�ú��QûNijI�ú��a÷´ùzúzøF���Oú��aøk÷�ü ý�ú��a÷G� øºüU�N�Òô¨ù¿�cócôT^%��ü øºúºÿ ó!ú°ÿ ù¶ýXÿ�øpÿ ù¶ú��a÷Wùº÷Z�S�aøkôQú�û�ü ý�ú��a÷G�����[ÿ�^Îÿ��÷�ó��Oøºû��cúºôQü7ó1ù��j�a÷Z^�÷�ô¨ù)�Oü7ó��O÷økóa÷Z�³V

8 9'C�>³E\;;:�,�>=<¬E8CADFEG;;>�E�?@9A,¿<CB=>JD{CÆ@

�aø[úzøº÷�ÿ�ú�^�÷�ó�ú�ü ý¿����a��öôM�gk°� ÷û1÷�ó��OøºûL��úºôQü7ó���ÿ�økÿ��M�Q÷Z�öùWüU�aø[úzøº÷�ÿ�ú�^�÷�ó�ú�ü ý��cøkôQõ�ÿ�úz÷¤k°� ÷û$÷�ó��OøkûL�cúºôöü7ó³V� óÒú��a÷\�cøkôQõ�ÿ�úz÷¤k°� ÷û_�ÿ ùz÷�INKp÷GQ�økùzú¿�7ÿ5õ ÷�ÿ��c÷SQ�ócôQúºôQü7óÎü ý#ùz÷Z�S�aøkôöú�ûÎÿ��7ÿ ôöócùºú��ô����a÷øºúz÷¤£Fú{k�ü7ó��QûÎÿ�úzúºÿ����Fùÿ ó�� ù��cü�Kp÷Z� ÿ��Oü7ócùºúzøF���OúºôQü7ó ü ý[ÿ¸ù��F�a÷Z^Î÷sùz÷Z�S�aøk÷:K�ôQú��møk÷�ù ��÷Z�OúÎúzüoú��côöù_�a÷SQ�ócôQúºôöü7ó³V+��÷sú��c÷�ó�Oü7ócùºôM�a÷øk÷Z��ÿè�a÷SQ�ócôöúºôQü7órü ýbùz÷Z�S�cøkôQú�û�ÿ��7ÿ ôöócùºú:�F�aü7ùz÷�óNk°���öÿ ôöó�úz÷¤£�úsÿ�úzúºÿ����Fù c=ú��cÿ�úsôöùSI�ùz÷Z�S�aøkôöú�û�ôöóú��a÷Îùz÷�ócùz÷�ü ý��Q÷ý=ú{k�ü ø k�økô��U��ú�ôöó��cô¨ùzúºôöóN�U�cô¨ù��cÿ�a�ôM�¨ôZú�û�ipÿ ó���ù��aüpK¶÷Z��ú���ÿ�ú�ú��côöù\Kpÿ ù�ÿ�· Ë »gµ�¹ Ë Ó´Þ�· Ë » Ñ�¶ ß ¸S»Ø¸ED�¶�µ Ë µãÑ�¶�VFWÿ�^�÷Z�öû�I°ú��a÷øº÷§÷¤£aôöùzú]ù��F�a÷Z^Î÷�ù�ùz÷Z�S�aøº÷�ÿ��7ÿ ôöócùºú=�ôM���a÷øºúz÷¤£Fú{k�ü7ó��Qû¸ÿ�úzúºÿ��F��ù_Kê�côM�j�eÿ�øk÷�a÷SQ�ócôQúz÷Z�öûÙ¶�Ñ Ë ùz÷Z�S�aøº÷´ôöó�ú��a÷´ùz÷�ócùz÷Wü ý�ôöó��côöùºúºôöóN�U�côöùF�cÿ�a�ôM�öôZú�û_cXÿ óÎ÷¤£aÿ�^%���Q÷Wü ý�ù����j�·ÿ�ùF�F�a÷Z^�÷´ôöù°ú��c÷ü7óa÷¤k�úºôM^�÷���ÿ���ijV��#�cÿ�úê�cÿ�����÷�ócù�ôöó§ú��c÷����Na��¨ôM�gk°� ÷ûÙ�ÿ ùz÷HG

X�÷Z�ÿ��M�¶ú��a÷�Ó�¸ ÊgË Ö Ñ�»gÖ�»gµ ß�ÌLË Ñ�» × ¹ZÓ�¸�ú��cÿ�ú�Kp÷��cùz÷Z� ÿ��öùzü�KG�a÷�ó �c÷SQ�ócôöóN�1ùº÷Z�S�aøkôQú�û�ü ý8�cøkôQõ�ÿ�úz÷¤k°� ÷û÷�ó��Oøºû��cúºôQü7ó³VÚ�8�côöù[ü økÿ��S�ö÷·ôöù�ôöó��a÷¤£�÷Z��a�û�ÿba�ôöúA�]ÿ ó��oÿ:���Na��öôM��� ÷û�õ�ö c=üU�aú ���cú�a�û�ùzüU^Î÷ª� ÷û� ÷�óa÷økÿ�úºôQü7óHÿ��M� ü økôQú���^Ëý=ü øÎÿû���Na��¨ôM�gk°� ÷ûo÷�ó��Oøºû��cúºôQü7ómùF�F�a÷Z^�÷pijV+�8�c÷�ü økÿ��S�Q÷JILKM�ON óJô úºÿ�� ÷�ùÎú�K¶üôöóN���aúºùZ½PILKM�ON óJô ce�:î�f � ² i]øk÷ú��aøkócù_ò�óJô�ce���FijI8Kê�a÷øk÷1ú��côöùÒ÷�ó��Oøºû��cúºôQü7óèôöù_�aü7óc÷$økÿ ó��cüU^ª�Qû�÷�ÿ��F�úºôM^�÷'ú��a÷Òü øTÿ��S�Q÷�ôöù)ÿ��S�O÷�ùkùz÷Z�³V4��ôQú�� ú��a÷Ú�c÷SQ�ócôQúºôQü7ó¸ü ý�ú��côöù)ü økÿ��S�Q÷§ôöó ���öÿ��O÷�I?Kp÷b^Îÿ�û��7ôöõ ÷'ú��c÷ý3üU�M�öü�K�ôöó��b÷Zþ��côQõ�ÿ��Q÷�ó�ú8�a÷SQ�ócôöúºôQü7ó'ü ý.ùº÷Z�S�aøkôQú�û'ÿ��7ÿ ôöó�ùzú8�ô����a÷økúz÷¤£�ú{k�ü7ó��öû'ÿ�úzúºÿ��F��ùS`Q Â�R�SUTWVHTeÃXSPÇZY º�Ò¤Óâµã¹SÖ�ä�¸¤Þ'¸¤¶�¹Z»jÞjÏ Ë µãÑ�¶_·¤¹ Ì ¸¤Ô_¸8c��=f ò�f\[ªi�µz·�cedgfFhjijÖ�·S¸�¹ZºÆ» ¸ ×Zß× µe¶N· Ë ¹Sµ Ï Ì ¸S» Ë ¸Fá Ë Ö Ñ�¶�ÓâÞ×�ËãË°× ¹¤äp·Rµ Ê?Ê Ñ�» × ÓzÓ × Ø&#�¸¤»j· × »gµã¸¤·Gm¬»gºÆ¶�¶�µe¶ ß µz¶ Ë µeÔª¸êd�%�¸ ÌÆ× #�¸^]

nnn �'ÿ �°ø�o�c�õ�ö§f`_Yö�i¿tv�=½a�8t ��÷ÆfZ����`UmAb�cHd�e f�g�hWi N i j�c�õ�ö�i�~��F���è� nnn ý�hSf% Ì ¸¤»�¸�m¬Ô × Þ�Ñ�¶�ÓâÞlkZº�¸S»jÞ ËeÌ ¸mILK�Ñ�» × ¹SÓM¸ × ·jµe¶ ß Ó�¸ Ë µeÔª¸�å�ü úz÷bú��cÿ�úGm ôöó$ÿ����côQúºôQü7ó§úzü��cÿ�õFôöóN�·ü økÿ��S�Q÷)ÿ��S�O÷�ùkùWúzünILK ôöùWÿ��öùzüÎ÷¤£Æ���öôT�ôQú��Qû_�7ôöõ ÷�ó§õ�ö ÿ ùôöóN���aúêa�÷Z�ÿ���ùz÷RKp÷bÿ�øk÷)ôöó$ú��a÷����Na��öôM�gk°� ÷û§ùz÷úzúºôöó��NV�c�mPo ù´ÿ��S�O÷�ùkùWúzüpILKoôöù´ÿ ùWÿ ë{a��öÿ��F�:a�üZ£NíÆI�ÿ ó��ú��L�cù\m �ÿ ócóaü ú[ÿ��aúzüU^Îÿ�úºôT�ÿ��M�Qû:�a÷úz÷øF^·ôöóa÷Rõ�ö�o ü øZI�ü ý��OüU�cøkùz÷�Iq���Cý3øküU^ ôQúºùWôöó�úz÷økÿ��OúºôQü7ó§K�ôQú��1ú��c÷ü økÿ��S�Q÷�½�ôQú�ô¨ù³ý3ü ø�ú��côöù�øº÷�ÿ ùzü7ó$ú��cÿ�úêõ�öËôöù�÷¤£Æ���öôT�ôQú��Qûª�7ôQõ ÷�ó1úzüÙm�V´i

�8�côöù�ó�ÿ�ú��aøkÿ��M�Qû§�Q÷�ÿ��cù'�cù[úzü$ÿb�c÷SQ�ócôQúºôQü7ó¯ü ý�ùz÷Z�S�aøkôöú�ûsô¨óæú��a÷Îùz÷�ócùz÷�ü ý��Q÷ý=ú{k�ü ø k�økô��U��ú�ôöó���ôöùzúºôöóÆk�U�côöù��cÿ�a�ôM�öôQú�û�V H ù[ôöó�ú��a÷��cøkôQõ�ÿ�úz÷¤k°� ÷û§�ÿ ùz÷�I�ú���ôöùG�c÷SQ�ócôQúºôQü7ó$ÿ��öùzü��Oü øºøº÷�ù ��ü7ó��cùWúzü§ùz÷Z�S�aøkôQú�û$ÿ��7ÿ ôöócùºú�F�aü7ùz÷�óNk°���öÿ ôöó�úz÷¤£�úpÿ�úzúºÿ����FùSIaÿ ó��Îÿ��öùºü�ùz÷Z�S�aøkôöú�û=Kê�a÷�óª^����QúºôM���Q÷�^�÷�ùºùºÿ�� ÷�ùpÿ�øº÷�÷�ó��OøkûL�cúz÷Z�Ù�có��a÷ø¾ú��c÷ùºÿ�^�÷�����a��öôM�G� ÷û�V

¨

Page 93: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Q Â�R�SUTWVHTeÃXSZrsY º�Ò¤Óâµã¹SÖ�ä�¸¤Þ�¸¤¶�¹Z»jÞjÏ Ë µãÑ�¶4·S¹ Ì ¸SÔª¸=c��=f ò¿f\[%i�µz·�cedjfFhgijÖl·Z¸�¹Zº�»�¸=µe¶ ËeÌ ¸R·S¸S¶N·S¸ªÑ Ê µe¶�Ø�µz·jÖË µz¶ ß ºÆµT· Ì�× Ò¤µeÓâµ Ë ÞÚµ Ê?Ê Ñ�» × ÓzÓ × Ø&#�¸S»F· × »jµ°¸g·�m »jºÆ¶�¶�µz¶ ß µe¶ Ë µzÔ_¸êd�%�¸ Ì�× #�¸t]

nnn �'ÿ �°ø�o�c�õ�ö§f`_Yö�i¿tv�=½a�8t ��÷ÆfZ����`Um b�c d�e f�g hWi N i j c�õ�ö�i�~��F���è� nnn ý�hSf

% Ì ¸¤»�¸)m Ô × Þ%Ñ�¶�Ó´ÞukZº�¸S»jÞ ËeÌ ¸�ILK2Ñ�» × ¹SÓM¸ × ¶bºÆ¶�ÓâµeÔ�µ Ë ¸�Ø%¶�ºÆÔªÒj¸¤»'Ñ ÊGË µeÔ_¸g·Pv Ñ Ê ¹jÑ�ºÆ»F·Z¸gÛ ËeÌ ¸�¶�º�Ô_Òj¸¤»Ñ Ê kSº�¸S»jµ°¸g·w%?µeÓzÓ�Òj¸ Ê ¸$%�¸¤» ËeÌÆ× ¶�dyx�å� �Q÷�ÿ�øF�Qû�I�ú��cô¨ùG�a÷SQ�ócôQúºôQü7ósôöùWócüÙKp÷�ÿ�� ÷ø�ú��cÿ ó¯ú��a÷%�a÷SQ�ó�ôQúºôQü7ó1ü ý¶ùz÷Z�S�aøkôQú�û$ÿ��7ÿ ôöócùºú��ôM���a÷øºúz÷¤£Fú{k�ü7ó��Qûÿ�úzúºÿ����FùSV � órýXÿ��OúSI´ÿ�ùzüU^�÷SKG�cÿ�ú1ù��aø���økôöùºôöó���øº÷�ù����Qú'ôöù'ú��cÿ�ú�ú��a÷�ùz÷æú�K¶üè�a÷SQ�ócôQúºôQü7ócù'ÿ�øº÷�ôöóèý=ÿ��Oú¸zkSºÆµ{# × ÓM¸¤¶ Ë -��8���cùSI�ÿ ó�û����Na��öôT�gk°� ÷û·÷�ó��Oøºû��cúºôQü7ó$ù��F�a÷Z^Î÷)ùz÷Z�S�aøk÷�ÿ��7ÿ ôöócùzúG�ô����a÷øºúz÷¤£Fú{k�ü7ó��Qû§ÿ�úzúºÿ����Fùôöù¶ÿ��cúzüU^Îÿ�úºôM�ÿ��M�QûÒÿ��öùzü)ùº÷Z�S�aøº÷[ôöóÎú��a÷[ùz÷�ócùº÷Wü ý#ô¨ó��côöùzúºô¨óN�U�côöù���ÿ�a�ôM�QôZú�û�V³��÷��7ôQõ ÷[ÿ)ý3ü øF^·ÿ���ùzúºÿ�úz÷Z^Î÷�ó!úü ýCú���ôöù�øº÷�ù����öúSIcÿ ó���ÿ�ý����M���cøºü�ü ýÄü ýCú��côöù�ÿ ùºùz÷øºúºôQü7óJIcóa÷¤£�ú�úºôM^�÷�V

Page 94: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

|~}����.�$�a�z�������.�����&�a�����.}��� �n� �����.�P��� }��\�a���������z����}l�\� � �z�����\�.�.�a�&� ���¡C�z�.�{�t�z�\�.��¢.��}��&�z���.}�£��&�\¤

¥Z¦¨§w©«ª­¬«¦ ®°¯

± ²�³;´uµz¶�·Mµ`³;¸~¹¨µa¶Xº;»0¼¨½a¾À¿Á¹¨¼¨½`µzÂUÃÅÄƾ�Ç ÈF³;Â�ÉMÇPÊP·qµzË"³

ÌÍ�.�\�3�z��Ψ�.Ï�Ð��"���^�.���;�Ñ�$Ò�}����z����}u�.�Í�\�t�$���a���y����}Ó�z���"�\�t}��\�~�.�U��}������z�z��}Ñ�������a���&Ô������(�����{�.��� �ÑÔ�������Õ�Ö.�$��t}����z�����z����}J�z�`�Ñ�tΨ�t�$×ÙØ0�������Ñ�$Ò�}����z����}p������ÚÑ�.���z���J�.}��.���.�.�������\�°�z�Ñ�Ó�Ñ�$Ò�}����z����}pÐ��Ó���t�.�l��}Û�z����$�.�\���.�3���`�����&�\��Õ�Ö.�$�Û�t}����z�Å���z����}q×ÝÜO}Þ�z�Ñ���$�.�\���.�«���a���&�&�\��Õ�Ö.�$�Û�t}����z�����z����}ßÏU��}������\�z��}Ñ�������z���&Ô��à�����y�Ð��.�~�\�\�a�����z���Ý�\�\�z��}Ñ�.�$���z���.}��\�t�$�Ñ�`���y�Ý�&���.��}��\���$�����Ñ�$�z�\��ÚL�áÕ���}����Ý�&�\�z�.�zÖL�$×�Ø������3���«}Ñ�.�«�z�Ñ���$�.�z���W�.����ÑÔ����à��Õ�Ö.�$�Ó�t}����z�����z����}ßÏ��.�«Ð4�P�z�Ñ�^Ðs�Ñ�$�z�.×�â��M�t�$��Ò �$�.�����XãäPåUæÅç èHæLé ê�ëÍìtí0îEï;ð\ñCð\òuóuôaì¨õAöX÷�ô$ø7ù�ú$û�ü.ì$ýÙì$þ ú$ÿ�ý`öXíEù��&þ���ú��Lì��Óì���ùEú���ù��mî �`ð���ó`û��tìzút÷�ÿzìuõ���õ&ù{þ���íú$ù ö��Lì�ÿ�íyì���íEû��.þ�ø7ýpõ.í�í�õ�ú$ü��������Lì�þ­îEï;ð\ñCð\òuó¨ù��¨î ��ð����`ó`û��$ìzút÷Lÿ\ìÓù{þ í �Lì�$ì$þ��$ì��� ¨ù{þ"!&ù���íEù{þ#�.÷Lù�����õ�ô$ùWø ùEíEý$ ��Lì$ÿ\ì;í �Ñì;õ%!'&&ì$ÿ��$õ&ÿ�ýÝù��Aõ(����÷)�uì�!lí*�Ùõ�úaú�ì+�,�Aí �Lì.-)/0�&ÿ\õ�ú$ø�ì1�ÓíEù �Óì+��23�Ø0�Å���$Ï �.������}Ñ�Ý�.� � �à���zÎu�.�à��Ï �.}�� � ���~Ð����z���à}l�z�t�.�\��} î �.�54����Ñ�a�z�.Ï Ð��m�.��Ð��^����Îm���\�~���t�.� �.67�zó Ï�z�Ñ�m��4a�Ñ�tΨ�;���~�\��4$�Ñ�z�;��}��z���P�\�t}��z�A�.����}������\�z��}��������z���&Ô �������y�.×4Ø������84$�.�Í���.�����t�~Îu�������~ÔX� �:9<;#=�>@?�.}�� �A6B;#C > �.���\� î �$�.�t}���� � �à�«ÎP�D4`�����������$� ó Ï���Ψ��������}��A�z���&� ��� �à�0�\�z�����M�z�DEF4$���t}Å�z�����zÎÓ�.����×G èHçqç"H I �����z�H�.���z�Ñ�~�z���$�.�z�tÎ �W�.���z�Ñ�J4$�.�\� �:9K; ÏÑ�.}��l���t�^�.���z�Ñ�~�.�t}��$�a�.�L4$�.�\���\�P�z�Ñ�"�z�t�.�Ñ�$�^×M��.�\���z���&�¨�$�.�t}!�z�Ñ�N4$�.�\� �O9P; �à�¨�.���z�t�.�Ñ�F�p���.�\�u�àΨ���z�H�.�tΨ�t}��¨�H�.�$�Ó�z�Ñ�Ù���a���&�&�\��Õ�Ö.�$�Q4$�.�\�.ÏÐ��Ñ�$�z�0�z���0��}Ñ��Õ��z��Ψ�����.� î �{�.����ÚÑ�.Ψ����� ó Ð3�.� î ��ð�R�ó Õ �z��4$�Ñ�z���&���.��}��\�54$�������$�z�\��Ú��áÕ���}����m�&�\�z�34aÖ��tÏ�Ô �Ñ�}Ñ�.� î ��ð�S:TU�`ó Õy�z��4$�Ñ�z� î �{�.�~�.}Å� �WVXR�ó �à}l�z�Ñ���\�t}��\�A�.� ��}������\�z��}��������z���&Ô �����(�y�.Ï��$�.�t}��{�.� �:9Y; ×

Ìß�$�5Z ÔM�«�.}u�.�Ñ�.�$�a�a�&�z�¨�&�\�z�34zÖL��}Ñ�A�z���«�t}D4��z�����z����}¨��4`�Ñ�tΨ����};�z�����\�t}��\�«�.�M��}������\�z��}Ñ�������z���&Ô��à�(Õ���y�.ÏM�.}���ÎÓ�&Ö���}��u�yÐ��\[Å�Ñ�$�`���t�«�\�Ý�z�Ñ� -)/ �.�a�34$���.×0Ìß�$� î ] C ð@]\^ C ó �.}�� î ]N_&ð@]\^_ ó �Ñ�t}Ñ�.�\�m�z�Ñ�A���.���a��.�"Ψ�t�a�z�&�.�t�¨�z���&�`Z �z�ÑÔ�Îu���z�P�\�Û�z��� -)/ �.�a�34$��� î ��×��.×�Ï î ] C ð@] ^ C ó �&�z�Ù�z�Ñ��Ψ�t�z�a�&�.�t�u�z�ÑÔ�ÎÓ���\�\�t��z�Ñ�ÙÒ��a�\�;�z��Îu�l�.}�� î ]N_&ð@]\^_ ó �&�z�Ù�z�Ñ�ÙΨ�t�z�z�&�.�t���z�ÑÔ�ÎÓ���\�\�t�F�z�Ñ���\��4���}��­�z�àΨ� ó ×FØ��Ñ�t}@Ð4���&�a���}��\�$�a�t�\�\�t����}lÔM����}�����}Ñ�P�z�Ñ���{�������^Ð~��}Ñ�Ñã

aaa ;cb ¡C��d îfehg�ð,igJókj ïml�n:jpo�RLð�S(q ãrZts+uwv*x y#z5{}|�~�| � îfetgJó�9Kn��"TBS aaa9 aaa ¡C�(d Z s+uw� x yrz {}|�~�| � îfetgJó�9�R(�"T ¡��wd Z s+u%� x yrz {�|�~�| � îfetgJó�9�R(�

aaa9 aa ¡C�(d Z îfetg°ð\ñ)���mî ] C ó`ð\ñ)���;î ] _ ó\ó�9�R(�"T ¡��wd Z îfetg�ð\ñ)���;î ] ^ C ó`ð\ñ����mî ] ^_ ó\ó�9�R(� aa ð

Ð��Ñ�$�z�PÐ4�¨���^�.�mÔM�$�t}°�z�������Å�z���Ù��}Ñ�{�.�aÎu�.� î ��}�� �&�z�z�84$�����&�^Ï îfetg°ð,igJó �&�z�P�`�.}��Ñ��Îu���l�.�t}Ñ�$�`�&�\�t�Û��}�t�34a�°��Ú��M�$�a��Îu�t}��$Ï��.}�� ñ ��� î ]°ó �z�$�{�$�a�«�\�Ó�¨�`�.}��Ñ��Î �t}D4��z�����z����}��.� Ψ�t�a�z�&�.� ]°ó ×

���$�W�.�z�;�����L��}Ñ���z�Ñ�;�Ñ�$�z�.�����~�.�C�z�Ñ�;���z���.�OÏ Ð��P���z�^�L���Ñ�m���������LÕy���$�.�t�U�^�.�$�z�L���$Ð�×AM~�.�\�¨�z���&�~�z���Ò�}��.�M��ÚL���z�t�z�z����}Ù�&ÔX�H�.�P���3��[����.�q�\�Ñã

aa ¡C��d Z îfetg°ð\ñ����¨î ] C ó`ð\ñ)���mî ] _ ó\ó�9�R(�"T ¡��wd Z îfehg�ð\ñ)���¨î ] ^ C ó`ð\ñ����;î ] _ ó\ó�97R(�� ¡C�(d Z îfetg°ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ]N_^ó\ó�9�R(�"T ¡��wd Z îfetg�ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ] ^_ ó\ó�9�R(� aa î�S^ó

6 aa ¡C��d Z îfetg°ð\ñ ��� î ] C ó`ð\ñ ��� î ]N_^ó\ó�9�R(�"T ¡��wd Z îfehg�ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ]N_^ó\ó�97R(� aa î�;�ó� aa ¡���d Z îfetg°ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ]N_^ó\ó�9�R(�"T ¡C��d Z îfetg�ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ] ^_ ó\ó�9KR(� aa�� î���ó

S

Page 95: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

|~�z�à}Ñ�A�z�Ñ�"�E�34����z���&���z���~�t}D4��a�Å���z����}Ó��4`�Ñ�tΨ�w�����\��4$�Ñ�z���&���.��}��\��4$�������$�z�\��Ú��áÕ���}����u�&�\�z�34zÖL�$Ï�Ð��~Ð~�����ÔX����}��\��ÚL���z�t�z�a����}�� î�;�ó �.}�� î���ó ×

I �c4���}��\�\�a�D4����.}��.�Ñ�.�$�`�z�&�z�Z ^ Îu����}��z�à}Ñ�A�h4$�����Ñ�$�z�\��ÚL�áÕ���}����¨�&�\�z�34zÖÝ�&���.��}��\�4�z�Ñ�0�t}D4��a�Å���z����}��4a�Ñ�tÎu�.×k�~�$�z�.Ï�Z ^ �à�3�����.�t}��4$��� �Ñ�$�z�\��ÚL�1� Ð����}4a�Ù���«�t���z�Ñ�$�0�.}��t}D4��a�Å���z����}l�.� ] C �.�0�.� ] ^ C ã

Z ^ îfetg°ð � ó4���Ψ�����\��� _Wj ñ ��� î ]N_^ó�î }Ñ�.�\�P�z���&�WZ ^ 4$�.}°�Ñ�¨�z�����0�a��}D4��w���«ÖL}Ñ�^Ð�� etgJó�a��}\Z îfetg°ð � ð � _Hó���Ñ�\�����«Ð����&�\�$�.�$�~�à�3���Ñ�\���Ñ�«Ô���Z

�4�Ù�Ñ�$Ò }����z����}Ý�.��Z ^ ãaa ¡��wd Z ^ îfetg°ð\ñ ��� î ] C ó\ó�9KR(�"T ¡C��d Z ^ îfetg°ð\ñ ��� î ] ^ C ó\ó�9�R(� aa9 aa ¡C��d Z îfetg�ð\ñ ��� î ] C ó`ð\ñ ��� î ]N_^ó\ó�9KR(�"T ¡C��d Z îfetg°ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ]N_^ó\ó�9�R(� aa6 �$ð

Ð��Ñ�$�z���z�Ñ��Ò�}��.�&��}Ñ��[����.�������~�Ñ���à���q�z��}D4����z�����t}D4��z�����z����}w��4`�Ñ�tΨ����� î ��ð��`ó Õy�z��4$�Ñ�z�4�&���.�à}��\��4$��� �Ñ�$�z�\��ÚL�áÕ��}����l�&�\�z�34zÖL�$×

I �n}��^Ð�4���}��\�\�a�D4��Ó�.���.�$�a�z�&�z�XZ ^ ^ Ï«�.���\� Ψ����}Å�z��}Ñ�J�U4$�����Ñ�$�z�\��ÚL�áÕ���}����@�&�\�z�34zÖ �&���.��}��\�u�z����t}D4��z�����z����}Ý��4a�Ñ�tÎu�.×5���$�a�.Ï)Z ^ ^ ���4�����.�t}Ý��4$��� �Ñ�$�z�\��ÚL��� Ð����}4a�Ó�à���t���z���$���.}Ý�t}D4��z�����z����}Ó�.� ]N_ �.�] ^ _ ã

Z ^ ^ îfetg°ð � ó4���Ψ� �Ñ�\��� C j ñ ��� î ]\^ C ó�î �&���.��}ßÏ�Z ^ ^ 4$�.}����¨�z�����0�z�à}D4��w���«Ö�}��^Ð�� ehgÛó�a��}\Z îfetg°ð � C ð � ó���Ñ�\� �Ñ�«Ð����&�\�$�.�$�"���3���Ñ�\���Ñ�«Ô���Z

�4�Ù�Ñ�$Ò }����z����}Ý�.��Z ^ ^ ãaa ¡��wd Z ^ îfetg°ð\ñ ��� î ]N_tó\ó�9KR(�"T ¡C��d Z ^ îfetg°ð\ñ ��� î ] ^_ ó\ó�9�R(� aa9 aa ¡C��d Z îfetg�ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ]N_^ó\ó�9KR(�"T ¡C��d Z îfetg°ð\ñ ��� î ] ^ C ó`ð\ñ ��� î ] ^_ ó\ó�9�R(� aa6 �$ð

Ð��Ñ�$�z�.ÏM�&���.��}qÏX�z�Ñ�AÒ }��.�Í��}���[Å���.�à���y�Ù�Ñ���������z��}�4����z�Ñ�m�t}D4��z�����z����}���4a�Ñ�tΨ�;��� î �`ð���ó Õy�\��4$�Ñ�z�¨�&���.��}��z�4$�����Ñ�$�z�\��ÚL�áÕ���}����l�&�\�z�34zÖL�$×

Ø0�����$ÏMÔM�.�z�O�CÚ����a�t�z�z����}�� î�;�ó �.}�� î���ó �&�z�PÔM����}��Ñ�t��Ô�� � Ïq��Îu�����L��}Ñ�u�z���&�J��ÚL���z�t�a�z����} î�S^ó �à�ÔX����}��Ñ�t�lÔ�� ;3� �.}��Ù���z�H���à}Ñ�¨�z�Ñ�w�z�Ñ�$�.�z�tÎ�×

� }­�àΨ�M�.�z�z�.}��4��.�z���à���&�z�J�.�~�z�����m�z�Ñ�$�.�z�tÎ �à�P�z���&�¨��}D4��ÙÐ4�Ù���^�.���p�\��4$�Ñ�z�l����Ô����84�Õ�Ö.�$�J�t}ÑÕ4��z�Å���z����}p��4a�Ñ�tÎu�m�{�.�wΨ�t�z�z�&�.�t���.�����t}Ñ�.�z� � ÏXÐ��¨Îu�t�n��ÎuΨ�t�����&�\�t�������\�m�z�Ñ�;��4a���tΨ�;�\�Ù�t}D4��z������&�zÔ����\�a�&�a��Õy���t}��.�z� Ψ�t�a�z�&�.�t�;ÔÅ�ÛÔ��z�t�&Ö��à}Ñ�°Î¨�t�a�z�&�.�t�;�\�pÔX���t}D4��z�����\�t�@��}Å�\�°�p�\��[��Ñ�t}D4��Ý�.� � Õ�Ô ���Ô����)4zÖL� î ���.������}Ñ�w���q}Ñ��4��t�z�z�&�a� ó �.}��u�t}D4��z�����z��}��w�t�34a��Ô����)4zÖ¨�\�$���&�a�&�\�t��� î ���z��}Ñ�w�{�z�t�z�¨�`�.}��Ñ��Îu}Ñ�t�a��t�34a�n�z��Ψ� ó ×�M~�.�\�A�z���&�~�z�������à�`�á��[Å�������.���t}����Ó�\���\��[��Ñ�t}Å�z���.�ß�t}D4��a�Å���z����}��«�.� � Õ�Ô ����Ψ�t�z�a�&�.�t�$ÏX�.}�����«�z�Ñ�$�z�$�{�.�z�A�z��4$�Ñ�z�wÔ����z�Ñ�P�&ÔX�H�.���z�Ñ�$�.�z�tÎ�×

I �A}Ñ�.�\���z���t4��a�D4$���.�M�����M�$�z�t}D4��wÔM�$��Ð4�$�t}Ù�z���w���a���&�&�\��Õ�Ö.�$��4$�.�z�A�.}��l�z�Ñ�w����Ô����84�Õ�Ö.�$�F4$�.�z�.×�Üá}�z�Ñ�A���a�Å�.���&ÔM�^�.�.Ïq�.�Ñ�.�$�a�a�&�a���t�WZ ^ �.}��OZ ^ ^ 4$�.}°�.�t}Ñ�$�a�&�\� î �a�.}�����Î ó �t}D4��z�����z����}��0�.� ] _ �.}�� ]\^ C Ï�z�t�\�M��4��z���.�t���.Ï ôaìaúaõ&÷��$ìÙí �Lì�ýÛõ&ÿzì�ì���ö ø ùEútù{íEø ý���ù�&.ì�þ@í �LìAö ÷�ô$ø7ù�ú�ü.ì$ý�etg ×ÙØ0�Ñ�Ó�à� þ"�.í �z�Ñ��4$�.�\���W�.����a���&�&�\��Õ�Ö.�$� �t}�4��z�Å���z����}qÏ Ð��Ñ�$�z�l�z�Ñ�l�.�Ñ�.�$�a�a�&�z� �Ñ���t� þ"�.í �.�$�;�\�J���t�&�a}F�z�Ñ�ÝÖ.�$�F�.}�� �z���$�z�$�W�.�a�4$�.}�}Ñ�.�0�.�t}Ñ�$�`�&�\�A�t}D4��z�����z����}��3�.� �.�z���$��Ψ�t�z�z�&�.�t�t×

;

Page 96: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�����  �¡5¢U£�¤L¥�¦�¢Q§�¨� �¡�¢�§�©%¢�ª �« �Ñ�$�.�z�tÎ S �����.�$�z�J���\�$�E�����{�.�A���a�^�L��}Ñ���z�Ñ�Ý�\��4$�Ñ�a���y�p�.�«���ÑÔ��à�84�Õ�Ö.�$�°�t}�4��z�Å���z����}Þ��4`�Ñ�tΨ�t�$×O¬w�Ñ������z��Îu�&�\�Ó�.���.��Ð��à��� �.��Ð��^���PÔM�¨�\�O4���}��z�\�a�D4��P�.} ��}����à�\�z��}Ñ�����à�z���&Ô����A�t}D4��z�����z����}J��4`�Ñ�tΨ�.×�­C�$�;��}�.}��.����¤t��}Ñ� î �.}�� ���z�^�L��}Ñ�J�z��4$�Ñ�a����� �.� ó �a�D4a�Á�J��4a���tΨ�.Ï3Ð4�n}Ñ�$�t�­��}����F���z�H�.���\��4$�Ñ�`���y�­�&���.��}��z�4$�����Ñ�$�z�\��ÚL�áÕ���}����n�&�\�z�34zÖL� � �lÎm�D4a�J�a��Ψ�����$�~�z�.�\ÖX׬�}�4��mÐ��u���^�.�u�Ñ��}Ñ�¨�\�ÑÏß�Ñ�HÐ4�$�.�$�tÏUÐ4�uÎÓ�t���ÎuΨ�t���à�&�\�t���Þ�&������� « �Ñ�$�.�a�tÎ S �\�J�z�Ñ�HÐ �z���&�u�z�Ñ����4a�Ñ�tΨ���à�¨��}@�{�34����\��4$�Ñ�z����}F�z�Ñ���z�t}��\�Ù�.���}������z�z��}Ñ�������a���&Ô������(���.× « ���à�«Îu�&Ö.�t�0�z�Ñ�����t�z����}Ù�.� ���z�H���&Ô�����Õy�\��4$���z����4a���tΨ�t�0�t�.�z���$�t×

® ¯ ÇP¼;Éßµa´ ÈF³;Â�ÉMÇPÊP·qµzË"³

I �m���^�.�¨}Ñ�^Ð �\�$�t}°�yÐ��Ý�\��4$���z�A����Ô����84�Õ�Ö.�$�Ý�t}D4��z�����z����}°��4a�Ñ�tÎu�t�$×�Ìß�$�����~���Å�.ÖÙ�&�"�z�Ñ�A��E�4$���t}D4���.� �t�34`�q×

° « ��� ��4a�Ñ�tΨ��Ô��.�\�t�w��}�[Å���.���a�&�z�84 �z�t�z�����Ñ���z������Ð3�.�ß�.�a�����à}��.�����0�Ñ�$Ò }Ñ�t����}����"�W�.�Í�t}D4��z�����z����}��.�S Õ�Ô����ÍΨ�t�z�z�&�.�t�t×�3�Ñ� ���U�a�Ñ�������wÔM�54$���t�&� î �z�à}D4��.Ï&ÔÅ� « ���$�.�z�tÎ S Ï.�z�Ñ����4a���tΨ�4���U�\��4$�Ñ�z����}��z����\�t}��\���.�U��}������\�z��}Ñ�������z���&Ô��à�����y�A�.}��Ù�Ñ�t}D4��w�\��4$�Ñ�z��Ð����t}ÝÎm�����z�������~Îu�t�z�z�&�.�t�0�&�z���t}D4��a�Å���\�t� ó�z���&� � Õ�Ô�����Ψ�t�z�z�&�.�t�c4$�.}°ÔX�w�t}�4��z�Å���\�t��Ô��l�a��Ψ������4���}D4$�&�\�t}��&�z��}�� î �a�.}�����Î ó �t}D4��z�����z����}���.�4�t�34a�J�.���z�Ñ�u��}������L�������.�qÔ ���z�$×mM��.�\�u�z���&�w�t�34a�J�t}�4��z�Å���z����}n�.���Ù�z�à}Ñ�����mÔ������z�t�z�����z�w��}p�± Õ�Ô����14$�������$�z�\��Ú�� î Ð��Ñ�$�z� ± ���0�z�Ñ�;���t}Ñ�.�z�°�.�C�z�Ñ�;Ψ�L���������:² ó ÏqΨ�t�.}���}Ñ�Ó�z���&�~�t}D4��a�Å���z�à}Ñ��.} � Õ�Ô����~Îu�t�z�z�&�.�;�z�t�z�����z�~��}n� ± � Õ�Ô ���14$��� �Ñ�$�z�\��ÚL�$×"Üá}°�\�$�aÎu�"�.�k4���Ψ� �Ñ�z�&�z����}��.� ��EF4$���t}�4��.Ï�t}D4��a�Å���z����}p�.���t�34`�ÞÔ �����z��[Å�����z�t� S,³�; Îu���������&��ÎP�����z�������}4$�&�z����}��~�t�34a� �z�&Ö���}����z��Ψ��´ î ± _ óî �z���à��4$�.}�ÔM����Ψ���a�^�.�t�qÏÑÔ����3���0���0}Ñ�.�«�z�t���$���.}Å���Ñ�$�z� ó ×

° « ���`���¶µw�.Îu�.�C�t}D4��a�Å���z����}J��4a���tΨ�����.� ��Ψ���a�^�.�t�·4���ÎuÎm��}��}4$�&�z����}J��EF4$���t}D4��.×\M~�.Ψ�t���.Ï�t}D4��a�Å���z�à}Ñ�«� ± Õ�Ô����ßΨ�t�a�z�&�.���z�t�a�����\�t�w��}��W4$�����Ñ�$�z�\��ÚL�q�.�Ñ���t}Ñ�.�z� ; ± Ï��{�.�Í�.}���ÚL���.}��z����}��{�34��\�.��.����}���� ; × � ��Ψ� �Ñ�z�&�z����}��.�����.Ï4�Ñ�HÐ4�$�.�$�tÏ��z�Ñ�°��4`�Ñ�tΨ�n���u}Ñ�.�ÝÎP�D4`�!�.���.}Á��Ψ���z�H�.�tΨ�t}Å��H�.�$�Ù�z�Ñ�p���z�$�L�������Ý��4a�Ñ�tÎu�.×=ÜO} ���&�z�z�}4$�����&�tÏ«�t}D4��a�Å���z�à}Ñ� � ± Õ�Ô����lΨ�t�z�a�&�.�n�a��[Å�����z�t�Ó��Ð4���ÚL�M��}Ñ�t}��z�à�&�z����}��P�t�34a�F�z�&Ö��à}Ñ�°�z�àΨ�\´ î ±)¸ ó × « �Å���tÏU�z�Ñ�l�.Ψ����}Å�;�.�:4���Ψ���Ñ�z�&�z����} ö�ì�ÿ�ô�ùEí���m�z�����������J�z�Ñ���a�.Ψ���.�¨��}F�z�Ñ�Ù���z�$�L�������;��4`�Ñ�tΨ�.× î M��.�\�.ã�ÜO}F�E�34��$Ï4�z���à�4���Ψ���&�a���z��}F�à��z�à�������z���;�à}��34�4$�Ñ�a�&�\�.ÏÑ�z��}�4��~�����M�$�z�t}Å��Ö.�$�Ý�z��¤$�t� ± Îu�������4ÔX�����\�t���{�.�4�z�Ñ�������M�$�z�t}Å�4��4`�Ñ�tΨ�t�$×�~�^Ð��$�.�$�tÏ��z�����z�Ñ�a���z���.� �z�Ñ���&�a����Ψ�t}Å�����:4$���t�&�^× ó

ÜO}Û�&Ô��z�����Ñ�\�¨�\�$�aÎu�tÏÍ���CÐ��`4���Ψ���&�z�Ó�z�Ñ�;��EF4$���t}D4$���t���.�4����Ô����84�Õ3�.}��p���a���&�&�\��Õ�Ö.�$�n�t}D4��z�����z����}pÐ���\�$�p�z���&�����a���&�&�\��Õ�Ö.�$�!�t}D4��z�����z����} î �z�t�.Ï����z�à}Ñ�Þ� Ô����)4zÖX4$�������$� ó �����z���Ñ������� S�R3R3R �z��Ψ�t���{�.�\�\�$��z���.}Û���ÑÔ ���84�Õ�Ö.�$���t}D4��a�Å���z����}q× � ���.��}qÏÍ�z������������}����n�l�z���Ñ���p�t�\�z�àÎu�&�\�.ÏU�.���������$�X�t}����"��}ÛÐ����84`����ÑÔ����}4�Õ4�.}��°���a���&�&�\��Õ�Ö.�$����4`�Ñ�tΨ�t�w�&�z�mÔM�t��}Ñ�F4���Ψ� �&�z�t�q×1­C�$�����"�����{�.���~�\�Ù�z�^���z���&�����a���&�&�\��Õ�Ö.�$��t}D4��z�����z����}����3�a���Ñ������� � �.�a�Ñ�$�a�0�.� Îu�&��}����z���Ñ�w�E�.�\�\�$�0�z���.}����ÑÔ����}4�Õ�Ö.�$�Ó�t}D4��z�����z����}ß×

� ���t�&�a���.Ï��z�Ñ�t}qÏ�Ð4�PÐ��.}Å�~�\�Ý�t�.�����n���z�à}Ñ�·�á���ÑÔ��à�84�Õ�Ö.�$�\4��z�Å���\�.�.�a�&���Å���Ó�\�Ó�\�a�.}��aÎu���0�.�$�z������}Ñ�Ψ�t�z�z�&�.�t�$×��3�Ñ���Ñ�HÐX4$�.};Ð4�0�����z�"Ð~�������4�z�$�z�.��}���}����z�Ñ�«ÔM�t}Ñ�$Ò��z� �.� ���ÑÔ����}4�Õ�Ö.�$�w�t}D4��z�����z����}"¹�M~��Ú���z��Ψ�.Ï�Ð4�"������4$���a� ��ý�ô�ÿ�ùf!Óì$þ ú$ÿ�ý`öXíEù��&þ Ð����}4a�u���4�AΨ�$�z�����Ó�W�.�4�.Ô��z�.��}���}��w�z�Ñ��ÔM�t�\�C�.�MÔM�.�z�uÐ��.�a�����$×

Page 97: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

º1»D¼8½3¾�¿�À�¼ÂÁ�Ã�Ä3Å�ÆÈÇ'¿�Ã�É8Ç3»�ÊË ÆÍÌ Ë¶ÎrÏ3Ð�ÑÓÒ »rÁ@¿�Ä�ÊDÔ�Õ�Á�¼ÂÄ%»�Á@Ä Ë ¿�Ã�ÖDÁ@Ä3×3¿�Ç'Ö"ØrÃÙ�¿�Ä3Å ¾�À�À@Ä3¿JÚ3Ä%»DÇ'Á�ØDÇ3»ÈÛtÇ'Á@Ü

Ý�ÞßAà�áãâ�Þ äÈä

å æèç�émê�ë�ìîíQïmð�êLç�ñ�ò�ë�ócï

ô:¾�Õ�Ç3É8É"Ä%Ô�¿5×3Ä%Ç3ÉõÅ ¿�Ä%ö÷É}Ç3À@Á5Á�¼8ö`¾3øùÁ@Ä�×3¾�ÁkÁ�ØD¾1Ç3Ê�½'Ç3»rÁ�Ç'×3¾�À�Ä3Å�Ö"Ô�ú�É8¼8Õ+û*ü3¾�Ãm¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»Èýf¼�þ�¾3þÂÿ�»�Ä�»D¾�¾�ÊÁ@ÄhÀ�ØDÇ'¿�¾:À�¾�Õ�¿�¾�Á�À5¼8»Ç3ÊD½(Ç3»DÕ�¾3ÿ#¾�Á�Õ'þ��mÇ3»DÊmÁ�Ø�¾W¾��FÕ�¼Â¾�»DÕ�Ã.Ä3Å"Ö�¿�¼Â½'Ç'Á@¾�û*ü3¾�Ã.¾�»�Õ�¿�Ã#Ö�Á�¼ÂÄ%»�þ���¾W»�Ä��0À�Ø�Ä��Ç��¶Ç�Ã�Á�ØDÇ'Á:Á�¶ÄF¾�»�Õ�¿�Ã#Ö�Á�¼ÂÄ%»\À�Õ�ØD¾�ö¾�ÀWÕ�Ç3»Èúõ¾tÕ�Ä%ö�ú�¼8»D¾�Ê\öÄ�Ê�ÔDÉ8Ç'¿�ÉÂÃ�¼8»\ÇÖD¿�Äw½(Ç'ú"ÉÂÃrû�À�¾�Õ�Ô�¿�¾��Ç�Ã3þ��¾�Átý �����������kúL¾cÇ3»rÃFÀ�¾�Õ�Ô�¿�¾���Ö�Ô�ú�É8¼}Õ+û*ü3¾�Ã.¾�»�Õ�¿�Ã#Ö�Á�¼ÂÄ%»�À�Õ�Ø�¾�ö¾JÇ3»DÊ�ɾ�ÁAý�������������úõ¾cÇ.ÖD¿�¼Â½'Ç'Á@¾�û

ü3¾�ÃU¾�»DÕ�¿�Ã�ÖDÁ�¼ÂÄ%»QÀ�Õ,Ø�¾�ö¾\À@¾�Õ�Ô�¿�¾�Ç'×%Ç3¼8»DÀ�Á.Õ�¼ÂÖ"Ø�¾�¿�Á@¾��)Á�û*Ä%»DÉÂÃ��FÇ'Á@Á�Ç3Õ�ü)À�þ���¾ �W¼}É8É5Ç3À�À�ÔDö¾FÁ�Ø�Ç'Á.Á�ØD¾ÖD¿�¼Â½'Ç'Á@¾�û*ü3¾�ÃÈÀ�Õ,Ø�¾�ö¾�Ô�À@¾�ÀWü3¾�Ã)À1Ä3Å�ɾ�»�×3Á�Ø"!NÇ3»DÊOÕ�Ç3»È¾�»DÕ�¿�Ã#ÖDÁcö¾�À�À�Ç'×3¾�ÀcÄ3Å�ɾ�»D×3Á�Ø$#:þhý���¼ÂÁ�Ø�Ä%ÔDÁÉÂÄ%À�À¶Ä3Å�×3¾�»�¾�¿,Ç3É8¼ÂÁ�Ã3ÿ%�¶¾AÇ3À�À�ÔDö`¾1Á�ØDÇ'Á�Á�Ø�¾JÖ�ÔDú�É8¼8Õ+û*ü3¾�þ�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»�À�Õ,Ø�¾�ö¾tÕ�Ç3»�úL¾1Ô�À@¾�Ê�Á@ľ�»DÕ�¿�Ã�ÖDÁö¾�À�À�Ç'×3¾�À�Ä3ŶÇ'¿�ú�¼ÂÁ@¿�Ç'¿�ÃÍɾ�»�×3Á�Ø Ñ �k¾FÀ�Ç&� É}Ç3À@ÁAÁ�¼8ö¾`Ø�Ä'�÷Á�ØD¼}ÀAÕ�Ç3»·úL¾`Ê�Ä%»�¾3þ��(��¾�Õ�Ä%»DÀ@Á@¿�ÔDÕ�Á�Ç»�¾��YÖ�Ô�ú"É8¼8Õ+û*ü3¾�ÃF¾�»�Õ�¿�Ã#Ö�Á�¼ÂÄ%»\À�Õ�ØD¾�ö¾�ý �)� �����*� � ���+� �,�¶Å Ä3¿Wö¾�À�À�Ç'×3¾�ÀWÄ3Åɾ�»�×3Á�Ø$#BÇ3À:Å Ä%É8ÉÂÄ'�WÀ�ø- ÛA¾�Ã�×3¾�»�¾�¿�Ç'Á�¼ÂÄ%»ãÇ3É8×3Ä3¿�¼ÂÁ�ØDö.� � � ×3¾�»�¾�¿�Ç'Á@¾�À`Ö�Ô�ú�É8¼}Õü3¾�Ã$/0!�Ç3»�ÊQÀ�¾�Õ�¿�¾�Áü3¾�Ã(12!QÔDÀ�¼8»�×Oü3¾�Ã×3¾�»�¾�¿,Ç'Á�¼ÂÄ%»NÇ3ÉÂ×3Ä3¿�¼8Á�ØDö3�Qþ

-54 ľ�»DÕ�¿�Ã�ÖDÁ:ö¾�À�À�Ç'×3¾+6 ÔDÀ�¼8»�×mÖ�ÔDú�É8¼8Õ1ü3¾�Ã+/7!õÿ�Ç3ÉÂ×3Ä3¿�¼8Á�ØDö8� � � Ê�Ä#¾�À:Á�Ø�¾AÅ�Ä%É}ÉÂÄ��W¼}»�×�ø9 þ:12! �%;=<�> � 92?�@A þCB ; �2D @ ýE12! � �F þCB �%; � �G @IH ýE6J�

-54 Ä`Ê�¾�Õ�¿�Ã�ÖDÁ:Õ�¼ÂÖ�ØD¾�¿�Á@¾���Á.ýKB:�IB � �,ÿDÇ3ÉÂ×3Ä3¿�¼8Á�ØDöL� � � Ê�Ä�¾�À�Á�Ø�¾tÅ Ä%É8ÉÂÄ'�W¼8»�×�ø9 þ1ºcÀ�¼8»�×À@¾�Õ�¿�¾�Á:ü3¾�Ã$1'!õÿDÕ�Ä%ö`Ö�Ô�Á@¾�12! �%M � G @ ýKBN�A þ1ºcÀ�¼8»�×O12!P��ÿ"Õ�Ä%öÖ�Ô�Á@¾N6 M �+�G @ H ýKBC�Q�

Ò Át¼8ÀJ¾�Ç3À�ÃÍÁ@ÄȽ3¾�¿�¼ÂÅ ÃÈÁ�ØDÇ'ÁtÊ�¾�Õ�¿�Ã#ÖDÁ�¼8Ä%»�¼8ÀAÇ3É,��Ç�Ã)ÀtÖL¾�¿�Å Ä3¿�ö¾�Ê Õ�Ä3¿�¿�¾�Õ�Á�ÉÂÃXýfÇ3À�À�ÔDö`¼8»�×�Ê�¾�Õ�¿�Ã�ÖDÁ�¼ÂÄ%»�¼}ÀÇ3É,�¶Ç�Ã�ÀtÖõ¾�¿�Å�Ä3¿�ö`¾�ÊÍÕ�Ä3¿�¿�¾�Õ�Á�ÉÂÃO¼}»OÁ�ØD¾.Ä3¿,¼Â×%¼8»DÇ3ÉÀ�Õ,Ø�¾�ö¾�ÀI�,þ���¾F»�Ä�� À�Ø�Ä�� Á�ØDÇ'ÁAÁ�Ø�¼8ÀJÕ�Ä%»DÀ@Á@¿�ÔDÕ�Á�¼8Ä%»¼8À:À@¾�Õ�Ô�¿�¾3þR�SUTWVYX'T�Z []\_^ ý �$���`�����ba�cbdfe7gih�j�aEk�lnmpo�qro�sYk&tIqIe7u aEv�swc�k�x%o�yzo${|x}a k�x(a_cNý�~I�����Ilc&o�k�g�t�o$d��}d�a�s%c�uk�a�e�x�o�t�uo��}u l�vpsij�q�dpuEuKd�k�m�c�d�s7�.ý�� � ��� � ��a_c�d�e7tIa��2dpuo�lm�o�q+o�sYk�t�qIe7u aEv�sfc�k�x%o�yzo�{|x}aEk�x�a�cWý�~���� � �Ilnc�o�k&g�t�od&�}d�a�s%c�u�k&a�eYx�o�t�uo��}u l�v�s�j�q dpuEuKd�k�m'c���u�x�o�s�u�x�o�x}q}h�t�aE�Oc�k�x�o�yzomý � � � ��� � � ��� � � �NkIv�s%c�u tIgik&uo���d}h�vp��o�a_c�deYg�h�j�a k&lm�o�qfo�sYk�t�qIe7u a vps"c�k�x%o�yzo�{|x}aEk�xba_ctý�~�� A �|�����Q�Ilc&o�k&gPt�o�d&�Wd�a_sic�u�k�a�e�x�o�t�uo���u l�v�sij�q dpuEuKd}k�m�c���E�����K��� ����� ¡�¡K¢�£��¤¡K¥���¡�¦¨§�©¤ª2«'¬2��¢�­¯®�°��¯±��¯²2­¯©E±�ª'¡K¢�§I²³�E­¥2��£��¯��´Y�E�¯­�«'©K¢µ¡�±z��¶���¢�²'� ¡�­�¢�ª2¥2�n©E¡K�¯·&¡E®_§I²2�µ±���¡E¡�I­°&��¢����¸&«'¢�¹������¯²&¡Y¡K§º�E�¯­�«'©K¢µ¡�±�¢�²�¡K¥'�U�E��²'�E��§�¦P¢�²�»�¢�� ¡K¢�²'¶�«'¢��E¥2�I¬'¢���¢µ¡�±�´��E§`­��I����¢�²2¶¼�`ª'«2¬2��¢�­¯®�°��¯±¼��²2­n©E±'ª�¡K¢�§�²��E­¥2��£���½��E��­¯«'©K��¾¢��*��²2§I«2¶I¥W¿�À2§�©*¡K¥'�|­�§I²2­¯©K�¯¡K�Á�E�¯­�«'©K¢µ¡�±:�I²2�I�µ±��E¢��*¢�²Â0¥'��§I©K�¯£ÄÃI´'�7�|��¢����W¬p�Á£�§I©K�|­���©K�¯¦�«2�P¢�²�§�«'©�¡K�¯©K£�¢�²'§���§�¶�±�¿Å À2§I©�ª�©K¢�¹I��¡K�n®_°��¯±¤��²2­n©E±'ª�¡K¢�§�²:�E­¥2�¯£�����´'�7�Á²2����»�¡K§�£�§�©K�|­���©K�n¦¨«2���µ±C�Eªp�¯­�¢µ¦�±�¡K¥2�|²'��­����E�K��©E±:¡�±�ªp�|§�¦%�E��­�«�©K¢µ¡�±�¿

9

Page 98: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Æk¾�Å Ä3¿�¾m×%¼Â½)¼8»�×�Á�Ø�¾�Ö�¿�Ä#Ä3Å ÿ7�k¾m»�Ä3Á@¾mÁ�Ø�¾.Å�Ä%É8É8Ä��W¼8»D×F¼8ö`ö¾�Ê�¼8Ç'Á@¾�Õ�Ä3¿�Ä%É}É8Ç'¿�Ã�ýfö`Ç'ü3¾À�ÔD¿�¾hÃ3Ä%Ô ÔD»DÊD¾�¿@ûÀ@Á�Ç3»DÊ��WØ#Ã�Á�ØD¼8À�Õ�Ä3¿�Ä%É8É}Ç'¿�Ã�Å Ä%É8ÉÂÄ'�WÀI�,øÇ V�X2VYÈ È�ÉiX'ÊÄ[ÌË sY�}o�t¤u�x�o¤c�d�yzo:d2c�c�g�y�e0u a v�sic:d2c�a_s u�x�oÁeYt�o��'a v�gPc�u�x�o�v�t�o�y+��u�x�o¤x}q}h�tIaE�¤eYg�h�j�a k&lm�o�qo�sYk&tIqIe7u aEv�s�c�k�x%o�yzotý �)� � ����� �����+� �,�Ca�cAý�~I��Í%ý A �Y���������Ilc&o�k�g�t�oa�s$u�x�oc�o�s%c�oNv ^ j,o ^ u l�v�t�lnt�a¨�pxWuÁa�sY��a_c�u a�sil�pg�a�cIxPd}h�a_j�a�u q2�`{|x%o�t�o�d�s�d����po�tIc�d�t�q y�dpqfÎ�g�o�tIqfu�x�o+Ï�ÐÑv�t�d�k�jQo�dpuºyOv2c�uUÍzu a�yOo�c��4 Ø�¾1Õ�Ä3¿�Ä%É8É8Ç'¿�Ã`¼8ÀkÇ�É}¼ÂÁ@Á�ɾWÀ�ÔD¿�ÖD¿�¼8À�¼8»�×cúõ¾�Õ�Ç3Ô�À@¾��¶¾1Ä3ú�Á�Ç3¼8»�ÇhÖ�Ô�ú�É}¼8Õ+û*ü3¾�Ãm¾�»DÕ�¿�Ã�ÖDÁ�¼ÂÄ%»FÀ�Õ�Ø�¾�ö¾JÀ@¾�Õ�Ô�¿�¾¼8»Á�Ø�¾cÀ@¾�»DÀ@¾:Ä3Å�¼8»�ÊD¼8À@Á�¼8»D×%ÔD¼8À�ØDÇ'ú"¼8É8¼�Á�ÃJ¾�½3¾�»�Á�Ø�Ä%Ô�×%Øz�k¾cÀ@Á�Ç'¿�Á@¾�ʳ�W¼ÂÁ�Ø`ÇAÖ�¿�¼Â½'Ç'Á@¾�û*ü3¾�Ã`À�Õ�Ø�¾�ö`¾1À@¾�Õ�Ô�¿�¾Ä%»DÉÂ÷Ç'×%Ç3¼8»DÀ�ÁmÕ�¼8Ö�Ø�¾�¿�Á@¾��)Á�û*Ä%»DÉÂ÷Ç'Á@Á�Ç3Õ�ü�ÀÈý��1ØD¼8Õ�Ø�ÿùÅ Ä3¿mÖD¿�¼Â½'Ç'Á@¾�û*ü3¾�ÃU¾�»�Õ�¿�Ã#Ö�Á�¼ÂÄ%»�ÿ�¼8À.»�Ä3Á�¾&Ò�ÔD¼Â½'Ç3ɾ�»#ÁÁ@Ä�À@¾�Õ�Ô�¿,¼ÂÁ�Ãã¼8»ãÁ�ØD¾OÀ@¾�»DÀ�¾NÄ3Åt¼8»DÊD¼}À@Á�¼8»�×%ÔD¼}À�ØDÇ'ú�¼8ɼ�Á�Ã%�IÓ 4 ØD¾N¿�¾�Ç3À@Ä%»BÅ�Ä3¿�Á�ØD¼8ÀF¼}ÀÁ�ØDÇ'ÁFÁ�ØD¾Nü3¾�ÃãÅ�Ä3¿Á�Ø�¾tÖD¿�¼8½(Ç'Á@¾�û*ü3¾�ÃNÀ�Õ,Ø�¾�ö¾.¼8À:Õ�ØDÄ%À@¾�»È»�¾��1ÉÂÃ�Ç'ÁW¿�Ç3»DÊ�Ä%ö ¾�½3¾�¿�Ã\Á�¼8ö¾hÇ`»D¾�� ö¾�À�À�Ç'×3¾.¼8À�¾�»DÕ�¿�Ã#ÖDÁ@¾�Ê�þ4 Ø#Ô�À�ÿ�Ç3»\Ç3Ê�½3¾�¿�À�Ç'¿�Ã�Õ�Ç3»D»�Ä3ÁOÔ�Å Ä3¿�Õ�¾&ÕÁ�Ø�¾AÀ�Õ,Ø�¾�ö¾AÁ@ľ�»DÕ�¿�Ã#ÖDÁ�öÄ3¿�¾AÁ�Ø�Ç3»�Ä%»D¾tö¾�À�À�Ç'×3¾�W¼8Á�Ø�Ç3»#ÃÖ�Ç'¿�Á�¼8Õ�ÔDÉ}Ç'¿.À@¾�Õ�¿�¾�Áü3¾�Ã512! � ý ¾�½3¾�»ãÁ�ØDÄ%Ô�×%ØQÁ�Ø�¾\Ç3Ê�½3¾�¿�À�Ç'¿�ÃQØDÇ3ÀÇ3Õ�Õ�¾�À�ÀÁ@Ä Á�Ø�¾�Ï�Ð0Ä3¿�Ç3Õ�ɾ�Å�Ä3¿Á�ØD¾Ö�Ô�ú�É8¼}Õ+û*ü3¾�Ã�À�Õ,Ø�¾�ö¾'�,þÖ X'V×VYØ ý Ä3Å 4 ØD¾�Ä3¿�¾�ö 9 � ÙDÄ3¿hÇ3»rÃ�Ç3ÉÂ×3Ä3¿,¼ÂÁ�ØDöÛÚ Ç3»DÊ·ö¾�À�À�Ç'×3¾�À�ÜbÝ2��Ü � ÿ��¶¾FÇ'¿�¾`¼8»#Á@¾�¿�¾�À@Á@¾�Ê�¼8»úõÄ%Ô�»DÊD¼8»�×�ø ÞÞ Ù5¿�ß Ú.ý�/0!7��� � �D @ ý�ÜbÝ���� M 9�à7á Ù�¿�ß Úmý�/7!7��� � �D @ ý�Ü � ��� M 9�à

ÞÞ�âý 4 Ø�¾mÖD¿�Ä3ú�Ç'ú"¼8É8¼ÂÁ�¼Â¾�À1Ç'úLÄ�½3¾`Ç'¿�¾mÁ�Ç'ü3¾�»�Äw½3¾�¿t¿�Ç3»DÊ�Ä%ö Õ,Ø�Ä%¼8Õ�¾Ä3Å5Ö"Ô�ú�É8¼8Õhü3¾�ÃO/7!OÄ%Ô�Á@Ö"Ô�ÁJú#Ã$� � � ß ¼}»Ç3ÊDÊD¼ÂÁ�¼ÂÄ%»NÁ@ÄFÁ�Ø�¾�¿�Ç3»DÊ�Ä%ö`»D¾�À�À�Å Ä3¿�Ú ¼ÂÁ�À@¾�É8Å à ÿDú�Ô�Á�Å Ä3¿1ú�¿�¾�½�¼8Á�Ãf�k¾�Ä%ö`¼ÂÁ:Á�ØD¼}À:Å ¿�Ä%ö Á�Ø�¾�Ê�¾�À�Õ�¿�¼ÂÖ�Á�¼ÂÄ%»Ä3ÅùÁ�ØD¾A¾���ÖL¾�¿�¼8ö`¾�»rÁ�þ��mºcÀ�¼8»D×.Á�ØD¾hÊ�¾�ã�»D¼8Á�¼ÂÄ%»�Ä3ÅU��� �fÿ�Á�ØD¼8À:¾���ÖD¿�¾�À�À�¼ÂÄ%»\¼8À�¾&Ò�ÔD¼Â½'Ç3ɾ�»#Á�Á@Ä�øÞÞÞ Ù�¿'ßµ12! � ;=<�> � 92? @ ø�Úmý�/7!7���2D @ ýE12! � �I��� �G @�H ý�ÜbÝ���� M 9�à

á Ù�¿�ßµ12! � ;ä<�> � 92? @ ø}Úmý�/7!7��Ú.ý�/0!7���2D @ ýE12! � �I��� �G @IH ý�Ü � ��� M 9�àÞÞÞ â ý 9 �

å À5Ç3»Ç'Á@Á@¾�öÖDÁ5Ç'Á�ÇJÖD¿�Ä#Ä3Å ÿ%¼ÂÁ�À@¾�¾�ö`À�»DÇ'Á�ÔD¿�Ç3É)Á@ÄAÁ@¿�ÃhÁ@ÄtÔ�À@¾�Ç3ÉÂ×3Ä3¿�¼ÂÁ�ØDöæÚ�ÿrÇ'úLÄ�½3¾3ÿ#Á@ÄtÕ�Ä%»DÀ@Á@¿�Ô�Õ�ÁÇ3»ãÇ3ÉÂ×3Ä3¿�¼ÂÁ�ØDö.Ú��5úD¿�¾�Ç'ü�¼8»D×OÁ�Ø�¾�ÖD¿�¼8½(Ç'Á@¾�û*ü3¾�ÃQ¾�»DÕ�¿�Ã�ÖDÁ�¼ÂÄ%»QÀ�Õ�ØD¾�ö¾Uý���� �����Q�,þr��¾�Á�ç ÀÀ@¾�¾NØ�Ä��èÁ�ØD¼}Àö`¼Â×%Ø#Á:�¶Ä3¿�üõþ���¾`Õ�Ä%»DÀ@Á@¿,ÔDÕ�ÁÚ � ý��WØ�Ä\¼8À1×%¼Â½3¾�»�Ç�Õ�¼ÂÖ�Ø�¾�¿�Á@¾��)ÁNB � �1ØD¼8Õ�Ø ¼8À1¾�¼ÂÁ�Ø�¾�¿tÇ3»Í¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»Ä3Å�Ü Ý Ä3¿�Ü � ��Ç3À�Å�Ä%É}ÉÂÄ��WÀ�ø

Ú � ýKB � �ý�/7!7��1'!i� ; �è1'! � ;ä<�> � 92? @B ; ��D @ ý è12! � �ôWÔD»bÚmý�/7!7�IB:�IB����Ç3»DÊ\Ä%Ô�Á@Ö"Ô�Á��WØDÇ'Á@¾�½3¾�¿:Ú Ä%Ô�Á@Ö"Ô�Á�À

å Á`ãD¿�À�Áù×%É}Ç3»DÕ�¾3ÿ#Á�Ø�¼8À�À@¾�¾�ö`ÀkÉ8¼Âü3¾�ÇAÖL¾�¿�Å ¾�Õ�Á5À�¼8ö.ÔDÉ8Ç'Á�¼ÂÄ%».Ä3ÅLÁ�Ø�¾:½)¼Â¾��ã¾��)ÖL¾�Õ�Á@¾�Ê`ú�Ã�Ú�þÁ��¾�Á�ç ÀkÇ3»DÇ3ÉÂÃ�Ü�¾Á�Ø�¾hÀ�ÔDÕ�Õ�¾�À�À�ÖD¿�Ä3ú�Ç'ú"¼8É8¼ÂÁ�Ã`Ä3ÅÁÚ � öÄ3¿�¾tÕ�Ç'¿�¾�Å ÔDÉ}ÉÂÃ3ÿDØ�Ä��¶¾�½3¾�¿�øÞÞ Ù�¿2ß Ú � ý�� �G @ H ý�Ü Ý ��� M 9�à7á Ù�¿�ß Ú � ý�� �G @ H ý�Ü � ��� M 9�à

ÞÞM

ÞÞÞ Ù5¿�ßµ1'! � � è12! � ;ä<�> � 92? @ ø�Ú.ý�/0!7���2D @ ý è12! � �I��� �G @ H ý�Ü Ý ��� M 9�àá Ù�¿�ßµ12! � � è12! � ;ä<�> � 92? @ ø�Úmý�/7!7���2D @ ý è12! � �I��� �G @�H ý�Ü � ��� M 9�à

ÞÞÞéMÞÞÞ Ù5¿�ßµ1'! � ;ä<�> � 92? @ øWÚ.ý�/7!7����D @ ýE1'! � �I��� �G @�H ý�ÜbÝ&��� M 9�àá Ù�¿�ßµ12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ýE12! � �I��� �G @ H ý�Ü � ��� M 9�à

ÞÞÞ �

A

Page 99: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�WØ�¾�¿�¾.Á�Ø�¾�À@¾¾��)ÖD¿�¾�À�À�¼ÂÄ%»DÀcÇ'¿�¾ÊD¼¨êL¾�¿�¾�»rÁcúL¾�Õ�Ç3ÔDÀ@¾3ÿ�¾���Õ�¾�ÖDÁC�W¼ÂÁ�ØÍÖD¿�Ä3ú"Ç'ú�¼8É8¼ÂÁ�à AWë�@ ÿ×�¶¾ØDÇ�½3¾z12! � éMè1'! � þC�7Ø#ÃOÇ'¿�¾mÁ�Ø�¾�À�¾.ÊD¼,êõ¾�¿�¾�»#ÁJ¼8»ÈÄ%ÔD¿1¾��)ÖL¾�¿�¼8ö¾�»#Ác¼}»r½3Ä%ɽ)¼8»�׳Ú���ì5í1Ä3Á@¾Á�ØDÇ'ÁÚ��¼��v�o�c�sYv�u�m'sYvp{Á�Ø�¾1ü3¾�Ã�ÔD»DÊ�¾�¿��1ØD¼8Õ�Ø B � ��Ç3À¶¾�»DÕ�¿�Ã�ÖDÁ@¾�ÊOýf¼8»�Ê�¾�¾�Ê�ÿ#¼8ÁkÕ�Ç3»D»�Ä3Á¶ü)»�Ä��7Á�ØD¼8À5ü3¾�ÃFÀ�¼8»DÕ�¾c¼ÂÁ5¼8À¶Ç'Á@Á�Ç3Õ�ü)¼8»�×ÇmÖD¿�¼Â½'Ç'Á@¾�û*ü3¾�ÃF¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»�À�Õ�Ø�¾�ö¾'�,þ 4 Ø�ÔDÀ�ÿ�Á�Ø�¾�¿�¾A¼8À�»�Ä��¶Ç�Ã�Á@Ä�Ô�ãi��Õ.Ä%Ô�¿:Ê�¾�ã"»D¼ÂÁ�¼ÂÄ%»FÄ3Å�Ú��"Ç'úLÄ�½3¾Ñ Á�Ø�¾�¿�¾1¼8ÀkÀ�¼8öÖ�É8Ã�»�Ä��¶Ç�Ã`Á�ØDÇ'ÁºÚ � Õ�Ç3»�Õ�Ø�Ä�Ä%À@¾cü3¾�ó1'! � À�ÄhÁ�ØDÇ'Á¶¼ÂÁ��W¼8É}É)úõ¾1¾&Ò#Ô�Ç3ÉDÁ@ÄhÁ�ØD¾Wü3¾�ÃFÔDÀ�¾�ÊÁ@Ä`¾�»DÕ�¿�Ã�ÖDÁ�B � þ 4 ØD¾A¾���ÖD¿�¾�À�À�¼ÂÄ%»DÀ�Ç'¿�¾t»�Ä3Á:¾&Ò�ÔDÇ3É�ÿDÇ3»DÊ��¶¾AÁ�Ø�¾�¿�¾�Å Ä3¿�¾h»D¾�¾�Ê\Á@ÄO�¶Ä3¿�ü\ØDÇ'¿�Ê�¾�¿�Á@Ä`×3¾�ÁÇÖD¿�Ä�Ä3Å Ä3ÅÀ@¾�Õ�Ô�¿,¼ÂÁ�Ã3þ

Ò »�À@Á@¾�Ç3Ê�ÿ��k¾$�W¼8É8É�Õ�Ä%»DÀ�Á@¿�ÔDÕ�ÁFÇ·À�¾&Ò#Ô�¾�»�Õ�¾NÄ3ÅAÇ3ÉÂ×3Ä3¿�¼ÂÁ�ØDöFÀ`Ç3»DÊãúLÄ%ÔD»DÊ ¾�Ç3Õ,ØBÄ3ÅJÁ�Ø�¾�¼8¿FÀ�ÔDÕ�Õ�¾�À�ÀÖD¿�Ä3ú�Ç'ú�¼}É8¼ÂÁ�¼Â¾�À�þ×��¾�ãD¿,À@Á Õ�Ä%»DÀ�Á@¿�ÔDÕ�ÁÇ3»�Ú��rÇ'Á@Á�Ç3Õ�ü)¼8»�×AÁ�ØD¾ÁeYg�h�j�aEk�lnmpo�qJ¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»�À�Õ,Ø�¾�ö¾Aý �$���`�����,þ��¾`Ê�¾�ã�»�¾+Ú � ý��WØ�Ä�¼8ÀW×%¼8½3¾�»ÍÇ�Õ�¼ÂÖ"Ø�¾�¿�Á@¾��)ÁCBî�WØD¼}Õ�Øͼ8ÀW¾�¼8Á�Ø�¾�¿cÇ3»Í¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»ÈÄ3ź12! � Ä3¿ è12! � ß Å�Ä3¿¿�Ç3»DÊ�Ä%öÌ12! � � è1'! � à ��Ç3À:Å Ä%É8ÉÂÄ'�WÀ�ø

Ú � ý�/7!7�IB��BC� ; ���G @�H ý�ÜbÝ��ôWÔD»bÚmý�/7!7�IB:�IB � ��Ç3»DÊ\Ä%Ô�Á@Ö"Ô�Á��WØDÇ'Á@¾�½3¾�¿:Ú Ä%Ô�Á@Ö"Ô�Á�À��¾�Á�ç ÀWÇ3»�Ç3ÉÂÃ#Ü�¾hÁ�Ø�¾hÀ�ÔDÕ�Õ�¾�À�À�ÖD¿�Ä3ú�Ç'ú"¼8É8¼ÂÁ�Ã`Ä3ÅÁÚ � øÞÞÞ Ù5¿'ß Ú � ý�/7!7����D @ ýE1'! � ��� M 9�àYá Ù�¿�ß Ú � ý�/7!7����D @ ý è1'! � ��� M 9�à

ÞÞÞM

ÞÞÞ Ù�¿�ßµ12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ýE12! � �I��� �G @�H ý�ÜbÝ���� M 9�àá Ù�¿�ßµ12! � � è12! � ;ä<�> � 92? @ ø�Úmý�/7!7����D @ ý è12! � �I��� �G @�H ý�ÜbÝ���� M 9�à

ÞÞÞYï ��� ý A ��WØ�¾�¿�¾AÁ�Ø�¾Nã�»DÇ3É�¼8»�¾&Ò�ÔDÇ3É8¼8Á�Ã�ØDÄ%É8ÊDÀ¶úL¾�Õ�Ç3ÔDÀ@¾��¶¾hÇ3À�À�Ô�ö¾AÁ�ØDÇ'Á�ý ���������O��¼8Àtý�~����I��û�À@¾�Õ�ÔD¿�¾3þíW¾��)Á�ÿ��¶¾`Õ�Ä%»DÀ@Á@¿�Ô�Õ�ÁJÇ3É8×3Ä3¿�¼ÂÁ�ØDöðÚ � Ç'Á@Á�Ç3Õ�ü�¼}»�×\Á�Ø�¾ÖD¿�¼8½(Ç'Á@¾�û*ü3¾�ÃÍÀ�Õ�Ø�¾�ö`¾Èý�� � ��� � �,þN��¾`Ê�¾�ã"»�¾Ú � ý��WØ�¼8Õ�ØN¼}À�×%¼Â½3¾�»NÇ`Õ�¼ÂÖ"Ø�¾�¿�Á@¾��)Á�BC�õÁ�ØDÇ'ÁW¼8À�¾�¼8Á�Ø�¾�¿WÇ3»\¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»�Ä3Å�ÜbÝcÄ3¿�Ü � ��Ç3À�Å�Ä%É}ÉÂÄ��WÀ�ø

Ú � ýKBC�Q�ý�/7!7��1'!i� ; �è1'! � ;ä<�> � 92? @B ; ��D @ ý è12! � �ôWÔD»bÚmý�/7!7�IB:�IB � ��Ç3»DÊ\Ä%Ô�Á@Ö"Ô�Á��WØDÇ'Á@¾�½3¾�¿:Ú Ä%Ô�Á@Ö"Ô�Á�À

4 ØD¼8À�Ç3ÉÂ×3Ä3¿,¼ÂÁ�ØDö ¼8À�¾���Ç3Õ�Á�ÉÂÃ�Á�Ø�¾hÀ�Ç3ö¾hÇ3ÀWÇ3É8×3Ä3¿�¼ÂÁ�ØDöLÚ��fÿ�Ç'úLÄ�½3¾3þ å À:úL¾�Å Ä3¿�¾3ÿ�Á�Ø�¾�»�ÿ�Á�Ø�¾tÇ3Ê�½'Ç3»rÁ�Ç'×3¾Ä3Å|Ú � ¼8À ÞÞÞ Ù5¿'ßµ12! � � è12! � ;ä<�> � 92? @ ø}Úmý�/7!7���2D @ ý è12! � �I��� �G @IH ý�ÜbÝ&��� M 9�à

á Ù5¿�ßµ1'! � � è1'! � ;ä<�> � 92? @ øWÚ.ý�/7!7����D @ ý è1'! � �I��� �G @�H ý�Ü � ��� M 9�àÞÞÞ�ï � � � ý F �

�WØ�¾�¿�¾AÁ�Ø�¾ã"»DÇ3É�¼8»�¾&Ò�ÔDÇ3É8¼ÂÁ�Ã�¼8À�»DÄ�� ÊDÔD¾AÁ@Ä`Á�Ø�¾tÀ@¾�Õ�Ô�¿,¼ÂÁ�Ã�Ä3Å:ý�� � ��� � �¶Ç3ÀWÇÖ�¿�¼Â½'Ç'Á@¾�û*ü3¾�Ã�¾�»DÕ�¿�Ã#ÖDÁ�¼8Ä%»À�Õ�Ø�¾�ö`¾3þÙ¼8»DÇ3É8ÉÂÃ3ÿ'�k¾kÕ�Ä%»DÀ@Á@¿�ÔDÕ�Á�Ç3É8×3Ä3¿�¼ÂÁ�ØDöÄÚ:ñ�Ç'Á@Á�Ç3Õ�ü)¼8»�×1Á�Ø�¾�Ö�ÔDú�É8¼8Õ+û*ü3¾�ÃcÀ�Õ�Ø�¾�ö`¾1ý �$���`������Ç3»DÊAÊ�¾�ã�»D¾�Ê

Ç3À�Å Ä%É8ÉÂÄ'�WÀ�ý Ú ñ ¼}Àh»�Ä'� ×%¼Â½3¾�» ÇOÕ�¼ÂÖ�Ø�¾�¿�Á@¾��)Á�B Á�ØDÇ'Á.¼8À�¾�¼ÂÁ�Ø�¾�¿�Ç3»�¾�»DÕ�¿�Ã�ÖDÁ�¼ÂÄ%»UÄ3Å�12!P��Ä3¿ è12! � ß Å�Ä3¿¿�Ç3»DÊ�Ä%öÌ12! � � è1'! � à �,ø

F

Page 100: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Ú ñ ý�/7!7�IB��B �i; � �G @ H ý�Ü � �ôWÔD»bÚmý�/7!7�IB:�IB����Ç3»DÊ\Ä%Ô�Á@Ö"Ô�Á��WØDÇ'Á@¾�½3¾�¿:Ú Ä%Ô�Á@Ö"Ô�Á�À

å »DÇ3É8Ã#Ü�¼8»D×�Á�ØD¼}Àc¼}»OÁ�ØD¾mÀ�Ç3ö¾��¶Ç3ÀAÇ3À�¶¾ÊD¼8ÊOÅ�Ä3¿�Ú � ý��WØD¼8Õ,Øͼ}À1Á�ØD¾mÀ�Ç3ö¾`Ç3ÀÚ ñ ¾���Õ�¾�ÖDÁAÅ�Ä3¿AÄ%»D¾ü3¾�Ã�Ê�¼¨êõ¾�¿�¾�»DÕ�¾'�,ÿi�¶¾tÄ3úDÁ�Ç3¼8»ÞÞÞ Ù5¿'ß Ú ñ ý�/7!7����D @ ýE1'! � ��� M 9�àYá Ù�¿�ß Ú ñ ý�/7!7����D @ ý è1'! � ��� M 9�àÞÞÞ

MÞÞÞ Ù�¿�ßµ12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ýE12! � �I��� �G @ H ý�Ü � ��� M 9�àá Ù�¿�ßµ12! � � è12! � ;ä<�> � 92? @ ø�Úmý�/7!7����D @ ý è12! � �I��� �G @�H ý�Ü � ��� M 9�à

ÞÞÞYï � â ý Î �å Á Á�Ø�¼8À�ÖLÄ%¼8»#Á�ÿ��¶¾kÀ@Á@¾�Ö�ú�Ç3Õ�ütÇ3»DÊtÁ@¿�ÃJÁ@ÄW¿�¾�Õ�Ç3É8ÉrÄ%Ô�¿�Ä3¿�¼8×%¼8»DÇ3É3×3Ä%Ç3ÉEÓ���¾`��Ç3»#Á@¾�ÊhÁ@ÄcÔ�ÖDÖL¾�¿@û*úLÄ%ÔD»DÊ

¾���ÖD¿�¾�À�À�¼ÂÄ%»·ý 9 �,ÿõÇ3»DÊ��¶¾hÕ�Ç3»N»�Ä'� Ê�Ä`À@Ä�Ç3À�Å�Ä%É}ÉÂÄ��WÀ�ýfÔDÀ�¼8»�×m¾&Ò�ÔDÇ'Á�¼ÂÄ%»DÀ�ý A �Kòõý Î ���,øÞÞÞ Ù5¿�ßµ1'! � ;ä<�> � 92? @ ø}Úmý�/7!7����D @ ýE1'! � �I��� �G @�H ý�ÜbÝ&��� M 9�àá Ù5¿�ßµ1'! � ;ä<�> � 92? @ ø�Úmý�/7!7��Úmý�/7!7����D @ ýE1'! � �I��� �G @ H ý�Ü � ��� M 9�à

ÞÞÞM

ÞÞÞ Ù5¿�ßµ1'! � ;ä<�> � 92? @ ø}Úmý�/7!7����D @ ýE1'! � �I��� �G @�H ý�ÜbÝ&��� M 9�àá Ù5¿�ßµ1'! � � è12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ý è12! � �I��� �G @�H ý�ÜbÝ���� M 9�à� Ù5¿�ßµ1'! � � è12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ý è12! � �I��� �G @ H ý�Ü Ý ��� M 9�àá Ù5¿�ßµ1'! � � è12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ý è12! � �I��� �G @�H ý�Ü � ��� M 9�à� Ù5¿�ßµ1'! � � è12! � ;=<�> � 92? @ ø}Ú.ý�/0!7���2D @ ý è12! � �I��� �G @�H ý�Ü � ��� M 9�àá Ù5¿�ßµ1'! � ;ä<�> � 92? @ ø�Úmý�/7!7����D @ ýE1'! � �I��� �G @�H ý�Ü � ��� M 9�à

ÞÞÞï �|�5� � ��� M A �U��� � �

Õ�Ä%öÖ�ɾ�Á�¼8»D×Á�Ø�¾tÁ�Ø�¾�Ä3¿�¾�öNþ

Î

Page 101: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ó�ôiõQöp÷�ø�ù�õ,únû³üpýÁþ$ÿ�ø�û��Qÿpô��� þ�� ������ �� ô}ú�ø�ü�������ú�õ,ü�ô ú�ü � ø�û��iú�ü��pø�ÿ����}û��ø�üpý�÷&ù�ù�üpø��pü�ôiÿ�ú��iÿpô���ÿ�ú �

!#"%$'&)(+*)" ,.-

/ 021%3%465�798;:=<?>A@CBD7?EF>�G�HI5�JKBMLNEIJKO PQ<RBD<REIST4VUXW�WF1%OYG�HI5�JZBDW

[ �;��ù ý�ÿ�ø]\Cú��%÷�ù�÷]�^�%ø�õ,únûwüpý�ÿ��_�Cú��%÷`���ba��Qõ_�dcfep÷�ûÑ÷&ô���ø�û��iú�õ,ü�ô]ù��6�%÷]g�÷&ùih¼÷`�iÿ�öp÷rù�÷�÷&ôQj õfkµ÷�kl\:ú��i÷m ün�_�bh�ÿpù�ù�÷�ø cnþ�õ_��ÿ��Qõ�÷&ô���ø�û��iú�õ,ü�ô ù��6�%÷]g�÷oa�ÿpù�÷]� ü�ô#p;��ÿ��%ø�ÿ�ú�õ_�$ø�÷&ù�õ_���iü�ù�õ,úû�ÿpô�� ú��%÷oqR� m ÿ�gOÿ��÷&ô���ø�û��iú�õ,ü�ô)ù��r�%÷]gO÷�a�ÿpù�÷]�$ü�ô�ú��%÷DsKs�t ÿpù�ù���g%�iú�õQü�ô�u�ÿ�ø�÷Na�ÿpù�÷]��ü�ô`vxwby�z]{ |�z�ô;��g}a0÷�ø cKú��%÷�üpø�÷�ú�õ_�ÿpù�ù���g%��ú�õ,ü�ôiù^k � ô³ú���õQù~�,÷]��ú��%ø�÷�\�h¼÷��%÷^��ô%÷:ú��%÷'g�üpø�÷��p÷&ô%÷�ø�ÿ��×ôiüpú�õ,ü�ô üpýUÿ+ú�ø�ÿ��I�%üPüpø~�7÷�ø6g��%ú�ÿ�ú�õQü�ôI\��õ,öp÷)ù�üng�÷�÷��%ÿ�g%���Q÷&ùzüpý�j ÿpù�ùr��g�÷]��u�ú�ø�ÿ��I�%üWüpø.�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôiù^\�ÿpô�� ù��%ü�h��iü�h ú�üC��ü�ôiù�ú�ør����ú ÿù�÷]�^�%ø�÷����ba��Qõ_�dcfep÷�û³÷&ô���ø�û��iú�õ,ü�ô�ù��6�%÷]g�÷�ý�ø�üng3ÿpôWû ú�ø�ÿ��I�%üPüpø)�7÷�ø6g��%ú�ÿ�ú�õQü�ôIk

���Wû+�%ü�h¼÷���ÿ�ø�÷$ÿ�a0ün�%úOÿ"ø�÷&ùr���,ú�üpýCú��iõQù�ù�üpø�ú6�Y�%ø�üngäÿ"ú��%÷�üpø�÷�ú�õ_��ÿ����0ü�õQô}ú�üpý:öPõQ÷^h�\ºõ,úOõ�ùõQô}ú�÷�ø�÷&ù�ú�õQôb��ú�ü��r��ÿ�ø�ÿ���ú�÷�ø�õl��÷����ba��Qõ_�dcfep÷�û�÷&ô���ø�û;��ú�õ,ü�ô.a�ÿpù�÷]� ü�ô�h��iÿ�ú)��ø�õ_gOõ,ú�õQöp÷&ù`ÿ�ø�÷�ô%÷]��÷&ù�ù�ÿ�ø�û õQôüpør�%÷�ø+ú�ü���ü�ôiù�ú�ør����úN�iø�ü�ö�ÿ�a��,û;cù�÷]�^�%ø�÷³ù��6�%÷]g�÷&ù^k [ �iõQù}��ÿpô(ÿ��Qù�üo�,÷&ÿ��rú�üV�pø�÷&ÿ�ú�÷�ø���ô��%÷�ø�ù�ú�ÿpô��iõQôb�üpý�����a��Qõ_�dcfep÷�û�÷&ô���ø�û��iú�õ,ü�ôI\Ih���õ_�r��gOÿ�û�û�õ,÷]�_�)üpú��%÷�ø�a0÷&ô%÷^�iú�ù^kD�%ø�üng ÿ�g�üpø�÷%�iø�ÿ���ú�õ���ÿ����7ü�õ�ô}úCüpýöPõ,÷^h}\|ÿV�p÷&ô%÷�ø�õ��zø�÷&ùr���,úNüpý�ú��i÷³ý�üpørg���ú�ø�ÿ��I�%üPüpø��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôiù�õ_g%���Qû)ú��i÷³÷��%õQù�ú�÷&ô���÷füpýT����a��Qõ_�dcep÷�û ÷&ô���ø�û;�iú�õQü�ô(ùr�r�%÷]g�÷&ùr�)ÿ����,ü�h�ù%��ù�ú�ü�õQô�ù�ú�ÿpô}ú�õ�ÿ�ú�÷fú��i÷fú�øIÿ��F�iüWüpøD�0÷�ørgN�%ú�ÿ�ú�õ,ü�ô5õQô(ÿ�ô;��g�a0÷�øüpý��iõ��0÷�ø�÷&ôWú�h�ÿ&û�ù^k � ôwüpú��i÷�ø.h¼üpør�iù]\Th��%÷&ô ù�üng�÷�ü�ôi÷��%÷�öp÷]�Qü���ù ÿ(ôi÷^h�j���ÿpô��iõ_�iÿ�ú�÷�uOú�ø�ÿ��I�%üPüpø�7÷�ø6g��%ú�ÿ�ú�õQü�ôI\�h¼÷%�%ü ô%üpúK��ÿ&öp÷�ú�ü���üng�÷%�b�Vh�õ,ú���j ÿpô����iø�ü�öp÷�ù�÷]�^�%ø�÷�u�ÿ ô%÷^h���ü�ôiù�ú�ør����ú�õQü�ô�üpýºÿ���ba��Qõ��dcfep÷�û�÷&ô���ø�û;�iú�õQü�ôOù��r�i÷]g�÷��Põ�ôiù�ú�÷&ÿ��I\;hº÷�ù�õ_g%���,ûN�iù�÷�ú��%÷Z�p÷&ô%÷�ø�õ_�¤ø�÷&ù����,úÁú�ü�õ_g�g�÷]�iõ�ÿ�ú�÷]�,û���õQöp÷ÿ�ù��r�i÷]g�÷¤ú��iÿ�ú�h¼÷�ú��%÷&ô%e�ô%ü�hÑõQù�ù�÷]�^�iø�÷�k?��üpú�÷¤ú���ÿ�ú�a�ÿpù�õQô��:ø�÷&ù����,ú�ù|ü�ôOÿpùR�p÷&ô%÷�ø�ÿ��iÿpôzÿpù�ù���g%��ú�õ,ü�ôÿpùT�0ü�ù�ù�õla��,÷:ÿ���ù�ü%�iø�üpú�÷]��ú�ùZ�iù¤õQô���ÿpù�÷�\Yý_üpø�÷���ÿ�g%���,÷�\%ý�ÿ���ú�üpøIõQôb�Oú��%øIôiù�ün�%ú¤ú�ü�a0÷C÷&ÿpù�û¡ 

¢I£x¢ ¤K¥n¦F§T¨�©�©ª¥�«N¬�¥­¯®?°�¦¡°n±f©³²�´� ô%ý�üpørgOÿ����,û�\�ø�÷]��ÿ��_�×ú��iÿ�úKµ�õ�ù¤ÿ�ü�ô%÷�cfh�ÿ&û��0÷�ørgN�%ú�ÿ�ú�õ,ü�ô�õ,ýUú��%÷�ý�ün�_�,ü�h�õQôb�%�%ün�_�ª¶

· µ)õQù¤ÿD�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôªk

· µ)õQù�÷^¸¹��õ,÷&ô}ú��Qû.��üng%���%ú�ÿ�a��,÷�k· µ)õQùT��ÿ�ør� ú�üzõQôWöp÷�ø�ú}j�ü�ô$ÿOø�ÿpô��%üng��Qûcx�6�%ü�ù�÷&ôbõQô����%ú6u6k

º¼»�½ ¾ w�¿À]À ½ �7÷�ø6g��%ú�ÿ�ú�õQü�ôÂÁ�h�õ��_��a7÷i�i÷^��ô%÷]� ù�õ�gOõ_�Qÿ�ør�Qû�h�õ,ú���ü�ô%÷$÷�����÷^�iú�õQü�ôI¶$ÿ��,ú��%ün���n�+Á�h�õ_�_�a7÷V�iÿ�ør� ú�ü(õQôWöp÷�ø�ú�{ÄÃÂÅ�y�áy ½Æ¾�Ç \¤ú��i÷�ø�÷�õQùzù�üng�÷ »�½ ¾ w�¿À]À ½ {ÄÃ]È�À ½6É�¾�» {ÊÀ�ÃËhZ�iõ_�r� ÷&ôiÿ�a��,÷&ùz÷^¸���õ,÷&ôWúõQô}öp÷�øIù�õ,ü�ô�üpý�Á�k?̼÷]��ÿ��iù�÷ºüpý%ú��%÷ºÿ�ö2ÿpõ��Qÿ�a�õ_�QõQúû:üpýPú�øIÿ��F�iüWüpø�õQô%ý�üpørgOÿ�ú�õ,ü�ôI\'ün�%ø*ú�ø�÷&ÿ�ú�g�÷&ô}ú�üpý�ú�ø�ÿ��I�%üPüpø�7÷�ø6g��%ú�ÿ�ú�õQü�ôiùTh�õ��_�Ia0÷Nù��Qõ_�n�}ú��,û��iõl�7÷�ø�÷&ôWú¤ý�ø�üng ün�%ø�ú�ø�÷&ÿ�ú�gO÷&ô}úCüpýÁü�ô%÷�cfh�ÿ&ûV�7÷�ø6g��%ú�ÿ�ú�õQü�ôiù^\�ÿpùKh¼÷h�õ_�_�0ù�÷�÷Nõ�ô ú��%÷�ý�ün�_�,ü�h�õQôb�D�%÷^��ôiõ,ú�õQü�ôIk

Í�ÎÏ?Ð9ÑÓÒ�ÑÄÔ¡ÐÖÕ#×ØjÄÙdÚrÛ6uxcKú�ø�ÿ��I�%üPüpø}�0÷�ørgN�%ú�ÿ�ú�õ,ü�ô�{Óv ¾V»�½ {Üw Ç yiÀÆÈ'w�À ÇÞÝ�ßÆ» { É y ¾�Ç ÅÀ ½ { »ÄàÉ v�j�áDÚrµ¡Ú]â�ã�ä;uv6å�z ài»Äà�¾�»fæ

ç

Page 102: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

· áY{�v ¾i½ ¾ Ã�¿À É {lè�y�¿�ény Ý Å;y^áy ½ ¾�» {ÊÀ�à ¾�Ç ÅÀ ½ { »Äà;Éëê�à {Êz à À�å » w�å » v ¾ w ¾ { ½ jíì�Ú îíïbu ê�à y ½ y�îÆï�{�vz ¾�ÇÓÇ y�¿ ¾ ú�øIÿ��F�iüWüpø�ð

· µ¯{�v ¾ ¿y » y ½dÉ {ÄÃ�{Óv » {Êz ¾�Ç ÅÀ ½ { »ÄàÉØê�à {Êz à�»f¾ énydv ¾ v}{ÄÃ�w�å »~¾ é�y Ý ì ¾ Ã�¿ ¾ Ã�{ÄÃ�w�å »³ñoòió�ôiõfê y½ y�È]y ½�» À ó�ô�¾ v »Äà y��%üngzÿpõQô(üpý�ep÷�û�ì6ö�÷N{ »�½ y » å ½ Ã�v ¾ Ã�À�å » w�å »)ø¯ù µ?jíì�Ú ñ u ò#ó�ô ð õ�ú y¿y^Ã�À » y�µ?jÊû¡Ú^üÞuNý Ý µ ô jíüÞu�ðþö

· â�ã�ä »f¾ é�y�v ¾ v�{ÄÃ�w�å »�»Äà y »�½Æ¾ wb¿nÀ]À ½ îÆï ¾ Ã�¿ ¾ Ão{ÓÃ�w�å »³ø�¾ Ã�¿ ½ y » å ½ Ãbv ¾¹ÿ�¾�Ç å�y ñ ð�Få ½d»Äà y ½dÉ À ½ y��)j�áDÚrµ¡Ú]â�ã�ä�u�v ¾�» {�vÄ|�ydv »Äà y�È�À ÇÄÇ À ê {ÓÃ�Å æ

· �³À ½�¾�ÇÄÇ jíì Ú îÆïbuiÀ�å » w�å » ý Ý á ¾ á¿ ¾�ÇÄÇ?ñ�ò ó�ôVê y à�¾�ÿ y æ â�ã]ä�jÄîíï�Úrµ ô j ñ u u ùØñ ð õ �ªÀ ½�»Äà {�v½ y ¾ v^À�Ã�� ê y�v6{ É w ÇÞÝ ¿y^Ã�À » y�â�ã�ä�jÄîíï�Ú^üÞu�ý Ý µ����ô jíüÞu�ý^å » { » {�v�{ É w�À ½�»f¾ à »�» À ½ y É y É ýry ½o»Äà�¾�»y� z^{fy�à »�Ç Ý zrÀ É w�å » {ÄÃ;Å%µ����ô {�v}À�à ÇÞÝ wbÀ vrvd{Êý Ç y ê { »Äà îÆï�ðþö�6Ã%w ¾�½�» {Êz^å Ç�¾�½ � »Äà {�v'{ É w Ç {fydv »Äà�¾n» È^À ½¾�ÇÄÇ ì�À�å » w�å » ý Ý á�� µ ô {Óv ¾ wby ½dÉ å »f¾n» {�À�ÃCÀ�à ó�ô ð

· µÂ{ÓvNy � z]{Êy^à »�Ç Ý zrÀ É w�å »f¾ ý Ç y�ð�� À ½ y�È^À ½dÉ�¾�ÇÓÇ Ý ��Ån{ ÿ y�à ¾ Ã Ý ì.À�å » w�å » ý Ý á ¾ Ã�¿ ¾ à Ý�ñVò�ó�ô �{ » {Óv~w�À v6v6{fý Ç y » À�y � z^{fy�à »�ÇÞÝ zrÀ É w�å » y�µ ô j ñ u�ð

· µ {�v à�¾�½ ¿ » Ài{Óà ÿ y ½�» h�õ,ú��%ün�%ú¤ú��i÷�ú�ø�ÿ��I�%üPüpø õ y ÿ y^à ê�à y�ÃVÅn{ ÿ y�Ã�ìdö�ð�� ¾�É y ÇÞÝ �IÈ�À ½D¾�ÇÄÇ?¾�Ç Å;À ß½ { »Äà;É v�� ½ å�Ã�Ã�{ÄÃ;Å.{Äà » { É yZÙ ê y à�¾�ÿ y æ

��ø��ljíì�Ú îÆï�u���áD� ñ � ó�ô � ø�ù µ ô j ñ u�¶��Djíì�Ú ø u ù�ñ���� Û��õ ×RÅ ¾ {ÓÃ���h�õ,ú���ú��%÷�ú�øIÿ��F�iüWüpø~µ ô {�vKy ¾ v ÝN» À}{Äà ÿ y ½�» � {ÓÃiÀ »Äà y ½Tê À ½ ¿ v!�³µ ���ô {ÓvZy ¾ v Ý�» ÀDzrÀ É w¡å » y"å�v6{ÄÃ;Å ¾�Ç ÅÀ ½ { »ÄàÉ â�ã]ä�ð ö

� ú�õQù�h¼üpø�ú�� ô%üpú�õQô��Nú��iÿ�ú¼÷�öp÷�ø�ûOú�ø�ÿ��I�%üPüpøR�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôfõQùºÿ��Qù�ü�ÿ�ü�ô%÷�cfh�ÿ&û¹�7÷�ø6g��%ú�ÿ�ú�õQü�ôIk���÷K��õQöp÷ÿ�ý_÷^h ÷���ÿ�g%���,÷&ù�üpý|ú�ø�ÿ��I�%üPüpø)�7÷�ø6g��%ú�ÿ�ú�õQü�ôiù�ú��iÿ�úTh¼÷}�iÿ�öp÷�ÿ��,ø�÷&ÿ��iûfù�÷�÷&ôªk#%$'&�(�)�*,+.-�/103254 ��÷N�iÿ&öp÷N�iõQù��^��ù�ù�÷]� ú��%÷76�� º �7÷�ø6g��%ú�ÿ�ú�õQü�ô�a0÷�ý_üpø�÷�\�a��%ú)h¼÷�ô%ü�h ÷������Qõ_��õ,ú��Qû�iø�÷&ù�÷&ôWúzõ,úOõQôÑÿ�ý_üpø6g ��üng%��ÿ�ú�õla��,÷�h�õ,ú��5ú��%÷�ÿ�a7ü'öp÷V�%÷^�Yôiõ,ú�õ,ü�ôIk98*÷�ú�áØa0÷bÿpôÑÿ��_�püpø�õ,ú���g ú���ÿ�ú�r�%üPü�ù�÷&ù+ÿbø�ÿpô��iüng g�ü������_�iù;: jÄh��iõ��r��õQù�ú��%÷��iø�ü�������úüpý�ú hºüV�iø�õ_g�÷&ùdu:ÿpô���ÿ��Qù�üV�r�%üPü�ù�÷&ù3<�Ú>=ù����r�"ú���ÿ�ú7<?= ù ç g�ü��A@�j :�u6k [ �%÷Dep÷�û�ì%��ü�ôiù�õQù�ú�ùCüpýKj :VÚ!<�u�h��iõ_�,÷�ú��%÷�ú�ø�ÿ��I�%üPüpø�îíïbõ�ùù�õ_g%���Qûj :�Ú>=�u6kR��÷���ÿ&öp÷ ó�ô~ù�ó7BDC E�ùGF%HB ÿpô��

· µ ô j ñ u ù µ B�C E j ñ u ù�ñ E g�ü��I:�k

· â�ã�äbjÄîÆï�Ú ø u ù â�ã�äbj j :VÚ>=�u6Ú ø u ù#øKJ g�ü��L: ù µ ���BDC E j ø u6k

#%$'&�(�)�*,+NM /L2PO�Q'&�RPSUT�V ��÷�ÿ��Qù�üfù��%ü�h �%ü�hJú��%÷�6�ÿ�a�õQô$ùrp;�iÿ�øIõQôb���0÷�ørgN�%ú�ÿ�ú�õ,ü�ô)ù�ÿ�ú�õQù ��÷&ù�s:÷�ý�cõQôiõ,ú�õ,ü�ô ç k�8*÷�ú�: ùXW Y hZ�%÷�ø�÷ W Ú Y ÿ�ø�÷N�iø�õ_gO÷�ÿpô�� W�ùZYDù\[ g�ü�� � kZ��÷�ô%üpú�÷]�"õQô)ÿ���ø�÷�öPõQün�iù�,÷]��ú��%ø�÷ú��iÿ�ú�ú��%÷ýÄ��ô���ú�õ,ü�ôiµ B j ñ u ù�ñ5] gOü��L: õQù�ÿD�0÷�ørgN�%ú�ÿ�ú�õ,ü�ô ü'öp÷�ø_^a` B k��Püb\%ün�%ø�ep÷�û.�p÷&ôbc÷�ø�ÿ�ú�õ,ü�ô)ÿ��l�püpø�õ,ú���g á h�õ��_�Fa0÷Nÿpô�ÿ��_�püpø�õ,ú���gÌú��iÿ�úZ�6�%üWü�ù�÷&ù�ÿ.g�ü������_�iùa:¼h���õ_�r��õQù¤ÿ¹�iø�ü�������ú�üpý�iø�õ_g�÷&ù W Ú Y h�õ,ú�� W�ùbY�ùG[ gOü�� � k [ �%÷'ep÷�ûoìCõ�ù�ù�õ_g%���Qû_: \�ú��%÷ú�ø�ÿ��I�%üPüpø¤õQù�ú��%÷�ý ÿ���ú�üpø�õl�&ÿ�ú�õ,ü�ôj W Ú Y u:üpýa: \*ÿpô�� ó�ô}ùXó B ù ^a` B k%��÷¹�iÿ�öp÷zÿ��,ø�÷&ÿ��iû��%÷^�Yô%÷]�"ün�%ø�ý��iô���ú�õ,ü�ô`µ B jíüÞu6��hº÷Oÿ��Qù�üô%üpú�÷³ú���ÿ�ú+õQôWöp÷�ø�ù�õQü�ô ��ÿpô�a7÷.�%ü�ô%÷³÷^¸���õ,÷&ôWú��,û j õÊkµ÷�kl\Áù�p��iÿ�ø�÷zø�üWüpú�ùD��ÿpô�a7÷³÷^¸���õ,÷&ôWú��,û���üng%���%ú�÷]�g�ü������,üI:ou~h��%÷&ô ú��%÷�ý ÿ���ú�üpø�õl�&ÿ�ú�õQü�ôbüpýc: õQùTePô%ü�h�ôIk

d

Page 103: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

¢I£Ue «¹®Dfhgx±ji,k�l�¬�mon%²Di�¥pm?§R°�±x©³²rqí¥©³­ ¤K¥n¦F§T¨�©�©ª¥�«D¬�¥­¯®�°�¦¡°n±f©9²R´� ú�gzÿ&û ù�÷�÷]g ��ü�aPöPõQün�iù��5ú��iÿ�úbú�øIÿ��F�iüWüpø��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôiù�ÿpô��#���ba��Qõ��dcfep÷�ûÑ÷&ô���ø�û��iú�õ,ü�ô]ùr�r�%÷]g�÷&ùÿ�ø�÷ ÷&ù�ù�÷&ôWú�õQÿ��_�,û�÷]p��iõ,ö�ÿ��,÷&ôWú^kZs à {Óv¹{Äà » å�{ » {ÊÀ�Ã2{�v3t_u Ô�Ð�v³  �iüpøOü�ô%÷ ú��iõQôb�b\)áÀ » y ÿ y ½dÝ w�å�ý Ç {�z ß ény Ýy�Ã�z ½6Ý w » {ÊÀ�Ã�v�z à y É y%{Óv�ý ¾ v]y�¿�À�à ¾�»�½Æ¾ wb¿nÀ]À ½ wby ½dÉ å »f¾n» {�À�Ã�k � ô)ý�ÿ���ú^\�ô%÷&õQú��%÷�øCüpý`ú��%÷�ù��6�%÷]g�÷&ù�h¼÷�iÿ&öp÷fù�÷�÷&ô�ù�übý ÿ�øNÿ�ø�÷�a�ÿpù�÷]��ü�ô�ú�øIÿ��F�iüWüpø'�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôiù]¶ú��%÷��iÿ�ør��ô%÷&ù�ùCüpý~�%÷]��õ_�iõQô���p��iÿ��%øIÿ�ú�õ_�ø�÷&ù�õ_���iü�ù�õ,úûN�^�,÷&ÿ�ør�,ûN�%üP÷&ùÁô%üpú?��õ,öp÷¤ÿCú�ø�ÿ��I�%üPüpø?�0÷�ørg��iú�ÿ�ú�õ,ü�ô%�iõ,ø�÷]��ú��Qû � ÿpô���ú��%÷Ts�s�t ÿpù�ù���g%��ú�õ,ü�ôõQùÁÿ��Qù�ü��iôiø�÷]�Qÿ�ú�÷]��ú�üú�ø�ÿ��I�%üPüpø?�0÷�ørg��iú�ÿ�ú�õ,ü�ôiù^k � ô�ý ÿ���ú^\WõQú|õ�ù�a7÷]��õ,÷�öp÷]��ú��iÿ�ú?���ba��Qõ_�dcfep÷�û�÷&ô���ø�û;�iú�õQü�ôõQù ê y ¾ ény ½ ú��iÿpô ú�ø�ÿ��I�%üPüpø)�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôiù^\%õQô ú��i÷�ù�÷&ôiù�÷Cú��iÿ�ú¤ú��i÷ý_üpørgO÷�øTgOÿ&ûf÷���õ�ù�ú�÷�öp÷&ôbú��%ün���n�ú��%÷}�Qÿ�ú�ú�÷�øK�%üOô%üpú^k

�P÷]��ü�ô����Qû�\Uõ,úõQù�ôiüpúú�ø�õ,ö�õQÿ���ú�ü���ü�ôiù�ú�ør����úNÿ.w ½ À ÿ ¾ ý Ç Ý ß v^yrz^å ½ y.���ba���õ_�dcfep÷�ûb÷&ô���ø�û��iú�õ,ü�ôrù��r�%÷]gO÷ý_ø�üng ÿpôWû�ú�ø�ÿ��I�%üPüpø}�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôIk [ �%÷ ��ü�aPöPõ,ün��ù���h�ÿ&ûrüpý��%ü�õQôb��ù�üb\�õQô`hZ�iõ_�r��g�÷&ù�ù�ÿ��p÷Awäõ�ù÷&ô���ø�û��iú�÷]�.aWû���üng����%ú�õQôb�%µ ô jxwVu�j ÿpô����%÷]��ø�û;�iú�õQü�ôzüpýzy õ�ù��iü�ô%÷ZaPû��iÿ�öPõ�ôb��ú��i÷�ø�÷]��÷&õQöp÷�ø hZ�%ü�iÿpùÁú��%÷�ú�ø�ÿ��I�%üPüpø ��üng%���%ú�÷�µ ��� j{y}u u�õ�ù?��üng%���,÷�ú�÷]�,û+õQôiù�÷]�^�iø�÷�¶�ú��iõQùÁõ�ù�ÿ'�i÷�ú�÷�ørgOõQôiõ�ù�ú�õ_�`ùr�r�%÷]g�÷�\ù�ü���ÿpôiô%üpú��7ü�ù�ù�õla��,û%a0÷�ù�÷]�^�iø�÷Nÿpù¤ÿ%���ba��Qõ_�dcfep÷�û³÷&ô���ø�û��iú�õ,ü�ô�ù��6�%÷]g�÷�k

̼÷�ý_üpø�÷Nh¼÷���õ,öp÷+ÿzù�÷]�^�iø�÷N��ü�ô�ù�ú�ør����ú�õ,ü�ô$üpý�ÿ�����a��Qõ_�dcfep÷�û ÷&ô���ø�û��iú�õ,ü�ô�ù��r�i÷]g�÷Ný�ø�üng ÿpô}û�ú�ø�ÿ���c�%üWüpø%�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôª\Rh¼÷�ø�÷]��ÿ��_��ú��%÷�ô%üpú�õ,ü�ô üpý�ÿ à�¾�½ ¿ ß zrÀ ½ y`ý^{ » ú��iÿ�ú%h¼÷bù�ÿ]h ��ø�÷�öPõQün�iù��,ûrõ�ô ú��i÷��ü�ô}ú�÷���úNüpý¼ü�ô%÷�cfh�ÿ&û��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôiù^k38�÷�úNµ a0÷�ÿ ü�ô%÷�cfh�ÿ&û��0÷�ørgN�%ú�ÿ�ú�õ,ü�ô"ü'öp÷�ø�ù�üng�÷¹�%üngOÿpõQô ó k��÷zù�ÿ&û)ú��iÿ�ú7|C¶ ó~}���� Ú ç�� õQùÿ��iÿ�ør��cx��üpø�÷NaYõ,úCý_üpø}µ(õ,ýZj õ�ô%ý_üpø6gOÿ��_�,û�u¤õQúCõ�ù�� �iÿ�ør���fú�ü��iø�÷]��õ_��úú��%÷�a�õ,ú%|9j ñ u�jÄh�õ,ú����iø�ü�a�ÿ�aYõ_�Qõ,únû�g����r��a0÷�ú�ú�÷�ø�ú���ÿpô ç���d u~��õ,öp÷&ôbü�ô��,û�µ?j ñ u6��g�üpø�÷Ný_üpørgzÿ��_�,û¡¶Í�ÎÏ?Ð9ÑÓÒ�ÑÄÔ¡Ðb�r�³y » µ ý6y ¾ w�y ½dÉ å »f¾�» {ÊÀ�Ã�À ÿ y ½ v^À É yD¿nÀ É�¾ {Äà ó ð ú y�v ¾�ݹ»Äà�¾n» |i¶ ó�}���� Ú ç�� {�v¾ jÄÙ6ÚrÛduxcx�iÿ�ør��cx��üpø�÷}a�õ,ú�ý�üpøKµÂ{ È���È�À ½�¾�ÇÄÇ9¾�Ç ÅÀ ½ { »Äà;É v;� ½ å�Ã�Ã�{ÓÃ;Å.{Äà » { É y�Ù ê y à�¾�ÿ y æ

� ��ø�� ñ � ó � ø�ù µ?j ñ u�¶��Nj ø u ù |9j ñ u �P� ç���d � � Û?���÷�gzÿ&û���õ,öp÷Cÿpôf÷&ô}ú�õQø�÷]�,û�ÿpôiÿ��,ü��pün�iù~�%÷^�Yôiõ,ú�õ,ü�ô�ý�üpø�ÿ���ÿ�ør��cx��üpø�÷Za�õ,ú`üpýUÿ »�½ ¾ w�¿nÀ]À ½ �7÷�ø6g��%ú�ÿ�ú�õQü�ôIk��÷��%üô%üpú?��õ,öp÷�ú��%÷�ý�üpørgOÿ��b�i÷^��ôiõ,ú�õ,ü�ôN�%÷�ø�÷�\a��%ú|õQô%ý�üpørgOÿ��_�,û'h¼÷¤ù�ÿ�û�ú��iÿ�ú?ep÷�ûp÷]�zýÄ��ô���ú�õ,ü�ôI|�jÄh�õQú��| ô ¶ ó�ôI} ��� Ú ç�� uNõ�ù�ÿV��ÿ�ør��cx��üpø�÷�aYõ,ú�ý�üpø�ú�ø�ÿ��I�%üPüpø��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôCµ�õ,ý�õ,úNõQù��iÿ�ør��ú�üo��üng%���iú�÷| ô j ñ u~jÄh�õQú��}�iø�ü�a�ÿ�a�õ_�QõQúû'g����r�Da0÷�ú�ú�÷�øÁú��iÿpô ç���d u ��õ,öp÷&ô%µ ô j ñ u�ÿpô��.ì�\�a��%ú h�õ,ú��%ün�iúUú��%÷�ÿpù�ù�ü���õQÿ�ú�÷]�ú�ø�ÿ��I�%üWüpø�îíïzý_üpø�ì�k

��÷bù�ÿ]hLõQô(ÿV�iø�÷�ö�õ,ün�iù}�Q÷]��ú��%ø�÷fú��iÿ�ú�ý_üpø�÷�öp÷�ø�û�ü�ôi÷�cfh¼ÿ�û��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôµ \|ú��%÷�ø�÷�õQù+ÿV�iÿ�ø6��c��üpø�÷%a�õ,ú�|"ý�üpø'µ k [ �%÷+ù�ÿ�g�÷�ø�÷&ù����QúZ�%ün�_��ù�ý�üpøCú�ø�ÿ��I�%üPüpøK�0÷�ørgN�%ú�ÿ�ú�õ,ü�ô�ù^k���÷Oô%ü�h ù��iü�h �%ü�hæú�ü��ü�ôiù�ú�ør����ú�ÿi���ba��Qõ��dcfep÷�ûb÷&ô���ø�û��iú�õ,ü�ô�ù��6�%÷]g�÷�ý�ø�üngÛÿpôWû)ú�øIÿ��F�iüWüpø��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôIk38�÷�ú¹j�áDÚrµ¡Ú]â�ã�ä;ua7÷Oÿbú�ø�ÿ��I�%üWüpø'�0÷�ørgN�%ú�ÿ�ú�õ,ü�ôª\*ÿpô����,÷�ú�|`a0÷Oÿ�ep÷�ûp÷]��ý��iô���ú�õ,ü�ôohZ�iõ_�r��õ�ùÿi�iÿ�ør��cx��üpø�÷%a�õ,úý�üpø�µ k[ �%÷÷&ô���ø�û��iú�õ,ü�ô�ùr�r�%÷]g�÷+õQù)�%÷^��ôi÷]��ÿpù¤ý�ün�_�,ü�h�ù^¶

· [ �i÷�ep÷�û.�p÷&ô%÷�ø�ÿ�ú�õ,ü�ô ÿ��l�püpø�õQú���g ør�iôiù�á ú�ü�ü�aiú�ÿpõ�ôfÿN��ÿpõQø�jíì�Ú îÆï�u6k [ �i÷Z����a��Qõ_�)ep÷�û.��ü�ôiù�õQù�ú�ùüpý�ì�ÿpô�� j ÿ��i÷&ù���ø�õl�iú�õQü�ô³üpý6u%|I�iú��%÷�ù�÷]��ø�÷�úTep÷�û�õQù�ú��i÷ú�ø�ÿ��I�%üWüpø)îÆï�k

· [ ü ÷&ô���ø�û��iúÿfù�õQôb�n�,÷Na�õ,ú��}��ù�õQôb�¹���ba��Qõ��'ep÷�û W û ù jíì�Ú!|�u6\ �r�%üPü�ù�÷Oÿfø�ÿpô��iüng�� ò ó�ô ÿpô��ù�÷&ô����ʵ ô jx�u6Ú!| ô jx�u'�9���6k

· [ ü.�i÷]��ø�û;��úK��õl���%÷�ø�ú�÷���ú��{y � Ú�y ] ���iù�õQôb�³ú��%÷�ù�÷]��ø�÷�ú�ep÷�û�\ª��üng%���%ú�÷I� ù y ] ��| ô jʵ����ô j{y � u u6kj���üpú�÷Nú��iÿ�ú�ú��iõQù)��ÿpôia0÷'�%ü�ô%÷÷^¸¹��õ,÷&ô}ú��Qû ù�õQô���÷Cú��%÷Nù�÷]��ø�÷�úTep÷�û���ü�ô}ú�ÿpõQô�ùTîÆï�kÞu�{�����P��� �¢¡�£�¤{¥���¦�§�¨�©�¡ª¨!¤{«­¬�¡{��®«¯©�§�� ®«°¤x±²¨!®®§�³a´�¤{«¯��µ%©�����® «­³a´�¶°±·¤{¥�¨!¤5¸¹¨�¬¢¤{�!¡{«­µ�ºD«¯® ¥�¨>¡ª©»£��'¥�«­¬ª¥%º «­����®¼¡{«¯®��¤{�z¤{¥��

½ ¨�¾�«­µ�¤¡ª¨�´�©�����¡�´p�¢¡{³·§�¤ª¨!¤{«­� µ�©��¢®¬¢¡{«­¾p��©;¨�¾p�>����¿5ÀP¥�«­®�«­®'³a��¡{�z�!¸�¨�µAÁ ¨�¬¢¬�«¯©���µ�¤{·¤{¥�¨�µ;¨!µ�±�¤{¥�«­µ�º·�¢¶¯®��¿

[

Page 104: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

� ú�õ�ù�ô%üpú��iÿ�ør� ú�ü³ù�÷�÷�ú��iÿ�ú¤ú��i÷�÷&ô���ø�û��iú�õ,ü�ô�ù��6�%÷]g�÷Nÿ��_h¼ÿ�ûPù���õ,öp÷&ù���üpø�ø�÷]��ú��%÷]��ø�û��iú�õ,ü�ôIk

Ã�Ä Î;Ô u�ÎÆÅ ÕGÓȲ|C{Óv ¾�à�¾�½ ¿ ß zrÀ ½ y¹ý�{ » È^À ½}»�½ ¾ w�¿nÀ]À ½ w�y ½dÉ å »f¾�» {ÊÀ�Ã�j�áDÚrµ¡Ú]â�ã�ä;u »Äà y^à »Äà y ¾ ý�À ÿ y�{Óv ¾v^y�z]å ½ yTw�å�ý Ç {Êz ß é�y Ý y�Ã�z ½6Ý w » {ÊÀ�Ãov^z à y É y�ð

º �,ú��iün�b�n� h¼÷o�%ü ô%üpú.��õ,öp÷)ÿrý����_�)��ø�üWüpýüpýú���õQùzú��i÷�üpø�÷]gA�i÷�ø�÷�\Th¼÷��%ü��0÷bú��i÷�ø�÷&ÿ��i÷�ø¹h�õ��_�)a0÷��ü�ô}ö�õQô���÷]�¹aWû�ú��i÷�ý�ün�_�,ü�h�õQôb�Nõ�ô%ý_üpø6gOÿ��iÿ�ø��n��gO÷&ô}ú^k�6�÷]��ÿ��_�Yõ�ôOú��%÷K��ÿpù�÷�üpýUÿ�ü�ô%÷�cfh�ÿ&û¹�7÷�ø6g��%ú�ÿ�ú�õQü�ôµoh�õ,ú����iÿ�ø6��cx��üpø�÷}aYõ,ú²|bú���ÿ�ú�ú��%÷�ý��iô���ú�õQü�ô�Ç%jx�ucÈ�ÉÊù µ?jx�u�Ëa|³jx�u¤õQù¤ÿ%��ù�÷]���%üpø�ÿpô��iüng=�p÷&ô%÷�ø�ÿ�ú�üpø]k[ �iõQùNg�÷&ÿpôiù+ú��iÿ�ú�ý�üpø�ÿ)ø�ÿpô��%üngÌ��\Áú��i÷fù�ú�øIõQôb��µ?jx�u�Ë1|³jx�ui� �QüWü�e�ù�ø�ÿpô��%üng��Cj õQô(ÿ�h¼ÿ�û�gOÿ��i÷ý_üpørgzÿ��ªaPûfú��i÷}�%÷^��ôiõQú�õ,ü�ô üpý���ù�÷]���%üpøIÿpô��%üngOô%÷&ù�ù6u6k º ��üng%���Q÷�ú�÷]�,û�ÿpôiÿ��,ü��pün�iù�ø�÷&ù����,úZ�%ün�_�iù~h��i÷&ôµ õQùOÿ�ú�ø�ÿ��I�%üPüpø��0÷�ørgN�%ú�ÿ�ú�õ,ü�ôIk#�Püb\�ÿ ��ù�÷]�^�%ø�÷]�`h¼ÿ�û�ú�ür÷&ô���ø�û��iúOÿ`gO÷&ù�ù�ÿ��p÷Íw hºün�����Âa7÷$ú�ü��üng%���%ú�÷ÎwÏ�Xjʵ ô jx�u�Ë7|³jx�u u6k �#�iõ_�,÷³ú��iõQùNh¼üpø�ePù�ý�üpøD�iø�õ,ö�ÿ�ú�÷�cfep÷�û(÷&ô���ø�û��iú�õ,ü�ô�h��%÷&ôÐ�)õ�ù���ø�÷�cù��iÿ�ø�÷]�`j õQô��%÷�÷]�I\%ú��iõQù�h�ÿpù¤ü�ô%÷üpý�ú��%÷'�iøIù�ú�ù�÷]�^�%ø�÷�ù��6�%÷]g�÷&ù�hº÷�ù��iü�h¼÷]��u6\iú��iõQù)�iüW÷&ù�ô%üpú)h¼üpø�efý_üpøú��%÷%��ÿpù�÷�üpý����ba��Qõ_�dcfep÷�û�÷&ô���ø�û��iú�õ,ü�ôª\¡hZ�%÷�ø�÷+ú��%÷+ù�÷&ô��%÷�øÿpô��)ø�÷]��÷&õ,öp÷�ø��%ü ô%üpú��iø�÷�cùr�iÿ�ø�÷���k�t�ü�hú�ü�hº÷}�b�fú���õQù6�

� ô³÷��0÷]��ú^\bhº÷:ÿ��_�,ü�h ú��%÷:ø�÷]��÷&õ,öp÷�ø�ú�üD��üng%���%ú�÷���aWûOù�÷&ô��iõQôb�Nµ?jx�nu�ÿ��,ü�ô%÷Dj õÊkµ÷�kl\�h�õ,ú��%ün�%ú��Püpø?Ñ�õ�ôb�µ?jx�nu�h�õ,ú���ÿpôWûo��ÿ�ø�ú�üpý�ú��%÷�gO÷&ù�ù�ÿ��p÷�u6k [ �%÷Où�÷&ô��%÷�ø���ÿpô ô%ü�hX�Püpø�ú��%÷ ç cfa�õQú}g�÷&ù�ù�ÿ��p÷¹h�õ,ú���ú��i÷��ôiÿ���a�õ,ú7|9jx�u:üpý�ú��%÷`�Æ��ù�÷]���iüpø�ÿpô��%üng ù�ú�ø�õQôb�n�iµ?jx�u6Ú!|³jx�u6k � ô�ú��%÷¹��ÿpù�÷Oüpý~��ù�÷]���iüpø�ÿpô��%üngOôi÷&ù�ù^\hº÷Cù�ÿ]h ú��iÿ�ú��iÿ&ö�õQôb�Dµ?jx�nu?�iõ_�Oô%üpú¼ÿ��_�,ü�hwÿpôfÿ��%öp÷�ø�ù�ÿ�ø�û�ú�üN�iø�÷]��õ_��úh|³jx�u�jÄh�õ,ú�����ø�ü�a�ÿ�a�õ_��õ,úû�a7÷�ú�ú�÷�øú��iÿpô ç���d u6k~�PüNú�üPüb\b�%÷�ø�÷�\�ÿpô³ÿ��%öp÷�øIù�ÿ�ø�û%h��%ü+ù�÷�÷&ù~µ ô jx�uR��ÿpôiô%üpúR��ø�÷]�iõ_��úh| ô jx�nuZjÄh�õ,ú����iø�ü�a�ÿ�aYõ_�Qõ,únûa7÷�ú�ú�÷�ø�ú��iÿpô ç���d u6k)�PüÒ| ô jx�u���ÿpôia0÷'�iù�÷]�$ÿpù�ÿ���ü�ô%÷�cKú�õ_gO÷}��ÿ�����ú�üz÷&ô���ø�û��iú�ú��%÷ ç cfa�õQúTg�÷&ù�ù�ÿ��p÷�k¢I£UÓ n%²Di�¥pm?§�°n± ²�Ô�Õ�©³²�Ô³¬�¥×Ö=¬�´�´ ¦5Ô³¬�´��÷��iÿ&öp÷³ÿ��,ø�÷&ÿ��%û"ù�÷�÷&ô�õQô��iø�÷�öPõ,ün��ù'�,÷]��ú��%ø�÷&ùCú��iÿ�ú�ÿ�ù�÷]�^�%ø�÷%����a��Qõ_�dcfep÷�ûb÷&ô���ø�û;��ú�õ,ü�ô�ù��r�i÷]g�÷�ý_üpøç cfa�õ,ú¹gO÷&ù�ù�ÿ��p÷&ùfÿ��_�,ü�h�ùO÷&ô���ø�û��iú�õ,ü�ôÑüpý�ÿ�ø�a�õ,ú�øIÿ�ø�ûcx�Q÷&ôb�pú��¯g�÷&ù�ù�ÿ��p÷&ù³ÿpù¹hº÷]����jÄaPû¯��ü�ô���ÿ�ú�÷&ôiÿ�ú�õQôb�÷&ô���ø�û��iú�õ,ü�ôiùüpý�÷&ÿ��6� aYõ,úüpý�ú��%÷�g�÷&ù�ù�ÿ��p÷�u6kiÌ~�%ú�ú��iõQùNõQù'�pü�õQôb�$ú�ü�a0÷zõQô%÷^¸¹��õ,÷&ô}ú^\9a0üpú��rõQô���üngDcg��iô�õ_��ÿ�ú�õ,ü�ô2j�ú��i÷�ø�÷¹h�õ_�_��a0÷.ûLØ ç a�õ,ú�ù�üpýZ��õl���%÷�ø�ú�÷���ú��0÷�ø}aYõ,úNüpý�ú��%÷.g�÷&ù�ù�ÿ��p÷�\?hZ�%÷�ø�÷.ûrõQù�ú��i÷õQôb���%ú�Ù'ün�iú ���%úZ�Q÷&ôb�pú��"üpý�µ ô u�ÿpô��o��üng%���%ú�ÿ�ú�õ,ü�ô¯j�ü�ôi÷+÷�ö�ÿ��_�iÿ�ú�õQü�ô"üpý�µ ô õ�ù�ø�÷]p;�iõQø�÷]�$ý�üpøC÷&ÿ��r��aYõ,úüpý¼ú��i÷�g�÷&ù�ù�ÿ��p÷�u6k [ �%÷��Wû;aiøIõ_�"ÿ����iø�ü�ÿ��6��hº÷³ù�ÿ]h õQô�ÿ��iø�÷�ö�õ,ün�iù'�,÷]��ú��iø�÷%h�õ_�_���%÷]�_�ª\Ia��%úü�ô��Qû�õ,ýhº÷N�iø�ù�úZ�iù�÷Nú��i÷}���ba��Qõ_�Kep÷�û$ù��6�%÷]g�÷+ú�ü³÷&ô���ø�û;�iú�ù��b¸���õ,÷&ôWú��,û;cxgOÿpôWû�a�õ,ú�ù�j õfkµ÷�kl\Y÷&ô%ün���n�$ú�ü.�iù�÷�ÿpùÿ%ep÷�û ý_üpø�ÿ��iø�õ,ö�ÿ�ú�÷�cfep÷�û�ù��r�%÷]gO÷�u6k

t�ü��0÷�ý����_�,û�\0ú��%÷%�iõ�ù��^�iù�ù�õQü�ôbüpý`ú��i÷N��ø�÷�öPõQün�iù�ù�÷]��ú�õ,ü�ô¯j���üng%��ÿ�ø�õ�ôb�³ú��i÷+÷&ô���ø�û;��ú�õ,ü�ô"ù��r�%÷]gO÷�ú�üú��%÷���ü�ôiù�ú�ø6����ú�õ,ü�ô�üpýCÿ`�D6 m u%g�üpú�õ,ö�ÿ�ú�÷&ùzú��%÷�ý�ün�_�,ü�h�õQôb�b\�g�üpø�÷�÷^¸���õQ÷&ô}ú���ü�ô�ù�ú�ør����ú�õ,ü�ôªk � ô�ú��i÷��ÿpù�÷�üpý��D6 m ù^\bh¼÷�ù�ÿ]h ú��iÿ�ú¤ú��%÷'��ü�ôiù�ú�ø6����ú�õ,ü�ô�Ç%jx�u ù µ¼Ú^jx�uPËh|9jʵ¼Ú ��� jx�u u�Ë�ü^ü^ü�Ë·|³jx�u¼õQù�ÿ%�h6 mù�ú�ø�÷�ú��r��õQôb��õQú�ù�õQô����%ú'aPû�Û�aYõ,ú�ù^k���÷.��ÿpôC�iù�÷³ú��%÷³ù�ÿ�g�÷zú�÷]�6�iôiõ_p��%÷zý�üpøN���ba��Qõ��dcfep÷�û�÷&ô���ø�û;�iú�õQü�ôI ��ÿ�g�÷]�Qû�\×ú�üf÷&ô���ø�û;��ú:ÿpô�Ûdcfa�õ,ú�g�÷&ù�ù�ÿ��p÷L� � � ] ü^ü^ü � Ú \Fhº÷D��õ���e$ÿfø�ÿpô��iüngÜ��\I��üng%���%ú�÷Ly � ù µ ô jx�u6\ÿpô��

y ] ù j| ô jʵ ì Ú ��� jx�u u'Ë%| ô jʵ Ú � ]ô jx�u u�ËKü^ü^ü�Ëa| ô jx�u u'� j� � ü^ü^ü�� Ú u��[ �%÷zú�øIÿpôiù�gOõ,ú�ú�÷]����õl���%÷�ø�ú�÷��PúNõ�ù�y � Ú�y ] k j��÷�û��p÷&ô%÷�øIÿ�ú�õ,ü�ô õ�ù}�%ü�ô%÷fÿpù}a0÷�ý�üpø�÷�\|ÿpô�� �%÷]��ø�û;�iú�õQü�ôù��%ün���_��a0÷Cü�aPöPõ,ün��ù^kÞu

t�ü�h ÷^¸¹��õ,÷&ô}ú:õQù¤ú���õQù¤ù��6�%÷]g�÷���ó:ô%ý�üpø�ú��iôiÿ�ú�÷]�Qû�\iú��%÷���üng����%ú�ÿ�ú�õ,ü�ô$ø�÷]p;��õ,ø�÷]�bõQù�ù�ú�õ_�_�0ü�ô%÷�÷�ö2ÿ��lc�iÿ�ú�õ,ü�ô�üpý'µ ô �0÷�øDa�õ,ú�üpý:ú��%÷ig�÷&ù�ù�ÿ��p÷�\�a��%ú�ú��%÷i��õl���%÷�ø�ú�÷��Pú��,÷&ô��pú���õQù�ô%ü�h�ûÒØÝÛ ý�üpøOÿpôNÛ6cfaYõ,úg�÷&ù�ù�ÿ��p÷�j õQô)ú��%÷�üpø�õ_��õQôiÿ���ù��r�%÷]gO÷�\�ú��%÷%��õl���%÷�ø�ú�÷��Pú�h¼ün���_���iÿ&öp÷%a0÷�÷&ô¯jÊû3Ø ç uªÛNa�õ,ú�ù��,ü�ôb� ý�üpøCÿpôÛ6cfa�õQúTg�÷&ù�ù�ÿ��p÷�u6k

Page 105: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Þ�ß�àâápã�ä!å>àçæjèÒépêzëÍì�ä!èKíâìpß¼îï ëñð ï·ò»ópô�õ÷ö ß»æ�ä!éKî�ø¼ù�æ>àçé�ßÎæ�é ï ä>èKú�æ�épûpä!ì�ú ü»èý�ä>épêxã?å>å�épä;þpé�ß�ì�æ>ü�ìpßÍÿ1ì�æ��

���������� ��

� ��������������� �!�#"%$��'&!�'(

)'ã�æ+*­åAä>ã�áÆàçã+,-,�ü�ì�æAù�ä!è,ú�æ�épûpä�ì�ú¼ü�àâù ú¼ä>épæ�éKù�é�í å.,·ã ü�ì?ápã ù�é�ápã�ä>ã?îbæ>üKø�åAê ì�ä0/ ö ßXæ>ü�ã ù�é�ß,æ�ã21KæÎépêå�è4353Iã�æ�ä!àâù7698pã�è�:xépä�ú�ä!àçá�ì�æ�ã2698pã�è<;;ù�ä>èKú�æ�épûpä!ì�ú¼ü,è>=?,·ã3ü¼ì?ápã_î�à å>ù�ø�å>å�ã?î�ã?ß�ù�ä>èKú�æ>àçé�ßñú�ä!épæ�éKù�é�íâå�ìpß¼î3Iã?å>å>ì�ûpãLìpø�æ>ü�ã?ß»æ>àâù�ì�æ>àâé�ßñå!ù!ü�ã03Iã?å0/#@%ü�ã�êUépäA3Iã�ä;ú�ä>é�áÆàâî�ã5B+CED+F�C�D+G�,²ü�àâíâã1æ>ü�ãIíâì�æ�æ�ã�ä;ú�ä>é�áKàâî�ã?å�HJI<KL C9M>F7H L GN/'O×ã;ß�ã21Kæ î�àâå>ù�ø¼å>å�ã?îLú øQP¼íâàâù7698pã�è3ã?ß�ù�ä!è,ú�æ>àâé�ß5,²ü¼àâù!üÒàâåDæ>ü�ã�î�àâä>ã?ù�æhù�é�ø�ß,æ�ã�ä>ú¼ì�ä!æ·épê�ú�ä!àçá�ì�æ�ã268pã�èñã?ß¼ù�ä>è,ú¼æ>àçé�ß ='ìpíçæ>ü�é�ø�û�ü×àçæ1ìpíâíçéR,²å�3Lìpß,è�úPã�épú¼íçã3æ�é å�ã?ù�ä!ã�æ>íçèñù�éS353�ø�ß¼àâù�ì�æ�ãT,²àçæ>üñé�ß�ãIúPã�ä!å>é�ß:J,²ü�éLü�ìpå ú¼ø<P¼íâàâå>ü�ã?î_ì3ú¼øQP íâàâùU8pã�è<;hä!ì�æ>ü�ã�ä²æ>ü�ìpß�VXW�ø¼å�æ�YIìpíâíçéR,²àâß�û�æZ,·éLúPã�épú¼íçãT:J,²ü�éIü�ì�ápã7å>ü�ì�ä!ã?îì58pã�èÍà ß�ìpî�á�ìpß�ù�ãR;%æ�éAå>ã?ù�ä>ã�æ>íçèÍù�éS3T3�ø�ß�à ù�ì�æ�ã>/[Obü�ì�æ;àâå%æ>ü�ã3ù�é�ø�ß,æ�ã�ä>ú¼ì�ä>æ�épê'3Iã?å!å>ì�ûpã�ìpø�æ>ü�ã?ß»æ>à\6ù�ì�æ>àçé�ß^]

ö ß æ>ü�ã_ù�ìpå�ãÒépê3Iã?å>å>ì�ûpãAìpø�æ>ü�ã?ß,æ>àâù�ì�æ>àçé�ß =_,hãÒå>ì0, æ>ü�ì�æ7æ`,hé�ú5ã�épú íçã�,²ü�éÍü¼ìpî�å>ü�ì�ä>ã?î ìa8pã�èàâßÒìpî�á�ìpß�ù�ã�ù�ìpß ø�å>ã�ìb3Iã?å>å>ì�ûpã1ìpø�æ>ü�ã?ß»æ>à ù�ì�æ>àçé�ßÎå>ù!ü�ã03Lã�æ�éIî�ã�æ�ã�äA3Làâß�ã[,²ü�ã�æ>ü�ã�ä ê ø�æ>ø�ä>ã�3Iã?å>å>ì�ûpã?åæ>ü�ã�è ä!ã?ù�ã?àçápã ì�ä>ã àâß�î�ã�ã?î êxä>éS3 ã?ìpù�ü.épæ>ü�ã�ä�: ìpß�î.ß�épæ�3_ìpíâàâù�àçé�ø�å!íçè�àâß�å�ã�ä!æ�ã?îcPKèNå�éS3Iã�é�ß�ã ã?íâå>ãR;A/@%ü�ã;ú¼øQP¼í àâù7698pã�èLìpß�ìpíçépû�ø�ã�àâå ìedfH\M>H L9gfh B7HiMSI g>Lkj F�ClB+D2mnC+oTC+/'p�ã�ä>ã>=�é�ß�ã;úPã�ä!å�é�ßq: ù�ìpíâíçã?îÎæ>ü�ãbBAH\M>I�C2FS;ú¼øQP¼íâà å>ü�ã?åDì3ú øQP¼íâàâùU8pã�è.,²ü¼àâù!ü =Æìpå%ø�å!ø�ìpír=Æàâå·æ>ü�ã?ß ìpå>å>ø<3Lã?îÎæ�é�P5ã�8Æß�és,²ßtPKè_ã�ápã�ä!èpé�ß�ã1ã?íâå�ã>/u@²ü�ãå>àçû�ß�ã�äv8pã�ã�ú¼å ìpß_ìpå>å�éÆù�àâì�æ�ã?îÎå�ã?ù�ä>ã�æ�8pã�è�,²ü�à ù!ü_ü�ã²ù�ìpßAæ>ü�ã?ß_ø�å>ã²æ�é�å!àçû�ß53Iã?å!å>ì�ûpã?å+/'@%ü�ã%ä>ã?å>ø�íâæ>àâß�ûå>àçû�ß�ì�æ>ø�ä!ã_é�ßÐæ>ü�ã.3Lã?å>å>ì�ûpã àâå1æ>ü�ã?ß ápã�ä�à\w¼ìfP¼íçãqx2G g I<GSyfIzCa{|m4y~}RIzyf{|B L mnCtBAH\M>IzC+Fn� B�� j x h HrD�}>C+Gf/O×ã7û�àçápã7ìLêUépäE3_ìpí�î�ã+w¼ß�àçæ>àâé�ßAìpß¼î æ>ü�ã?ß ìIä!ã�ú�ä>ã?å�ã?ß,æ>ì�æ>àçápã1ã21�ì>3Iú¼íçã>/�a�������X�R�J�z����� L F7H�� h Cay��l�ny h GfK L HJo5C gfh M�yfF7H L m�obBt:k���E�z� �Q�������\�X��;�HXB g BAH\M>I gSLkj F�C.B+D2mnC+oTC��`yf�>C2FB2yfo5Cbo5C7BAB g M4C#B9� g DAC¡ -¢TH � L mnCv�2y hXh yf{�HJI£M�m4y h dS¤¥S¦ �-H§B g F g I^dSyfo�H©¨sCEd gfh M�yfF7H L m�oª{|m�HkD+m«y jQL � jQL B g � j x h HkD#}>C+GU¬z­ g I^d g B+CED+F�C L }>C+G�®R­ ¦¯£¦ �z� �Q�aH§B g �i�QyNBEB7Hrx h G7¢tF g IzdSyfo�H\¨RC�d gfh M�yfF7H L m£o!{|m�HrD2m L9g }SC7B g B+C�D0F�C L }>C+G�®N­ g I^d g oTC2BEB g M£C

°!±   g I^day jQL � jQL B g BAH\M>I gSLkj F�C¡² ¦t³ Cbd�C+I^y L C L m�HXB�x+G#²«´µ�z� �Q�s¶r·s:¹¸�; ¦º�¦ ���\�J��HXB g d£C L C2F7o�HXI»H§B L HkD gfh M�yfF7H L m£o¼{|m�HrD2m L9g }>C2B g � j x h HrD�}SC2Gv¬z­^½ g o5C7BAB g M4C °!±  ¾½ g I^d

g �\� j F9�QyfF L CEd7¢aBAH\M>I g>Lkj F�C#² g I^dey jnL � jnL B ¥ yfF5¿ ¦À³ C�d�C+I^y L C�C�Á4C�D jQL HryfIcy�� L mnC gfh M£yfF7H L m�ox+G����©�X�SÂ�·N:¹¸#��²%; ¦ � ¥ HXIzdfHkD g>L C7B L m g>L�L mnC5BAH\M>I gSLkj F�C.HXBÒá�ìpíâàâî g I^d g ¿«HJI^dfHrD g>L C2B L m g>L¡L mnCB7HiMSI g>Lkj F�C2BlH§B3àâß,á�ìpíâà î ¦

³ C#F�CEà j HXF�C L m g>L �+yfF gfhJh :Ĭz­z�E®R­<;�y jQL � jQL x2G#�b½ gfhJh °!±  ¾½ g I^d gfhJh ²Åy jQL � jQL x2Gb�z� �Q�R¶k·R:¹¸¡;E½'{vCm g �fCÆ���\�J�>Â�·N:¹¸���²%;'ÇÉÈ ¦@%ü�ã�w¼ß�ìpí5ù�é�ß�î�àçæ>àçé�ßlW>ø�å�æ�3Iã?ìpß�å·æ>ü�ì�æaìpíâí5å>àçû�ß�ì�æ>ø�ä!ã?å·à å>å>ø�ã?î.P,è_æ>ü�ã�íâã�û�àçæ>à©3Lì�æ�ã�å>àçû�ß�ã�ä·ì�ä>ã;ä>ã?ù�épûf6ß�à\��ã?î ìpåaá�ìpíâàâîtPKèÎìpß»èpé�ß�ã�,²àçæ>ü ìpù�ù�ã?å>å�æ�éLæ>ü�ã�ú øQP¼íâàâù[8pã�èÎêUépä²æ>ü�ã7å>àçû�ß�ã�ä0/

O×ãLû�àçápãLìtP�ä�àçã�ê�ã21�ì>3Iú¼íçã3épê ü�és,�ì å>àâû�ß�ì�æ>ø�ä>ãIå>ù�ü�ã03IãLù�ìpß�PPãIø�å�ã?î /IðÆì?è ìtP¼ìpßQ8~Ê~ü¼ìpå�ìú¼øQP¼íâà ù#8pã�èqËlÌ.ÍÎ:xæ>ü�ã�è×ù�ìpß×ú¼øQP¼í àâù�à\��ã7æ>ü¼àâå¡8pã�è×àâß é�ß�ãIépê·æ>ü�ã�,·ì�èKå#,hãÒî�àâå>ù�ø�å!å�ã?îñã?ì�ä!íâàçã�ä0='í à\8pã

Ï

Page 106: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ú¼ø�æ�æ>àâß�û3àçæ·àâß_æ>ü�ã;ß�ã+,²å�ú¼ì�úPã�ä7;A/�Ð�å>å>ø<3Iã�æ>ü�àâå�P¼ìpßQ8Iæ>ü�ã?ß., ìpß,æ>å æ�éLù�ã�ä>æ>àçêxè_ì�å>æ>ì�æ�ã03Iã?ß»æ ° épê'æ>ü�ãêUépäE3ÑV�Ð�åDépê�þ»ìpß / Ï =QÒfÓ>ÓSÒ4=QÔ ü�ìpåÕ Ï Ó>Ó�à ßLü�ã�ä ìpù�ù�é�ø�ß»æ�Yn/v@%ü�ã�èLù�ìpßAø�å�ã²æ>ü�ã?àçähå>àâû�ß�ì�æ>ø�ä>ã�å>ù!ü�ã03Lãæ�éLûpã?ß�ã�ä!ì�æ�ã7ìIå!àçû�ß�ì�æ>ø�ä>ãl² é�ßÎæ>ü¼àâå3Iã?å>å>ì�ûpã>/�@%ü�ã¡P¼ìpßQ8Aù�ìpß û�àçápã1æ>ü¼àâåaå>àçû�ß�ì�æ>ø�ä>ã;æ�é�Ô¾,²ü�é3ù�ìpßß�és,bæ>ìf8pã�àçæhìpß¼î_ø�å�ã;àçæ�æ�éIù�é�ß,áKà ß�ù�ã g I»GSxEy0dfG�C h B0CT: ìpå íçé�ß�û�ìpåDæ>ü�ã�èT8Kß�és,Ýæ>ü�ã[P ìpßQ8^*­åhú¼øQP íâàâù�8pã�èËlÌ.Í'; æ>ü�ì�æ%æ>ü�ã�P¼ìpß<8Aàâß¼î�ã�ã?î ù�ã�ä>æ>à\w�ã?îÍå�æ>ì�æ�ã03Iã?ß,æ ° /

Ö ê�ù�é�ø�ä!å�ã>=�æ>ü�ã ú�ä>ã?ù�ã?î�àâß�û ã21�ì>3Iú¼íçã×î�éKã?å>ß *°æa3Lìf8pã×3�ø�ù�üGå�ã?ß�å�ã×ø�ß�íâã?å>å�,hã×ìpíâå�é.î�ã+w ß�ã×ìß�épæ>àçé�ßÎépêcå�ã?ù�ø�ä!àçæjè_êxépä²ìIå>àçû�ß¼ì�æ>ø�ä>ã�å>ù!ü�ã03Iã>/[: ö ßÎú¼ì�ä>æ>àâù�ø�í ì�ä0=Kàâê%Ô ù�é�ø¼íâîÎü�ì?ápã1êxépä>ûpã?î æ>ü�ã�P¼ìpßQ8z*­åå>àçû�ß�ì�æ>ø�ä!ã>=aæ>ü�ã?ß�å!ü�és,²à ß�û ìÐå>àçû�ß�ì�æ>ø�ä>ãÍú�ä!é�ápã?åÎß�épæ>ü�àâß�ûQ/Ä; Ö ø�äAî�ã+w¼ß�àçæ>àâé�ß.épê7å�ã?ù�ø�ä�àçæªè�,²à íâíPPãìpß�ìpíçépûpé�ø�å�æ�éñæ>ü�ãÒé�ß�ã5,·ãÒû�ì�ápã êUépä�3Iã?å>å>ì�ûpãÎìpø�æ>ü�ã?ß,æ>àâù�ì�æ>àçé�ßNå>ù�ü�ã03Iã?å+=�ìpß¼î ä>ã�ú¼ä>ã?å�ã?ß,æ>å�ì�ápã�ä>èå�æ�ä>é�ß�û�ß�épæ>àçé�ßLépê'å>ã?ù�ø�ä!àçæjè>/ ö ß�êxépäE3Lìpí íçè>=»àçæDå!ü�é�ø�íâî�PPã%æ>ü�ã�ù�ìpå�ã�æ>ü�ì�æ·ä>ã�û�ì�ä!î�íçã?å!å�épê ,²ü�ì�æv3Iã?å>å>ì�ûpã?åì�ä>ã7å>àçû�ß�ã?îtP,èAæ>ü�ã7å>àâû�ß�ã�ä0=�ß�éIé�ß�ã1ã?í å�ã5:J,²àçæ>ü�é�ø�æaæ>ü�ã1å�ã?ù�ä>ã�æU8pã�è<;aå!ü�é�ø�íâîÎß�épæUPPã�ìfP¼íçã�æ�é_êUépä>ûpã�ìá�ìpíâà îÍå>àçû�ß¼ì�æ>ø�ä>ã7é�ß g I<G ß�ã+,Ø3Iã?å!å>ì�ûpã�à ß æ>ü�ã3ß�ì>3Iã�épê�æ>ü�ã�å!àçû�ß�ã�ä0/UO×ã�3IéÆî�ã?í�æ>ü�àâåUPKè î�ã+w¼ß¼àâß�ûìeB7HiMSI<HXI4M«yfF g D h C.�z� �Q�R¶k·R:¹ÙÄ;U,�àçæ>üe,²ü¼àâù!üñìpß�ìpî�ápã�ä!å>ì�ä>èñù�ìpßñà ß»æ�ã�ä!ìpù�æ+Ú�,²ü�ã?ßÍæ>ü�ã3ìpî�ápã�ä!å!ì�ä>èÍå�ã?ß�î¼åìpß»èÀ3Lã?å>å>ì�ûpãAæ�é�æ>ü�à å1épä!ìpù�íçãÒæ>ü�ãAépä!ìpù�íçã_ä!ã?å�úPé�ß�î�ål,�àçæ>ü×æ>ü�ãÎå>àçû�ß�ì�æ>ø�ä>ãLé�ß æ>ü�ì�æb3Lã?å>å>ì�ûpã>/�@²ü�ãìpî�ápã�ä!å>ì�ä>è ù�ìpß�àâß,æ�ã�ä!ìpù�æ�,²àâæ>ü�æ>ü¼àâå�épä!ìpù�íçãLìpå;épêxæ�ã?ß�ìpå1ü�ãIíâà\8pã?å0=Pìpß¼î æ>ü�ãLìpî�ápã�ä!å>ì�ä!è^*­å;ûpé�ìpí�à å;æ�éù�éS3IãLø�úq,²àçæ>ü ìAá�ìpí àâî�å!àçû�ß�ì�æ>ø�ä>ã�²Ðé�ß×ìt3Iã?å>å>ì�ûpã ° L m g>L { g B�IzC2�>C2F5C�Á2� h HkD0H Lkh G«Ã j C2F7H9C�d L y L mnCyfF g D h C«:xä>ã�ú¼ä>ã?å�ã?ß,æ>àâß�ûLæ>ü�ã�êxìpù�æ�æ>ü�ì�æ�æ>ü�àâå[3Iã?å>å>ì�ûpãb,·ìpå�ß�ã�ápã�ä�å>àçû�ß�ã?îePKè æ>ü�ã7ä>ã?ìpí�å>àçû�ß�ã�ä2;A/�ë épä!ãêUépäE3_ìpíâíçèzÛ�a�������X�R�J�z��Ü��µBAH\M>I g>Lkj F�C×B2D2mQC2o5CÀ:k���E��� �Q�������\�J�S;×HXB�:JÝ7�EÞA;AK`B+CED j F�CÀH ���2yfF g>hXhlg df�>C2FAB g F7H9C7B«ßF j I<I<HJI£M�HJI L HXo5CÆÝl{vClm g �>C0¤

ý�äsà\:Ĭz­z�E®N­<;u´á��Ús: ° ��²%;v´âß�ã2ä åAæ`çÄè¹éXê ëE:Ĭz­<;�Ûf���\�J�>Â�·N:¹¸���²%;'ÇÉÈvìq¸îí±Àï�ð%ñ Þ0�{|mnC2F�C�ò d�C+I^y L C2B L mnC#B+C L y��#oTC2BEB g M£C2B�B j x2o�H LrL C�dex2ó�ß L y��z� �Q�R¶k·R:¹ÙÄ; ¦ô²épæ�ã æ>ü�ì�æ3æ>ü�ã ìpî�ápã�ä�å>ì�ä>è à å7ã21Æú¼íâàâù�àâæ>íçè û�àçápã?ßNæ>ü�ãAú¼øQP¼íâà ù�8pã�è ìpß�î?=Dàâß ú¼ì�ä>æ>àâù�ø�í ì�ä0=zù�ìpß.ìpí\,·ì�èKåápã�ä!àçêxè�,²ü�ã�æ>ü�ã�ähì�ú ì�ä>æ>àâù�ø�íâì�äu²Íàâå�àâß�î�ã�ã?îIì�á�ìpí àâîIå>àçû�ß�ì�æ>ø�ä>ãaé�ß_ìl3Lã?å>å>ì�ûpã ° /�@%ü¼àâåzàâåzàâßLù�é�ß,æ�ä!ìpå�ææ�éÒæ>ü�ã�ù�ìpå�ã�épêu3Iã?å>å>ì�ûpã�ìpø�æ>ü�ã?ß,æ>àâù�ì�æ>àçé�ß�,²ü�ã�ä>ã�æ>ü�ã�ìpî�ápã�ä�å>ì�ä>è ß�ã�ápã�ä;íçã?ì�ä!ß�å²æ>ü�ã�å>ã?ù�ä>ã�æÆ8pã�èÍìpß¼îæ>ü�ã�ä>ã�êxépä>ã7î�éKã?å%ß�épæ%ã�ápã?ß«8Kß�és, àçê_,²ü�ì�æ%ü�ã�é�ø�æ�ú¼ø�æ>å%à åaìIêUépä!ûpã�ä>èÎépä²ß�épæ+/

õ öZõ ÷×ø9ù�ørú>û ü�ý|øZù%þuûzúSÿ����������4ö������Rû�ù ����ÿuú����»þ�úSø��nûzúSø��%þö æ�àâåU,·épä>æ>üñå�æ�ä>ã?å>å>à ß�ûLæZ,·éÎà©3IúPépä>æ>ìpß,æ�î�à��Pã�ä>ã?ß�ù�ã?åÆPPã�æZ,·ã�ã?ß�î�àâû�àçæ>ìpí�å>àâû�ß�ì�æ>ø�ä>ã?å�ìpß�î«3Iã?å!å>ì�ûpã�ìpøQ6æ>ü�ã?ß»æ>à ù�ì�æ>àçé�ß Û�������k���! ^�#"f�§��$%�|�&�J�X�('*);àçû�àçæ>ìpí�å>àâû�ß�ì�æ>ø�ä>ã?å ì�ä>ã[� j x h HrD h óT�fC+FAH + g x h C+/'@%ü�ì�æaàâå+=Æìpß,èpé�ß�ãl,²ü�éb8Æß�és,²å

ìAø�å�ã�ä0*­å�ú¼ø<P¼íâàâù¡8pã�èñù�ìpß ápã�ä!àçêxèa,²ü�ã�æ>ü�ã�ä;æ>ü�ì�æ;ø�å�ã�ä�ü�ìpå;å>àçû�ß�ã?î�ìÒú¼ì�ä>æ>àâù�ø¼íâì�ä�3Lã?å>å>ì�ûpã>/ ö ßù�é�ß,æ�ä!ìpå�æ+=_3Iã?å>å>ì�ûpãLìpø�æ>ü�ã?ß»æ>àâù�ì�æ>àâé�ß�å>ù!ü�ã03Iã?å�é�ß�íçèñìpíâíçéR, æ>ü�ã3ä>ã?ù�àçú¼àçã?ß,æ;æ�é ápã�ä!àçêxèÍæ>ü�ì�æ�æ>ü�ã3Iã?å!å>ì�ûpã1épä!àçû�à ß�ì�æ�ã?î êxä>éS3 æ>ü�ã1å�ã?ß¼î�ã�ä0Ú�épæ>ü�ã�ä!å²ü�ì?ápã�ß�é�,·ì�è épê�ápã�ä�àçêUèÆàâß�û3æ>ü�à å êxìpù�æ+/

,«�^� -."N�0/���1|��$Q�R�J�^�2);àçû�àçæ>ìpí�å>àçû�ß�ì�æ>ø�ä!ã?å%ú�ä>é�áKà î�ã�I^yfI�F�C`� j dfH g>L HryfI àâßÎæ>ü�ã�å�ã?ß�å>ã1æ>ü�ì�æ%æ>ü�ã�èaP àâß�îæ>ü�ã å!àçû�ß�ã�äLæ�é�æ>ü�ãÍå>àçû�ß�ã?î 3Iã?å>å>ì�ûpã>/Å@%ü�ã å>àçû�ß�ã�ä_ü�à©3Lå>ã?íçê²ù�ìpß�ß�épæAíâì�æ�ã�ä_î�ã?ß,è : å>ì?è>=aæ�é ìW>ø�î�ûpãR;%å>àçû�ß�àâß�ûIæ>ü�ãb3Lã?å>å>ì�ûpã>Ú'ü�àâå�å>àçû�ß�ì�æ>ø�ä>ã«V�P¼àâß¼î�å�YLü�à§3ÉW>ø�å�æ�ìpå�ìÒä>ã?ìpí�å!àçû�ß�ì�æ>ø�ä>ã#,hé�ø�í îé�ß ì�íçã�û�ìpíhî�éÆù�ø<3Iã?ß,æ+/ Ö ß�æ>ü�ãAépæ>ü�ã�ä3ü�ìpß�î =_3Iã?å>å>ì�ûpãÎìpø�æ>ü�ã?ß,æ>àâù�ì�æ>àçé�ßÐî�éKã?å�ß�épæ7ú�ä>é�áKà î�ãæ>ü�à å g>L�g>hXh /43�épä�é�ß�ãLæ>ü�àâß�ûQ='æ>ü�ã�ä>ãLàâå7ß�éa,·ì�è×æ�é ã�ápã?ß ù�é�ß,áÆàâß�ù�ã_ì�W>ø�î�ûpãIæ>ü�ì�æ�ìe3Iã?å>å>ì�ûpã, ìpå1ìpø�æ>ü�ã?ß,æ>àâù�ì�æ�ã?îc,²àçæ>ü�é�ø�æ�ä!ã�ápã?ìpíâàâß�û æ>ü�ãÒå�ã?ù�ä>ã�æl8pã�è>/65�ápã?ß�ì�æ�æ>ü�ì�æ�úPé�àâß»æ+=�æ>ü�ã�ä>ã_àâå�ß�é

Ò

Page 107: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

, ì?è æ�é«8Æß�éR, æ>ü�ì�æ1æ>ü¼àâå1å�ã?ù�ä>ã�æ�8pã�è�,·ìpå�ìpù�æ>ø�ìpíâíçèñæ>ü�ã_ì�ûpä!ã�ã?în6ªø�úPé�ß×8pã�è æ>ü�ì�æ�æ>ü�ã_å�ã?ß�î�ã�äìpß�îñä>ã?ù�ã?àâápã�ä1ø�å�ã?î /�3càâß¼ìpíâíçè>=5ã�ápã?ß�àçêDì#W>ø�î�ûpãIî�ã+w¼ß�àâæ>àçápã?íçèa8Æß�és,²å�æ>ü�ãLå>ã?ù�ä>ã�æ¡8pã�èñø�å>ã?îñìpí íæ>ü�à åzú�ä>é�ápã?å·à åzæ>ü�ì�ælC2H L mnC2F�æ>ü�ã�å�ã?ß�î�ã�ähépähä>ã?ù�ã?àâápã�ä·ìpø�æ>ü�ã?ß»æ>àâù�ì�æ�ã?îAæ>ü�ã�3Iã?å>å>ì�ûpã�: å>àâß¼ù�ãUPPépæ>üæ>ü�ã?å>ã_ú¼ì�ä>æ>àâã?å�ü�ì�ápãÒæ>ü�ãÒå>ì>3IãÎå�ã?ù�ä>ã�æ3àâß�êxépäE3Lì�æ>àâé�ß =�æ>ü�ã�ä!ãAàâå�ß�é«, ì?è æ�é�æ�ã?íâí',²ü�àâù�ü�ú¼ì�ä!æªèìpù�æ>ø�ìpí íçèAìpø�æ>ü�ã?ß»æ>à ù�ì�æ�ã?î æ>ü�ãl3Iã?å!å>ì�ûpãR;A/

õ ö87 9:�%��;<���4úSü>=@?sý����Qÿ��A�c÷×øZù%ørú>û ü¡ý�ø9ù%þ'û^úSÿB�����ACï ìpß úPã�ä>êxã?ù�æ>íçè£6ªå�ã?ù�ø�ä>ã1å>àâû�ß�ì�æ>ø�ä>ã7å!ù!ü�ã03Iã?å²ã21Æà å�æA]�O×ã�å>ìs,ræ>ü¼ì�æ²àâß æ>ü�ã�ù�ìpå�ã7épê�3Iã?å>å!ì�ûpã�ìpø�æ>ü�ã?ßQ6æ>àâù�ì�æ>àçé�ß ='úPã�ä>êxã?ù�æ>íçè£6ªå�ã?ù�ø�ä>ãIå>ù�ü�ã03Iã?å�ù�é�ø�íâî~PPã3ù�é�ß¼å�æ�ä!ø�ù�æ�ã?î�ìpå7íçé�ß�û ìpå�,·ãLíâà©3Làçæ�æ>ü�ãLßKø<3#PPã�ä;épêìpø�æ>ü�ã?ß,æ>àâù�ì�æ�ã?î�3Iã?å>å>ì�ûpã?å3æ>ü�ãAìpî�ápã�ä�å>ì�ä>è ù�ìpßÐä>ãEDKø�ã?å�æ+/�ë�àçû�ü»æ3å�éS3Iã�æ>ü¼àâß�û�å!à©3Làâíâì�ä#,hépä�8 àâß æ>ü�ãù�ìpå�ã7épêcå>àçû�ß�ì�æ>ø�ä>ã?åA]

O×ã�å>ü�és,oæ>ü�ì�æ�Izy_å�épä!æaépê�úPã�ä>êxã?ù�æ>íçè£6ªå�ã?ù�ø�ä>ã1å>àâû�ß�ì�æ>ø�ä>ã1å>ù�ü�ã03Iã?å%ù�ìpßÍã21Æà å�æ+/ ö ß ú¼ì�ä>æ>àâù�ø�í ì�ä0=Kìpßø�ß�PPé�ø�ß¼î�ã?îÎìpî�ápã�ä!å>ì�ä>è ù�ìpß�ìpí\,·ì�èKå�êxépä>ûpã7ìIá�ìpíâàâî å>àâû�ß�ì�æ>ø�ä>ã�é�ß�ìpß»èa3Iã?å>å!ì�ûpã7épêzü�àâåaù�ü�é�àâù�ã7ã�ápã?ß,²àçæ>ü�é�ø�æ·å�ã�ã?àâß�û�ìpß»èIú¼ä>ã�áKàâé�ø�å>íçè3å>àçû�ß�ã?î�3Iã?å>å!ì�ûpã?å+/u@�é3å�ã�ã�æ>ü�àâå+=Kù�é�ß�å>àâî�ã�ähìpßAìpî�ápã�ä!å>ì�ä>è.ß�,²ü�é�à åû�àçápã?ß_æ>ü�ã%ú¼øQP¼í àâù�8pã�è�¬z­Iìpß�î5,·ìpß,æ>åhæ�é7êxépä>ûpã�ì�å>àçû�ß�ì�æ>ø�ä!ã%é�ß53Iã?å!å>ì�ûpã ° / ö ê�å>àçû�ß�ì�æ>ø�ä!ã?åDì�ä>ã>=Æå>ì?è>=­#P àçæ>åcíçé�ß�û�æ>ü�ã?ß3æ>ü�ãaìpî�ápã�ä!å>ì�ä!è�ù�ìpßIå>à©3Lú¼íçè;æ�ä>è7ã�ápã�ä!è7úPé�å>å>à\P íçãhå�æ�ä�àâß�û¡²_épê5íçã?ß�ûpæ>üT­�ø�ß,æ>àâí,àâæ�w¼ß¼î�åé�ß�ã�êUépä,�ü�àâù!ü����\�J�>Â�·R:¹¸#��²%;vÇÉÈf/'@%ü�ã?ßa²ñà åaì3á�ìpíâà îÒå>àâû�ß�ì�æ>ø�ä>ã;é�ß °GF ô�épæ�ã7ìIù�ä!ø�ù�à ìpí5î¼à��5ã�ä!ã?ß�ù�ãP5ã�æ`,hã�ã?ßAæ>ü�ã;ù�ìpå�ã�épê'î�àçû�àçæ>ìpí å>àçû�ß�ì�æ>ø�ä>ã?åDìpß�î�3Iã?å>å!ì�ûpã�ìpø�æ>ü�ã?ß»æ>àâù�ì�æ>àâé�ßAà å�æ>ü�ì�æ·æ>ü�ã�ìpî�ápã�ä!å>ì�ä>è_ù�ìpßápã�ä!àçêxè�ù�épä>ä>ã?ù�æ>ß�ã?å>å�épêDæ>ü�ã3å!àçû�ß�ì�æ>ø�ä>ã�² PPã�êUépä!ã3àâæ�é�ø�æ�ú ø�æ>å�²|Ú'æ>ü�àâå;ù�ìpß�ß�épæ�P5ã3î�é�ß�ã3êxépäl3Iã?å>å>ì�ûpãìpø�æ>ü�ã?ß,æ>àâù�ì�æ>àçé�ß å>ù�ü�ã03Iã?å+=»,²ü�àâù�ü àâå�,²ü»èAúPã�ä>êxã?ù�æ>íçè�6ªå>ã?ù�ø�ä>ã�å>ù!ü�ã03Iã?å²ì�ä>ã�úPé�å>å!à\P¼íçã�àâßAæ>ü�ì�æ²å�ã�æ�æ>àâß�ûQ/

õ ö8H IJ�B� ý�ÿuù?ù����Rú>ø��%þK� ;#÷Àø>LM�cû þBN*OP�»ü`ü�QÅû þ)�àSRLãñìpß¼î p�ã?íâí©3Lìpß�å>ø�ûpûpã?å�æ�ã?îÝæ>ü�ãñêxé�íâíçés,�àâß�û×, ì?è9æ�é.à©3Lú¼íçã03Iã?ß,æ_î�àçû�àâæ>ìpí²å>àçû�ß¼ì�æ>ø�ä>ã�å!ù!ü�ã03Iã?åP¼ìpå�ã?î.é�ß9ìpß,èNæ�ä!ì�ú�î�éKépäIú5ã�äA3�ø�æ>ì�æ>àâé�ß / )'ã�æ�:k���.Tz�.TVUXWA;bPPã ì×æ�ä!ì�ú�î�éKépäIúPã�äE3�ø�æ>ì�æ>àçé�ß / @%ü�ã?àçäå>ø�ûpûpã?å�æ>àçé�ß«,·ìpå%æ>ü�ã1êxé�íâíçés,�àâß�ûQÛ

Y ÿ�ã�èÎûpã?ß�ã�ä�ì�æ>àçé�ß�ù�é�ß¼å>àâå�æ>å²épê�ä�ø�ß�ß�àâß�û�� æ�é_ûpã?ß�ã�ä�ì�æ�ã�8pã�èGZ1ìpß�îÍæ�ä!ì�ú�î�éKépä\[<]Q/@%ü�ã1ú¼ø<P¼íâàâù8pã�è àâå^Z1ìpß�î æ>ü�ã1å�ã?ù�ä>ã�æ[8pã�èÎà å[_]»/

Y @�éLå>àçû�ße3Lã?å>å>ì�ûpã °!±�` ·s=�æ>ü�ã1å>àçû�ß�ã�ä#:J,²ü�é_ü�ìpåaæ>ü�ã�æ�ä�ì�úPî�é,épä7;·ù�éS3Lú¼ø�æ�ã?åƲ«ÇaT UXW· : ° ;A/

Y @�é�ápã�ä�àçêUè�å!àçû�ß�ì�æ>ø�ä>ãt²�é�ß 3Lã?å>å>ì�ûpã ° =�ù!ü�ã?ùE8À,²ü�ã�æ>ü�ã�ä:TS·N:k²%;cbÇ ° /~ô�épæ�ã æ>ü�ì�æ3ìpß»èpé�ß�ã,²àâæ>ü ìpù�ù�ã?å>å%æ�éLæ>ü�ã1ú¼øQP¼í àâùÆ8pã�è ù�ìpßÍápã�ä!àçêxè_æ>ü�àâå0/

ö ß ê ìpù�æ+=�æ>ü�à åaàâå ã21Æìpù�æ>íâèÎü�és,-V�æ�ã21Ææ�PPé,é>8nYed;ðnÐrå!àçû�ß�ì�æ>ø�ä>ã?å�,·épä�8zÛ�æ>ü�ã�ú¼ø<P¼íâàâù[8pã�è àâå3IéÆî�ø�íâø�ågfìpß�îLú¼ø<P¼íâàâùhã21KúPé�ß�ã?ß,æih�ìpß�îLæ>ü�ã²å�ã?ù�ä>ã�æv8pã�èLàâåzã21Æú5é�ß�ã?ß»æBj�å>ø�ù�üLæ>ü�ì�æghEj�Ç Ï 3IéÆî4k�:&fq;AÚ�å>àçû�ß¼àâß�ûìa3Iã?å>å>ì�ûpã ° ±mlonp àâå;î�é�ß�ã�PKè�ù�éS3Lú¼ø�æ>àâß�ûa² Ç °!q 3LéKîef : ìpß�î ápã�ä!à\w ù�ì�æ>àçé�ß�àâå;î�é�ß�ãLà ß�æ>ü�ãé>P,áÆàçé�ø�å, ì?è<;A/

Þ�ß�êUépä>æ>ø¼ß�ì�æ�ã?íçè>='æ>ü�à å�ì�ú¼ú�ä>é�ìpù!ü àâåbI^y L å�ã?ù�ø�ä>ã F O×ã_ß�éR, ã21Kú íçépä>ã�,²ü,è>/r3�épä7é�ß�ãIæ>ü�àâß�ûQ='ß�épæ�ãæ>ü�ì�æLß�é�3Lì�æ�æ�ã�äT,²ü¼ì�æ�æ�ä�ì�úPî�é,épä3úPã�äE3�ø�æ>ì�æ>àçé�ß.àâå3ø�å�ã?î?=Dìpß.ìpî�ápã�ä!å>ì�ä!è ù�ìpß.ìpí\, ì?èÆåIêxépä>ûpã á�ìpí àâîå>àçû�ß�ì�æ>ø�ä!ã?å7é�ßÐß�ã+,î3Iã?å>å>ì�ûpã?å�PKè�ù!ü�éKé�å>àâß�û�ì�ä�P¼àâæ�ä!ì�ä>èײ ±s` ·0=�ù�éS3Iú¼ø�æ>àâß�û ° ÇtTS·s:k²%;�:xä>ã?ù�ìpí íæ>ü�ãIìpî�ápã�ä!å>ì�ä>è�8Kß�és,²å:Z3å>àâß�ù�ã�æ>ü�àâå�àâå�æ>ü�ã3ú¼øQP íâàâù�8pã�è<;A=�ìpß�î ù�íâìpà©3_àâß�û_æ>ü¼ì�桲Nàâå�ìÎå>àçû�ß�ì�æ>ø�ä>ã�é�ß° /qô�és,#= ° 3_ì?è ß�épæ�ù�épä>ä!ã?å�úPé�ß�î æ�é×ì~3Iã?ìpß�àâß�ûpêxø�ív3Iã?å>å>ì�ûpã>='P¼ø�æ�{vCedSyqIzy L { g I L#L yqo g }>Cg I<ó g BAB j o[� L Hky>IQB g x�y jnL {|m g>L B2yfF L B5y�� L m£HXI£M>B g BAH\M>I gSLkj F�CTB2D+mnC2o5CT{�H hJh xEC j B0C�d L yaBAH\M>I</�ðÆéQ='ìêUépä>ûpã�ä!èAé�ß g I<ót3Iã?å>å>ì�ûpã�å�æ>àâíâí�V�ù�é�ø�ß,æ>å�Y_ìpå²ìIêUépä!ûpã�ä>è>=¼ìpß�î æ>ü�ã7å>ù!ü�ã03Iã7àâåaß�épæ�ì�æ²ìpíâí�å�ã?ù�ø�ä!ã>/

u

Page 108: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

@%ü�ã²å!àçæ>ø�ì�æ>àçé�ßIàâå�ã�ápã?ß5,·épä!å�ãU,�ü�ã?ß�B9�QCED+H +vD�æ�ä!ì�ú�î�éKépä�úPã�äE3�ø�æ>ì�æ>àçé�ß�åzì�ä>ã²ø¼å�ã?î /v3�épäDã21�ì>3Iú¼íâã>=àâß1æ>ü�ã ù�ìpå�ã·épê�d�ðQÐ ,hã ù�ìpß�ß�épæ�ã·æ>ü¼ì�æ�ìpß�ìpî�ápã�ä!å>ì�ä!è�,²ü�é²é>P¼æ>ìpàâß�å�å>àçû�ß�ì�æ>ø�ä!ã?å�êxä>éS3ræ>ü�ãDä>ã?ìpíKå>àçû�ß�ã�äù�ìpß êxépä>ûpã7ìLå>àâû�ß�ì�æ>ø�ä>ã;é�ßÍìIß�ã+, 3Iã?å>å>ì�ûpãty��¡m�HXBbD2mnyfHkD7C2/'@²ü,ø�å0=Æàâê�ìpß ìpî�ápã�ä!å>ì�ä>è�,·ìpß,æ>å�æ�éLêUépä!ûpãìLå>àçû�ß�ì�æ>ø�ä>ã�é�ß«3Iã?å!å>ì�ûpã ° æ>ü�ã�ìpî�ápã�ä!å>ì�ä>èÎù�ìpßÍî�éLæ>ü�ã1êxé�íâíçéR,²àâß�ûQÛ|w¼ß�î ° W � °!wl±xl np å>ø¼ù!ü æ>ü¼ì�æ° W °!w Ç ° 3IéÆîyfc/«Ð�å�8×êUépä3å>àçû�ß�ì�æ>ø�ä>ã?å#² W ìpß�îc² w é�ß 3Iã?å>å!ì�ûpã?å ° W ìpß�î °!w =cä>ã?å�úPã?ù�æ>àçápã?íâè>/@%ü�ã ìpî�ápã�ä!å>ì�ä!è é�ø�æ�ú ø�æ>å3êUépä!ûpã�ä>è ² Ç ² W ² w 3IéÆîef é�ß.æ>ü�ãa3Iã?å>å>ì�ûpã ° / @�é�å�ã�ã æ>ü�ì�æIæ>ü�à å�à åàâß�î�ã�ã?î ìIêxépä>ûpã�ä>è>= ß�épæ�ã1æ>ü�ì�æ

²Xz�Ç :k² W ² w ;_z�Ç�²XzW ²Xzw Ç ° W °!w Ç ° 3IéÆîyfx{

| }�~���(<���?��"_���.~[��~\� �#�u"u��� �!�@�#���A&!� ���E�[����� ��� �ª�#"%$��'& �u(

ö ßAê ìpù�æ+=Æù�é�ß¼å�æ�ä!ø�ù�æ>àâß�û�å�ã?ù�ø�ä>ã�å!àçû�ß�ì�æ>ø�ä>ã�å>ù!ü�ã03Iã?åaàâåhápã�ä>è_î�à�RÒù�ø�íçæ+Ú,àâæ·àâåhê ìpàçähæ�é3å>ì�è_æ>ü�ì�æaß�é�æ�ä!ø�íçèå>ì�æ>àâå�ê ìpù�æ�épä>è.�^F�yf� g x h ófK`B+CED j F�C�å>é�íâø�æ>àçé�ßÎàâå·èpã�æU8Æß�és,�ß /uÐ�åaìIå>æ>ì�ä>æ+=<,·ã1ù�ìpß ù�é�ß�å>àâî�ã�äaä!ã?íâìN1Æà ß�û�æ>ü�ãî�ã+w¼ß�àçæ>àâé�ßLépê�å�ã?ù�ø�ä!àçæªè_æ�é3å>ã�ã�,�ü�ì�æ·ù�ìpß�PPã;ìpù!ü�àçã�ápã?îÎæ>ü�ã?ß /�Ð�å àâßÒæ>ü�ã�ù�ìpå�ã;épê%3Iã?å>å!ì�ûpã;ìpø�æ>ü�ã?ß»æ>à\6ù�ì�æ>àçé�ß =z,hã�ù�é�ß�å>àâî�ã�ä�æ>ü�ã�ù�ìpå>ã#,²ü�ã?ßÍæ>ü�ã�ìpî�ápã�ä!å>ì�ä>è àâå�íâà©3Làâæ�ã?îÎæ�é_ûpã�æ�æ>àâß�ûÎì_å!àâß�û�íçã1å>àâû�ß�ì�æ>ø�ä>ã1é�ßì�3Lã?å>å>ì�ûpã3épêhü�à å²ù!ü�é�àâù�ã�P5ã�êxépä>ã3ü�ì?áÆàâß�ûAæ�éÎêUépä!ûpã3ìAß�ã+, å>àçû�ß¼ì�æ>ø�ä>ã>/1ðÆù!ü�ã03Lã?å¡,²ü�àâù�ü�ì�ä>ã3å�ã?ù�ø�ä!ãì�û�ìpàâß�å�æ%æ>ü¼àâå êUépäA3 épêzì�æ�æ>ìpù�8 ì�ä>ã7ù�ìpí íçã?î ¥ K L HXo5C#B7HiMSI g>Lkj F�C#B+D2mnC+o5C7BA/

@%ü�ãlw¼ä!å�æaú�ä>é�á�ìfP íçè�6ªå>ã?ù�ø�ä>ã Ï 6{æ>à©3Lã7å>àçû�ß�ì�æ>ø�ä>ã1å>ù!ü�ã03Iã�àâå²î�ø�ã�æ�é.)�ì>3IúPépä>æ+/v@²ü�àâåaå>ù�ü�ã03Iã�ù�ìpßP5ã�P ìpå�ã?î é�ß ìpß,èÎé�ß�ã269,·ì�è êxø¼ß�ù�æ>àçé�ß!T_=�ìpß�îa,·épä�8Æå%ìpå%êxé�íâíçéR,²å+Û

Y Ð�å>å>ø<3Lã�,hã.,·ìpß,æ7æ�é~PPãLìfP¼íçãLæ�éñå>àçû�ßÀ3Lã?å>å>ì�ûpã?å�épê%íçã?ß�ûpæ>ü��f/a@�é�ûpã?ß�ã�ä!ì�æ�ã�8pã�èKå+=cù!ü�é,é�å�ãÒ��1ã?íçã03Lã?ß»æ>å�� W�� � ��� W���W ��{�{�{s����� � � ����� ��W ø�ß�àâêUépäE3_íçèÒì�æ�ä!ìpß�î�éS3ÜêUä!éS3 æ>ü�ã�î�éS3Lìpàâß épêBT_/@%ü�ã?ßù�éS3Iú ø�æ�ã4� W�� � Ç�T�:�� W�� � ;A��� W���W Ç�T�:�� W���W ;A��{�{�{s����� � � Ç�T�:���� � � ;A����� ��W Ç�T�:��%� ��W ;A/�@²ü�ã_ú¼ø<P¼íâàâù8pã�è àâål:�� W�� � ��� W���W ��{�{�{0����� � � ����� ��W ;aìpß�îÎæ>ü�ã7å�ã?ù�ä!ã�æU8pã�è àâå#:�� W�� � ��� W���W ��{�{�{s����� � � ����� ��W ;A/

Y @�éLå>àçû�ße3Lã?å>å>ì�ûpã ° Ç�� W Ù+Ù+Ù����+=¼å>à©3Lú¼íçè_íçã�æaæ>ü�ã1å>àçû�ß�ì�æ>ø�ä!ã�P5ã:�� W�� �_� ��� w � ��� ��{�{�{R����� � ��� ;�{

Y @�éIápã�ä!àçêxèAì_å>àçû�ß�ì�æ>ø�ä!ã5:�� W ��{�{�{s���%�E;·é�ß«3Iã?å>å>ì�ûpã ° Ç�� W Ù+Ù+Ù����2=�ù�ü�ã?ù�8a,²ü�ã�æ>ü�ã�ä

T�:�� W ;6bÇs� W�� � � Ú.T�:�� w ;�bÇ�� w � � � Ú+Ù+Ù+ÙsÚ.T�:��%�E;6bÇM��� � ��� {: Ö ß¼íçèÒàçê�ìpíâíPépêcæ>ü�ã?å�ã7ì�ä!ã�æ�ä!ø�ã7àâå æ>ü�ã7å>àçû�ß�ì�æ>ø�ä>ã�ù�é�ß�å>àâî�ã�ä!ã?îÎá�ìpíâà î /Ä;

O×ã#,²àâí í5å>ã�ã7ì3ú�ä>éKépê�épêcå�ã?ù�ø�ä�àçæªèÎêxépä%æ>ü�àâåaå>ù�ü�ã03Iã7ß�ã21Kæ%æ>à§3Iã>/

ò

Page 109: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���% S¡£¢�¤.¥� �¦¨§J©£ªv«x¬�¤.§#­S¬£��®¯ «±° ¯g²A³£´¶µ¸· �A¦�¤.©#®%¹�º�¦� �©���¦�© ¯ ¤�§#»%¦�©£¼£¤.¬�»�½A§¾�¤�©£ª�¢E¥�¥�©£¤^¿£©��%¬�¦�½%¬£�xÀÁ¬�¦�Â

ÃMÄyÅÇÆÉÈËÊÉÄ ÌxÍ

Î ÏMÐ:ÑÒ!ÓoÔÖÕJ×�ØÚÙÛÎ�Ü#Ï�Ý�ÔÞÑà߶Ý.á�â:ÓgÙXã:ØXÑàß�ä�Ð:ÑBÔåÑ

æG¢èç�¤. �¢�é%§r¤�¢�¡ê �¢�ëc¦�½�¢èì�¬£íy»î©£¤�¦èï(ð�¦� Síy¢è¥� �¼��%¬�¦�¹�¤.¢\¥�º.½%¢Eíy¢:ñ�ª�©£¤íy¢E¥�¥�¬�¼£¢E¥g©£ª ­�¢E��¼£¦�½4òôó�ª�¤�©�í*­S¬£¥�¦¦� Síy¢£õBö\¢Eº�¬£­S­X¦�½%¬�¦�÷ùø�úôûêüEï�ýEþ ÿ úôûêüEï�ý��� S¥\¬y©���¢�ð�ë¬E§�ª&¹%�%º�¦� �©��Xõï�õ�ÀÇ¢�§ ¼£¢E��¢�¤�¬�¦� �©��sº�©���¥� S¥�¦�¥y©£ªÁº�½�©#©�¥� S��¼���ò�¢E­�¢Eíe¢E�A¦�¥6¬�¦4¤.¬£��®�©�í ª�¤�©�í�úôûêüEï�ý þ ñ& >õ�¢£õ��ɦ�½%¢®�©�í4¬£ S�G©£ª�÷ ó�õ��o½#¹%¥�Vëi¢�º.½%©0©�¥�¢� ��� ��ü�������Eü���Eü��� � � ü��� ����� úôûêüEï�ýEþrõ��%©£¤:¬£­ ­���ü��Ëñ�ë\ S¦�½ï���� � ò�¬£�%®!�#"±úôûêüEï�ý�ógëg¢è¦�½%¢E�6º�©�íe»�¹�¦�¢%$'&(� )+*�÷vñ(,&(� )Eó�õ-�\½�¢�»�¹�ç�­S Sº/.£¢�§�021嬣��®J¦�½%¢¥�¢Eº�¤.¢�¦3.£¢�§5461 ¬�¤�¢�¬£¥Éª�©�­S­�©�ë\¥�ø

4617*98 ��� �:<;=� � >>>?@� � � �����A<;=���B>>>?@� ���DC 0217*98 $E��� �F$G;=� � >>>H$I� � �$E�����J$G;=���K>>>H$I� ���DC�#õL�V©4¥� �¼���¬£�!ò�ð�ç� �¦oíy¢E¥�¥�¬�¼£¢NMO*PMQ� >>>RM����¥� Síe»�­�§TS<»�  ºR.�©�¹�¦RUy¦�½�¢¶º�©£¤�¤�¢E¥�»î©��%®%  ��¼¶¢E�0¦�¤. �¢E¥ª�¤�©�íM¦�½�¢B¥�¢Eº�¤�¢�¦ .£¢�§Ç¬£�%®Á¥�¢E��®^¦�½�¢Eí�õ6�o½0¹�¥�E¦�½�¢B¥. �¼��%¬�¦�¹�¤�¢�ë\  ­S­�çÚ¢oñ(���� þ3V ü�<;=� þ/W ü���ôü��� � þ-X ó�õ�V©e S­S­S¹�¥�¦�¤.¬�¦�¢'�� �ª@ëi¢Ç묣�A¦�¦�©4¥� �¼���¬4íy¢E¥�¥�¬�¼£¢2MO*�û�ï6>>>�ïY�%ëg¢�¥�¢E��®�¦�½�¢Áçî©�Zê¢E®�¢E�0¦�¤. �¢E¥�ø[ ���� � <;=� � >>> @� � ������� <;=��� >>> @� ��� \

] õL�V©4¡£¢�¤. Sª8§!¬J¥� �¼��%¬�¦�¹�¤.¢6ñ( ��ü�^;�ü���ôü����óo©��ùíy¢E¥.¥�¬�¼£¢NMQ�_>>>RM���Úëi¢:¥� Síy»�­�§4¡£¢�¤. Sª8§�¦�½%¬�¦�ª8©£¤¬£­S­`�èñ�ë\ �¦�½ ï2�a�-� òôógëi¢Á½�¬E¡£¢r÷vñ(<&>ó�b*c$'&d� þ e õ

æG¢J��©�ë »%¤�©�¡£¢e¦�½�¢e¥�¢Eº�¹�¤� �¦�§±©£ªi¦�½� S¥Ç¥�º.½%¢Eíy¢e¬£¥�¬Gï(ð�¦� Síe¢e¥� �¼��%¬�¦�¹�¤.¢:¥.º.½�¢Eíy¢£õeì�¢�¦�¹%¥^¤�¢Eº�¬£­ ­ë\½%¬�¦�¦�½% S¥@íy¢E¬£�%¥�õ@æG¢o½%¬ô¡£¢o¬£�r¬£®%¡£¢�¤.¥�¬�¤�§¶ë\½�©è¼£¢�¦�¥�¦�½�¢»�¹�ç�­S  ºf.£¢�§N021g�£º�¬£�y¬£¥�.¶ª8©£¤v¬^¥� �¼��%¬�¦�¹%¤�¢©��G¬£�A§±íy¢E¥�¥�¬�¼£¢!M  �¦Çº.½�©#©�¥�¢E¥�X¼£¢�¦�¥Á¦�½�¢y¥� �¼��%¬�¦�¹�¤.¢'���®ù¦�½%¢E��¦�¤. �¢E¥è¦�©�ª8©£¤�¼£¢e¬6¡�¬£­S S®±¥� �¼��%¬�¦�¹%¤�¢©�� ¬���¢�ë íy¢E¥�¥�¬�¼£¢DMih3j*kMGõræG¢yëg¬£�0¦Á¦�©�çî©�¹%�%®ù¦�½�¢e¥�¹�º�º�¢E¥�¥Ç©£ªi¬£�0§�¬£®�¡£¢�¤.¥�¬�¤�§±©£ªB¦�½� S¥^¦�§#»î¢£õæG¢Áë\ S­S­�®�©:¦�½% S¥É S�J¦�½�¢Á¥�¦�¬£��®%¬�¤.®6ëg¬ô§�øBëi¢Á¥�½%©ôë�¦�½%¬�¦o¬£�0§J¬£®�¡£¢�¤�¥�¬�¤�§6ë\½�©yº�¬£��ª8©£¤.¼£¢Á¥� �¼��%¬�¦�¹�¤.¢E¥ë\ �¦�½6½% �¼�½6»%¤�©£ç�¬�ç� S­S �¦¨§rº�¬£��çî¢^¹%¥�¢E®6¦�©y S�A¡£¢�¤.¦i¦�½�¢^©���¢�ð�ë¬E§6ª&¹%�%º�¦� �©���÷!ë� �¦�½J½� �¼�½4»�¤�©£ç�¬�ç� S­  �¦�§'�#¬º�©��A¦�¤.¬£®� Sº�¦� �©��XõlNmonqp<rsnEt uwvdx ÷HydzT{�ñ(|(ü~}�ó����Y�^����-{'� x=� �^����y��Y�����(�E���k�(�E�Q��{Y�L�E�Y����z�y��'�<{'� � �R�Tz���E���D��ydz�{ñ(|�ü~��ò�}�ó���z�~� � �����Y����y(�D��z�y��'�<{G� � �R��z���E���D���� rsp_p<� � ¥�¥�¹%íe¢Jëg¢6½�¬E¡£¢!¬£� ¬£®�¡£¢�¤.¥�¬�¤.§T� ë\½�©±ª�©£¤�¼£¢E¥y¥� �¼��%¬�¦�¹%¤�¢E¥�ë� �¦�½P»�¤�©£ç�¬�ç� S­  �¦�§ ��õGæG¢¥�½�©ôëa½%©ôë�¦�©y¹%¥�¢%�a¦�©e S�0¡£¢�¤�¦g¦�½�¢^©���¢�ð�ë¬E§�ª�¹��%º�¦� �©���÷@õ ¯ ©��%¥�¦�¤.¹%º�¦É¬£­�¼£©£¤� �¦�½%í?�+h�ñ�ë\½% Sº�½J¼£¢�¦�¥\¬¡�¬£­S¹%¢�$�"�úôûêüEï�ý � ¬£��®�¦�¤. �¢E¥É¦�©�¡��%®�¬£�iQ"±úôûêüEï�ý � ¥�¹%º.½!¦�½%¬�¦�÷vñ(îóf*c$�󬣥oª8©�­S­S©ôë\¥�ø

ï

Page 110: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�Lh>ñ($%ó��¢ � úAï�ü����ü<ò�ýG£q�q¢ � úôûêüEï�ý$'&¥¤~� )�¤¦*P$ª�©£¤\¬£­S­`�.ü��4ë� �¦�½Gï2���-�Ëò^¬£�%®#�D"�úôûêüEï�ý:¬£�%®Pñ(��ü��#ó2j* ñ(��¢�ü��§¢Eó�ø &(� ) � úôûêüEï�ý þ £,$ &d� ) *�÷vñ( &(� ) ó0�19*78 $E��� �:$G;=� � >>>H$I� � �$ ����� $ ;=��� >>>H$I� ��� C¤�¹%�i�rñ�0�1Góɹ%�A¦�  ­Ú S¦É¤�¢�¨0¹%¢E¥�¦�¥\¬y¥� S¼��%¬�¦�¹�¤�¢Ç©��iMO*cMQ� >>>RM�� Sª�Mi& ¤ *��§¢I��¬�çÚ©£¤.¦£%©£¦�½�¢�¤�ë\  ¥�¢'��¤�¢�¦�¹�¤.��¦�½%¢Áº�©£¤�¤�¢Eº�¦�¥. �¼��%¬�¦�¹�¤�¢Ç¦�©#�¢�¡£¢E�0¦�¹%¬£­S­S§'�,� ©�¹%¦�»�¹�¦�¥o¬yª�©£¤�¼£¢E®ù¥� �¼��%¬�¦�¹�¤.¢yñ(��Eü���ôü�@�.óg©���Mih� >>>~Mih� Sª�M h& ¤ j*��§¢Á¬�çÚ©£¤.¦£�©£¦�½�¢�¤�ë\ S¥�¢'��©�¹�¦�»�¹�¦¦<& ¤

· �Pëi©£¤.®�¥�o� h ®�©0¢E¥�¦�½�¢4ª�©�­S­�©�ë\ S��¼�øÁ S¦2¡%¤.¥�¦Áº�½�©#©�¥�¢E¥¶¬x¤.¬£�%®�©�íå S�%®�¢�ZQ��¢y¬£�%®P¬!¤.¬£�%®%©�í ç� �¦+�q¢�õ�o½% S¥®�¢¡��%¢E¥g¬:»î©�¥� S¦� �©��J  �J¦�½%¢Ç»�¹�ç�­S  º¦.£¢�§6¬�¦Éë�½% Sº.½©�Lh�ë� S­S­%»�­S¬£º�¢Ç¦�½�¢^¡�¬£­ ¹�¢%$4¦�½%¬�¦o �¦ë¬£�A¦�¥o¦�© S�A¡£¢�¤.¦�õ�ñ�ª�¬£íe¢E­�§'�@� h ¥�¢�¦�¥/$'& ¤ � ) ¤ *c$�õ ó«�\½�¢�¤.¢Eíe¬£ S�%®�¢�¤g©£ª�¦�½%¢Ç»�¹�ç�­S  º¦.£¢�§J S¥g¼£¢E��¢�¤�¬�¦�¢E®!½�©���¢E¥�¦�­S§£õ�o½% S¥Áíe¢E¬£�%¥Á¦�½%¬�¦��+h%�~{Y�¬z=y­�G�a{Y�@�5�D��z~z�{��q�#�®�'�^�=z���¯°�T�R±q�����^�%�(�®�Iz�©�fy(�(��Mi& ¤ *k� ¢ õ©ª�©ôë��6�Lh¤.¹%�%¥/�Ûñ�¼� �¡#  ��¼!�a¦�½�¢Ç»�¹%ç�­S Sº+.£¢�§J¦�½%¬�¦3� h »%¤�¢�»�¬�¤�¢E®ÚóB¹��A¦� S­`� ¤�¢�¨0¹%¢E¥�¦�¥o¬y¥� �¼���¬�¦�¹�¤�¢^©���íy¢E¥�¥�¬�¼£¢MGõ � ¥Éëg¢¶��©£¦�¢E®_�<� h º�¬£�x¼£¢E��¢�¤.¬�¦�¢r¦�½% S¥É¥� �¼���¬�¦�¹�¤�¢�¬£¥�­�©���¼e¬£¥LM & ¤ j*a�§¢�õ¦²è¦�½�¢�¤�ë� S¥�¢'�@� h ¥� Síy»�­S§¬�çÚ©£¤.¦�¥o¬£�%®�¼� �¡£¢E¥�¹�» õ� ¥.¥�¹%íe S�%¼N� h ½%¬£¥g��©£¦g¬�çî©£¤�¦�¢E® �ê S¦B¼� �¡£¢E¥g¦�½�¢^¥� �¼��%¬�¦�¹%¤�¢\¦�©!�a¬£�%®6º�©��0¦� S�0¹%¢E¥i¤�¹%�%�% S�%¼2�a¹%�0¦� S­ �¦É¤�¢�¦�¹�¤.��¥g¬eª8©£¤�¼£¢�¤.§�ñ(��Eü���Eü����óg©���íy¢E¥�¥.¬�¼£¢2Mih� >>>RMih� õ ¯ ©��%®% �¦� S©���¢E®6©���¦�½�¢Çª&¬£º�¦o¦�½%¬�¦3�Lhî½�¬£¥��©£¦o¬�çî©£¤�¦�¢E®_��ëg¢Áíe¬E§���©£¦�¢Ç¦�½%¬�¦\¦�½% S¥ S¥É¬:¡�¬£­S S®6ª8©£¤.¼£¢�¤�§Jë� �¦�½�»%¤�©£ç�¬�ç� S­S �¦¨§D��õ � ¡�¬£­S S®�ª�©£¤�¼£¢�¤�§� S�»�¬�¤�¦� Sº�¹%­ ¬�¤èíy¢E¬£��¥^¦�½%¬�¦�÷vñ( &¥¤ ó+*k$ & ¤ � þ´³e ¤ ¬£�%®G¬£­S¥�©6¦�½%¬�¦�M h j*µMGõ:°#©��X¬£¥�¥�¹%íe  ��¼J¦�½�¬�¦è¦�½� S¥^ S¥^¬¡�¬£­S  ®�ª8©£¤�¼£¢�¤.§6ëg¢Á½%¬ô¡£¢�ª�©�¹%�%®�¦�½�¢�  �A¡£¢�¤.¥�¢Çª8©£¤3$�¬£¥\­S©���¼e¬£¥¦Mih& ¤ *�� ¢ ñ&¥� S�%º�¢�$'&¥¤�� )�¤/*P$�ó�õ

æG¢o¶�¹%¥�¦���¢�¢E®¶¦�©è¬£�%¬£­S§0Â�¢i¦�½�¢B»%¤.©£ç�¬�ç� S­S S¦�§è¦�½%¬�¦o� h ¤�¢�¦�¹�¤��%¥ ¦�½�¢gº�©£¤�¤�¢Eº�¦�¬£�%¥�ëg¢�¤Eõ6�o½% S¥ ½�¬�»%»î¢E�%¥¬£¥o­�©���¼y¬£¥¦�½�¢Çª�©�­S­�©�ë\ S��¼¶¦�½�¤�¢�¢Ç¦�½% S�%¼�¥g½�¬�»%»î¢E�Xøoñ<ïôó �+hî S¥¬�ç�­�¢^¦�©y¤�¢�¦�¹�¤��6¬:º�©£¤�¤.¢Eº�¦\¥� �¼��%¬�¦�¹%¤�¢^¦�©�àñ& �õ�¢£õ��oMi& ¤ j*µ�§¢Eó��Éñ���ó��2©�¹�¦�»�¹�¦�¥�¬�¡�¬£­S S®�ª�©£¤�¼£¢�¤�§'��¬£�%®�ñ ] ó^¦�½�¢eª�©£¤�¼£¢�¤�§G½%¬£¥�M h& ¤ *µ�q¢�õ��\½�¢»%¤�©£ç�¬�ç�  ­S �¦�§�©£ª�¢�¡£¢E�0¦eñ<ïôó� S¥�¥. Síy»�­�§�ï�·Y��ñ&¥� S�%º�¢��§¢Áëg¬£¥Çº�½�©�¥�¢E��¬�¦^¤.¬£�%®�©�í���¬£�%®� S¥� S�%®�¢�»î¢E�%®�¢E�0¦©£ªB¦�½�¢:¡ê �¢�ë ©£ª �^ó�õ ¯ ©��%®% �¦� �©��%¢E®!©���¦�½�¢rª�¬£º�¦Ç¦�½�¬�¦è¢�¡£¢E�0¦4ñ<ïôó\©#º�º�¹%¤.¥�X¦�½�¢r»%¤�©£ç�¬�ç� S­S �¦¨§6©£ªi¢�¡£¢E�A¦ñ���óo S¥�¢�Z�¬£º�¦�­�§5�G�î¦�½�¢r¬£¥�¥�¹%íe¢E®x»%¤.©£ç�¬�ç� S­S S¦�§4©£ªvª�©£¤�¼£¢�¤�§xª8©£¤¸��õ+�� S��¬£­S­�§'��º�©���®% �¦� �©���¢E®!©��x¦�½�¢¶ª�¬£º�¦¦�½%¬�¦É¢�¡£¢E�A¦�¥¶ñ<ïôóg¬£�%®Gñ���óiçÚ©£¦�½6©êº�º�¹�¤��êëg¢�.#��©�ë�¦�½%¬�¦ëi¢Çír¹%¥�¦É½%¬ô¡£¢�Mih¹j*PMGõi°ê©r¦�½%¢�¤�¢^¢�Zê S¥�¦�¥É¬�¦­�¢E¬£¥�¦o©���¢��i¥�¹%º�½�¦�½%¬�¦3Mi&/j*PM h& õ · ªV¦�½%  ¥¦�6*P��¢è¦�½�¢E��ëg¢�¬�¤�¢�®�©��%¢4ñ&¥� S�%º�¢Ç¢�¡£¢E�0¦rñ<ïôóÉ©#º�º�¹�¤.¤�¢E®�ëg¢.#��©�ëK¦�½%¬�¦LM &¥¤ j*c�§¢¶¥�©�M h& ¤ *P�q¢Eó�õÇ°ê  �%º�¢�M h   ¥Éò�ç� S¦�¥\­�©���¼��Ú¦�½�¢�¤�¢: S¥�»%¤�©£ç�¬�ç� S­S �¦¨§6¬�¦Ç­�¢E¬£¥�¦:ï�·ôò¦�½%¬�¦3�6*c� ¢ õ

¾B¹�¦�¦�  ��¼4¢�¡£¢�¤�§#¦�½% S�%¼4¦�©£¼£¢�¦�½�¢�¤��Xëg¢:¥�¢�¢¶¦�½�¬�¦¸� h  S�0¡£¢�¤�¦�¥^÷Gë\ �¦�½x»%¤�©£ç�¬�ç� S­S �¦¨§ �; >I�¸> �� *º�Y·Y��ò�õ°ê S�%º�¢ùëi¢».#�%©ôë ¦�½%¬�¦�÷  S¥ùñ(|(ü~}�ó�ð�©��%¢�ð�ëg¬ô§'�èëg¢�í¶¹%¥�¦J½%¬ô¡£¢ �Y·Y��ò¼�½}6¬£�%®s¦�½�¢�¤�¢�ª�©£¤�¢ �¾�¿��ò}��º�©�íy»�­�¢�¦� S�%¼y¦�½�¢Á»%¤�©#©£ª_õ

À Á%ÔåÕ:ØX״¶Ý�â:áÞÙXÐ:ÑÒ!ÓoÔåÕ6×�ØÚÙ*ß�ä�Ð:ÑBÔåÑ

æG¢ríe¬ô§x��©£¦�¢�ñ&¬�¦è­S¢E¬£¥�¦�ó¦¨ëi©�­S Sí4 �¦�¬�¦� �©��%¥©£ªv¦�½�¢:ì�¬£íe»Ú©£¤.¦\¥�º.½%¢Eíy¢'�Ú¬£��®xëg¢¶¥�¹%¼£¼£¢E¥�¦è½%¢�¤�¢¶¥�©�íe¢ëg¬ô§#¥Ç¦�©!¬ô¡£©� S®�¦�½�¢Eí�õ · ��¥�¹%ç�¥�¢�¨#¹�¢E�A¦^­�¢Eº�¦�¹�¤.¢E¥�Xëi¢yë� S­S­�¥�¢�¢r©£¦�½%¢�¤�� çî¢�¦�¦�¢�¤Çëg¬ô§#¥Á¦�©�¬ô¡£©� S®G¦�½�¢E¥�¢»%¤�©£ç�­�¢Eí4¥�õÃ<Ä(ÅoÆ_ÄdÆoÅÈÇ¸É Ê�Ë^Ä(Ì@ʧͬǸÍ,Î�Î�Ï_ÅoÍ,ΧР�o½%¢eì�¬£íy»î©£¤�¦eï(ð�¦� Síe¢4¥� �¼��%¬�¦�¹%¤�¢e¥�º.½%¢Eíy¢4 S¥Ç¢�Z�¬£º�¦�­�§�¦�½%¬�¦�ør �¦©��%­�§s¬£­ ­�©ôë\¥�¥�¢Eº�¹%¤�¢G¥� �¼��% S�%¼m©£ªr¬Ë¥� S��¼�­�¢±íy¢E¥�¥�¬�¼£¢£õ ñ · ¦! S¥6¢E¬£¥�§s¦�©�¡��%®s¬�¦�¦�¬£º~.#¥ùë\½%  º.½M¬£­S­�©�ë

Page 111: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

ª8©£¤�¼£¢�¤.§� SªV¦¨ëi©J©£¤�íy©£¤�¢¶íy¢E¥�¥�¬�¼£¢E¥�¬�¤�¢¶¥� S¼���¢E®Xõ ó ¯ ¬£�!ëi¢�  íy»%¤�©�¡£¢Á¦�½% S¥�Ñ»²^��¢�¦�½% S��¼:ëg¢¶º�¬£�x�%©£¦�¢ Síeíy¢E®%  ¬�¦�¢E­�§6  ¥É¦�½%¬�¦� SªVëg¢¶��¢�¢E®x¦�©4¥� �¼��»Ò íy¢E¥�¥�¬�¼£¢E¥:ñ�¢E¬£º.½±©£ªv­�¢E��¼£¦�½!òEógëg¢¶º�¬£�!¶�¹�¥�¦o¼£¢E��¢�¤.¬�¦�¢Ò  S�%¥�¦�¬£�%º�¢E¥\©£ªv¦�½�¢�ìV¬£íy»î©£¤�¦\¥�º�½�¢Eíy¢�ñ�©£¤��� S�!ª�¬£º�¦��A§ ï(ð�¦�  íy¢¶¥� S¼��%¬�¦�¹�¤�¢�¥.º.½�¢Eíy¢�ó�õ3�\½% S¥É¼� �¡£¢E¥¬e��¢�ë ¥� S¼��%¬�¦�¹�¤�¢Á¥�º�½�¢Eíy¢�®%¢¡���¢E®�¬£¥oª�©�­S­�©�ë\¥�øï�õ�ÀÇ¢�§y¼£¢E�%¢�¤.¬�¦� �©��6»%¤�©êº�¢�¢E®%¥gç0§y¤�¹%�%�% S�%¼D{Y�@��.£¢�§e¼£¢E�%¢�¤.¬�¦� �©���¬£­�¼£©£¤. �¦�½�í*ª�©£¤g¬6ï(ð�¦� Síy¢^¥� �¼���¬�ð¦�¹�¤.¢¥.º.½�¢Eíy¢3Ò ¦� Síy¢E¥@¦�©Ç¼� �¡£¢�ñ�021��Eü�461i�(ó�üôñ�0�1D;£ü�461#;ôó�ü���ôüôñ�021�Ó±ü�461#Ó�ó�õf�o½�¢»�¹%ç�­S Sº.£¢�§i021  S¥É¥�  íy»�­�¢yñ�021i�Eü����üR021#Ó!óg¬£��®�¦�½�¢�¥�¢Eº�¤.¢�¦3.£¢�§� S¥�ñÔ461��Eü���Eü�461#Ó!ó�õ�#õL�V©6¥. �¼���¥�¢Eº�¹�¤�¢E­�§xëi¢:��¢�¢E®±¦�©!íe¬£ S�0¦�¬£ S�±¬6º�©�¹��A¦�¢�¤Á S��®% Sº�¬�¦� S��¼4ë\½% Sº�½5.£¢�§ùëi¢y¥�½�©�¹�­S®x¹%¥�¢£õ�o½%¢º�©�¹��A¦�¢�¤ S¥v S�% S¦� S¬£­S �Â�¢E®¶¦�©4ïo¬£�%®4 S¥v S�%º�¤�¢Eíe¢E�A¦�¢E®y¢E¬£º�½4¦�  íy¢Éëi¢�¥� �¼��Xõ · ª`Õ S¥v¦�½�¢oº�¹%¤�¤�¢E�0¦¡�¬£­S¹�¢g©£ª�¦�½�¢º�©�¹%�0¦�¢�¤���¦�½�¢E�:¦�©Ç¥� S¼��ríe¢E¥�¥�¬�¼£¢¦M�Öñ�ë\½% Sº�½¶ S¥@¦�½�¢¦Õ�×dØ^íy¢E¥�¥.¬�¼£¢gëg¢o¬�¤�¢É¥� �¼��� S��¼0óëg¢�¥�  íy»�­�§:¥� �¼��6¹%¥� S�%¼�.£¢�§�461DÖi¦�©r¼� �¡£¢�¥� S¼��%¬�¦�¹�¤�¢¸Ù,ÖB¬£��®4©�¹�¦�»�¹�¦èñ�Õ�üRÙ,Ö.ó�õvæG¢Ç S��º�­S¹%®�¢3Õ� S�¦�½�¢y¥� S¼��%¬�¦�¹�¤�¢+¶�¹�¥�¦^¦�©�í4¬Y.£¢y¡£¢�¤. �¡�º�¬�¦� �©���¢E¬£¥. �¢�¤Eõ�ª�©£¦�¢e¦�½%¬�¦^ëg¢eí¶¹%¥�¦è¤.¢�ª�¹%¥�¢¶¦�©!¥� S¼��±¬£�0§íy©£¤.¢Áíy¢E¥�¥�¬�¼£¢E¥\©��%º�¢NÕ%Ú¼Ò*õ] õL�V©\¡£¢�¤. Sª8§^¬\¥� S¼��%¬�¦�¹�¤�¢�ñ�Õ�üRÙ�ó ©��¶íe¢E¥�¥�¬�¼£¢-M ë\ �¦�½Ç¤.¢E¥�»î¢Eº�¦ ¦�©+.ê��©ôë��Á»�¹�ç�­S Sº¹.£¢�§�021g��¥� Síy»�­S§¤.¹��J¦�½%¢Á¡£¢�¤. �¡�º�¬�¦� S©��!¬£­�¼£©£¤. �¦�½�íÛª8©£¤o¦�½%¢4ï(ð�¦� Síy¢�¥. �¼��%¬�¦�¹�¤�¢Ç¥�º�½�¢Eíy¢¶©��iMÞ¹%¥� S�%¼r»�¹�ç�­S Sº+.£¢�§021#Ö�õ

· ¦�¥�½�©�¹�­S®M��©£¦�çÚ¢�®% �ÛJº�¹%­�¦4¦�©�º�©��0¡ê S�%º�¢�§£©�¹�¤.¥�¢E­Sª�¦�½%¬�¦�¦�½% S¥� S¥� S��®�¢�¢E®M¥�¢Eº�¹�¤�¢�ª�©£¤!¥� �¼��% S�%¼gÒíy¢E¥�¥�¬�¼£¢E¥�õ�ÜB¡£¢E�P¥�©��@¦�½�¢�¤�¢6¬�¤�¢4¥�¦�  ­S­v¬x�#¹%í¶çÚ¢�¤�©£ª­  Síe �¦�¬�¦� �©���¥eñ�ëi¢Jë\ S­S­�¤�¢Eíe©ô¡£¢�¥�©�íy¢e©£ªÉ¦�½�¢E¥�¢ S�!¥�¹�ç�¥�¢�¨0¹%¢E�A¦\­S¢Eº�¦�¹�¤�¢E¥�ó�ø^ñ<ïôó¦�½�¢��#¹%í¶çî¢�¤o©£ªvíy¢E¥�¥�¬�¼£¢E¥%Ò ¦�©eçî¢�¥� �¼��%¢E®!��¢�¢E®%¥\¦�©eçî¢2.#�%©ôë\�ù S�¬£®�¡�¬£��º�¢'��¬�¦^¦�½�¢:¦� Síy¢r©£ª-.£¢�§x¼£¢E�%¢�¤.¬�¦� �©��_£iñ���ó�¦�½%¢:¥�¢Eº�¤�¢�¦%.£¢�§±¬£�%®±»�¹%ç�­S Sº�.£¢�§±½%¬E¡£¢e¥. �Â�¢DÝeñ�ÒKóë\½% Sº�½:ë\ S­S­Açڢɹ%�0ë\ �¢E­ ®�§Á¬£¥-Ò ¼£¢�¦�¥i­S¬�¤�¼£¢'£îñ ] óV¦�½�¢o¥� �¼���¢�¤v��¢�¢E®�¥�¦�©�íe¬£ S�0¦�¬£ S�:¥�¦�¬�¦�¢¶ñ�¦�½�¢oº�©�¹��A¦�¢�¤(ó�õÃ<Ä(ÅoÆ_ÄdÆoÅcÊqÞoÆ�Å�Í,ßaÇLÍ<Î�ÎÏ ÅoÍ,ÎGÐ � ��©£¦�½�¢�¤^­S Síe �¦�¬�¦� S©���©£ª�¦�½�¢rì�¬£íy»î©£¤�¦è¥�º.½�¢Eíe¢r S¥\¦�½%¬�¦^ �¦� S¥\©��%­�§®�¢¡���¢E®yª�©£¤iíe¢E¥�¥�¬�¼£¢E¥�©£ªX¬�»�¬�¤�¦� Sº�¹%­ ¬�¤�­�¢E��¼£¦�½xñ�ë\½� Sº.½eír¹%¥�¦BçÚ¢¦.ê��©�ë\�e S�e¬£®%¡�¬£�%º�¢�ó�£#ª&¹�¤�¦�½�¢�¤.íe©£¤�¢'�¦�½�¢�»�¹�ç�­  Sº�à�¥�¢Eº�¤�¢�¦«.£¢�§ê¥e¬£¥yëg¢E­S­¬£¥y¦�½�¢!¥� �¼���¬�¦�¹�¤�¢� �¦�¥�¢E­�ªè½%¬E¡£¢!¥. �Â�¢áÝeñ8òôór¥�© ¥� �¼��� S��¼±¡£¢�¤�§ ­S©���¼íy¢E¥�¥�¬�¼£¢E¥� S¥¢�Z#¦�¤.¢Eíy¢E­�§� S��¢Û4º� �¢E�0¦�õvæG¢�íe¬ô§6�%©£¦�¢Ç¦�½%¬�¦\ �¦É S¥¢E¬£¥�§�¦�©e Síe»%¤�©ô¡£¢Ç¦�½� S¥��½�©ôëg¢�¡£¢�¤��%ç#§½%¬£¥�½% S�%¼Ç¦�½�¢�íy¢E¥.¥�¬�¼£¢\çî¢�ª�©£¤�¢\¥� �¼��� S��¼Á �¦�õ · õ�¢£õ��0¦�©r¥� �¼��eíy¢E¥.¥�¬�¼£¢LMâ¡%¤.¥�¦�º�©�íy»�¹�¦�¢¬ãMO*cä ñ(Mùóv¬£��®¦�½�¢E�4¥� S¼��åãM¬£¥B¹%¥.¹%¬£­>õ · ª_äÖøêúôûêüEï�ýs¢�ÿ úôûêüEï�ý � ¦�½%¢E�e¦�½% S¥B¬£­S­�©ôë�¥v¦�½�¢\¥� S¼��% S��¼^©£ªX¬�¤�ç� �¦�¤�¬�¤. S­�§0ð�­�©���¼íy¢E¥�¥�¬�¼£¢E¥Éë� �¦�½�©�¹�¦i¼£¤�©ôë\  ��¼¶¦�½%¢è»�¹�ç�­S Sº�à�¥�¢Eº�¤�¢�¦-.£¢�§J©£¤g¦�½%¢^¥. �¼��%¬�¦�¹�¤�¢^­�¢E��¼£¦�½ õ · �6»�¬�¤�¦� Sº�¹�­S¬�¤èñ&¬£�%®¦�½% S¥Çë\  ­S­�çî¢Eº�©�íy¢J Síy»î©£¤�¦�¬£�0¦^­ ¬�¦�¢�¤(ó��´�(�§ydz#{Y¯(¯­�'�6z�z�y��'�@y(�q�á�D�=z�z�{��®�=z��6�§y����T{Y�R�#¯­�'�q�q��D�(�®{Y���(�E�� �,æ ¯çy��NèG����õé¹�¦ië�½%¬�¦i»�¤�©£»î¢�¤�¦� �¢E¥B®%©¶ëg¢è�%¢�¢E®Jª�¤�©�í?äÞ S�4©£¤.®�¢�¤gª8©£¤¦�½% S¥i¦�©rçÚ¢�z�~� � ���YÑ · ¦¥�½�©�¹�­S®4��©£¦gç%¬�¤.®ù¦�©6º�©��0¡#  �%º�¢¶§£©�¹%¤.¥�¢E­�ª�¦�½%¬�¦è¦�½�¢:¥�º.½�¢Eíe¢r S¥�¥�¢Eº�¹�¤�¢: �ªv¬£�%®ù©��%­�§� �ª�ä  S¥�¬Jº�©�­ ­S S¥� �©���ð�¤�¢E¥� S¥�¦�¬£�0¦½%¬£¥�½ ª&¹%�%º�¦� S©��sñ&¬±®�¢¡��% �¦� S©���ëg¬£¥r¼� �¡£¢E�m S�P¬x»%¤�¢�¡ê �©�¹%¥�­S¢Eº�¦�¹�¤�¢�ó�õ · � ª&¬£º�¦�vë\½�¢E�Tä   ¥�º�©�­S­S  ¥� �©��¤�¢E¥� S¥�¦�¬£�0¦@¦�½% S¥�º�©���¥�¦�¤.¹%º�¦� �©��r­�©#©'.#¥V¢�Z�¬£º�¦�­�§�­S �.£¢g¬\º�©��%¥�¦�¤.¹%º�¦� �©��¶ëi¢g¼�¬E¡£¢É»�¤�¢�¡# S©�¹%¥�­�§^ S�Á¦�½%¢iº�©��0¦�¢�Z#¦©£ª�íe¢E¥�¥�¬�¼£¢�¬£¹�¦�½%¢E�A¦� Sº�¬�¦� S©��xº�©ê®�¢E¥�õ

]

Page 112: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

University of MarylandCMSC456 — Introduction to CryptographyProfessor Jonathan Katz

Lecture 37

1 Signature Schemes for Multiple Messages

Before continuing to discuss the construction of signature schemes, we summarize the resultsof the last lecture:

Theorem 1 A 1-time signature scheme can be constructed from any one-way function.

(This theorem is evident from the Lamport 1-time signature scheme we have discussed.)

Theorem 2 A 1-time signature scheme which can sign messages longer than the public keycan be constructed from (any one-way function and) any collision-resistant hash function.

(This theorem follows from the note at the end of the previous lecture. We note that anycollision-resistant hash function is already a [weak] one-way function and therefore collision-resistant hash functions are a sufficient assumption for the theorem.)

Theorem 3 For any (polynomial) M , a signature scheme for signing M messages can beconstructed from any 1-time signature scheme.

This theorem follows from the second-to-last note of the previous lecture. The given con-struction, however, had a number of drawbacks that we seek to remove in the next fewlectures. These included:

• It was required to know in advance the number M of messages to sign.

• The public and secret key grow in size as O(M).

• The signer was required to maintain and update state information.

1.1 A Path-Based Signature Scheme

We now give a scheme that avoids some of these problems. Our scheme will be based on anysecure 1-time signature scheme (K,Sign,Vrfy) which signs messages longer than the publickey, and works as follows:

1. To generate the public/secret key, simply run K to generate public key PK0 and secretkey SK0.

1

Page 113: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

2. We give a slightly informal description of the signing algorithm and assume the inter-ested reader could make it more formal. To sign the first message m1, the signer doesthe following: (1) run K to generate public key PK1 and secret key SK1 (we stressthat K must be randomized; note further that PK0 6= PK1 and SK0 6= SK1 withall but negligible probability [why?]). (2) Compute σ1 ← SignSK0

(m1|PK1) (where“|” denotes concatenation). Finally, (3) output signature 〈PK1, σ1〉. The signer alsostores the values (PK1, SK1,m1, σ1) as part of the internal state.

To sign the ith message mi, the signer will use PKi−1. The signer does the following:(1) run K to generate PKi and SKi; (2) compute σi ← SignSKi−1

(mi|PKi); and(3) output signature 〈PK1,m1, σ1, . . . , PKi−1,mi−1, σi−1, PKi, σi〉. The signer alsostores the values (PKi, SKi,mi, σi) as part of the internal state.

3. We discuss verification in the general case. Remember that all the verifier has is thepublic key PK0 (the “master” public key of the signer), a message mi, and a pur-ported signature 〈PK1,m1, σ1, . . . , PKi−1,mi−1, σi−1, PKi, σi〉. The verifier checks

that VrfyPK0(m1|PK1, σ1)

?= 1, . . . ,VrfyPKi

(mi|PKi, σi)?= 1. Only if all of these are

true does the verifier accept the signature as valid.

What is going on is that each time the signer signs a message mi, the signer also “refreshes”its key by generating (PKi, SKi) which will be used to sign the next message. However, wedo not want an adversary to be able to generate and use its own PK ′

i(if this occurs, then

the adversary can obviously forge a signature on any message it likes); to prevent this, thesigner certifies each public key with respect to the previous public key PKi−1. The verifierchecks that each public key has been so certified back to the “root” public key PK0. Weemphasize that it is essential here that the original scheme (K,Sign,Vrfy) allow for signingmessages longer than the public key (why?).

We do not give a rigorous proof of security here, although such a proof should largely beevident. Note that each public key generated by the signer is used to sign only one message;since (K,Sign,Vrfy) is assumed to be a secure 1-time signature scheme, the scheme as awhole is secure. A sketch of the proof is as follows: assume the adversary outputs signature〈PK ′

1,m′1, σ

′1, . . . , PK ′

i−1,m′i−1, σ

′i−1, PK ′

i, σ′

i〉 on message m′

iwhich was never signed by the

real signer. In particular, we have m′i6= mi (assuming without loss of generality that the

signer has signed at least i messages). Let 〈PK1,m1, σ1, . . . , PKi−1,mi−1, σi−1, PKi, σi〉be the ith signature issued by the real signer on message mi. If PKi−1 = PK ′

i−1 then theadversary has forged a signature with respect to public key PKi−1 — but this is impossibleby security of the 1-time signature scheme. If PKi−1 6= PK ′

i−1 but PKi−2 = PK ′i−2 then

the adversary has forged a signature with respect to PKi−2 — which is again impossible.Continuing in this way, we see that the adversary must have forged a signature with respectto some public key PK0, . . . , PKi−1; however, this should be impossible for any such keyby the security of the 1-time scheme.

Let’s see whether this scheme offers any improvements over previous schemes:

• The number of messages to be signed does not need to be known in advance. Thisscheme allows signing any arbitrary (polynomial) number of messages.

• The public key size is constant (i.e., O(1)), independent of the total number of mes-sages signed. Unfortunately, although the initial secret key is also constant size, the

2

Page 114: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

state maintained by the signer grows as O(M), where this is the number of messagessigned thus far. We may note, however, that most of the state information (i.e., allthe {PKi,mi, σi}1≤i≤M ) need not be secret. This may save on the cost of storing thisinformation.

• The signer still needs to maintain state when signing messages.

• We also note a new disadvantage of the present scheme: both the signature length andthe verification time grow as O(M). This is in contrast to the previous scheme whichhad essentially O(1) signature length and verification time. It is worth mentioning,though, that the signing time in the current scheme is essentially O(1).

1.2 A Tree-Based Signature Scheme

We may note that the signature scheme of the previous section essentially constructs apath from a root PK0 to a “current key” PKi; this path is certified using the 1-timesignature scheme and the current leaf PKi is used to sign the next message mi+1. Thisled to signature size/verification time O(M). It seems natural that we might do better byconsidering a tree-based approach instead of a path-based approach.

We sketch this generalization here. We now denote the signer’s public key by PK and theinitial secret key by SK. We will construct a tree in which every node has two children: thechildren of PK are PK0, PK1 and the children of any node PKw (for any binary string w)are PKw0 and PKw1. We use binary notation to represent signed messages; thus, the firstmessage is denoted m and the second, third, and fourth messages are denoted m0,m1,m01,respectively. To sign the first message m, the signer generates two public keys PK0, PK1,computes σ ← SignSK(m|PK0|PK1), and outputs (PK0, PK1, σ0). The signer also storesassociated secret keys SK0 and SK1, as well as m and σ, as part of the internal state. To signthe second message m0, the signer will use key SK0 and sign with respect to PK0. Thus,the signer generates public keys PK00, PK01, computes σ0 ← SignSK0

(m0|PK00|PK01),and outputs 〈PK0, PK1,m, σ, PK00, PK01, σ0〉.

Now, to sign the third message m1, the signer will use SK1 and sign with respect toPK1. Thus, the signer generates PK10, PK11, computes σ1 ← SignSK1

(m1|PK10|PK11),and outputs 〈PK0, PK1,m, σ, PK10, PK11, σ1〉. Note that the length of the signature doesnot grow linearly with the number of messages signed ! Instead, the signature length growslogarithmically. It should also be clear that verification time grows logarithmically as well.

Thus, the scheme has the following properties:

• The number of messages to be signed does not need to be known in advance.

• The public key is of size O(1). The internal state grows as O(M), but as mentionedin the previous section most of this state does not need to be kept secret. (Using animproved traversal of the tree it is possible to reduce the state to O(log M); however,since we show below how to reduce the state to O(1), we do not go into the details.)The signature length and verification time grows as O(log M). The signing time isessentially O(1).

• The signer still must maintain state, although we discuss below how this can beavoided.

3

Page 115: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

For completeness, we mention how the signer can reduce the size of its (total) storageto O(1), at the expense of increasing the signing time to O(log M). In the Lamport 1-timesignature scheme, the secret key is just a sequence of random values. Also, the public keyis a deterministic function of the secret key, and signatures are also deterministic functionsof the message and the secret key. So, to avoid storing all the public keys, secret keys, andsignatures we can do the following: Imagine a full binary tree as before, except that we nowmake two changes: (1) we imagine that the tree is fixed in advance (and thus assume forsimplicity that the number of messages to be signed is known in advance) and (2) we signactual messages only using the leaves of the tree (as opposed to before, where every nodeof the tree was used to sign messages). Note that using only the leaves still gives a schemewith signature length/verification time O(log M).

In a mental experiment, we can imagine that the signer stores the complete tree (thatis, for every node in the tree, the signer stores SKw; as mentioned above, given this thesigner can generate PKw and a signature σw on PKw0|PKw1 deterministically). It shouldbe clear that the signer can then sign messages as they come in, using the correspondingleaf of the tree (the signer will need to maintain state to remember the number of messagesit has signed so far). However, we note an easy way to avoid storing the entire tree: simplyhave the signer store a random seed s for a PRF F . Instead of choosing secret key SKw atrandom, simply set SKw = Fs(w). Now, the secret keys can be dynamically generated asneeded. And the storage has been reduced to O(1) — simply a seed for a PRF. The secretkeys are now no longer random, but pseudorandom; however, since F is a PRF, this willnot affect the security of the scheme.

We mention that in the above description, the signer must know the number of messagesto be signed in advance and must also maintain state (in particular, the number of messagessigned thus far). It is possible to remove all of these drawbacks (if you are interested, pleaseask me); we state this as in a theorem summarizing what we have so far.

Theorem 4 Based on any 1-time signature scheme which can sign messages longer thanthe public key, we can construct a signature scheme for signing any (polynomial) number ofmessages M with the following properties (k represents a security parameter for the scheme):

• The number of messages to be signed does not need to be known in advance.

• The public key and secret key have size O(1). The signature length, verification time,and signing time are O(log2 k).

• The signer is not required to maintain any state.

Since we know how to construct 1-time signature schemes that can sign messages longerthan the public key based on any collision-resistant hash function, this shows that CRHFsare sufficient to construct a signature scheme with the above properties. We will see in thenext class how to base the construction on any one-way function.

4

Page 116: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

University of MarylandCMSC456 — Introduction to CryptographyProfessor Jonathan Katz

Lecture 38

1 Signature Schemes From Any One-Way Function

At the end of the last lecture, we saw how a secure signature scheme for arbitrarily manymessages could be constructed from any 1-time signature scheme which allows signing mes-

sages longer than the public key. So far, however, the only construction of a 1-time signaturescheme of this type was based on collision-resistant hash functions (CRHFs). We mentionedearlier in the course that the existence of CRHFs seems to be a strictly stronger assumptionthan the existence of one-way functions; in fact, it is believed to be impossible to constructCRHFs from an arbitrary one-way function. On the other hand, based on our experiencewith the Lamport 1-time signature scheme it may seem intuitive that one-way functionsshould be sufficient to construct a secure signature scheme or many messages. In fact, thiscan be done; we explore this matter here.

1.1 Universal One-Way Hash Functions

We first recall the definition of a collision-resistant hash function. We say H is (t, ε)-collisionresistant if for all algorithms A running in time at most t we have:

Pr[(x, y)← A : H(x) = H(y) ∧ x 6= y] ≤ ε.

That is, the adversary A gets to output two values x and y, and succeeds if x 6= y yetH(x) = H(y).

A different notion is that of universal one-way hash functions (UOWHFs), which weintroduce here. In this case, the adversary does not get to choose both x and y; instead, theadversary is given a random x and must find a different y such that H(x) = H(y). Moreformally1, let H be a keyed hash function so that Hs : {0, 1}m → {0, 1}n for all s ∈ {0, 1}k .We say this function is (t, ε)-universal one-way if for all algorithms A running in time t wehave:

Pr[x← {0, 1}m; s← {0, 1}k ; y ← A(s, x) : Hs(x) = Hs(y) ∧ x 6= y] ≤ ε.

(Note: this definition is different than the one given in class since here we consider a keyed

hash function.) Note that here both s and x are fixed in advance; the adversary only hascontrol over y. This makes the adversary’s job harder, meaning that UOWHFs are weakerthan CRHFs. It should be immediately clear, for example, that any collision-resistant hashfunction is also universal one-way.

The following theorem was proved by Rompel [2], building on the work of Naor andYung [1].

1The reader should be aware that, for simplicity, I have used a weaker definition of UOWHFs thanappears in the literature. However, any hash function satisfying the present definition can be immediatelyconverted to one satisfying the stronger definition.

1

Page 117: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

Theorem 1 Universal one-way hash functions can be constructed from any one-way func-

tion.

We now show that UOWHFs are sufficient to construct secure signature schemes. Recallthe tree-based signature scheme from the previous lecture, and remember why CRHFswere needed: because at node w of the tree we sign PKw0|PKw1 using PKw. To do this(recalling the way we sign messages longer than the public key), we will first computeh = H(PKw0|PKw1) and then sign h using the regular Lamport scheme. Clearly, if H iscollision resistant then this is secure.

We may note, however, that collision resistance is stronger that what we actually needfor this application. The reason is that the adversary doesn’t choose PKw0|PKw1; instead,this is a “random” pair of public keys chosen by the signer. Thus, we don’t require that theadversary cannot find any collision in H; all we need is that the adversary cannot find a(different) PK ′

w0|PK ′

w1such that H(PK ′

w0|PK ′

w1) = H(PKw0|PKw1). Since PKw0|PKw1

is chosen independent of the adversary, having H be a UOWHF2 is enough!Thus, the previous theorem as well as our results from the last lecture show that:

Theorem 2 A secure signature scheme for arbitrarily-many messages can be constructed

from any one-way function.

We emphasize the amazing nature of this result: as first glance, it seems that some sortof “trapdoor” is necessary in order to sign. This intuition, however, is wrong: one-wayfunctions (which have no “trapdoor”) are sufficient to construct digital signature schemes.(This is in contrast to public-key encryption schemes, which are believed to be impossibleto construct from one-way functions alone.)

2 The Random Oracle Model

The preceding section ends our discussion of schemes which can be rigorously proven securebased on well-defined, standard cryptographic assumptions. But where do we go from here?The reader will notice that — with the possible exception of the El Gamal encryption scheme— all constructions of public-key encryption and digital signature schemes that we have given

are inefficient. In practice, the construction we have given are simply too inefficient to beused. What, then, is done? Do we simply use efficient schemes that have no argument infavor of their security?

The answer is (in general) no. When possible, we never want to use schemes which haveno argument in favor of their security. By using a scheme which “seems hard to break” weleave ourselves open to an adversary who is more clever than us, and who can break thescheme. It is always better to have some justification for believing a scheme is secure.

Unfortunately, in general we simply do not know how to construct efficient schemeswhich are provably secure based on standard cryptographic assumptions. And yes, some-thing must be done if we want to have efficient schemes. What is done is to introduce

2In fact, the careful reader will notice that having H satisfy Definition 1 is not quite enough for thisapplication, since PKw0|PKw1 is not uniformly distributed when the Lamport scheme is constructed froman arbitrary one-way function. However, the “real” (stronger) definition of UOWHFs in the literature is

sufficient for the present application. We hope the reader will forgive this simplification.

2

Page 118: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

the Random Oracle model, and to prove the security of cryptographic constructions in thismodel. Although, as we will see, a proof in this model does not guarantee security in the“real world”, it still provides a useful check that our construction is not inherently flawed.

In brief (we discuss this in more detail next lecture), the random oracle model assumesthe following:

• There is a public oracle that everyone (including all honest parties as well as theadversary) has access to.

• This oracle implements a truly random function in the following sense: the first timeanyone asks the oracle a query x, it chooses a completely random value y and returnsthis value. In the future, whenever someone asks query x again, the same answer y isreturned. Thus, consistency is maintained. Note that this is equivalent to having theoracle be a “black box” which stores a completely random function (as discussed inan earlier lecture).

• In the real world, of course, random oracles don’t exist and even if we wanted toimplement a random function we would be unable to for reasonable input/outputsizes. Thus, what we do is as follows: design a scheme in the random oracle modeland prove its security in that model. Then, to actually use the scheme in the realworld, we replace the random oracle by a cryptographic hash function (e.g., SHA-1or MD5).

• The intuition is that if the hash function is “good” and really “garbles” its input,then it “acts like” a random oracle. Thus, on an intuitive level, a scheme secure inthe random oracle model should also be secure in the standard model (i.e., the realworld, without random oracles) when the random oracle is replaced by a “good” hashfunction.

• We mention, though, that all the terms of the previous paragraph (“good”, “acts like”,etc.) are vague; in fact, there is no (known) way to replace a random oracle with any

hash function so that the resulting scheme is provably secure in the standard model.

References

[1] M. Naor and M. Yung. Universal One-Way Hash Functions and Their CryptographicApplications. STOC ’89.

[2] J. Rompel. One-Way Functions are Necessary and Sufficient for Secure Signatures.STOC ’90.

3

Page 119: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

University of MarylandCMSC456 — Introduction to CryptographyProfessor Jonathan Katz

Lecture 39

1 The Random Oracle Model

At the end of the last lecture, we introduced the random oracle model and distinguished itfrom the standard model in which we have been working until now. We review and extendthe discussion here.

• The random oracle model assumes a publicly-accessible oracle (“black box”) that ev-eryone can access. This oracle is assumed to implement a completely random function.

• In practice (random oracles do not exist in the real world!), we will instantiate therandom oracle with a cryptographic hash function.

• From a theoretical point of view: no (efficiently-computable) hash function can pos-sibly be a random function! (A true random function would have a description thatis exponentially large!)

• From a practical point of view: better to have a construction which can be provensecure in the random oracle model than to have a construction with no proof at all.

• In practice: if an adversary can break a scheme which was proven secure in the randomoracle model, then the hash function used to instantiate the random oracle must not be“good enough”. So a proof of security in the random oracle model gives us confidencein the scheme, since it seems that the only way to break the scheme (in the real world)involves finding some weakness in the hash function used to instantiate the randomoracle.

• In theory: no formal statement along these lines seems possible. In particular, it mightbe possible to break a scheme which was proven secure in the random oracle modelwithout finding any specific weakness in the hash function used. Furthermore, thereare (artificial) examples of schemes that can be proven secure in the random oraclemodel, but which are insecure when the random oracle is replaced by any specifichash function.

Hopefully, the above discussion captures some of the tension inherent in using the ran-dom oracle model (indeed, use of this model to prove security remains controversial amongcryptographers). To summarize: a proof of security in the random oracle model is no guar-

antee of any security in the real world; on the other hand, a proof of security in the randomoracle model is better than no proof at all, and the only efficient schemes we (currently)know how to construct are provably secure only in the random oracle model. So if we wantefficient schemes with some evidence (but not guarantee!) of their security, it seems wemust work in the random oracle model.

1

Page 120: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

2 An Efficient Signature Scheme in the RO Model

As an example of how the random oracle model can be useful, we show here that the originalsuggestion of Diffie and Hellman for constructing signature schemes can be modified andproven secure in the random oracle model. Let (K, f, Inv) be a trapdoor permutation. (Asusual, if td is the trapdoor associated with a particular key key, then we denote Invtd(·) byf−1

key(·), where it is clear that f−1

key can be efficiently evaluated only by someone who knowstd.) We construct a signature scheme as follows:

• To generate a public/secret key, run K to generate (key, td). The public key is key

and the secret key is td.

• Let H : {0, 1}∗ → Dkey (recall Dkey is the domain of fkey) be a hash function modeledas a random oracle. To sign message m, output σ = f−1

key(H(m)).

• To verify signature σ on message m, check that fkey(σ)?= H(m). (Remember that

everyone is assumed to have access to the random oracle H).

The following theorem shows that this construction is indeed secure. Pay careful atten-tion to the proof of security, which relies in an essential way on the fact that H is indeed arandom oracle. (The proof simply fails when H is any concrete function.)

Theorem 1 If (K, f, Inv) is a (t, ε)-secure trapdoor permutation, then the above signature

scheme is (t, qhε)-secure, where qh represents the number of queries an adversary makes to

the random oracle H.

Proof Assume there is some adversary A which outputs a forgery for the above construc-tion with probability δ. We use this adversary to construct an algorithm A′ that inverts thetrapdoor permutation. A′ is given a key key and a random element y, and tries to computex such that fkey(x) = y. It proceeds as follows:

A′(key, y)Set PK = key

Pick a random index i∗ ∈ {1, . . . , qh}Run A(PK)Answer the ith query of A to oracle H as follows (let mi denote this ith query):if i = i∗, return yotherwise, pick random ri ← Dkey, compute ansi = fkey(r), and return ansi

When A requests a signature on message m:find i such that m = mi (see discussion below)if i = i∗, abort; otherwise, return ri as the signature

When A outputs its forgery (m,σ)If m = mi∗ then output σ; otherwise, abort

In the above description, we make a few assumptions about A: (1) we assume that beforeA ever asks for a signature on message m, it has already queried H(m); (2) we assume thatA never submits the same message to H more than once (indeed, there is no reason for it to

2

Page 121: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

do so, as the answer it would get back remains the same); (3) we assume that if A outputsa forgery on message m, then at some point it has queried H(m). It is not hard to see thatthese assumptions may be made without loss of generality, since given any adversary A wecan modify it to get an adversary A that satisfies the above assumptions and has the sameprobability of forgery (although A might now make at most qh + qs +1 queries to H, whereqs is the number of signatures requested by A).

We note the following about the above simulation:

• As long as A′ does not abort, the simulation for A is perfect. The outputs of oracleH are random and independently distributed, as required for a random oracle.

• A′ sets things up so that it is able to answer all signature queries of A unless A ′ asksfor a signature on mi∗ . Note that the response of A′ to all other signature queries isindeed a correct signature.

• Since the output (m,σ) of A can only be a forgery if A never requested a signatureon m, and since m = mj for some j (by assumption), it must be the case that thereis at least one index j for which A never requests a signature on mj. Since i∗ ischosen at random, with probability at least 1/qh we have j = i∗. When j = i∗ (and Aoutputs a valid forgery) then σ is indeed an inverse of y and A′ succeeds in invertingthe trapdoor permutation.

To summarize the above discussion, the probability that A′ correctly outputs an inverse isat least 1/qh times the probability that A outputs a forgery; hence, A′ outputs the desiredinverse with probability at least δ/qh. Since the trapdoor permutation is assumed to be(t, ε)-secure we must have δ ≤ qhε, completing the proof.

Again, we stress that it was vital to the proof that A′ have the ability to “fix” theoutputs of H. This is fine in the random oracle model (as long as A′ fixes the outputs inan independent and uniform way) since the output is supposed to be random, anyway. Butit should be clear that the proof fails once we commit to any specific function H (and A ′

loses the ability to fix outputs as it likes).

3

Page 122: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

êLë@ìîí'ïð~ñRì�ò�ó#ô'õfö5÷Yð~ó®øî÷'ë,ùú ö»û ú ü§ý'þNÿ�� ë§ò�ð~ô®ù�����òRì�ôGë©ò�ô ú ðRó��@ò�ô�'ð~÷���§ó �ðRô'õ(ï�ñRñ�ô'ð��'ôGë@÷Yò��@÷'ë��2÷Yò��

������������� �!

" #%$'&�(*),+.-0/21436587�92$,:<;>=,+.1?5A@CB,10(D1E7F$G;>B,1IHKJ L -M3�10N

O�÷'ñ�ò¸òRìQP!ïRTS´ï8�@ðRï�ñ�ï�ëqò�ï�ùQ÷'ë5ïVUW�ì�ï�ë§ò%ñRìX�Gë,÷Yò���ðRï�ñ��F��ïYP!ïZS[�@ìQ�F����ô\�@ø¥ù^]`ïA�@ð~ô�í'ï�ëQñ�ïY�V��ðRï�ìîë�ò��@ïð~÷'ë@ù�ô\P¿ô'ð~÷�ø�ï_P!ô®ù@ï�øa`áûEìX�Gë@ì¥ëb�5÷�P�ï�ñRñR÷�'ï_c ðRïYd��@ì�ðRï�ùe�@÷'ñ��@ì¥ëb�fc ò�ôQ÷5ð�÷'ë@ù�ô\P7ï�øîïYP!ï�ë§ò�ô'õg8h�ij�@ñRìîë���ò���ï+ð~÷'ë,ù�ô\P ô'ð~÷�øîïlkf÷'ë@ùDò���ï�ëDìîëqí'ïðRòRìîë��2ò���ï+ò�ð�÷�`ù@ôqô'ðm�^ïðFP8��òR÷YòRìîôGë!ôGëDò��,÷Yò-ð~÷'ë@ù@ô\Pï�ø�ïYP!ï�ëqòVn¹ò����@ñVRoò���ï�ñRìX�Gë,÷Yò���ðRï!ôGëoc ìîñ�ñRìQP��,øîóqpCr.sh ijtuivcqk�kwivSx��ïðRïwtAìîñ�ò���ï�ð�÷'ë@ù�ô\P ô'ð~÷�ø�ïlkF`y ��ï2ñRìX�Gë,÷Yò���ðRï2ñ��F��ïYP!ï�ìîñ{zEë�ôlS+ë�÷'ñ3ò��@ï^|�}��@øîø�~��¸ô\P�÷'ì¥ë^�L÷'ñ�����i(ô'ðx}�����k/ñ����@ïYP!ï`

� ï�ñ~÷YSÈì¥ë©ò���ïM�@ðRïíEì�ô\�@ñ+ø�ïY��ò���ðRï2ò��,÷Yò+ì�õ¹ò���ï2ò�ð~÷�_ù�ô®ô'ð��`ïð�PZ��òR÷YòRì�ôGëfS/÷'ñ8iv�����Fk�~�ñ�ïY�V�@ðRïR<ò��@ï�ëò���ïA}����kñRìX�Gë@÷Yò���ð~ï2ñ�����ïYP�ïA��ôGë@ñ�ò�ð�����ò�ï�ù^],÷'ñ�ï�ù5ôGëiò��@÷Yòx�^ïðFP8��òR÷YòRìîôGëiìîñAiv�����Y�\�Fk�~�ñ�ïY�V�@ðRïR�S[�@ïðRï�Y�iðRïV�@ð~ï�ñ�ï�ë§òRñNò���ï�ë���P8]`ïð2ô'õ¦ô'ð~÷�ø�ï_d��@ïð~ì�ï�ñAPD÷'ù�ï�]®ó�÷'ëT÷'ù�í'ïð�ñR÷YðRó`^���@ìîø�ï!ò��,ìîñ�ìîñA�@ðRô�'ðRï�ñ~ññRìîë���ï�S ï2÷Yò+ø�ï�÷'ñ�ò��,÷�í'ïN÷�P�ï�÷'ñ���ðRï�ô'õ��@ðRôsíI÷],øîï2ñ�ïY�V��ð~ì�ò�óR�ò���ï2ðRï�ñ��@ø�ò/ìîñ3ë�ô'ò3÷'øîø`ò��,÷Yò��'ðRï�÷YòV`/ûEì¥ë���ï�Y����ô'ðRð~ï�ñ��`ôGë@ù@ñ¦ò�ô�ò���ïNë���P8]`ïð+ô'õfòRìQP!ï�ñ+ò���ï«÷'ù�í'ïð~ñR÷YðRóiïíY÷'øQ�@÷Yò�ï�ñLò���ïZ�@÷'ñ��áõv�@ë���òRì�ôGë^teR^ñRì¥ë���ïïíI÷'øQ�,÷YòRìîëb�et ì¥ñ�ò�ó��,ìQ�÷'ø¥ø�ó¬í'ïðRó�ïVU��ì�ï�ëqòVR+÷'ë@ù¾ñRìîë���ï�ïíY÷'øQ�@÷YòRìîôGë@ñDô'õMt��÷'ë�]`ï5ù�ôGë�ïq]qó�ò��@ï÷'ù�í'ïð~ñR÷YðRógô�T~�øîìîë�ï�i�÷'ë@ùuSLì�ò���ô\��ò«ò���ï�ñRìQ�Gë�ïðY� ñ,zEë�ô�S+øîï�ùb�'ïlkFR<���oP�ìX�\�qòZS ï�øîø{]`ï©í'ïðRó�øî÷Yð��'ï`��ù�ï�ù@ìQ�÷Yò�ï�ù�÷'ù�í'ïð~ñR÷YðRó�P�ìX�\�qòMS ï�øîø�]`ï�÷],ø�ï!ò�ô��,÷�í'ïW����������` � ë ò��@ì¥ñM�÷'ñ�ïR��@ñRìîëb�©ïí'ï�ë�÷ií'ïðRóñ�ïY�V��ðRï»ò�ð~÷�_ù�ô®ô'ð��`ïð�PZ��òR÷YòRì�ôGë�SLì�ò������ � r ��� S ô\�@øîùað~ï�ñ��@ø�òDì¥ë¼÷Të@ô'ò¡~Ôí'ïðRó¢~�ñRïY�V��ðRï ñRìX�Gë@÷Yò��@ðRïñ�����ïYP�ï_i�ñRìîë���ï8����Y�<�¤£¥¦kF`�§%õ���ô\��ð�ñ�ïR�S ïA�÷'ë5ñ~ìQP��,ø�ó_�@ñ�ï2÷Dò�ð~÷�_ù�ôqô'ð��`ïð�PZ��òR÷YòRì�ôGë^S+ì�ò���ø�ôlS´ïð�VR�]��@ò ò��,ìîñ�P�÷�ó©øîï�÷'ùiò�ôD÷!ø�ï�ñRñ3ïVU��ì�ï�ëqò+ñ����@ïYP!ï` s

�+ïð~ïRS´ï ñ���ô�Sgò��@÷Yò6ë�ô'òo÷'ø¥ø§ìîñoø�ôGñ�òV`C}@ô'ð¹ò��@ï0�,÷YðRòRì¨�V�@øî÷Yð©�÷'ñ�ï<S[��ï�ë�ò���ï-ò�ð�÷�`ù@ôqô'ðC�^ïðFP8��òR÷YòRìîôGë�@ñ�ï�ù¼ì¥ñ�ò���﫪¸û¬�­�`ïð�PZ��òR÷YòRì�ôGë.R¦÷®]`ïò�ò�ïðW�@ðRô®ô'õ¸ô'õ2ñRïY�V��ð~ì�ò�óaìîñ��`ôGñRñRìX]<ø�ï`¯� ï«°@ð~ñ�ò#ñ�òR÷Yò�ïáò��@ïò���ïô'ðRïYPfR@ò��@ï�ëf]@ð~ì�ïV±,ó#ù@ìîñ��V�@ñRñ/ò���ï2ìQPw�,øîìQ�÷YòRì�ôGë,ñVR®÷'ë,ùf°,ë@÷'øîøîów�Gì�í'ï�÷��,ðRôqô'õ²`³8´Cµ�¶�·lµ¬¸ ¹»º�¼�¼F½¬¾�¿8ÀvÁ�ÂÀÄÃ2Å�ºÇÆȼ� iv�����FkFÉ ¼V¿�ÊV½¬Ë�¿8ÀjË�ÂF̬Í\Î�ÎËxÌb¿ÏË�¾,½¬ÀÐÂ\ÀjÆjÎÑTÒWÓ©Á¬¿VÑ«ÀvÁ¬¿�Ô�Õ�Ö¤¼FÆX× ÉÑ�ÂÀj½¬Ë�¿�¼VÊÏÁ¬¿V¾w¿[ÆvÑb¼�ÀÐÂÑ�ÀjÆjÂ\À�¿�Í8ØÄÆvÀvÁAÃ2Å�ºÙÆȼ iv����ÚY�lÛ��FkFÉ ¼V¿�ÊV½¬Ë�¿�Ü©Ø�Á¬¿VË�¿ ��Û Æ¨¼�ÀvÁ¬¿�Ñ�½¬¾�Ý�¿ÏË[Î�Þm¼�Æ�×\Ñ�ÂÀj½¬Ë�¿�¼ÀvÁ¬¿,Â\Íß¿ÏËF¼ÏÂË�à�Ë�¿�áV½�¿�¼ÏÀv¼4ÞFË�ξ?ÀvÁ¬¿A¼FÆX×ÑT¿VË�âãÂÑ�Í Úx�»��ä¦å ÆȼMÀvÁ¬¿�Ý�Âæ¼Y¿,ΡÞAÀvÁ¬¿8ÑTÂÀj½¬Ë¡Âç�çXÎ�×¢ÂË�ÆjÀvÁ¢¾Z¼�èÒ

é ���,ñVR3÷'ë¾÷'ù@í'ïð~ñR÷YðRóa÷Yò�òR÷��z®ìîë��u}����ê],÷'ñRï�ù¼ôGë몸û¬�ê�,÷'ñ��@ð~ô],÷],ìîøîìîò�ógô'õ�õdô'ð��'ïð~ó¯��Û��#ð~÷Yò���ïðò��@÷'ëo�Y��%÷'ñ�S´ô\�,øîù�]`ï�ïÏì��`ïY��ò�ï�ùQõ(ðRô\P ò���ï,�@ðRô®ô'õ�ô'õ-ñ�ïY�V�@ð~ì�ò�óáõdô'ð%ò���ï��÷'ñ�ï«ô'õm�'ï�ë�ïð~÷'ø¹ò�ð~÷�_ù�ô®ô'ð�^ïðFP8��òR÷YòRìîôGë@ñV` � ëí�@ð~÷��òRìQ��ïR<� Û�î �Y�¬n ò�ô�ñ�ïïfS[�qóR-ë�ô'òRì¨��ïiò��@÷Yò���ô\P�����òRìîë��Qñ~ìX�Gë@÷Yò���ðRï�ñ�òR÷z'ï�ñø�ôGëb�'ïðD÷'ë@ù¯P!ô'ðRï5ìQPw�^ô'ð~òR÷'ë§òRø�óeP8�@ñRòw]`ï©ù@ôGë�ï^]®ó�ò���ï�ñ~ìX�Gë�ïðY` � òDìîñ�PZ������P!ô'ðRï�ù@ìQU��V�@ø�ò�õdô'ð÷'ëT÷'ù�í'ïð~ñ~÷YðRóáò�ôq�'ïò�÷�ñRìQ�Gë�ïð2ò�ô5ñRìQ�Gë�£VïïïqP!ï�ñ~ñR÷�'ï�ñ2ô'õ/ò���ï�÷'ù�í'ïð�ñR÷YðRóT� ñA���@ôGìQ��ïDò��@÷'ë�õdô'ð2ò��@ï÷'ù�í'ïð~ñR÷YðRóNò�ô�ïíI÷'ø¨�@÷Yò�ï3÷��@÷'ñ���õv�,ë���òRì�ôGë�£Vïïï%òRìQP!ï�ñV`�û®ôbR é ��ïô'ðRïYP�£¦ìîë@ù@ì¨�÷Yò�ï�ñ�ò��@÷Yòfõ(ô'ðÄ�@ð~÷��òRì¨�÷'ø����ð��`ôGñ�ï�ñ��@ñ~ìîëb��ª¸û¬�ðS+ì�ò��^�<�»� r ����ìîñ��`ïðRõ(ïY��òRø�óW°,ë�ï`ñ ·l¶.¶�ò � ï��Gì�í'ï�÷Z�@ìX�\��ø�ïí'ï�ø^ô�í'ïðRíEì�ïVScô'õ�ò���ï2�@ðRô®ô'õ.]`ïõdô'ð~ïx�@ð~ï�ñ�ï�ë§òRì¥ëb��ò���ï%ù�ïòR÷'ìîø¥ñV`Ä�Lñ{�@ñ��,÷'øaRS´ïWS+ìîøîø¹òR÷z'ï#÷'ë�÷'ù�í'ïð~ñR÷Yð~ó�óH÷Yò�òR÷��z®ìîë��5ò���ïDñ~ìX�Gë@÷Yò���ðRï�ñ�����ïYP!ï#÷'ë@ù��@ñRï!ò��@ìîñ�ò�ô���ôGë,ñ�ò�ð�����ò�÷'ëô�õCöb÷�ømù²úl÷�û�üæýþù�ÿ��þø���ù������þø��� �õ����������������������û��! #"æý$�Ïû��$�%!�'&������Ðù¡öaü(�Fø) *���,+.-/�10(2�354'67"#�8���#��÷�� ø#�þø�+7-9�!0(:�3

û��$�%!���Ðù�;!"#���Ðù<"æö��þø=�, lõ?>@����%A������BC�D�#���Cû��! �"æý$�Xÿ#>@%#�$&E%F>7�5"lýG H�ù�ýþù¡öaöCùEI�&��þù¡ø1�J4£

Page 123: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

÷'ù�í'ïð~ñR÷YðRóeóLK4Sx�@ìQ���gìîëqí'ïðRòRñ,ª¸û¬�8`C}@ô'ð!ò���ïW�,ðRôqô'õ¦ô'õ¦ò��@ïW�@ð~ïí®ì�ô\�,ñNø�ïY��ò��@ðRï«i(õ(ô'ð«ò���ï_�÷'ñRï�ô'õL÷�'ï�ë�ïð~÷'ø_ò�ð~÷�_ù�ô®ô'ð{�`ïð�P8�@òR÷YòRì�ôGë�kFR¬S´ïM�÷'ëiù@ï�ñ���ð~ìX]`ï¸ò���ï�ñ�ò�ð~÷Yò�ïV�'ó©ô'õCó K ÷'ñ3õdôGøîøîô�S+ñ1Mfø�ïò��Y�«ù�ï�ë@ô'ò�ïò���ï�ë���PA]`ïð�ô'õ{�@÷'ñ���d��@ïð~ì�ï�ñ�P�÷'ù�ï�]®ó�óA`, -ìQ��z»÷©ð�÷'ë@ù�ô\P9ìîë@ù@ïÏìONQPSR¢£�VäVäVäæ���Y�UTD÷'ë@ù ñ�ïò�ò��@ïô\��ò�����ò/ô'õCt½ìîë©ñ������©÷ZS/÷�ó�ò��@÷YòAi¡£�kmó K �÷'ëi÷'ë@ñ�S ïð¦ñRìX�Gë@÷Yò���ð~ï2d���ïð~ì�ï�ñ{��ô'ðRð~ï�ñ��`ôGë@ù@ìîëb�Nò�ô«ïí'ïðRód���ïð~ó ò�ôqt ¿CV�ÊF¿²ÌTÀ ò���ïWN�XDYqd���ïð~óQ÷'ë,ù�ia�\k�ì�õ�óHõdô'ð��'ï�ñ«÷áñRìX�Gë@÷Yò���ð~ï���ô'ðRðRï�ñ��`ôGë@ù,ìîëb�iò�ô»ò���ïZN�X[Yd���ïð~óiò�ô_teR,ò���ï�ë«óHK©��ô\P�����ò�ï�ñLò���ï�ù�ï�ñRì�ðRï�ùáìîëqí'ïð~ñ�ï`¸ûEì¥ë���ïAN/ìîñ[�F��ôGñ�ï�ëQ÷YòLð�÷'ë@ù�ô\P*i�÷'ë,ùáñRì¥ë���ïó»�÷'ë@ë�ô'ò´÷'ñ�z!õdô'ð ñRìX�Gë@÷Yò���ð~ï�ñ�ôGë�P�ï�ñRñR÷�'ï�ñ{��ô'ðRðRï�ñ��`ôGë@ù,ìîëb��ò�ô ÂçÈç d���ïð~ì�ï�ñ-ò�ô8tokFRqò���ïx�@ðRô],÷]<ìîøîì�ò�óò��@÷Yò[óºô\��ò�����òRñ+÷«õdô'ð��'ïð~ó©÷Yò+ò���ï�ù�ï�ñRìîðRï�ù_�^ôGì¥ë§ò3ìîëi÷YòLø�ï�÷'ñ�òA£�\æ�Y�¬`

� ï8��ô\�@ø¥ù�ì¨P��@ðRôsí'ï�ò���ï��@ðRô],÷]<ìîøîì�ò�ó�ò��@÷Yò�óÈô\��ò�����òRñ3÷«õ(ô'ð��'ïðRó�õ(ô'ð+÷�P!ï�ñRñR÷�'ï�ò��@÷Yò���ï�øX�,ñ�ó Kì�õ©S ï�÷'øîø�ôlS óLK^ò�ô��F��ôqôGñRïAP8�,ø�òRìX�,ø�ï%ìîë@ù,ìQ��ï�ñ/ìîë/R¢£�VäVäVäl���Y�UT�÷Yò�Sx�@ìQ���©ò�ôe| ïYPA]`ï�ù��«ò���ï�íY÷'øQ��ï�ò��,÷Yòì�ò4S/÷'ë§òRñ-ò�ôNìîëqí'ïðRòV`<]{��ò-ìîë,�'ï�ë�ïð�÷'ø@ò��@ìîñ-ìîñ�ë�ô'ò4�`ôGñRñ~ìX],ø�ï(M õ(ô'ð�ïÏì�÷P��,ø�ïR®ì�õ>ó K ñ�ïòRñ_^«÷'ñ-ò���ï3ô\��ò�����òô'õmt ôGëoP�ô'ðRï«ò��@÷'ë»ôGë�ï!ìîëb����òLò���ï�ëot ë�ôiø�ôGë��'ïð2÷��òRñ�÷'ñ�÷©ð~÷'ë@ù�ô\P ô'ð~÷�ø�ï^i�ìîë«�,÷Yð~òRìQ�V�@øî÷YðYR>óñ���ô\�@øîù�ñ�ïï���ôGøîøîì¥ñRì�ôGë@ñoìîë,t S+ìîøîøqë�ïV�GøîìX�GìQ],ø�ï0�,ðRô],÷],ìîø¥ì�ò�ó�kF`�]{��ò6õdô'ðfò��@ï<�÷'ñ�ï¦ô'õTª¸û¬��S´ï Ê�ÂÑ ïYP8]`ï�ùô\��ð¸ì¥ë@ñ�òR÷'ë���ï�ìîë�P!ô'ð~ïNò��@÷'ëáôGë�ïZ�,øî÷��ï�÷'ë@ù5ò��@ïðRïV]qóáìîë���ðRï�÷'ñ�ïNô\��ð����@÷'ë���ï�ñ¸ô'õ-ñ����V��ï�ñRñV`x� ïZ�Gìîí'ïò���ï�ù�ïòR÷'ìîø¥ñ¦ë�ô�SM`�[�G÷'ì¥ë.R\S ï/÷Yð~ï��Gì�í'ï�ë!÷'øX�'ô'ð~ì�ò���P ó¯S[�@ìQ�F��õdô'ð��'ï�ñfñ~ìX�Gë@÷Yò���ðRï�ñ6õ(ô'ðÄ}��2�Pìîë@ñRòR÷'ë§òRìî÷Yò�ï�ù,SLì�ò��,ª¸û¬�

S+ì�ò��TñRô\P!ïW�,ðRô],÷],ìîø¥ì�ò�óa`\`q� ïf�@ñ�ïWóHò�ôo��ôGë@ñRò�ð�����òN÷'ë¬÷'øX�'ô'ð�ì�ò���P óHK4S[�,ìQ���Tò�ð~ì�ï�ñ2ò�ô ìîëqí'ïðRò�÷�Gì�í'ï�ë�ª%û¬�Èì¥ë@ñ�òR÷'ë���ï�i�ìa`�ï`XR��Gì�í'ï�ë9bq��Ú�J^TR,ò�ð~ì�ï�ñ/ò�ôW��ô\P�����ò�ïAc»ñ����F�iò��@÷Yòdc7egfh^�P!ôEù�b�kF`

ó K iibq��Ú�J^�kû®ïòdjAklf iibq��ÚlkFn<ð��@ëfó,iijAk®k����ï�ë^óº÷'ñ�z®ñ+õdô'ðxtíivc?mÐkFR,÷'ë@ñ�S ïðL÷'ñ3õdôGøîøîô�S+ñ1MS+ìîò����@ðRô],÷],ì¥øîì�ò�óWn*M�,ìQ��zWo m@prqdst ÷'ë@ù©ðRïò���ð�ë?o(em P!ôEùubij�÷'øîø.c m ô'õ¹ò��,ìîñ¦ñ�ô'ðRò ÀjàFÌb¿wv kS+ìîò����@ðRô],÷],ì¥øîì�ò�óq£dxyn_M�,ìQ��zWo!m prqdst ÷'ë@ù©ðRïò���ð�ë?^{z1o(em P!ô®ù�bij�÷'øîø.c m ô'õ¹ò��,ìîñ¦ñ�ô'ðRò ÀjàFÌb¿{| k

����ï�ë^óº÷'ñ�z®ñ+õdô'ðL÷�ñRìX�Gë@÷Yò��@ðRï�ôGë^P!ï�ñRñ~÷�'ïMc m Mì�õ�c m ìîñ/ò�ó��`ï�£R@ðRïò���ð�ë}o mì�õ�c m ìîñ/ò�ó��`ï8��R,÷]`ô'ðRòS[��ï�ëfóºô\��ò�����òRñ¦õ(ô'ð��'ïðRóoivc m �C~©k�Mì�õ�c m ìîñ/ò�ó��`ï�£R,÷]`ô'ðRòVn�ô'ò���ïð�SLìîñ�ïR�ô\��ò����@òd~�\#o m

� ïAP�÷�ó©ë�ô'ò�ï�ò���ï�õ(ôGøîø�ôlS+ìîëb�8M�i¡£�k ÷'ñ3øîôGëb�!÷'ñ�ó K ù�ô®ï�ñ¦ë�ô'ò3÷]`ô'ðRòVR�ò���ï2ñRìQPZ�@øî÷YòRìîôGë#ì�ò{�@ð~ô�íEìîù�ï�ñ õdô'ðó ìîñx�^ïð~õdïY��òV` � ë��,÷Yð~òRìQ�V�@øî÷YðYR<ò���ï�ô\��ò�����òRñ¸ô'õ0t ÷Yð~ïZ�@ë,ì�õdô'ðFP�ø�ó©÷'ë@ùáì¥ë@ù�ïV�`ï�ë@ù�ï�ëqòRø�ó©ù@ìîñ�ò�ð~ìQ]���ò�ï�ùi(õdô'ð¦ò�ó��`ï,£xc?m²REò��@ìîñ ìîñ{�ø�ï�÷YðYnEõ(ô'ð ò�ó��`ï��8c?m�ì�ò õdôGøîøîô�S+ñ õ(ðRô\P?ò���ïLõ�÷��ò/ò��@÷Yò=o(em P!ôEù�b½ìîñ´ð~÷'ë@ù@ô\Pñ�ôWPZ�@ø�òRìX�<ø�ó®ì¥ëb�8]®ó?^iñ�òRìîøîø>�Gìîí'ï�ñL÷Dð~÷'ë@ù�ô\PâíI÷'øQ�@ïlkF`{}���ðRò���ïð�P�ô'ðRïR�ia�\k+ì�õ�ó K ù�ô®ï�ñ¸ë�ô'ò+÷]`ô'ðRòL÷'ë,ùì�õÄókô\��ò�����òRñL÷DíY÷'øîìîù�õdô'ð��'ïð~óR<ò���ï�ë«óHK ô\��ò�����òRñ+ò���ï,��ô'ðRðRïY��ò%ìîë§í'ïð�ñ�ï�ô'õ<^T` é �@ì¥ñ3ìîñ+ñRô�ñRì¥ë���ï�ì�õÄóô\��ò�����òRñ3÷�õdô'ð��'ïð~ó©ìîò{P�ï�÷'ë@ñ3ò��@÷YòH~ e f�tuivc m k_fh^{z1o em P!ô®ùZb¿ñ�ô!ò��,÷Yò8ii~�\#o m k e f�^ZP!ôEùube`�Lø¥ø,ò��@÷Yò/ðRïYPD÷'ìîë@ñ ìîñ-ò�ô«ù�ïò�ïð�PDìîë�ï¸ò���ïx�,ðRô],÷],ìîø¥ì�ò�ó�ò��@÷Yò�óLK<ù�ôqï�ñ/ë�ô'ò/÷]`ô'ðRòV`*�-÷�F�iñRìX�Gë@÷Yò��@ðRï

d���ïð~óNô'õ>ó��÷'ëw]`ï¦÷'ë@ñ�S ïðRï�ù�]®óZó K S+ì�ò��,�@ðRô]<÷],ìîøîì�ò�ó2ïÏì�÷��òRø�óuníi�ñ~ìîë���ï�ó K �÷'ëD÷'ë@ñ�S ïð�ò���ï[d���ïðRóôGë@ø�óQì�õ ìîòM��ô'ðRðRï�ñ��`ôGë@ù@ñ%ò�ô»÷iò�ó��`ï�£wP!ï�ñ~ñR÷�'ïlkF`W����ï�ë®ó?ô\��ò�����òRñ2ì�òRñ�õ(ô'ð��'ïðRóR¹ò��@ìîñ�|���ï�øX�,ñ��fóHKi�÷'ë@ùoóLK¹ù�ô®ï�ñ¸ë@ô'ò%÷]`ô'ðRòFk[S+ìîò��q�,ðRô],÷],ìîø¥ì�ò�ó©ïÏì�÷��òRø�óe£Fx�n0`A m��ò�òRìîëb��ò��@ìîñ¸ò�ô�'ïò���ïð�ñ���ô�S+ñ%ò��,÷Yòò���ï2ò�ô'òR÷'ø �,ðRô],÷],ìîø¥ì�ò�ó�ò��,÷Yò[ó K ù�ô®ï�ñ3ë�ô'ò3÷]`ô'ðRòLìîñdn��E�li¡£dx�n�kF`� ï+ë�ôlS�P�÷æìEì¨P�ìX�ï ò��@ìîñ��@ðRô]<÷],ìîøîì�ò�ó` é ÷z®ìîë��¸ò��@ï/ù@ïð~ì�íY÷YòRì�í'ï¦÷'ë@ù!ñ�ïò�òRìîëb�%ïYd��@÷'øEò�ô��ïðRô��Gì�í'ï�ñ!M

Page 124: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

� Û x�ij� Û)� £�k�naf�ï¬R'ô'ð_naf�� Û \¬ij� Û(� £�kF`� ´øQ�b��Gìîë��Lò��@ì¥ñ¹ì¥ëNñ��@ô�S+ñ6ò��@÷Yò�ì¥ë�ò��@ì¥ñ��÷'ñRï¦ò���ï��@ðRô],÷]<ìîøîì�ò�óô'õ6ë@ô'ò3÷]^ô'ð~òRìîëb��ìîñ1M � ��Û�lÛ � £8� � � z £

�lÛ � £ f £��Û� £Lx £

��Û � £8� � ��� s � Ú r.s��Û �

S[��ïðRï�ò��@ì¥ñ���ôGøîù@ñ/õ(ô'ð3ðRï�÷'ñ�ôGë,÷],ø�ó©øî÷Yð��'ïA��ÛAi�÷'ë@ùqÚA��ïðRï2ìîñ¦ò��@ï�],÷'ñ�ï2ô'õ¹ò���ï�ë@÷Yò��@ð~÷'ø_ø�ô�G÷Yð~ì�ò���PWkF` 0��ò�òRì¥ëb�!ïí'ïðRó®ò��@ìîëb�!ò�ô�'ïò���ïð�R�S ï�ñ�ïï2ò��@÷Yò+ò���ï��@ðRô],÷]<ìîøîì�ò�ó�ò��@÷Yò[óHK_ìîë§í'ïð~òRñ¦ò���ï��Gì�í'ï�ëqª¸û¬�ìîë@ñ�òR÷'ë���ï´ìîñ0i�÷Yòoøîï�÷'ñ�òFk.Ú r.s `)\æ��Û<i�ìa`�ï`XRYò���ï4�@ð~ô],÷],ìîøîìîò�ó3ò��@÷Yò�ó K õ(ô'ð��'ï�ñ�P8�@øîòRìX�,øîì�ï�ùx]®ó¸ò��@ï4�@ðRô],÷]<ìîøîì�ò�óò��@÷YòAóHK6ù@ôqï�ñ2ë@ô'ò�÷]`ô'ðRòFkF`�ûEìîë���ï�ò��,ìîñ��÷'ë®]`ï�÷YòAP!ôGñRòA�MS ï!ô]@òR÷'ìîëy`/� ÚY��ÛF�VR©��ô\Pw�,ø�ïòRìîëb�iò��@ï

�@ðRô®ô'õã`

Page 125: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

�H�����(�1���C���E���(�<�O�)���U���(���� �¡  �g¢¤£(¥{¦¨§ �¤�J���U��©�ª��C���«�}�J� � �C�U¬��J�(­(���)¬.®¤�¯*�C�(�[�!�C�J�(�±°(�«���)�C®��(�O²A�)�J³´�µ�¶Q·�¸º¹�µ »½¼

¾ ¿ÁÀ�Â�Ã�ÄCÅÇÆÉÈËÊ<ÌÎÍ�ÏwÅ<ÐÑÌ{Ò{Ó�ÄCÔFÏÕÄ�ÏÎÓÑÖwÊØ×ÚÙ Û ÔAÜ�Ê_àU���[�)��Ý8Þß�A®����(�à�J�1�!�}��á8�(â�¬����!�d�(�gã[�(�1�C��äg�1åZª1���!�É�d�C��­«���)�C©8�C�Q�Cª�®��!â��!�d���}�C®8�A�C�)æçª1�(�����!�}���(�����«â�(���(ª1���WãièdéAädâ��'�8�!��ê � �(�9Þg�{©��C�A�C®8�àèdéëâ��U���!���J��ª��«���C�J��©�ª��L�1å�ª1���!�¤�L¬�©8ì�����ª5æ�í(�1���!��ª����ɬ��C���«��Cª�®8�!âu�!�L�(�dÞg�!���Dîï �Çâ��(�C���)�)�C���«��Ý#Þß�*���!âu�����±�C®8�<���!�(�8�1���C®��)���C®8�*â��«�J�@�1å�ª1���!�¤��¬�©8ì�����ª5æ�í(�1�L�!��ª��C�U¬��C���«�à�Cª�®8�!âu�ì��(�J�!�a�«�a�J���)¬Ñ���É�(�±¬Ñ�1��âð©8�C�)�C���«���uã[�(ê�­8ê�Ý�èF  ï äF���¡�C®8�Z�C�C�(�����)���aâ��'�8�!�,Þg�(�CíU�à�(�Q�D�«������ÞL�!ñF�C®���C�!ª��!���(�1�à­(�!���1���)�J�!�à�}í(�1��ò�ó5ô9�(���a�(�C�J�'ª1���)�J�!�y�J���)¬��8�É�(�±õ�ö9�[�(�A�}�J���)¬��8�U�(�Q¬7�1��â{©8�C�)�C���«��êð÷L®8�¬�©8ì�����ªßí(�1������ò�ó5ô{�(���u�C®8�F�J�!ª��C�1�_í(�1�����*�C��âu¬����àõ�ö�ê<÷d®��1�C�L���ß�(���J�à�Wã[¬�©8ì�����ª1���¤æ�í'�8�#ÞL�.äÇ®��)���'æçª��(���ì����Fø¡ñ'ù{ú�û�üQýÿþ����������D�(�F�C®8�{�J���)¬��8�É�(�H¬Ñ�1��â{©��C�)�C���«��êL÷Ç�Z�!��ª����ɬ��H�C®8�w�C���8­«���Qì.����±�C®��{�J�!�����1�ª�®8�U�«�J�!�W�½���(�����«â�'Ý=ª��«âu¬�©8�J�!� ��� ���«úiû�ü«ã�¤ä��(��������� ø�ã�¤ä����#Ý��(���Á�J�!���������������)ê�÷Ç��8�!ª��C�U¬��1Ý��C®8�Z�C�!ª��!���(�1�������J�à�C�!ª��#�(�1����9�[�C�«â����Wãi©��C���8­9�C®��Z�J���)¬��8�U�(�5äQ�(���½�C®8�!���C�!ª����(�1���{�C®��â��!�C�C�)­(�ðìÉ�9ª��«âu¬�©8�C���8­���� øÇã�¤ä �!� � ê#"H�(�J�{�C®��)�L�C®����d�C��$É©����C�!�d�«���à�1�)�(��©��)�C���«�O�(�%� �D�(�H�!�(ª�®ì����ð�(�H�C®8�}�(����­«�����(�gâ��!�C�C�)­(�(ê § � ª1���(���wÞß�?�(���J�a�����Cª1©��C�J�!���!��ª��C�U¬��C���«� �Cª�®8�!â��!��ÞH®���ª�®��C�!��©�ª���C®8�Fª1��¬�®8�1�C�J��á'�g���!�8­(�C®¡ã[�[�(�����«�8­(�1��â��!�C���)­(�!��ä_ì�©8�_�C®��!�J�F��ª�®8�!â��!��®��!�(�±�C®8�±�C�(â��'&)(+*-,/.103240657(+8/2+9�1åZª1���!��ª��?�(�d�)ìÑ���(�}ãi��ê��(ê�Ý��«�8�à�1��(��©��)�C���«�?�(�:�¡¬7�1��ì.���=�(�,�(����­«�����(�Ñâ��!�C���)­(�#ä�ê�H���D�(�C�C©����)�J�!���(Ý�Þg�àª1�(�9�!���J�!�¤�C���(���������u�8�uìÑ�1�J�J�1�d�C®��(�O�C®���� � ©��C����­w�)��ì����J���)�C�W�J���)¬��8�U�(��¬Ñ�1�Jæâ{©8�C�)�C���«���w���½�C®8�W�J�C�(�����)����â��'�8�!��ê § �w���à�(���C�/�8�(�ðíU����ÞL��ÞL®8�1�C®8�1�w���à���à¬Ñ�«�C�C��ì������J� �8�/�C©�ì8æ�J�C�(�¤�C���(������ìÑ�1�J�J�1��©��C���8­8ÝÉ�(ê�­8ê�Ý�èF  ï ���u�C®8�±�J�C�(�����)����âu�U�8�!��ê�÷L®����_®��(�gâu�(�C����)�J�!�W�C�!�C�!�)��ª�®8�1���=�J�ª��«���C���8�1�LÞL®��)�d�J�(�C���(���1åZª1���!��ª��WâZ��­«®¤�Lì7�±¬Ñ�«�C�C��ì����±���}�C®8�A���(���8�«â �(���(ª1���àâu�U�8�!��ê

;a�y�8�#Þl¬��C�!�J�!�É�?�º�C��â�¬.���/�(�����(�1���Á�1åZª1���!�¤�9�Cª�®8�!â��(ê ï �9ì7�1�[�(�C�(ݱ���1�½ã6<=�)�>��?A@�BÉä�ìÑ�a��J���)¬��8�É�(�_¬7�1��â{©8�C�)�C���«�w�[�(âu�����(ݤ�(���u���1�DC ñ8þ��������FEFýÎþ��������HGßìÑ���Q���(�����«â �(���(ª1���(ê*÷d®��d�Cª�®8�!âu�Þß�(�Cí'�L�(�d�D�«������ÞL�1ñI ÷Ç�?­(�!�8�1���)�J�uí(�1�'�1Ý@�C®����C�!ª��!���(�1�Q��©����< �J�O­(�!�8�1���)�J�u¬�©�ì�����ªQí(�1� ò�ó5ô9�(��� �J�!ª����1�Aí(�1�/�C®���(�C�C�Uª1���)�J�!�9�J���)¬��8�É�(�Lõ�ö�êI ÷Ç���!��ª��C�ɬ��±âu�!�C�C�)­(�=JLK þ��������HG�Ý��C®8�w�J�!���8�1�Qª�®8�É�«�C�!�±���(���8�«âMNKaù{úiû�üw�(��� ª��«â�¬�©8�J�!�� � �O�«ú�û�ü(ã�¤ä�ê_÷L®8�A�J�!���8�1�H�(���J�uª��«â�¬�©��J�!�-� � �PC ã�«ä �QJ �(���?�J�!�����-� � ��� � ê

I ÷Ç�A�8�!ª��C�U¬��<ª1��¬�®8�1���J��áU�%���������Ý«�C®8���C�!ª��!���(�1�_ª��«â�¬�©8�J�!�Hãi©��C���8­F�C®8���J���)¬��8�U�(�5ä R�O�TS �úiû�ü ã3���5ä�(���u�C®8�!�u�(ì��C�(�����*�C®8�Hâ��!�C�C�)­(�F�(�U� � �PC ã�¤äV�W� � êdãièL�!ª1�(�����C®��)�%C ���X,/.ZY[9\57&[9^]=2V&)&�_H`)`a57Y[9b_ìU���C®��Q�C�!ª��!���(�1�!Ý7�(�¤�W¬Ñ�«�C�C��ì����F�J�!���8�1���1Ý��(���9�C®8�A�(�8�(�1���C�)���}�(�dÞß�!����êGä§ �_��®8�«©����wìÑ�dª1���!�)�ß�C®��)�*�C®����<��ª�®8�!â��F®��(�*ª��(�C���!ª��ß�8�!ª��C�U¬��C���«�@ê,÷d®���¬��)���(�����!�'ÞH���C®w�C®8�d�!��ª����ɬ��C���«��Cª�®8�!âu�=���à�C®8�=�J�C�(�����)���{âu�U�8�!�Ñã[­«���(�!�ð¬��C�1�'���«©������8ä��C®8�«©����{�(���J�FìÑ�ß�(ìU�U���«©��1êTc������(�����(Ý�C®8���Cª�®8�!âu�dfe�g�hji3k�g�l7mXnjk�gpoqh3mDrts[i3muh3mul6i3mwvVxFy/njr{z{z�|4gar{s[ijga}�iji3k+o)iji3kFmuh3m:o)h3mD~pmui3kFg�vFl�njr�i3k�l7z{r{�ak[i3z���4m�i7i3muh mu�p��r{m�sF�u��F}Fi

mu�Hm�s-i3kFm�l7mDvFg�s�g�i���r{m�ztv�o�i7h3}Fz��-|�hfo��ui3r{�wo)z�l7�fk�m�~pm��

Page 126: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���g�[�(���������1åZª1���!�É�1ñ*��������� ¥ �w�C®8�!�}©8¬}�J� � ¥ �{ì.���C�g�(�@�C®8�Qâ��!�C�C�)­(�Aª1�(�WìÑ�F�!��ª��C�U¬��J�!�}©��C���8­{�«��������C���8­«���à�1�)�(��©��)�C���«�O�(���,êðã6"L�(�J�u�(���C�Z�C®��)�F�!��ª��C�U¬��C���8­W� ¥ �Wì����C�L���L�!�8�«©�­«®O�J�?�(�����#Þ �C®��ð©��C�à�(�®¤�Uì������ �!��ª��C�ɬ��C���«�½�[�(�{â��!���C�)­(�!�{���«�8­(�1�à�C®��(��� ¥ �Oì.���C�1ÝÇ������ª��u�C®8�Q��¬������)�)�J�Zí(�1���O�J�1�ð©8¬����1�!��«�����Wì7�����4��ì����C�����«�8­u�(�É�UÞg���(êGä�g�1�D�(�C�w­«���'���8­��C®8�ð¬��C�U�(�<�(�*�J�!ª1©8�����E�9�[�(�H�C®8�w�)ìÑ���(���Cª�®8�!â��(Ý�Þg�{­«���(�w�J�«â��w���É�C©����C���«�/�(���¡�ª1�!�(�!�)�1êpc8�(�L�C®��A���¤�C©����C���«�W�(�d�J��ÞL®É���C®��A�Cª�®8�!âu�A�����J�!ª1©8�C�(Ý.�8�(�J�Q�C®��)�dÞL®��!�?�(�9�(�8�(�1���C�)���W�J�1�!�ª1��¬�®8�1�C�J��á'������������ã[�!��ª��C�U¬��J�!�?©����8�1�=¬�©�ì�����ªLí(�1�/ò�ó5ôUäß�C®��±�(���(�1���C�)�C�}®��(���8�����8�[�(��âu�)�C���«�W�)�d�(����)ì7�«©��ß�C®8�F©����8�1�����U���8­Aâ��!�C�C�)­(�±©������!�C�_���ßíU����ÞL��C�ã�¤äHã[ÞL®8�1�C�-��a�3��M� S �ú�û�ü ã3� � äJä�ê ï ���uìÑ�!ª1�(©��J�FÞg�â��U���!�jC �(�H�����(�����«âË�(���(ª1���(Ý��C®��à�(�8�(�1���C�)���}®��(�d�8�u���8�[�(��âu�)�C���«�?�)�L�(���@�)ìÑ�«©8�#C ã�¤ä=©������!�������®��(�A��á'¬�����ª1���C����$U©8�1�����!��CÕ�)�à¬Ñ�«���É��UêN�H��Þg�1�(�1�!Ý*�J�/���O�J�O�C®��Z�(�8�(�1���C�)���aÞß�«©����½®��!�(�}®��(�y�J�ª��«â�¬����1�J�!�����8�1�J�1��âu������A�[�C�«â����ßÞL®���ª�®Z���_��$U©����)�(���!�É�_�J�����É�(�1�C�C���8­A�C®��H�J���)¬��8�U�(�g¬7�1��â{©8�C�)�C���«��ê '����ª����C®8�Z�Uª1ª1©8���AÞL���C® �«�����¡��âu�(���<¬����(ì��)ì��������ç�(Ý��C®8�Z��ª�®8�!â��W���A®8�!��ª��Z�J�!ª1©8�C�¡ã[Þß�u­«���(���9â��(����D�(��âZ�(��¬��C�U�(�ÇìÑ�!���#ÞFä�ê� ®8�?ª1�!�(�!�)���(�H�C®��}ª��«���J�J��©�ª��C���«� ãi�(��� �C®8�}¬��C�U�(��äà���{�C®��)� 0�¡�_H¢q_£5¥¤�8/(§¦f¢w_)2+9p¨%(+¢a9A©>ª ¡�2�¤)¡

« .�8/&�0656(+8 « (+¢ ¨X¡V57&H¡¬0�¡�_£2­Yq(+®4_�&�(+81¤H06¢�.¯&�0657(+8P5°¤R,/¢w(+®+2VYH9^]�¤�_q&�.�¢q_�ê � ®��}ª��«���J�J��©�ª��C���«�ºª1�(�±(+8¯9^]ì7�}¬������(�!�º�J�!ª1©8���WÞH®8�!�³² ���ð�J�C�!�)�J�!�S�(��� ���(���8�«â �(���(ª1���(ê § ��¬����(ª��C��ª��WÞg�9âu�����8�(��Þg�(�C�C��)ì7�«©��d�C©�ª�®}�C®����8­«���C����ª��±���u�(ª��C©��(�@�)�J�C�(ªCí/�)­«�(�����J���C®8�A��ª�®8�!â��A���=íU�8�#ÞL�?ÞL®��!�?©��C���8­wª��«âuâ��«�®��(�C®½�i©���ª��C���«���±�D�(��²yêN�=©8�A�[�C�«âl�O�C®8�1�(�C�1�C��ª1�(�_¬7�«���¤�Q�(���'���1Þ �C®����A�J�C�)�J�Z�(����´��(�����{���A�9�����J�C���©����J�1�J�C�����8­8êµ�¶T·­¸/¹F·�º »O¼ « ã6½=�)�>��?A@�BÉä�5¥¤'29ã�¾a�)¿�ä�À�¤[_qÁ�.�¢w_�06¢�2�,1©Â(�(+¢Ã,1_H¢a*=.�032Â0656(+8¯Ä-0�¡�_H8Q0�¡�_�2VY)(+®+_�Á)(481¤a06¢a.¯Á[À0656(+8Å( « 2�,/.ZYH9^56Á�ÀfÆÂ_H]�_[8/Á[¢a]�,>0656(48£¤[ÁH¡�_[*�_R5°¤àã�¾��)¿5ä�Àf¤�_qÁ[.�¢q_R5°8W0�¡�_ä[_H8¯¤[_�( « 5°8>©+5¥¤H065°8�Ç4.�5¥¤�¡�2VY[5°9^5�06]�5°80�¡�_�¢�248/©Â(+*È(+¢w2ÂÁ[9É_�*�(�©V_[9AÊ

Ë ¹F¸�¸/Ì ï �C��©�â��QÞg�à®��!�(�ð�(�9�(�8�(�1�����)�C� Í �(���Oâ��!�C�C�)­(�!��J ÎF�wJ��d�[�(�LÞH®���ª�®@ñÏ ¯*��Ð�ã3����������äDÑ�Ò�úiû�ü(ã�J��5ägñÂÍwã�ò�ó5ô¯������������ä:���aÓ

Ô ¯*��Ð�ã3����������ä:ÑÕÒ�úiû�ü(ã�J Î�äßñVÍwã�ò�ó5ô¯������������ä:���aÓ Ï �OÖã[ÞL®8�1�C�A�C®��A¬��C�(ì��)ì.�������C���!�g�)�C�A�C�)í(�!�/���(�1�F���(�����«âu���Z­(�!���1���)�J�!�O¬.©8ì�����ªHí(�1� ò�ó5ôUä�ê%;a�ð�C®8�#Þ ®8��Þ�J�uª��«���J�J��©�ª����(�?�(�8�(�1���C�)�C��Í-×�ì��C�!�)íU���8­{�C®��±�C�!ª1©8�����E�Z�(�@�C®8�±�J���)¬��8�U�(�=¬Ñ�1��â{©��C�)�C���«��êDÍ-×�ÞL������ìÑ�­«���(�!�O�w���(���8�«âËí(�1� ò�ó5ôZ�(���9�u���(���8�«â �!���!âu�!�¤�#Ø7Ý��(���?ÞH�����7�J�C�W�J������� � S �ú�û�ü ã�Ø�ä�êÍ × ã�ò�ó5ô¯�wØ�ä U�1�������Ø����ÑÎþ�������ÙHGèL©�� Íðã�ò�ó5ô¯������������ä�Ý.�(���JÞg�1�����8­'²Oæ��(���(ª1���R$U©8�1�����!�d�(�ÚÍÚ�(���[�«�����#ÞL�1ñé±�9�C®8�ÃÛ7Ü�Ý=$U©8�1�C��ÞZß�Ý�ª�®��!ªCí?ÞL®8�1�C®��1�-�«úiû�ü«ã�ÞZßçä:��Ø>à����,�J�8Ý��«©8�J¬.©8�pÞZß��(���9�J�J�(¬éF�C®��1�CÞL���J�(Ý�����â�¬�����ª�®8�U�«�J�A���(���8�«â�Ø ß ÑÎþ�������ÙHGH�(���?�C�1�C©����NØ ßã�;a�W�(�C��©�â��uÞL���C®8�«©8�à���«���Q�(��­(�!�8�1���(�����ç� �C®��)��Í �8�1�(�1�w�(�Jí'�A�C®8�W�C�(â���²9æ��(���(ª1����$U©8�1�C� �çÞL��ª��4à�8�(�J�A�C®��)�L����­«�(�����d�8�(�C®����8­wìÉ�}�8�«����­w�C�8êGä � ���!�)�����(Ý8���TÍ-×Ñ�«©8�J¬�©8�C�d�(�É�U�C®����8­w�C®8�!�9���d�C©�ª1ª��1�!���H���

����������­� S �úiû�ü ã�Ø8ä�ê ï ���ð�C®8�=¬��C�(ì��)ì��������ç�Q�C®��)�XÍ × �«©��J¬�©8�C�<�(�É�É�C®����8­F���Ç��á8�(ª��C���{�C®8�=¬��C�(ì��)ì��������ç�Q�C®��)�Í �1�(�1��$U©8�1�����!��² �«�?���8¬�©���� S �úiû�ü ã�Ø8ä�êD;a�{âu�!�W�(���J���8�(�J�Q�C®��)�#Í-×.¬��C���'���8�!�=�ð¬Ñ�1�C�D�!ª����C��â{©����)�C���«��D�(�#Í ãi©8¬?©��¤�C�����C®8�±¬Ñ�«���É� ¦ ���@�(�É� ¦ �C®��)�UÍ × �C�J�(¬���ä�ñ*�C®8�±�«©8�J¬.©8�ß�(�Ú²l���=�������1¬7�!���8�!�¤�C���w�(���©������[�(��âu���ð�����J�J����ì�©8�J�!��Ý«�(���u�C®8�Fª1��¬�®8�1�C�J��á'�ß�(���u¬�©8ì.����ªgí(�1�u®��!�(�F�C®8�H�C��$É©����C�!�u�����J�J����ì�©��C���«�9ã3���

Page 127: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���F��$É©��(�,�J�£�«ú�û�ü(ã�¤äd�D�(�A�C�«â��w���(���8�«â�Ð ©��8í'�8�#ÞL��ÓX��(���á� � ���H©������[�(��âu���?�����C�J����ì�©8�J�!�OÞL®���ª�®/������8�A�(�L���«�8­u�(�#ÍÚ®��(�d�8�(���(�1�$É©��1�����!� ² ã�«äJä�ê

â@�1�Åã1ä^@#ö�ìÑ�/�C®��/�1�(�!�É�}�C®��)� Í �1�(�1�?��áU¬.����ª1���C���³$U©8�1�����!��²�ã�¤ä�ÝHÞL®8�1�C�¡�(�W�)ìÑ�#�(�aÞß� ®��!�(��H�7��å� S �úiû�ü ã3���5ä�êX"H�(�J�L�C®��)�1ݤìU�ðÞH®��)�<Þß�H�C�(�����)ìÑ���(�(ÝU�C®8�d¬����(ì��)ì��������ç�A�C®��)��Í�×8���É�(�1�C�C�:Øw������á8�(ª��C���¯*�FÐAã¯ä\@#ö4Ó3àd�C®É©��1Ý_Þg�9â{©��J�u®��!�(�/¯*�FÐAã¯ä\@#öÂÓ�æç¿uìÉ���C®8�9�J�!ª1©������ç���(�F�C®8�?�J���)¬��8�U�(��¬7�1��â{©8�C�)�C���«��ê� �«�������C���«�����8­ð�«�?�1�(�!�É��ã1ä^@#öZ­«���(�!�1ñ¯*��Ð�ã3����������ä%Ñ�Ò8ú�û�ü«ã�J��5ägñÂÍðã�ò�ó5ô¯������������ä:���aÓ

� ¯_��Ð�ã3����������äDÑ�Ò�úiû�ü«ã�J��5äßñVÍwã�ò�ó5ô¯������������ä:��� Ï ã¯ä\@#öÂÓZè!¯*��ÐAã1ä\@ö+Óé ¯*��Ð�ã3����������ä:ÑÕÒ�úiû�ü(ã�J��5äßñVÍwã�ò�ó5ô¯�����������!äD��� Ï ã¯ä\@ö4ÓZè!¯_�FÐ ã1ä^@#ö+Ó3�

ÞL���C®?�u�C��âu�����)�=��á'¬��C�!�C�C���«�}�D�(�d�C®��àª1�(�J�A�(���!��ª��C�U¬��C����­�J Îê � ®8�Qí(�1�}¬7�«���¤�����1ñ¯_�FÐ�ã3����������ä:Ñ�Ò�úiû�ü¤ã�J��5äßñVÍwã�ò�ó5ô¯�����[������äD��� Ï ã¯ä\@#öÂÓ��¯_�FÐ�ã3���������!äDÑ�Ò�úiû�ü¤ã�J Î!ägñÂÍwã�ò�ó5ô¯������������äD��� Ï ã¯ä\@#öÂÓ3�

ÞL®8�1�C�9ãi�(�à�����Cª1©��C�C�!�/���¡�C®8��¬.�)���)­(���)¬�® ¬����!ª��!�����8­}�C®8�w¬��C�U�(��äH�C®����±���±ìÑ�!ª1�(©��C�=² ���±�}���(�����«â�(���(ª1���H�(���u�J�à�C®��L�(�8�(�1���C�)���w®��(�_�8�à���8�D�(��âu�)�C���«�u�)ìÑ�«©8�D² ã�¤ä<�(�_���«�8­à�(�_���*®��(�_�8�1�(�1�g��á'¬�����ª1���C���$É©8�1�����!��² ã�¤äàãi�(���O�C®8�1���1�D�(�C��J Î��!²�ã�¤äL���d��$U©��(�����9�����J�J����ì�©8�J�!�?�J��J��X�!²�ã�¤äTêC©��J�±�(�±���9�C®��ª1�(�J�à�(�,�C®8�A�«�8��æ��C��â��A¬��(�.ä�ê*¯ß©8�J�C���8­��1�(�1���É�C®����8­��J�(­(�1�C®8�1�F­«���(�!�1ñ

Ö� Ï ¯_��Ð�ã3����������äDÑ�Ò�úiû�ü¤ã�J���ägñÂÍwã�ò�ó5ô¯������������äD��� Ï ã¯ä\@ö4Ó�¯*��ÐAã1ä\@ö+Ó

Ô ¯*��Ð�ã3����������äDÑ�Ò�úiû�ü(ã�J Î�ägñÂÍwã�ò�ó5ô¯������������ä:��� Ï ã¯ä\@ö4Ó�¯*��ÐAã1ä\@ö4Ó Ïæ ¯_��ÐAã1ä^@#ö4Óæ ¿�ë

� ®�����ª��«â�¬.���1�J�!�d�C®8�Q¬��C�É�(��êì í8Ü�Ê_ÏFÓ�Äwî�ï:ðgÓ�ÄCÔHÏòñ Ð�Ô�ÓÑÔÃï<ÔHÃ)ó;a�ß®����(�g�8�#Þ��������C®8�!�FÞL���C®A�«©8��ª����(�1���)­(�g�(�'�C®8�ß�!�C�J�!�É�C���(�«ª��C�U¬��J�(­(���)¬.®���ª<¬�����âu���C���(�!�!ñ.�!��ª����ɬ��C���«��(���wâ��!�C�C�)­(�d�(©8�C®��!�¤�C��ª1�)�C���«�ð���à�C®8�=¬�������)�J��æ�í(�1�ð�J�1�J�C���8­8Ý«�(���{�!��ª��C�ɬ��C���«�{�(���ð����­«���C�(�U�C��­«���)�C©8���!����u�C®8�F¬�©8ì.����ª5æ�í(�1�w�J�1�J�C���8­8êD�H��Þg�1�(�1�!Ý8�C®8�!�J�±�)�C�±�8�(�_�C®��=(+8Z9\]u©��C�!�ß�(�����)¬�¬�����ª1�)�C���«���*�(�@ª����ɬ��J�(­)æ���)¬�®É�(êAé±�D�J�!�/�C®8�1�O�[�(��â ��â�¬Ñ�(�C�C�(�É�Lì�©�����������­ðì.���Uª�íU�F�[�(�F���)�C­(�1�Q�(���/âu�(�C��ª��«â�¬�����á9¬����(�J�Uª��«���!ê;a�àì������[ô��u��á8�(â�¬����A�«���A�C©�ª�®O�)¬�¬�����ª1�)�C���«�?®8�1�C�(ê� �«���C���8�1���C®8�*¬����(ì����!â��(�8�J�!ª1©8�C�*���8�!�¤�C�É��ª1�)�C���«��êÚ�L�1�C�(ÝÞg�*ÞL�����)�(�C��©�â��*��¬������(�1� õ ÞL®8�dÞ=�(�¤�C��J��¬������(��®����H���8�!�É�C���ç�?�J�}�J�«â��ð�(�1���b���1�-öHê-;a����âu�)­«���8�à�C®��)�-õ �(����ö �)�±�J�«â��à¬Ñ�«���É�H�C�!ª1©8�C�!����!�J�C�)ì�������®/�}ª��«âuâ��«�@ÝÑ�C�!ª��C�1�Qí(�1�W÷7ê ï �A�J�«â��w���)�J�1�Q¬Ñ�«���É�H���/�C��â��(Ý�õlã[ÞL®8�}���±ª��«âuâð©�����ª1�)�C���8­���(�1���A�8�1�EÞß�(�Cí�ä*Þg�(�É�C�*�J�ðª��«�¤�'����ª��-ö��C®��)�ß���*���ß�����8�1�!��®���â?ê�ã § �Z�[�(ª��1ÝU�C®����*�C�(�C�*�(�Ñ�C®����8­à�Uª1ª1©8����(���@�C®8�ð�C��â��(ñ�ª��«�������8�1�LÞH®��)�L�(�«©¡�8�ZÞH®8�!�9�(�«© ���(­��«�O�J�W�(�«©8�F�!âu�(�����(ª1ª��«©��¤�1êGä ;h®��)�F�J�(���L�(��J�!ª1©8�����E�WâZ��­«®¤�LÞß�A�8�1�!�/�(���?®8�#Þ âu��­«®É��Þg�à�8�!�C��­«�?�Z�J�!ª1©8�C�Q¬����(�J�Uª��«���[�(�L�C®����=�C�(�Jí�îé±�8�?��âuâu�!�����)�J�W�C��$É©����C�!â��!�É�ð���{�C®��)�w�(�º�(���(�1���C�)�C��ã[ÞL®8� �8�É�!�����(�ðí'�8�#ÞË�C®8�?�J�!ª��C�1�wí(�1��C®��)�C�!�}ìU�øõ �(����ödäg�C®8�«©����ZìÑ�Q©����)ì����H�J�u��âu¬7�1���J�«���)�J�õØã[�C®��)�d���1Ý'�J���i�(���J�!���Zª��«�É�'����ª���öÁ�C®��)�

ù

Page 128: UMD Department of Computer Science | - lecture3jkatz/crypto/f02/lectures/crypto... · 2005-02-03 · Þ'ß ´ 3à |á QÏ ¶~ Mê vê A ~© /²~§m¡-¤6¶ ¡ £7¨ª¸Ò¥ ¡ £7¼|¤*¡t¡-

���Q���±�J¬Ñ�!�)íU����­�ÞL���C®�õ{ä�ê � ®����L¬����(¬7�1���ç�O�C®��«©����¡�)�A�?âu������âð©�â ®8�«���¡ÞL®8�!�¡�C®8���(�8�(�1�����)�C�¡®��(��8�1�(�1�<�(ì��J�1�C�(�!���(�É�à���É�J�1���(ª��C���«����ì7�1�EÞß�1�!�=õ �(����öFê § �<���Ç�(�1�C�à�!�(�J�{�J�Qª��«â��d©8¬ðÞL���C®ð�H¬��C�(�J�Uª��«�â��1�1�C���8­W�C®����F�8�[�������C���«�9�(�_�C�!ª1©8�����E�.ñd�C®��ð¬��C�(�J�Uª��«��ª��«�������J�C�F�C��â�¬����W�(�%õ �C�!�������8­Z�C®8�{í(�1��÷/�J�ö-à:ö �C®8�!��ª�®��!ªCí'�{ÞH®8�1�C®8�1�à�C®����Qí(�1�yâu�)�Cª�®8�!�ð�C®8�u�«�8�W���{®��(�A¬��C�1�'���«©������/�C®��)�C�!�½ÞL���C®áõ�ê/éF�ª��«©8���J�(Ý��«©8�Q­«©��F�[�1�!������­W���H�C®��)�A�C©�ª�® �}¬��C�(�J�'ª��«���C®8�«©����¡�8�(�±ìÑ�ðª��«���C���8�1�C�!� �J�!ª1©��C�4ú § ���8�1�!��Ý�����(�y�(�8�(�1�����)�C� �(ì��J�1�C�(�!�{�O�C����­«�������¤�J�1���(ª��C���«�½ìÑ�1�çÞg�1�!�áõ �(���ÅöFÝÇ�C®��!�a�C®8�Z�(�8�(�1���C�)�C�a���!�)������÷�(���Zª1�(�W���)�J�1�ß��â�¬Ñ�1���J�«���)�J�põ �)�ß�(�É�u���)�J�1�_¬Ñ�«���É�*�����C��âu�4ú � ®É©��1ݤ�(���C®8�«©�­«®u�C®8�L���1�(�!���(���J�!ª1©8�����ç�â��!�¤�C���«�8�!�9�)ìÑ���(�ð����ª��1�C�C�(�������W�8�!ª��!���C�)�C�(Ý����d���d�8�[�������J�!��� 8/(40���©8åZª1���!�É�1ê§ ���J�J�!�(��Ý(Þg�g�8�1�!�ð�J�±�(�������C���«���(�����F�C��$U©����C�*�C®��)���(�ð�(�8�(�1�����)�C�à�C®8�«©����AìÑ�ß©����)ì����ß�J�F��âu¬7�1���J�«�'æ�)�J��õû_H®+_[8�ü « 0�¡�_'2©+®4_H¢�¤H2+¢a]£Á)248�(VYa¤�_H¢a®+_Ný�_q2+®4_a¤H©4¢�(�þÿ(+8�� 2�¤R*�2+8¯]£_���_qÁ�.�06ü7(+81¤=( « 0�¡�_-þ/¢�(Â03(�Á)(+9Y)_[06¨�_�_H8�õ 2+8>©�ö±2�¤ü�0T9\ü°Æ4_H¤�ê § �ß�C®8�«©����wìÑ�dª1���!�)�_�[�C�«â �C®��L�����Cª1©��C�C���«���)ìÑ���(�H�C®��)�_�C®��d¬��C�!ª��!�����8­¬��C�(�J�'ª��«���8�U�!��8/(40=�C�)�C���J�D�W�C®��������(�C���«�?�(�<�J�!ª1©8�����ç�(ê

�L�#Þ ª1�(�¡Þß�w�8��ÞÚ�8�!�C��­«�/��¬��C�(�J�'ª��«�Ç�C�!ª1©8�C�w©����8�1�H�C®����F�8�[�������C���«�.î�éQ�8�{�C©�­(­(�!�J�C���«�¡���L�C®���D�«������ÞL����­8ñ ���1�¡ã�ÒD�Jùuä�ìÑ�¡�y¬�������)�J��æ�í(�1� �!��ª��C�U¬��C���«�h�Cª�®8�!âu�(ê � �������!�¤�C���[�³õ�ÝL�C®8�/�(�1���b���1��öª�®8�U�«�J�!�ð�}���(���8�«â �)�(��©8���¡�(���½�J�!�����A���Q�J� õ�ê � ®8��¬��C�#�(�1�Rõ �!��ª��C�ɬ��C���/©������8­ ÷ �J�9�(ì��C�(���ª1��¬�®8�1�C�J��á'�=� ãi��ê��(ê�Ý*ª��«â�¬.©8�J�!�=�çÑ Ò��Uã�8äJäA�(�����J�!�������Ø�(�{���C�à�C�1¬.���(ê � ®8�Z�(�1���b���1�{���!ª��C�ɬ��C�� �(���aª�®��!ªCí'�AÞL®8�1�C®��1�F�C®����C�!�C©����C���8­��8�!ª����ɬ��C���«�a���F��$É©��(���J��/à��«�����9�C®8�!�½�8�U�!�Q���±�(ª1ª��1¬��1ê § ��C©�ª�®y�}¬��C�(�J�'ª��«�<�J�!ª1©8�C�#î �9�(�C��¬��C�!ª1���J�!���(Ý@ª1�(�½Þß�u­«���(�u���)�C©8���(�<ª��«�������C���«���F�«� �C®8���!��ª����ɬ��C���«��Cª�®8�!âu�F�C©�ª�®��C®��)��248¯]��!��ª��C�U¬��C���«�W�Cª�®��!â��F���)�C���J�[�U����­A�C®8�«�J�Fª��«�������C���«���*ÞL������âu�)í(�F�C®����_�ð�J�!ª1©8������8�!�É�C�b��ª1�)�C���«�}¬����(�J�Uª��«�Dî

;a�W��®8��Þ ®8�1�C�u�C®��)�{�(���C®8�«©�­«®½�C®8��¬��C�(�J�'ª��«�_âu���½�J�1�!â �C�!ª1©8�C�(Ý��C©�ª�®��)¬�¬Ñ�!�)���(��ª��!�{�)�C������æª��!���U���8­8ê § �W¬��)�C�C��ª1©����)�!Ý«�1�(�!�?�(�W�!��ª����ɬ��C���«���Cª�®8�!â��±�J�!ª1©8�C�±�����C®8�±�J�!���J�F�(�����������J�C���8­«©����C®��)ì��������ç����'8/(40±�!���«©8­«®½�J�OâZ�)í(�Z�C®8�Z�)ìÑ�#�(���9�J�!ª1©��C�Z���8�!�É�C�b��ª1�)�C���«�a¬��C�(�J�'ª��«��ê � ®����Q�C®8�#ÞL�A�C®8����â�¬Ñ�(�Jæ�C�(��ª��±�(������­(�(�C�«©��߬��C�U�(�i�*�(���J�!ª1©������ç�u�(���Z�(�����(�ß���1�J�C����­��J���É�C©����C���«�Z�QìÑ�L�«©8�g­«©����8�L�D�(���(ª1ª��1¬��C����­ª��«���J�J��©�ª��C���«���d�(�L�C�!ª1©8�C�(êßèL�!ª1�(�����C®8�à�!��ª��C�U¬��C���«�?�Cª�®8�!â��{���}ÞL®���ª�®�ë�����u¯ßè�cº�(���O�uâ��!�C�C�)­(�J ���{�!��ª��C�U¬��J�!��ìU�½ª�®8�U�«�C���8­ �¡���(���8�«â O�(��� �J�!�������8­ ��������'ã�¤äD�PJ���ê¡èL�!ª1�(���g�[©8���C®8�1�{�C®��)�Þß�u¬��C���(�!�a�C®����Q�Cª�®��!â��Z�J�!ª1©8���u��� �C®8�Z�J�!���J���(�g���������J�C���8­«©����C®��)ì��������E�(êÃ�g©��Aª��«���C���8�1�A����Þ ÞL®��)�®��)¬�¬Ñ�!���=ÞL®8�!�}Þg�à©��C�Q�C®������Cª�®8�!â��{�(�d¬��)�C���(���C®8�A���8�!�É�C�b��ª1�)�C���«�?�Cª�®8�!âu�{�Jí(�1�Cª�®8�!�O�)ìÑ���(�(ê

;h®��!�S�C®8�O�(���(�1���C�)�C� �!�!�(�!���8�C�(¬����«� �½�����8­«���?��á'�!ª1©8�C���«� �(�F�C®��9¬��C�(�J�'ª��«��Ý=�C®8�O�(�8�(�1���C�)�C����!�)�������}�(���?�C®��A�C�!�J¬Ñ�«���J�����������ê�;a�àª1���(��â �C®����d���=�!���«©8­«®9�J�Z�(������Þ �C®��à�(�8�(�1���C�)���W�J����â�¬Ñ�1�Jæ�J�«���)�J�õ ���Z�C®8�L�i©8�C©8�C���'���à�C®8�L�[�«�����#ÞL���8­A�)�J�C�(ª�í.ñD;�®8�!��öS���)�J�1�ß�C�!�������­×8�J�{�C®��H�(�8�(�1�����)�C�(ÝÉ�C®���(�8�(�1���C�)�C�����!�J¬Ñ�«�����<ÞL���C®���F���Å������� × ��ê��*�«©�ª1�(�Wª�®8�!ª�íu�C®��)��öºÞL�����8�(��Þ=�!�'�g�(ª1ª��1¬��1ê � ®U©��1Ý«�C®�����8�!�É�C�b��ª1�)�C���«�W�Cª�®��!â��±­«���(�!�}�)ì7�#�(�A���-8/(40g�J�!ª1©8�C�FÞL®8�!�W©��C����­{�(�W�!��ª��C�U¬��C���«�}�Cª�®8�!â��Q�C�)�C���J�[�'���8­�(�¤�?�8�[�������C���«�W�(�<�J�!ª1©8�����E��Þg�à®����(�à�J�1�!�O�C®É©��d�[�)�!ê§ �W�i�(ª��1Ý�Þg�±âZ��­«®¤�=�8�(�C��ª��Q�C®��)�g�C®��1�C�Q���g�C�!�(�����u�8�w�C�!�(�C�«�}�J�ðìÑ�Q©��C���8­à�!��ª����ɬ��C���«���)�d�(����êÚ;a��8�à�8�(�_ª1�)���L�)ìÑ�«©8�*�J�!ª��C�!ª��u®8�1���4àÉ�(���8Þg�Lª1�)�C�H�)ì7�«©��_���<�(©8�C®8�!�É�C��ª1�)�C���«�@ê � ®É©��1ݤ���*���<âu�(�C�L���)�C©8���(��J�uª��«���C���8�1��©������8­w��â��!�C�C�)­(�A�(©8�C®��!�¤�C��ª1�)�C���«�?�Cª�®8�!âu�A�(�=�«©8��ì�©����������8­Qì����Uª�íZ�[�(�dª��«���J�J��©�ª��C���8­���J�!ª1©8�C�u�����!�¤�C�b�.ª1�)�C���«�¡¬��C�(�J�'ª��«�7à�Þg�u�8�?�C®����±�(�Q�[�«�����#ÞL�1ñ�â@�1�Zã������X���! #"[ô¤äHì7���?�J�!ª1©��C�uâ��!�C�C�)­(��(©8�C®8�!�É�C��ª1�)�C���«�?�Cª�®8�!â��Zã[�[�(��¬7�«���U�8�«âZ���(�����Éæçâu�(�É�ðâ��!���C�)­(�!��ä�ê � ®8�H¬��C�(�J�Uª��«���C�C�)�C�C�=ìÉ�W®��!�'���8­�öª�®8�U�«�J�Z�W���(�����«â ��(��©��$�/�(���a�J�!���a���±�J� õ�ê � ®8�w¬��C�#�(�1�ÃõË�C®��(�aª��«â�¬.©8�J�!�±õ�%'& �(�����)�Éã�'ä�(���9�J�!������õ�%'&u�J�øöHê%c������(�����(Ý'�C®8�A�(�1���É���1�d�(ª1ª��1¬��C�F���Ç�(���9�«���������*�+ #"[ô � ã�Z�Jõ�%'&'ä:���)ê;a�Q���!�!�(�±���*�J�ð�C®8�L���!�(�8�1�ß�J�ð�(�1�����[�w�C®��)�=�(�ß���«��­{�(�,�w���_ª�®��«�J�!�W�D�C�«â �{���)�C­(�L�!�8�«©�­«®Z�J¬��(ª��(Ý�C®����d�����!�¤�C�b�.ª1�)�C���«�}¬��C�(�J�'ª��«�@�����J�!ª1©��C�(ê

-