+ All Categories
Home > Documents > Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè Ù...

Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè Ù...

Date post: 11-Jan-2020
Category:
Upload: others
View: 10 times
Download: 0 times
Share this document with a friend
34
Transcript
Page 1: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

������������ ���������������� ������! #"%$'&("%�� �)����* +"%$,$'�����-�.�0/1)2�����3��"54��76��- ����89�.� :��<;=&�&������> ?�-��"52�A@

BDC>EGF=HJI�KLE�MNEPO:Q R�SGTVU�TVWYX=E>ZN[]\AF?^_TV\�Q []\a`�EGF?b:c

d3e-fhgjilknm9gop9qrqrsutvtvwyx{z�|n}~s���s~p9���r�����9������wj����o>z:�?��s~�r�(����wj���n}>��������tvp��9�����n�ns����n��s~q�s�tvs~�9w2tv���lt���p�q�|ns~�nwh��tv��w

���n����tvs~p9���9}�s�tvs~wh�Gp��D|nsut:��p9qrqrs�tvqrwj��t��9��x�p9|n}~s~��s~p9���.tv����������wj�y�� V��t��ns~�?�����¡wj�y¢�£.w5s���t��vp�x����jw%����w�¤�t�wj��x�wyx�9wh���vs�p��2p9�o>z:�Y�¦¥.o>z:�?�.£>�ns����§�9xnx�s�tvs~p9����}~}�����}~}�pl£?�N�n��p�p��¨�>p9���vwh}~��tvs~p9���>�9q�p��n����p�q�qrs�t�t�whx�|ns�t��h¢¡�9��x£w��jp9����tv������t����©w�ª��js�wh��t%���vp9tvp���p�}�tv����t���wy���n��wj}~�«��wh�9}�s~¬jwy�5���©¥.o>z:�­���n����tvs~p9���9}�s�t®�«s~�¯tv��w��n�ns~�9wh������}�°��p�q��¡p�����|�s�}~sut®�±�¨²³oN�.�����9qrwj£.p9��´�s���t��nw���p�qrq�p����vwj��wj��wj���jw³��tv��s~�n�µ��o.¶G·��>qrp�x�wj}��Nz��n����p9���®t��v����tvs~p9��s��qrp9��w:wjª���s~wj��tNtv���9�����vwh��s~p9���:�¨�np9�n°¸²³oN�N��p9���®t��v����tvs~p9���.p9�Ao>z:�³¢�s~���9p9}~��s�����p��n}��2���jp9����t��9��t.���nq�|�wh�>p��w�¤��¡p9�nwh��t�s~��tvs~p9��������xr��p�qrq��n�ns��j��tvs~p9�r��p9�n��x��j�D²G�vs����³tv�nwG¥No>z:�¹��������t�s�p�����}~sut®�r�9���³|n�ns~}~x�s~�n�%|�}�p���´º¢9£.w��p����®t��v����t?w�ª��js�wh��t:²:o»�n�vp9tvp���p�}~�.��p��?�9wh�nwj����}ºt®£.p�°¸���9��t®�3�9��x�q��n}�tvs�°¸�����vt®�����n����tvs~p9����}~sut�s�wy�%��s~��tv�nwro.¶G·qrp�xnwj}¼��¢nwy�9�������lt�w³�vwy½���s���s������r��p�����t�����t>���nq�|¡wj�Gp��-¥No>z:�³¾ �h�

¿ À�Á�Â-Ã�Ä3ŵÆ�Ç.ÂDÈ�Ä�ÁɳÊn˵˵̼Í�Í�Î�ÏÑÐrÒPÓ�̼Ժ̼ÊnÕ�Ör×ØjÙ�Ú�Ö�Û�ÎlØ(Ü®É%Ðr×�Ý�Þ:Ù�Ö�Ì�Ú�Í�Ø�Ê¡ÏPÕ�ßàÎ�Ï]Ò�áJÉ:غâÎlãDÎ�Ù�Õ­ä�å�æ�ç�ܸÕ�Ú�Ï�ÎlØ�Í�è_ΫÚ�Ù�˧ÎÑé�ê>ÎlØj̼ëPÙ�ÒPÓ�ÎÐrÒPÓ�̼Ժ̼ÊnÕ�Ör×ØjÙ�Ú�Ö�Û�ÎlØjì�Ý�Ù�Ö2ÙLÚ�Ù�Í�Õ�ØjÙ�ÓGßàÊnË2ÒPÌ�Ú�Ù�Í�Ì�ÊnÚ»Ê�Ûµí®î ïvðlñ�Ð�ÒPÓ�̼Ժ̼ÊnÕ�Ö�×ØjÙ�Ú�Ö�Û¦ÎlØ(äóò�ò�ç�Ù�Ú�ÏÑô:Ì�Í�É:Ên˵˫̼Í�˧Î�Ú�Ílõö Í�Í�è_Î3Ö�Í�Ù�ØjÍ�Ê�ÛNÍ�è_Î3ßàÊn˧ã�Õ_Í�Ù�Í�̼ÊnÚ ö Ó�Ì�ßàÎ�̨Ö%ßàÊn˵˵̼Í�Í�Î�Ï÷Í�ʵÒPÌ�Í�Ö%ø�ù3Ù�Ú�Ϲø ï Ù�Ú�Ϲô:Ê�Ò÷̨Ö%ßàÊn˵˵̼Í�Í�Î�Ï÷Í�ʵÒPÌ�Í�ú9ûÙ�ÍrÍ�è_Î�Î�Ú�Ï{ô:Ê�Ò]̨Ö�ßàÊn˵˫̼Í�Í�Î�Ï{Í�ʯø¡ü�Ù�ÚPÏLýºÚ_Ê9Þ�Ö�Ú�Ê�Í�è�Ì�Ú_þ±Ù�Ò-ÊnÕ�Í�ø�ÿü�� Þ�è�Ì�ӼΠö Ó�Ì�ßàÎ�Ó¼Î�Ù�ØjÚ�Ö�Ú_Ê�Í�è�̨Ú_þ(Ù�ÒDÊnÕ_Írú9õÐ�Ú_Î3ßlÙ�Ú{Ö�ÎlÎ3Í�è�Ù�Í�Í�è�Ì�Ö%Ù�Ó¨Ó¼Ê9Þ�Ö�Î�Ù�ßjè{ãPÙ�Ø�Ívá¯Î�Ú_þnÙ�þ�Î�Ï]Ì�Ú¹Ù�Ú¹Ê�Ò�Ó�̼Ժ̼ÊnÕ�Ö³Í�ØjÙ�Ú�Ö�Û¦ÎlØ�Í�Ê«ÒDÎ3ßàÎlØ�Í�Ù�Ì�Ú÷Í�è�Ù�Í�Í�è_Î�Ê�Í�è_ÎlØã�Ù�Ø�Í�á{Ì�Ö�ãDÎlØ�Û�Ê�Øj˵̨Ú_þµÍ�è_Î�Ê�ÒPÓ�Ì�ԡ̼ÊnÕPÖ%Í�ØjÙ�Ú�Ö�Û¦ÎlØrÊ�ãDÎlØjÙ�Í�̼ÊnÚ{ÊnÚ{Í�è_Î�̼ØrÏ_Î�ßlÓ�Ù�Ø�Î�Ï]Ì�Ú�ãPÕ_Í�Ölõ ï ×%è�Ì�Ö�è�Ù�Ö�ÒDÎlÎ�Ú{Öjè_Ê9Þ�ÚÍ�Ê�ÒDÎ�Õ�Ö�ÎlÛ�Õ�ÓPÌ�Ú÷ä�å���ç � Þ�è�Ê2ßàÊnÚ�Ö�Í�ØhÕ�ßàÍ�Ù2ã�Ø�Ê�Í�ʺßàÊnÓPÛ�Ê�سþ�Î�Ú_ÎlØjÙ�Ó-Ö�Î�ßlÕ_Ø�Î�Ë2ÕPÓ¼Í�Ì�ñ®ãPÙ�Ø�ÍváµßàÊn˧ãPÕ_Í�Ù�Í�Ì�ÊnÚ(Ì�Ú«Í�è_Îr˧ʡÏ�Î�ÓÊ�Û5äóò�æ�ç�ÕPÖ�Ì�Ú_þ«É%Ðr×�õ

� Ú¯Í�è�̨Ö�ã�Ù�ã-ÎlسÞ:Î�Ö�è_Ê�Þ è_Ê9Þ�Í�ʵÌ�˵ã�Ø�Ê9Ô�ÎrÊnÚ¯ã�ØjÎlԡ̼ÊnÕPÖ:ßàÊnÚPÖ�Í�ØjÕ�ßàÍ�̼ÊnÚPÖ³Ê�ÛNÉ%Ðr×�̨Ú(Í�è�ÎrÙ�ØjÎ�Ù�Ö5Ê�ÛÎ��(ßl̼Î�Ú�ßàáÙ�ÚPÏJÕ�Ú�Ì�Ô�ÎlØjÖ�Ù�Ó?ßàÊn˧ãDÊnÖ�Ù�Ò�Ì�Ó�̼Ívá�õ � ÚÑÍ�ÎlØj˵Ö�Ê�Û5Î��(ßl̼Î�Ú�ßàá � Í�è_Î(ã�Ø�Ê�Í�ʺßàÊnÓGÞ:Î(ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ3Û�Ê�صÉ%Ðr×�ÕPÖ�Î�Ö�ÊnÚ�Ó�á]ÙßàÊnÚPÖ�Í�Ù�Ú�Í�Ú¡Õ�Ë2Ò-ÎlØ�Ê�ÛGÎ��ºãDÊnÚ_Î�Ú�Í�Ì�Ù�Í�̼ÊnÚ�Ö�Ù�Ú�Ï{ßàÊn˵Ë2ÕPÚ�Ì�ßlÙ�Í�̼ÊnÚ{Ø�ÊnÕ�ÚPÏ�Ö�ãDÎlØ�Í�ØjÙ�Ú�Ö�Û�ÎlØ�õ î � Ú{ßàÊnÚ�Í�ØjÙ�Ö�Í � Í�è_Î�˧ÊnÖ�ÍÎ��(ßl̼Î�Ú�Íã�Ø�ÎlԺ̼ÊnÕ�Ö�Ó�á%ý¡Ú�Ê9Þ�Ú2ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ì�ÊnÚ�Ê�ÛPÉ%Ðr×­ä�å���ç¡Õ�Ö�Î�ÖµÜ���ÝD̨Ú�Ô�ʺßlÙ�Í�̼ÊnÚ�Ö�Ê�Û�Ðr×­Ü�Í�è¡Õ�Ö�Ì�˵ãPÓ¼áºÌ�Ú_þ³Ù�ÍÓ¼Î�Ù�Ö�ÍÍ�è�γÖjÙ�˧Î%Ú�ÕPË3ÒDÎlØNÊ�Û�ã�Õ_ÒPÓ�Ì�ßyñ®ý�Îlá3Ê�ã-ÎlØhÙ�Í�̼ÊnÚ�ÖNÕ�Ö�Ì�Ú�þ�ýºÚ_Ê9Þ�Ú§ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ì�ÊnÚ�ÖhÝÙ�Ú�Ï�ÒP̼ÍNßàÊn˵˵Ì�Í�˧Î�Ú�Í�Ö � Ù�Ú�Ï µÜ���ÝØjÊnÕ�Ú�Ï�Ö � Û�Ê�Ø��(Ù�Ö�Î�ßlÕ�Øj̼Í�á(ãPÙ�ØjÙ�˧ÎlÍ�ÎlØ9õ���Õ_Ø�Í�è�ÎlØj˧Ê�Ø�Î � Þ:Î�Ö�è_Ê9Þ�Í�èPÙ�Í:ÊnÕ�Ø5ã�Ø�Ê�Í�ʺßàÊnÓ�Ö�Î�ßlÕ_Ø�Î�Ó¼á«Ø�Î�Ù�Ó�Ì��lÎ�Ö5Ù�Ú©Ì�Ï�Î�Ù�ÓÉ�Ðr×�Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á¯Ì�Ú©Í�è_Î3Ø�Î�ßàÎ�Ú�Í�Ó�á�ñ®ã�ØjÊ�ã-ÊnÖ�Î�Ï��������������� "!$#&%"'�()%*�� ,+��-!.�0/01¯Ü02�É%Ý:Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý©Ò¡á¹É5Ù�Ú_ÎlÍ�Í�Ì�ä�å�3�ç �̨Ú3Í�è_Î4#�%�'5'6%"�7�8�09����8���:#��;��/0���-�=<�Ü®É?>A@PÝA˧ʺÏ_Î�ÓVõ$>%Î�ßlÙ�Ó�Ó¡Í�è�Ù�ÍNÍ�Ê�Ï_Îlë�Ú_Î�Ö�Î�ßlÕ�Øj̼Í�á�Ì�Ú2Í�è�Ì�ÖÛ¦ØjÙ�˵ÎlÞ�Ê�Ø�ý � ÊnÚ_Î:ë�ØjÖ�ÍÖ�ã-Î�ßlÌ�ë�Î�ÖGÙ�Ú é�Ì�Ï_Î�Ù�ÓPÛ�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá¡ì3Ï_Î�Ö�ßàØj̼Ò�Ì�Ú_þ�Í�è_ÎrÏ_Î�Ö�̼ØjÎ�ϧÒDÎ�è�Ù�Ժ̼Ê�Ø:Ê�Û�Í�è_Î�ã�ØjÊ�Í�Ê¡ßàÊnÓDÕ�Ö�̨Ú_þ�Ù3Í�ØhÕ�Ö�Í�Î�Ï(ãPÙ�Ø�Í�á �BDCFEHGJI�KLGDE*MNGOM6P&Q*R0K�STP&UJKWV�XYKLZ*[\R?]*P�]"GJS?P�]N]"GOP&SLGOMH[\E KLZNG_^a`�bOc8d$e&fLgihkjNlLb&`nmob0prq)`Tmns*gtbLu�`�vnsNj�mwqxb�yOpDln`Llny"cTlnz|{}V�R0KLV�E,z�~_C�z

� GDQNSL�*P&S��A���������� {:GD�\�F�=P&Q*RDzY�k�NUDGDE�K;�=GJUnZNE*V��\V���[\GDRDzF�����6~AV��NE�KTP&[\EwCF��G�z|~A��S�STPO� �F[��\��z��$�6�������&�����8���������)�-������ D¡���¢�£&¤���¥�¦�¥�����¢O�x§-¨�¥� � �©

 ��&¨N¦=§0¢�ªD¡a§«J¬ V�­W]N�NKLGJS¯®�UD[\GDENUDG±°GJ]�P&S�KL­WGJE�KOz ¬ P&SLE*GJ��[\G²~AGD�\�\V�E´³FE*[\��GJSLR�[µK-��z·¶������ � V&SLQ"GDR¯CF��G��\z¹¸x[µK�KLR�QN�NSL��Z,zº¸,C¼»O¶���»D½��

���&¾���¿�¥�¤�¢�¦=§0¢J¡�Àx§�¥�Á�Àx§Â �aZ*[\R|UDV�E�K�STP&R0KLR�Ã|[\KLZiR0KTP�E*M*P&STM�V�QN��[\��[\V��NRYK�STP&E*R0XÄGnSOzNÃ|ZNGJSLGÅR�V�­WGV&KLZ*GJS|­WGnKLZ*V�M4Æ�]"GJSLZ�P&]*R|P�ENV�KLZNGJS|UJS���]NKLV���STP�]NZ*[\UÅQ*�N[\�ÇM�[\E*�

Q*�\V�UnÈ"z�V&S���GJSL[µÉ�UDP&KL[\V�E�P&KaR�V�­WGYZ*[\��ZNGJSa��PD��GJS:]NSLV�KLV�UJV��ÇÊ:[\R}SLGOË��*[µSLGOM?KLV����*P&STP�E�KLGDG|KLZ�P8Ka]*P&S�KL[\GDR�P8SLG$]"GJS�X.V�SL­W[\E*��KLZ*GD[µS:]�P&S�K:V�X,KLZ*GK�STP�E*R0X.GJSYV�EAKLZNGD[µSFMNGDUD��P8SLGOMA[\E*]N�NKLRD�Ì ®�GJUD�NSL[µK���[\R|]NSLV8��GOM��*E*MNGJS|R�V�­WGR0KTP�E*M*P&STMiE��N­ÍQ"GJS}KLZ*GDV&SLGJKL[\U�P�R�R��*­W]�KL[\V�E*RDzNM�[�R�UJ�*R�R�GOM_��P&KLGnSO�

å

Page 2: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Ù�ÚPÏ Í�è_Î�Ú ÊnÚ_ίã�Ø�Ê�Ô�Î�ÖµÍ�è�Ù�ͧÙ]ãPÙ�Ø�Í�̨ßlÕ�Ó�Ù�Ø�ã�Ø�Ê�Í�ʺßàÊnÓ³Ê�ãDÎlØjÙ�Í�Ì�Ú_þ�Ì�Ú Í�è_ΩØ�Î�Ù�Ó³Þ:Ê�ØjÓ�Ï7é�Ö�Î�ßlÕ�Ø�Î�Ó¼áJØjÎ�Ù�Ó�Ì��lÎ�Ö�ì]Í�è�̨Ǫ̈Ï_Î�Ù�ÓAÛ¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�á � Ò�á÷Ö�è_Ê9Þ�Ì�Ú_þµÍ�è�Ù�Í�Ú_Ê é�Î�Ú�Ժ̼Ø�ÊnÚP˧Î�Ú�Í�ì(Þ:ÊnÕ�Ó�ϹÒDÎ�Ù�ÒPÓ¼Î3Í�Ê©Ï�̨Ö�Í�Ì�Ú_þnÕ�̨Ö�è�Ü�å9Ý�Ù�Ú{Ù�Ï_Ô�ÎlØjÖjÙ�Ø�áÊ�ãDÎlØjÙ�Í�̨Ú_þ2Ì�Ú«Í�è_Î�Ø�Î�Ù�ÓPÞ:Ê�ØjÓ�Ï«Þ�̼Í�è«ãPÙ�Ø�Í�̼Î�Ö�ØjÕ�Ú�ÚPÌ�Ú_þ�Í�è�Ì�ÖGã�Ø�Ê�Í�ʺßàÊnÓPÛ�Ø�ÊnË�ÜVònÝ�Ù�ÚÑé�Ì�Ï�Î�Ù�Ó�Ù�Ï_Ô�ÎlØhÖ�Ù�Ø�áºì3Ê�ãDÎlØjÙ�Í�Ì�Ú�þ̨ګÙ�Ú±Ì�Ï_Î�Ù�Ó_ã�ØjÊ¡ßàÎ�Ö�Ö:ßàÊnÚ�Ö�̨Ö�Í�Ì�Ú_þ�Ê�ÛAÏPÕ�˵Ë2á�ã�Ù�Ø�Í�̼Î�Ö?Í�èPÙ�Í:Ö�̨˧ãPÓ¼á�ßlÙ�Ó�Ó�Í�è�Î�Ì�Ï_Î�Ù�ÓPÛ�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á�õ ö ˵Ù�Ì�ګԡ̼ØjÍ�Õ_ÎÊ�ÛPÍ�è�Ì�ÖAÛ�ØjÙ�˧ÎlÞ:Ê�Ø�ý�Ì�ÖÍ�è�Ù�Í.Í�è_γÖ�Î�ßlÕ_Øj̼Ívá�Ê�Û_ã�Ø�Ê�Í�ʺßàÊnÓ�Ö.Í�è�ÕPÖÏ_Îlë�Ú_Î�Ï3Ì�Öã�Ø�Î�Ö�ÎlØ�Ô�Î�Ï2ÕPÚ�Ï_ÎlØÙ�þ�Î�Ú_ÎlØjÙ�ÓºßàÊn˧ãDÊnÖ�̼Í�Ì�ÊnÚÊ�ãDÎlØjÙ�Í�Ì�ÊnÚ÷ßlÙ�Ó�Ó�Î�ÏYé�Õ�ÚP̼Ô�ÎlØjÖ�Ù�ÓßàÊn˧ãDÊnÖ�̼Í�̼ÊnÚ � ì(Þ�è�̨ßjè¹Î�ÖjÖ�Î�Ú�Í�̨Ù�Ó�Ó¼á¹Ë§Î�Ù�Ú�Ö�Í�è�Ù�Í�ã�Ø�Ê�Í�ʺßàÊnÓ�Ö�Ø�Î�˵Ù�Ì�Ú{Ö�Î�ßlÕ_ØjÎ2ÎlÔ�Î�ÚÞ�è_Î�Ú©ßàÊn˧ãDÊnÖ�Î�ϹÞ�̼Í�è¯Ê�Í�è_ÎlØ�ãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö5Í�è�Ù�Í�˵Ù9á¯Ò-Î�ØhÕ�Ú�Ú�Ì�Ú�þ�ßàÊnÚ�ßlÕ�Ø�Ø�Î�Ú�Í�Ó¼á(Ì�Ú¹Í�è_Î�Ö�Ù�˧Î3Ö�á¡Ö�Í�Î�˹õ���Î3þn̼Ô�ÎÙL˧Ê�Ø�ΫÏ_ÎlÍ�Ù�̨ӼÎ�Ï]Ø�ÎlԺ̼ÎlÞ Ê�Û³Í�è_Î 2�É=Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý�Ó�Ù�Í�ÎlØ3Ì�Ú]Í�è_ΫãPÙ�ãDÎlØ�õ���Î(Ú_Ê�Í�ΫÍ�è�Ù�Í2Ù¹ÖjÌ�˵Ì�Ó�Ù�Ø�Û�ØjÙ�˧ÎlÞ:Ê�Ø�ýÞ³Ù�Ö�Ì�Ú�Ï_ÎlãDÎ�Ú�Ï_Î�Ú�Í�Ó¼á¹ãPØ�Ê�ãDÊnÖ�Î�ϻҡá��?ë�ÍO��˵Ù�ÚPÚJÙ�Ú�Ï��»Ù�Ì�Ï�Ú_Îlدä��� � ��æ�ç®õ � Ú�Í�ÕP̼Í�̼Ô�Î�Ó¼á � Í�è_Î�Ö�ΫÍvÞ�Ê{Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý¡ÖÙ�ØjÎ�Ö�̨˵Ì�Ó�Ù�Ø � Ù�Ó�Í�è_ÊnÕ_þnè(Í�è_ÎlØ�ÎrÙ�Ø�Î�Ù�Ú¡Õ�Ë3ÒDÎlØ�Ê�ÛAÍ�Î�ßjè�ÚPÌ�ßlÙ�Ó�ÏPÌ�-ÎlØjÎ�Ú�ßàÎ�Ölõ���Î�ßjè_Ê¡ÊnÖ�Î�Í�Ê�Õ�Ö�Î�Í�è_ÎA2rÉ Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý̨گÍ�è�Ì�Ö³ãPÙ�ãDÎlØ�õ�

ÐrÕ_صã�ØjÊ�Í�Ê¡ßàÊnÓ�Ù�ßàÍ�Õ�Ù�Ó�Ó¼á Ø�Î�Ù�Ó�Ì �lÎ�ÖµÙ�Ú Î�Ú�èPÙ�Ú�ßàÎ�Ï+É%Ð�× Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á � Þ�è�Ì�ßhèaÞ�Î÷ßlÙ�Ó�Ó�������� � Þ�è_ÎlØ�ÎL̨ÚÙ�ÏPÏ�̼Í�̼ÊnÚ]Í�ʹÊ�Ò�Ó�̼Ժ̼ÊnÕ�Ö�Í�ØjÙ�Ú�Ö�Û�ÎlØ � ÊnÚ_ΫßlÙ�Ú»ãPØ�Ê9Ô�ΫßàÎlØjÍ�Ù�Ì�Ú»Ø�Î�Ó¨Ù�Í�̼ÊnÚ�Ö�Ù�˧ÊnÚ�þLßàÊn˵˵̼Í�Í�Î�ÏJÒP̼Í�ֵܸÌ�Ú»ãPÙ�Ø�Í�Ì�ßlÕ�Ó¨Ù�Ø �Ù�˵ÊnÚ_þ÷Í�è�Ø�ÎlÎ(Ò�̼Í�ÖhÝhõ÷×Ê]Ï_Î�˧ÊnÚ�Ö�Í�ØhÙ�Í�Î(Í�è_ίÕ�Ö�ÎlÛ¸Õ�Ó�Ú�Î�Ö�Ö�Ê�Û5Í�è�̨Ö3Û¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�á � Þ:αÖjè_Ê9Þ*Í�è�Ù�ͧÕ�Ö�Ì�Ú�þ��:É%Ð�×Ù�Ö�Ù�ÒPÕ�Ì�Ó¨Ï�Ì�Ú_þ2ÒPӼʺß�ý � Ù�Ú�á¯Þ:Î�Ó�Ó�ñ®Û�Ê�Øj˧Î�ϯÍ�Þ:Ê�ñ®ãPÙ�Ø�ÍváLÙ�Ú�ϹË�Õ�Ó¼Í�̼ñ®ãPÙ�Ø�Í�á(Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á(ßlÙ�ÚLÒDÎ�Ö�Î�ßlÕ_Ø�Î�Ó¼á(Ø�Î�Ù�Ó�Ì��lÎ�Ï���·#��L���x/0!.1�̨ڵÍ�è_ÎrÕ�Ú�̼Ô�ÎlØhÖ�Ù�ÓPßàÊn˧ãDÊnÖ�Ù�Ò�Ì�Ó�̼Ívá�Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý-õ��GÓ�Õ�þ�þnÌ�Ú_þ3Ì�Ú«ÊnÕ_سã�Ø�Ê�Í�ʺßàÊnÓPÛ�Ê�سØ�Î�Ù�Ó�Ì���̨Ú_þ�Í�è_Î��:É%Ð�×̨Ú�Í�Ê{Í�è�Ì�Ö�ßàÊnÚ�Ö�Í�ØjÕPßàÍ�̼ÊnÚ � Þ:αè�Ù9Ô�ΩÙ�ÚÑÎ��(ßl̼Î�Ú�Í�ã�ØjÊ�Í�Ê¡ßàÊnÓ:Û�Ê�Ø�Ù�Ú�á»Þ:Î�Ó�Ó�ñ®Û�Ê�Øj˧Î�ÏÑÍ�Þ:Ê�ñ®ãPÙ�Ø�Ívá Ù�Ú�Ï Ë�Õ�Ó¼Í�Ì�ñ®ãPÙ�ØjÍ�áÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá(Ì�Ú¯Í�è_Î2ÉW>A@¯Ë§ÊºÏ_Î�ÓVõ

ɳÙ�Ú_ÎlÍ�Í�Ìo��/r �!��rä�å�ò�ç�Þ:ÎlØ�ιÍ�è_ÎLë�ØjÖ�ͧÍ�ÊJÖjè_Ê9Þ Í�è�Ù�Í«Ö�Õ�ßhèaÛ¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�̼Î�Ö�Ù�Ø�Î÷̨Ú�Ï_ÎlÎ�Ï Ø�Î�Ù�Ó�Ì ��Ù�ÒPӼΩÌ�ÚaÍ�è�̨Ö˵ʡÏ_Î�Ó � ÎlÔ�Î�Ú Õ�ÚPÏ_ÎlØ(þ�Î�Ú�ÎlØjÙ�Ó�ßàØ�á¡ã�Í�Ê�þ�ØjÙ�ãPèPÌ�ß÷Ù�ÖjÖ�Õ�˧ã�Í�Ì�ÊnÚ�Ö¯Ù�Ú�Ï+Ø�ÎlþnÙ�ØjÏ�Ó¼Î�ÖjÖ¯Ê�Û3Í�è_λڡÕ�Ë2Ò-ÎlدÊ�Û�ßàÊ�Ø�ØjÕ_ã�Í�Î�Ïã�Ù�Ø�Í�̼Î�Ölõ��¹Ê�Ø�Î�Ö�ãDÎ�ßl̼ë�ßlÙ�Ó�Ó¼á � ä�å�ò�çrÛ�ÊnÓ�Ó¼Ê9Þ�Ö«Í�è_Î{þ�Î�Ú_ÎlØjÙ�Ó±é�Í�Þ:Ê�ñ®ãPè�Ù�Ö�Î�ìaÙ�ã�ã�Ø�ÊnÙ�ßhè#Ê�Û«äóò��çrÊ�Û�ë�ØjÖ�ͱÏ_Î�Ö�̼þnÚ�̨Ú_þÙ­Ö�ÊnÓ�Õ_Í�̼ÊnÚ#Û�Ê�عÍ�è_λßlÙ�Ö�ÎJÊ�Û�è_ÊnÚ_Î�Ö�Í�ñ®ÒPÕ�Í�ñ�ßlÕ_Øj̼ÊnÕ�Ö¯ã�Ù�Ø�Í�̼Î�Ö � Ù�Ú�Ï�Í�è�Î�Ú�Í�Õ_ØjÚ�̨Ú_þ Ì�͹Ì�Ú�Í�ÊYÙaÖ�ÊnÓ�Õ_Í�̼ÊnÚ#Û�Ê�عÍ�è_ÎÙ�ßàÍ�Ì�Ô�Î�Ó¼á©ËµÙ�Ó�Ì�ßl̼ÊnÕPÖ5Ù�Ï_Ô�ÎlØjÖ�Ù�Ø�á � Õ�Ö�Ì�Ú�þ�ÙÑé�ßàÊn˧ãPÌ�Ó�ÎlØ�õ ì»×%è_Î�ßàÊn˵ãPÌ�Ó¼ÎlØ�Ù�ÏPÏ�Ö%Ù6�lÎlØ�Ê�ñ®ýºÚ_Ê�Þ�Ó¼Î�Ï_þ�Î2ã�Ø�Ê¡Ê�ÛÍ�Ê(ÎlÔ�ÎlØ�á˵Î�Ö�Ö�Ù�þ�Î � ãPØ�Ê9ÔºÌ�Ú_þ{Í�è�Ù�͵̼Í�Ì�Ö�ßàÊnÚ�Ö�Ì�Ö�Í�Î�Ú�ͧÞ�Ì�Í�èJÍ�è_ιèPÌ�Ö�Í�Ê�Ø�á»Ù�Ú�Ï Í�è�Î�ܸßàÊn˫˵̼Í�Í�Î�Ï�Ý2ã�Øh̼Ô�Ù�Í�ίÌ�Ú_ã�Õ_Í2Ù�Ú�Ï Í�è_ÎØhÙ�Ú�Ï_Ên˵Ú_Î�ÖjÖlõ ��Ê�Í�Ì�ßàαÍ�è�Ù�Í2ÖjÌ�Ú�ßàεÍ�è_Î é�ßàÊnÚ�Ö�Ì�Ö�Í�Î�Ú�ßàá¡ì÷ã�Ø�Ê¡Ê�Û¸Ö�Ù�Ø�ΫÛ�Ê�Ø�Ø�Î�Ó�Ù�Í�̼ÊnÚ�Ö2Ì�Ú�Ô�ÊnӼԡ̨Ú_þ©Í�è_ΫÎ��ºÎ�ßlÕ_Í�̼ÊnÚJÊ�Û×.Õ_Øj̨Ú_þ2˵Ù�ßhè�Ì�Ú_Î�Ö � Í�è�Îlá«Ù�Ø�Î"!¡Õ�̼Í�ÎrßàÊn˧ãPÓ¼Î��(Ù�ÚPϱÌ�Í:Ì�Ö³Õ�ÚPÓ�̼ý�Î�Ó¼á�Í�è�Ù�ͳÍ�è_Îlá±Ù�Ï�˵̼Í:Î��«ßl̼Î�Ú�ͳã�Ø�Ê�Í�ʺßàÊnÓ�ÖlûºØjÙ�Í�è_ÎlØ �ãPØ�Ê�Ê�Û¸Ö�Û¦Ê�اþ�Î�Ú�ÎlØjÙ�Ó#��� Ö�Í�Ù�Í�Î�˧Î�Ú�Í�Ö«Ù�Ø�ιÕPÖ�Î�Ï Ü�Þ�è�Ì�ßhè Ì�Ú�Ô�ÊnÓ�Ô�ιÙ÷ØjÎ�Ï�Õ�ßàÍ�̼ÊnÚ Í�Ê»Ù�Ú$���>ñ�ßàÊn˧ã�Ó¼ÎlÍ�Ωã�Ø�Ê�ÒPÓ�Î�ËӨ̼ý�Î&%�Ù�˵̨ӼÍ�ÊnÚ�Ì�Ù�Ú É:áºßlÓ¼Î�Ý � ˵Ù�ý¡Ì¨Ú_þ]Í�è_ιßàÊn˧ãPÌ�Ó�ÎlصÙ�˵Ù('�Ê�Ø(Ö�ÊnÕ_ØjßàΩÊ�ÛrÌ�Ú_Î��«ßlÌ�Î�Ú�ßàá�õ­É³Ù�Ú_ÎlÍ�Í�Ìr��/ �!��­ËµÙ�ý�ÎÍ�è�Î2ã�ØjÊ�Í�Ê¡ßàÊnÓ.Ì�ÚÑäóò��çNÖ�Î�ßlÕ�Ø�Î�̨ڹÍ�è_Î52rÉ Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý©Ò�áLØ�ÎlãPÓ�Ù�ßlÌ�Ú�þµÍ�è_Î�ÒPÙ�Ö�Ì�ß3ã�ØjÌ�˵Ì�Í�̼Ô�Î�Ö3ܸÚ�Ù�˧Î�Ó�á � Ê�ÒPӨ̼ԡÌ�ÊnÕ�ÖÍ�ØhÙ�Ú�Ö�Û�ÎlØ � ÒPÌ�Í�ßàÊn˵˵Ì�Í�˧Î�Ú�Í � Ù�Ú�Ï �lÎlØ�Ê�ñ®ýºÚ_Ê9Þ�Ó�Î�Ï_þ�Î�Ý�Þ�̼Í�è¹Í�è�Î�̼Ø�Õ�ÚP̼Ô�ÎlØjÖ�Ù�Ó�Ó�á±ßàÊn˧ãDÊnÖ�Ù�Ò�Ó¼Î2ßàÊnÕ�Ú�Í�ÎlØ�ãPÙ�ØjÍ�Ölõ�×%è�ÎØjÎ�Ö�Õ�Ó¼Í�Ù�Ú�Í�ã�Ø�Ê�Í�ʺßàÊnÓ>ÒDÎ�ßàÊn˧Î�Ö�Õ�ÚP̼Ô�ÎlØjÖ�Ù�Ó�Ó�áLßàÊn˧ãDÊnÖ�Ù�ÒPӼΠ� ÒPÕ_Í�Ø�Î�˵Ù�̨Ú�ÖrØjÙ�Í�è_ÎlØ3̨Ú_Î��«ßl̼Î�Ú�Ílõ � Ú�Í�èPÌ�ÖrãPÙ�ãDÎlØ�Þ�ÎÛ�ÊnÓ�Ó¼Ê�Þ Ù»Ï�Ì�-ÎlØ�Î�Ú�͵Ù�ã�ã�ØjÊnÙ�ßjè�õ­ô:áaÌ�Ú�ßàÊ�Øjã-Ê�ØhÙ�Í�Ì�Ú_þ�Ö�Í�Ø�ÊnÚ_þ�ÎlرÖ�Î�ßlÕ_ØjÌ�Í�á Ì�Ú�Í�Ê�Í�è�ιÒPÙ�Ö�Ì�߯Ò�Õ�Ì�Ó�Ï�̨Ú_þLÒPÓ�Ê¡ßjý#ܸÌVõóÎ�õ ��³É%Ðr×�Ý � Þ:Î÷Ù�ØjÎ÷Ù�ÒPӼΩÍ�ÊÑÒPÕ�Ì�Ó¨ÏÑãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö«Ö�Î�ßlÕ_ØjÎLÙ�þnÙ�Ì�Ú�Ö�ͱÙ�Ï�Ù�ã�Í�̼Ô�Î÷Ù�Ú�ϭ˵Ù�Ó¨Ì�ßl̼ÊnÕ�Ö§Ù�Ï_Ô�ÎlØhÖ�Ù�Øj̼Î�Ö*)��-�8�8#�/0!Ç1 �Î�Ó¨Ì�˵Ì�Ú�Ù�Í�̨Ú_þ»Í�è_λÚ_ÎlÎ�Ï#Û�Ê�عÙ�ܸÚ_Ê�Øj˵Ù�Ó¨Ó¼á­Ì�Ú_Î��(ßl̼Î�Ú�ÍhÝ(ßàÊn˧ã�Ì�Ó¼ÎlØ�õ � Ú+Í�è�Ì�Ö±Þ³Ù�á � Þ�λÙ�ØjλÙ�ÒPÓ¼Î{Í�Ê ßàÊnÚ�Ö�Í�ØjÕ�ßàÍãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö5Í�è�Ù�Í�Ù�Ø�Î3Î��«ßl̼Î�Ú�Í�Ù�Ú�Ï©Ù�Í�Í�è�Î�Ö�Ù�˧Î�Í�Ì�˵Î�Î�Ú+'�Ê�á©Ùµè�̼þnè©Ó¼ÎlÔ�Î�ÓAÊ�Û.Ö�Î�ßlÕ�Øj̼Í�á�õ

,.-0/"/+1324-658742:9 ��Î�Ú�Ê9Þaã�Ø�Î�Ö�Î�Ú�Í?Ù�˧Ê�ØjÎ%Ï_ÎlÍ�Ù�Ì�Ó¼Î�ϧÙ�ßlßàÊnÕ�Ú�Í?Ê�Û�ÊnÕ_Ø?Ø�Î�Ö�ÕPÓ¼Í�Ölõ;��Î�Ö�Í�Ù�Ø�Í>Ò¡á�Ï_Îlë�Ú�Ì�Ú_þ�Ù�Ú<�:É%Ð�×Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá(Ü8=#>@?BADCÝ � Þ�è�Ì�ßhè � Ù�Ö?˧Î�Ú�Í�̼ÊnÚ_Î�ϧÙ�ÒDÊ�Ô�Î � Ù�Ï�Ï�Ì�Í�̼ÊnÚ�Ù�Ó�Ó¼á�Ù�Ó�Ó�Ê9Þ�ÖNÍ�è_Î5Ö�Î�ÚPÏ_ÎlØ>Í�Ê�ã�Ø�Ê�Ô�Î5Ø�Î�Ó�Ù�Í�̼ÊnÚ�ÖNÊnÚÍ�è�Ø�ÎlÎ�ßàÊn˵˫̼Í�Í�Î�Ï«Ò�̼Í�Ö>Í�Ê3Í�è�Î�Ø�Î�ßàÎ�̼Ô�ÎlØ9õ?×%è_Î�ÚµÞ:Î�ßàÊnÚPÖ�Í�ØjÕ�ßàÍ:Ù�ã�Ø�Ê�Í�ʺßàÊnÓ�Í�Ê�Ø�Î�Ù�Ó�Ì �lÎ%Í�è_ÎE�³É%Ðr×aÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá�õ×�è_Î2Ö�Í�Ù�ØjÍ�Ì�Ú_þ«ãDÊnÌ�Ú�Í%Û�Ê�ØrÊnÕ_Ø�ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�̼ÊnÚ{Ì�Ö�Í�è_Î�Ö�Í�Ù�Ú�Ï�Ù�ØjÏF�Î�Ï�ÎlØjÖ�Î�Ú]ßàÊn˵˵̼Í�˵Î�Ú�Í�Ö�ßjè_Î�˵ίä��:G�ç®õ�×%è_Î�ÚLÞ�ÎÒ�Õ�Ì�Ó�ϵÙ�ÚLÐ�×YãPØ�Ê�Í�Ê¡ßàÊnÓ-Ê�Ô�ÎlØ%Í�è_Î�Ö�Î�ßàÊn˵˵Ì�Í�˧Î�Ú�Í�Ö³Í�èPÙ�ͳ̨Ö:Ó�Ê�ÊnÖ�Î�Ó�áµÒPÙ�Ö�Î�ϱÊnÚ¯Í�è_ΧܸÚ_ÊnÚºñ�ßàÊnÚ�ßlÕ�Ø�Ø�Î�Ú�Í:Ô�ÎlØhÖ�̼ÊnÚ±Ê�ÛÍ�è�Î�Ý:Ð�×#ã�Ø�Ê�Í�ʺßàÊnÓ�Ê�ÛIHrÙ�ØjÙ9á¯Ù�ÚPÏJ�÷Ù�ßLK�Î�Ú)��Ì�Ϋäóò+M�çGÜ�Þ�è�Ì�ßhè±Ì¨Ú«Í�Õ_Øhڱ̨Ö�Ò�Ù�Ö�Î�ϯÊnÚ¯Í�è_Î3Ðr×+ã�Ø�Ê�Í�ʺßàÊnÓDÊ�Ûô:Î�Ó�Ó�Ù�ØjÎÙ�ÚPÏN�LÌ�ßlÙ�Ó¨Ì?ä M�ç�ÝhõPO-ÎlØ�Ê�ñ®ýºÚ_Ê�Þ�Ó¼Î�Ï_þ�Î(ÜQORK2ÝGã�Ø�Ê¡Ê�Û�Ö%Ù�ØjÎ�Ø�ÎS!�ÕP̼Ø�Î�ϯÌ�Ú¯Í�è�Ì�Ö�Ð�×#ã�Ø�Ê�Í�ʺßàÊnÓ � Ù�Ú�ϯÍ�è¡Õ�Ö5Þ:Î�Þ�Ê�Øjý±Ì�Ú©Ùè�á¡Ò�ØjÌ�ϧ˧ʺÏ_Î�ÓºÞ�Ì�Í�èµÌ�Ï_Î�Ù�ÓTORK�Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�̼Î�Ölõ0��Ù�Í�Õ_ØjÙ�Ó¨Ó¼á � Í�è_Î�ßàÊnÚPÖ�Í�ØjÕ�ßàÍ�̼ÊnÚPÖ?Û�Ê�Ø?ã�Ø�Ê�Ô¡Ì�Ú�þ�Ø�Î�Ó�Ù�Í�̼ÊnÚPÖ>ÊnÚ§Í�è_Ø�ÎlÎU �aZ*G�[�MNGDP���Vt]NSLV�UDGDR�RXWOSLGOP���V ÃaV&SL��MrXÄV&SL­Í�*��P8KL[�V�E V�XxR�GDUD��SL[µK�� P&E�MiKLZNG�R�[�­ �*��P&KLV&S�VtQ�P�R�GDM_]�P&STP�MN[\��­¹ÃaGnSLG�[\EN[\KL[�P8KLGOM Q���Y$V���M�SLGD[\UnZ

lngav[Z�d:\ ���^] � � SLV�­ KLZ*GJE�V�Ekz�KLZ*GnSLG$Z�PO��GFQ"GDGDE�­;P�E��WM�GJÉ�EN[µKL[�V�E*Ra[\E;KLZ*[\RÅÆ�E*VOà R0KTP&E�MNP&STM�Êx]�P&STP�MN[\��­_z�Ã|[µKLZ�GD­W]*Z*P�R�[\RaV�EiMN[�_ GJSLGJE�KP�R�]"GDUJKLRD� � V�S�P|E��*­ Q"GJS=V&X*GnI*P&­W]*�\GDRDz&R�GDG`Y$V���M�ÃaP�R�R�GJS}P�E*M��kGD��[\EE\ ½��a] z�~A[\UOP&��[�P&E�MIbFV���PDÃ�PD�c\ ½��^] z�{:GOPO��GJS`\ ��z&½a] z�P�E�M ¬ P�ENGJK�KL[:\ �a] zXÄV�S|KLZNGXÄV�SL­ �*��P&KL[\V�E*R�KLZ�P8K|]NSLGDUDGDMNGKLZ*G$³ ¬ X.STP&­WGJÃaV&SLÈ �

ò

Page 3: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

ßàÊn˫˵̼Í�Í�Î�Ï»Ò�̼Í�Ö�Ù�Ó�Ö�ÊLÕPÖ�ΧÍ�è_Î�Ö�αÌ�Ï_Î�Ù�Ó O`K Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Í�̼Î�Ö�õ5�NÌ�ÚPÙ�Ó�Ó¼á � Í�ÊLßàÊnÚPÖ�Í�ØjÕ�ßàÍ�Î��(ßl̼Î�Ú�Í3ã�Ø�Ê�Í�ʺßàÊnÓ�Ö�Í�è�Ù�ÍÖ�Î�ßlÕ_Ø�Î�Ó¼á�Ø�Î�Ù�Ó�Ì �lÎ�Í�è_Î�Ö�Î"O`K Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Í�Ì�Î�Ö � Þ:Î�ßàÊnÚ�Ö�Í�ØhÕ�ßàÍ:Ù2Ö�ãDÎ�ßlÌ�Ù�Ó�Ívá�ãDÎ�Ê�Û�è_ÊnÚ_Î�Ö�Í�ñ®Ô�ÎlØh̼ë�ÎlØ�ORK ã�ØjÊ�Í�Ê¡ßàÊnÓ�Û¦Ê�ØÎ�Ù�ßhè»Ï_Î�Ö�Ì�Ø�Î�Ï{Ø�Î�Ó�Ù�Í�̼ÊnÚ�Ö � Ù�ÚPÏ]Í�è_Î�Ú]Þ�εÕPÖ�Î�Ù¯ØjÎ�Ö�Õ�Ó¼Í�Ò�á�HrÙ�ØjÙ9á¹��/� �!��Gäóò:G�çNÍ�èPÙ�Í�Ö�è_Ê9Þ�Ö�è_Ê9Þ1Í�ʹßàÊnÚ�Ô�ÎlØ�Í�Í�è�̨ÖÖ�ã-Î�ßl̨Ù�ÓAÍ�á¡ãDÎ2Ê�Û�è_ÊnÚ_Î�Ö�Í�ñ®Ô�ÎlØjÌ�ë�ÎlØ ORK ã�Ø�Ê�Í�ʺßàÊnÓNÌ�Ú�Í�ʱÙ(Õ�Ú�̼Ô�ÎlØjÖjÙ�Ó�Ó¼á�ñ�ßàÊn˧ãDÊnÖ�Ù�ÒPӼΠO`K ã�Ø�Ê�Í�ʺßàÊnÓVõ�×%è_Î�Ö�Î�Ø�Î�Ö�Õ�Ó�Í�ÖÙ�ØjÎ�ã�Ø�Î�Ö�Î�Ú�Í�Î�ϹÌ�Ú @ºÎ�ßàÍ�̼ÊnÚ��¡õ

×%è_Î �³É%Ðr× Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá©ßlÙ�Ú÷ÒDÎ3ÚPÙ�Í�Õ_ØjÙ�Ó�Ó¼á¯Î��ºÍ�Î�Ú�Ï_Î�Ï{Ì�Ú�Í�Ê(ÊnÚ_Î2Í�è�Ù�Í�ã-ÎlØjÛ¦Ê�ØjË«Ö(í � ïvðlñ®Í�ØjÙ�Ú�Ö�Û�ÎlØjÖ�ܸÌ�Ú�Ö�Í�Î�Ù�ÏÊ�Û¹í î ï�ð9Ý�Ù�Ú�ÏaãPØ�Ê9Ô�Î�Ö«Ø�Î�Ó�Ù�Í�̼ÊnÚ�ÖµÊnÚaÛ¦ÊnÕ�Ø«ßàÊn˵˵̼Í�Í�Î�Ï­ÒP̼Í�ֹܸÙ�Ö(Ê�ã�ãDÊnÖ�Î�Ï Í�ÊJÍ�è�Ø�ÎlÎ�Ýhõ ��Î÷ßlÙ�Ó¨Ó5Í�è�Ì�Ö§Î��ºÍ�Î�Ú�Ï_Î�ÏÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼ÍváE= �> ? ADC � Ù�Ú�Ï�Ö�è_Ê�Þaè_Ê9Þ Í�Ê�ßàÊnÚ�Ö�Í�ØjÕPßàÍ.̼Í.Õ�Ö�Ì�Ú�þ�Í�è_Î:Ê�Øj̼þnÌ�ÚPÙ�Ó:=#> ? ADC(Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼ÍvárÙ�ÖNÙ�ÒPÕ�Ì�Ó¨Ï�Ì�Ú_þÒ�Ӽʡßjý�õN� !�ÕP̼ã�ãDÎ�Ï]Þ�̼Í�è = �>@? ADC � Þ:ΫÍ�è_Î�Ú Ö�è_Ê�Þ*è�Ê9Þ*Í�Ê÷Ö�Î�ßlÕ_ØjÎ�Ó¼á�Ø�Î�Ù�Ó�Ì��lÎ(ÙLÍvÞ�Ê�ñ®ãPÙ�ØjÍ�áJÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá÷Í�è�Ù�ÍÞ:ΩßlÙ�Ó�Ó���%"�-�x/��W /T� �W�* �!.�� "/0�0%"�YÜ8=���D>AÝ � Þ�è�̨ßjè � Ù�֧̼Í�Ö�Ú�Ù�˧ΩÌ�ÚPÏ�Ì�ßlÙ�Í�Î�Ö � Ù�Ó�Ó�Ê9Þ�Ö�Í�Þ:Ê�ãPÙ�Ø�Í�Ì�Î�Ö2Í�Ê�Ö�Î�ßlÕ�Ø�Î�Ó¼áßàÊn˵ãPÕ_Í�Î�Ù�Ú�á¯ô:Ê¡ÊnÓ¼Î�Ù�Ú©Û¸Õ�Ú�ßàÍ�̼ÊnÚ¯Ê�Ô�ÎlØ�ÍvÞ�Ê«ÒP̼Í�Ö5Ö�è�Ù�ØjÎ�Ï©ÒDÎlÍ�Þ:ÎlÎ�Ú¹Í�è_Î�Ë©õ���Ö�Ö�Î�Ú�Í�Ì�Ù�Ó�Ó¼á � Í�è�Î�ã�Ø�Ê�Í�ʺßàÊnÓ�Ø�Î�Ù�Ó�Ì���̨Ú_þÍ�èPÌ�Ö³Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á«Õ�Ö�Î�Ö%Ù§ßàÊnÚ�Ö�Í�ØjÕPßàÍ�̼ÊnÚ¹Ö�Ì�Ë«Ì�Ó�Ù�Ø:Í�ʵÍ�è�Ù�Í%Ê�Û5äóò��çAÛ�Ê�Ø%Í�è_Î�ßàÊn˧ãPÕ�Í�Ù�Í�̼ÊnÚ©Ê�Û.Í�è_Î3Ë�Õ�Ó¼Í�̼ãPÓ¨Ì�ßlÙ�Í�̼ÊnÚþnÙ�Í�Î�õP%�Ê�Þ�ÎlÔ�ÎlØ � Ï�Ì�Ö�Í�̨Ú�ßàÍ�̼Ô�Î�Û�Î�Ù�Í�Õ_Ø�Î�Ö³Ê�ÛÍ�è_Îrã�Ø�Ê�Í�ʺßàÊnÓDÙ�Ø�ÎrÍ�è�Ù�Í5̼Í5Ï_Î�Ù�Ó�Ö³Ï�̼Ø�Î�ßàÍ�Ó�á§Þ�̼Í�è±Ù�Ï�Ù�ã�Í�̼Ô�Î�Ó¼á(˵Ù�Ó�Ì�ßl̼ÊnÕPÖã�Ù�Ø�Í�̼Î�Ö � Ù�Ú�ϩ̼Í�Ö%Î��«ßl̼Î�Ú�ßàá�>ÊnÚ�Ó¼á±Ù§ßàÊnÚ�Ö�Í�Ù�Ú�Í�Ú�ÕPË3ÒDÎlسÊ�ÛÎ��ºãDÊnÚ_Î�Ú�Í�Ì�Ù�Í�̼ÊnÚ�Ö5Ù�Ú�Ï©ßàÊnË«Ë2Õ�ÚPÌ�ßlÙ�Í�̼ÊnÚ©Ø�ÊnÕPÚ�Ï�Ö:ãDÎlØþnÙ�Í�Î2ÎlÔ�Ù�Ó¨Õ�Ù�Í�̼ÊnÚ�õ �nÊnÌ�Ú�ÍEH�Ù�Í�Î �?Ô�Ù�Ó�Õ�Ù�Í�Ì�ÊnÚ©Ì�Ö5ã�Ø�Î�Ö�Î�Ú�Í�Î�Ï¹Ì¨Ú @¡Î�ßàÍ�̼ÊnÚ&M_õ

�.̨Ú�Ù�Ó�Ó¼á � Þ�ÎLÕPÖ�Î&=���D>+Í�ÊÑÖ�Î�ßlÕ�Ø�Î�Ó¼á Ø�Î�Ù�Ó�Ì��lÎ�� Î��«ßl̼Î�Ú�Í�Ó¼á�� Ù�Ú�áaÙ�Ï�Ù�ãPÍ�̼Ô�Î�Ó¼á Þ�Î�Ó�Ó¼ñ®Û¦Ê�Øj˵Î�ÏaÍ�Þ:Ê�ñ®ãPÙ�Ø�ÍváÙ�ÚPÏ Ë�Õ�Ó¼Í�̼ñ®ãPÙ�Ø�Í�á�Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá � Þ�è�Ì�ßhèÑÌ�Ö�Î��ºã�Ø�Î�Ö�Ö�Î�ÏJÒ¡áÑÙ÷ÒDÊ�ÊnÓ¼Î�Ù�Ú ßl̼ØhßlÕ�Ì¼Í � Ì�Ú Ù{ÕPÚ�̼Ô�ÎlØjÖ�Ù�Ó¨Ó¼á�ñ�ßàÊn˵ã-ÊnÖjÙ�ÒPÓ¼ÎÞ³Ù�á�õ ö þnÙ�Ì�Ú � Ö�Ì�ÚPßàÎ�Í�è_Î3ØjÎ�Ù�Ó�Ì���Ù�Í�̼ÊnÚ÷Ì�Ö�Ï�Ì�Ø�Î�ßàÍ�Ó¼á±Û¦Ê�Ø�Í�è_Î2Ù�ßàÍ�Ì�Ô�Î�Ó¼á¹ËµÙ�Ó�Ì�ßlÌ�ÊnÕ�Ö%Ù�Ï_Ô�ÎlØjÖjÙ�Ø�á � Ù�ÚPϹҡá¯ËµÎ�Ù�Ú�Ö�Ê�Û?Ù�ÚÎ��(ßl̼Î�Ú�Í2ÒPÕ�Ì�Ó¨Ï�Ì�Ú_þ(ÒPӼʺß�ý � Í�è_ΫÊ9Ô�ÎlØjÙ�Ó�Ó:ßàÊn˧ãPÕ�Í�Ù�Í�̼ÊnÚ�Ù�ÓGßàÊn˵ãPÓ¼Î��_̼Í�á�Ì�Ö�Ù÷Ö�˵Ù�Ó�ÓGßàÊnÚ�Ö�Í�Ù�Ú�Í3Í�Ì�˧Î�Ö3Í�è_Î(Ú¡Õ�Ë2ÒDÎlØÊ�ÛGþnÙ�Í�Î�Ö�Ì�ÚLÍ�è�Î2Ø�ÎlãPØ�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ{Ê�Û>Í�è_Î2Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá � Ù�Ú�Ï÷Í�è_Î�Ú�ÕPË3ÒDÎlØ�Ê�Û?Ø�ÊnÕ�Ú�ÏPÖ�Ì�Ö�Ù(ßàÊnÚPÖ�Í�Ù�Ú�ÍrÍ�Ì�˵Î�Ö�Í�è_ÎÏ�Îlã�Í�è]Ê�Û�Í�è�εßl̼ØjßlÕ�̼Ílõ�×%è_Î�Í�Ø�Î�Ù�Í�˵Î�Ú�Í�Ê�Û:Í�Þ:Ê�ñ®ãPÙ�Ø�Ívá�Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�̼Î�Ö�̨Örã�Ø�Î�Ö�Î�Ú�Í�Î�Ï»Ì�Ú�@¡Î�ßàÍ�̼ÊnÚ G � Þ�è�̨ӼÎ3Í�è_ÎßlÙ�Ö�εÊ�Û�Ë�Õ�Ó¼Í�Ì�ñ®ã�Ù�Ø�Í�á÷Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Í�Ì�Î�Ö � Þ�̼Í�è{Í�è_ΧÊnÚ_Îàñ®Í�Ê�ñ�˵Ù�Ú�á�Î��ºÍ�Î�Ú�Ö�̼ÊnÚ�Ö�Ù�ÚPÏ]Ø�Î�Ù�Ó�Ì���Ù�Í�̼ÊnÚPÖrÊ�Û�Í�è_Î�Ø�ÎS!¡Õ�̼ØjÎ�ÏÒ�Õ�Ì�Ó�Ï�̨Ú_þrÒPӼʺß�ýºÖ � ̨Ö�Ï�̨Ö�ßlÕ�Ö�Ö�Î�ϱÌ�Úw@¡Î�ßàÍ�̼ÊnÚ��¡õ �GÕ�Í�Í�Ì�Ú_þ3ÎlÔ�ÎlØjá�Í�è�̨Ú_þ2Í�Ê�þ�ÎlÍ�è�ÎlØ � Þ�Î�ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ:Î��«ßl̼Î�Ú�ͳÙ�ÚPÏ(Õ�ÚPÌ�ñÔ�ÎlØhÖ�Ù�Ó�Ó¼á3ßàÊn˵ã-ÊnÖjÙ�ÒPÓ¼Î:Í�Þ:Ê�ñ®ãPÙ�Ø�Í�á§Ù�Ú�Ï�Ë�Õ�Ó¼Í�Ì�ñ®ã�Ù�Ø�Í�á�ßàÊn˧ã�Õ_Í�Ù�Í�̼ÊnÚ�ã�Ø�Ê�Í�ʺßàÊnÓ�Ö.Í�è�Ù�Í?Ù�Ø�Î5Ö�Î�ßlÕ�Ø�Î:Ù�þnÙ�̨Ú�Ö�Í>Ù�Ï�Ù�ã�Í�̼Ô�ÎÙ�Ï�Ô�ÎlØjÖ�Ù�Øj̼Î�Ö�Ì�Ú¯Í�è_Î7���D *���-�=<©Ë§ÊºÏ_Î�Ó � Þ�è_ÎlØ�Î�Þ:Î3Ù�Ó�Ó�Ê9Þ ãPÙ�ØjÍ�̼Î�Ö5Í�ʵÎlØjÙ�Ö�Î3ßàÎlØ�Í�Ù�Ì�ڹ̨Ú_Û¦Ê�Øh˵Ù�Í�̼ÊnÚ¯Ø�Î�Ó�̨Ù�ÒPÓ¼á�õ

ö Ö�ÙµÍ�Î�ßjèPÚ�Ì�ßlÙ�Ó.Ú_Ê�Í�Î � Þ:Î2Õ�Ö�Î�Í�è_Î3þnÙ�Í�Îàñ®Ò¡á�ñ®þnÙ�Í�εÙ�ã�ã�Ø�ÊnÙ�ßhèLÛ�Ø�ÊnË äóò��ç � Ù�Ú�Ï÷˵Ù�ý�Î2Ö�Õ�Ø�Î�Í�è�Ù�Í�Î�Ù�ßjè{þnÙ�Í�Î�̨ÖßàÊn˵ãPÕ_Í�Î�Ï]Î��«ßl̼Î�Ú�Í�Ó¼á�õ ��ΫÏ_Ê©Ú_Ê�Í�Õ�Ö�Î�Í�è_Î�é�Î�Ú�ßàØjá�ã�Í�Î�Ï�ßlÌ�ØjßlÕ�̼Í�ì¯Ù�ã�ã�ØjÊnÙ�ßjè�Ï�Õ_Î�Í�Ê��GÙ�ÊÑä M 3�ç � Þ�è�Ì�ßhè÷áºÌ¼Î�Ó�ÏPÖßàÊnÚPÖ�Í�Ù�Ú�Í�ñ®ØjÊnÕ�Ú�Ïaã�ØjÊ�Í�Ê¡ßàÊnÓ�Ö�Ò�Õ_͵Ì�Ö�ØjÙ�Í�è_Îlث̨Ú_Î��«ßl̼Î�Ú�ͧÌ�Ú Í�ÎlØj˵Ö�Ê�ÛrßàÊn˵Ë�Õ�Ú�Ì�ßlÙ�Í�̼ÊnÚ­ßàÊn˧ãPÓ¼Î��_̼Í�á � Ö�Ì�Ú�ßàίÊnÚ_ÎÚ�ÎlÎ�Ï�Ö:Í�Ê2ã�ØjÊ9Ô�Î�Ì�Ú �lÎlØ�Ê�ñ®ý¡Ú�Ê9Þ�Ó¼Î�Ï�þ�Î�Í�èPÙ�Í:Í�è�Î�Î�Ú�ßàØjá�ã�Í�Î�ϯßl̼ØhßlÕ�̼Í�̨Ö�ßàÊ�Ø�ØjÎ�ßàͳÙ�ÚPÏ(Í�è�Î�Ö�Î�ãPØ�Ê�Ê�Û¸Ö³Ù�Ø�ÎrÕ�Ú�Ó�̼ý�Î�Ó�á�Í�ÊÙ�ÏP˵̼Í5Î��«ßl̼Î�Ú�Í5ã�Ø�Ê�Í�ʺßàÊnÓ�Ölõ

� 1@5�� 741������T/���9 ��Î:Ù�Ó¼Ø�Î�Ù�Ï_ár˧Î�Ú�Í�̼ÊnÚ_Î�Ï3ã�Øj̼Ê�Ø�Þ:Ê�Ø�ýrÊnÚ2É%Ðr×Yä�å�æ � å���ç®õ ö Ó�Í�è_ÊnÕ_þnè�Í�è_ÎGã�Ø�Ê�Í�ʺßàÊnÓ�ÖAã�ØjÎ�Ö�Î�Ú�Í�Î�ÏÍ�è�ÎlØ�εÙ�Ø�εþ�Î�Ú_ÎlØhÌ�ߧÙ�Ú�Ï»è_Î�ÚPßàε˵Ù�á]ÒDεÌ�˧ãPÓ�Î�˧Î�Ú�Í�Î�Ï]Þ�̼Í�è]Ê�Ø�Þ�̼Í�è�ÊnÕ_Í�ßàÊn˧ãPÕ_Í�Ù�Í�̼ÊnÚPÙ�Ó?Ù�Ö�Ö�Õ�˵ã�Í�̼ÊnÚ�ÖµÜ�Î�õóþ_õ �ÕPÖ�Ì�Ú_þ�ã�ØjÌ�˫̼Í�̼Ô�Î�Ö:ÒPÙ�Ö�Î�Ï©ÊnÚ�!�Õ�Ù�Ú�Í�Õ�Ë ßhè�Ù�Ú�Ú_Î�Ó�ÖyÝ � Í�è_Îlá±Ù�ØjÎ3Ó¼Î�Ö�Ö³Î��«ßlÌ�Î�Ú�Í%Ò¡á¯Ù�Í%Ó¼Î�Ù�Ö�Í�Ù�Û¸Ù�ßàÍ�Ê�Ø�Ê�ÛÅ� � Þ�è_ÎlØjÎ �̨Ö�Í�è�Î�Ö�Î�ßlÕ_Øj̼ÍváLãPÙ�ØjÙ�˵ÎlÍ�ÎlØ � Ù�Ú�Ï]Û¸Õ_Ø�Í�è_ÎlØj˵Ê�Ø�Î � Í�è_Îlá÷Ù�Ø�ΧÚ_Ê�Í�Õ�ÚP̼Ô�ÎlØjÖ�Ù�Ó�Ó�á¹ßàÊn˧ãDÊnÖ�Ù�ÒPÓ¼Î�õ(Ü ö Ö�Ù©Ö�Ì�Ï�Î2Ú_Ê�Í�Î � Ùé�Ö�Í�Ù�Ú�Ϻñ�Ù�Ó¼ÊnÚ_Î�ì�Ô�ÎlØjÖ�̼ÊnÚ�Ê�Û-ÊnÕ_Ø �³É%Ðr× ã�Ø�Ê�Í�ʺßàÊnÓ_Þ:ÊnÕ�Ó�Ï�ÒDÎ5Ö�Õ_ÒPÖ�Í�Ù�Ú�Í�Ì�Ù�Ó¨Ó¼á3Ö�Ì�˵ãPÓ¼ÎlØ � Ì�Ú�ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�ØNÞ�̼Í�è�ØjÎ�Ö�ãDÎ�ßàÍÍ�Ê«Í�è_Î3Ì�˧ã�Ó¼Î�˧Î�Ú�Í�Ù�Í�̼ÊnÚ¯Ê�ÛÍ�è�Î3Ú_Î�ßàÎ�Ö�Ö�Ù�Øjá&O`K ã�Ø�Ê¡Ê�Û�Ölõ�Ý��Î�ßlÙ�Ú§Ù�Ó�Ö�Ê�ßàÊn˧ã�Ù�Ø�Î5Í�è_Î#�:É�Ðr× Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá�Í�Ê�Í�è_Î5Û¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�̼Î�Ö.Ï_ÎlëPÚ_Î�Ï�Ì�Ú©ä�å�ò�ç � Þ�è_Ê�Õ�Ö�Î%Ù¯é�Í�Þ:Ê�ñ

ã�è�Ù�Ö�Î�ì%Ù�ã�ãPØ�ÊnÙ�ßjè3Í�Ê�ßàÊnÚ�Ö�Í�ØjÕPßàÍÕ�Ú�Ì�Ô�ÎlØjÖ�Ù�Ó�Ó¼á�ßàÊn˧ãDÊnÖ�Ù�ÒPÓ¼Î>ÍvÞ�Ê�ñ®ãPÙ�ØjÍ�á! �Ë2Õ�Ó�Í�̼ãPÓ¼Îàñ®ãPÙ�ØjÍ�á�ßàÊn˧ãPÕ�Í�Ù�Í�̼ÊnÚ�ã�ØjÊ�Í�Ê¡ßàÊnÓ�Ö�õ� Ú�Í�è�ÎGë�ØjÖ�Í�ãPè�Ù�Ö�Î � Þ�è_ÎlØjÎ?Í�è_Îlá�ßàÊnÚ�Ö�Í�ØjÕPßàÍÙ%ãPØ�Ê�Í�Ê¡ßàÊnÓ¡Ö�Î�ßlÕ_ØjÎ�Ù�þnÙ�Ì�Ú�Ö�Í�Ö�Î�˵Ì�ñ�è�ÊnÚ_Î�Ö�ÍÙ�Ï�Ô�ÎlØjÖ�Ù�Øj̼Î�Ö � Ù�Ú2Ì�˧ãDÊ�Ø�Í�Ù�Ú�ÍÍ�Ê¡ÊnÓ_Ì�ÖNÍ�è_Î�Ðr× Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá�õ � Ú�Í�è_Î5Ö�Î�ßàÊnÚ�ϧãPèPÙ�Ö�Î � Þ�è_ÎlØ�γÍ�è_Îlá�Î��_è�̼ÒP̼Í.Ù©é�ßàÊn˧ãPÌ�Ó�ÎlØjì�Í�è�Ù�Í>Í�Õ_ØjÚ�Ö.ã�ØjÊ�Í�Ê¡ßàÊnÓ�Ǫ̈Ú�Í�è_εë�ØhÖ�Í�ãPè�Ù�Ö�Î(Ì�Ú�Í�ÊLÊnÚ_Î�Ö3Ö�Î�ßlÕ�Ø�ΫÙ�þnÙ�Ì�Ú�Ö�Í3˵Ù�Ó�Ì�ßlÌ�ÊnÕ�Ö�Ù�Ï_Ô�ÎlØjÖ�Ù�Øh̼Î�Ö � Ù�ÚJÌ�˧ãDÊ�Ø�Í�Ù�Ú�Í�Í�Ê¡ÊnÓGÌ�Ö�Í�è�ÎJé�ßàÊn˵˵̼Í�ñÙ�ÚPϺñ®ã�Ø�Ê�Ô�Î�ìrÛ�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá � Þ�è�Ì�ßhè�ã�Ø�Ê�Ô�Î�Ö>þ�Î�Ú_ÎlØjÙ�Ó ���JÖ�Í�Ù�Í�Î�˵Î�Ú�Í�Ölõ � Ú§Ö�Ên˧Î%Ö�Î�Ú�Ö�Î � Í�è_Î#�³É%Ðr× Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�áË«Ù�áLÒDÎ2ԡ̼ÎlÞ:Î�Ï]Ù�Ö�ÙÑé�ßàÊnË2ÒPÌ�ÚPÙ�Í�̼ÊnÚ�ì±Ê�Û?Í�è_ΫÐr× Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á¹Ù�Ú�Ï{Í�è_Î�ßàÊn˵˵̼Í�ñ�Ù�ÚPϺñ®ã�Ø�Ê�Ô�Î�Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá�õ%�Ê9Þ:ÎlÔ�ÎlØ � Þ�Î�Ö�Í�Ø�Î�Ö�Ö�Í�è�Ù�Í�ÖjÌ�Ú�ßàÎ"�:É�Ðr×�ÊnÚ�Ó¼á¯Ú_ÎlÎ�Ï�Ö�Í�Ê«ã�ØjÊ9Ô�Î�Ô�ÎlØ�á¹ÖjÌ�˧ãPÓ¼ÎrØ�Î�Ó¨Ù�Í�̼ÊnÚ�Ö3ܸÙ�˧ÊnÚ�þ«Í�è_Ø�ÎlÎ3ÒP̼Í�ÖhÝ � Ì�ÍßlÙ�ÚLÒDÎ�ØjÎ�Ù�Ó�Ì��lÎ�Ϲ˧Ê�ØjÎ�Î��«ßl̼Î�Ú�Í�Ó¼á�õ �"$# G�ENV�KLG�KLZ�P8K}PUDV�­W­W[µKL­WGDE�K=X.�*ENUJKL[\V�E�P&�\[\K-�ÍÃ|[µKLZ?KLZ*G�UDP�]�P&Q*[\�\[µK��ÍKLV$]"GJS�XÄV&SL­ ]�SLV�V�XÄRxV�EWUJV�­W­W[µK�KLGOM QN[µKLR�ÃaP�R}P��\R�V�]NSLV�]"V�R�GOM

Page 4: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

>%Î�ßàÎ�Ú�Í�Ó¼á ��� Ù�˧þ��Ù�ØhÏ«Ù�Ú�Ï ��̼Î�Ó¨Ö�Î�Ú¹äóòºåàçPã�ØjÎ�Ö�Î�Ú�Í�Î�Ï«Î��«ßl̼Î�Ú�ÍGÕ�Ú�Ì�Ô�ÎlØjÖ�Ù�Ó�Ó¼á2ßàÊn˧ãDÊnÖ�Ù�ÒPÓ¼Î�Ë�Õ�Ó¼Í�̼ñ®ãPÙ�Ø�Í�á�ßàÊn˧ãPÕ_ñÍ�Ù�Í�Ì�ÊnÚ©ã�Ø�Ê�Í�ʺßàÊnÓ�Ö5Õ�Ö�̨Ú_þ�Ù§ÏPÌ�-ÎlØjÎ�Ú�Í5Ù�ã�ã�ØjÊnÙ�ßjè�õ?×%è�Î�̼Ø5ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ì�ÊnÚ©Ì�Ö³ÒPÙ�Ö�Î�ϯÊnÚ©Ù�Ú¯Î��«ßlÌ�Î�Ú�Íc�F�:É+ãPØ�Ê�Í�Ê¡ßàÊnÓÒ¡á±É:ØhÙ�˧ÎlØ ��/� �!���ä�åSG�ç � Þ�è�Ì�ßjè±Ì�Ú±Í�Õ_ØjÚ±Ì�Ö�ÒPÙ�Ö�Î�ϯÊnÚ±Í�è_Ø�Î�Öjè_ÊnÓ�ϱè_Ên˧Ên˧Ê�Ø�ã�è�Ì�ß�ßàØjá�ã�Í�ÊnÖ�á¡Ö�Í�Î�Ë«Ölõ�É:Ên˧ãPÙ�ØjÎ�Ï(Í�ÊÊnÕ�Ø�Ø�Î�Ö�Õ�Ó�Í � Í�è_Î � Ù�˧þ��Ù�ØjÏ_ñX��̼Î�Ó¨Ö�Î�ÚJßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�̼ÊnÚ�Þ:Ê�Ø�ýºÖ3Ì�Ú�ÙLÖ�Ó�̼þnè�Í�Ó¼á]Ö�Í�Ø�ÊnÚ_þ�ÎlØ3˵ʡÏ_Î�Ó � Ú�Ù�˧Î�Ó�á÷Í�è_Î6(}�x+�!.�0#� ��1 �-��9��O *��/0����#�/0���O�LÜ � K � ݳ˧ʺÏ_Î�Ó � Þ�è_ÎlØ�Î3ÙµÍ�ØjÕ�Ö�Í�Î�ÏLãPÙ�Ø�Í�á¹Ú�Ê�Í%ÊnÚ�Ó¼á¯þ�Î�Ú_ÎlØjÙ�Í�Î�ÖrÙµßàÊn˫˧ÊnÚLØjÎlÛ¦ÎlØ�Î�ÚPßàÎ2Ö�Í�Øj̨Ú_þÜ�Þ�è�Ì�ßjèaßàÊnÚ�Í�Ù�Ì�Ú�ÖµÍ�è_ΩãPÕ_ÒPÓ¨Ì�ß«ý�ÎláºÖ§Ê�ÛrÙ�Ó�Ó³Í�è_ιã�Ù�Øj̼Î�ÖhÝ � ÒPÕ_ͧÙ�Ó�Ö�Ê»Ù]ã�Øh̼Ô�Ù�Í�ιÖ�Í�ØjÌ�Ú_þ]Û�Ê�اÎ�Ù�ßjè ã�Ù�Ø�Í�á ܸÙ�Ö§Í�è_Îã�Ù�Ø�Í�á�� Ö2Ö�Î�ßàØ�ÎlÍ3ý�Îlá�ÝhõLÐrÚ»Í�è_ΫÊ�Í�è_ÎlØ2è�Ù�ÚPÏ � Í�è_Î�Ì�Ø3ã�Ø�Ê�Í�ʺßàÊnÓG̨Ö�Ö�Î�ßlÕ_Ø�Î(Ù�þnÙ�Ì�Ú�Ö�Í�Ù�Ï�Ù�ã�Í�̼Ô�ΫÙ�Ï�Ô�ÎlØjÖ�Ù�Øj̼Î�Ö2Ì�Ú�Í�è_ÎÖ�Ê�ñ�ßlÙ�Ó�Ó¼Î�Ϻ�}%����8���O *�����k<Ñ˧ʺÏ_Î�Ó � Þ�è_ÎlØ�Î(Í�è_ΫãPÙ�ØjÍ�̼Î�Ö2Ù�ØjαÚ_Ê�Í�Ù�Ó�Ó¼Ê�Þ�Î�ÏÑÍ�Ê÷ÎlØhÙ�Ö�αÙ�Ú�á»Ì�Ú_Û�Ê�Øj˵Ù�Í�Ì�ÊnÚ � Þ�è�Ì�ӼΧÊnÕ_ØßàÊnÚPÖ�Í�ØjÕ�ßàÍ�̼ÊnÚLÌ�Ö5Ö�Î�ßlÕ_Ø�Î2Ì�Ú¯Í�è_Î�ÎlØjÙ�Ö�̨Ú_þµË§ÊºÏ_Î�ÓDÊnÚ�Ó¼á�õ

­Ã��� hÈ�� ÈjÁ��5ÃAÈ������%Á�Å������«Á§È�ÂDÈ�Ä�Á��ö Ó�Ó�ÊnÕ_Ø5Ø�Î�Ö�ÕPÓ¼Í�Ö³Ù�Ø�Î�Ì�Ú±Í�è_ÎH#&%�'5'H%�� �O��9����O���}#��4��/0�����k<]Ü®É?>A@PÝG˵ʡÏ_Î�Ó � Þ�èPÌ�ßjè¯Ù�Ö�Ö�ÕP˧Î�Ö³Í�è�Ù�ͳÍ�è_ÎlØ�Î�Ì�Ö³Ù§Ö�Í�Øj̨Ú_þÕPÚ�̼Û�Ê�Øj˵Ӽá¯þ�Î�Ú_ÎlØjÙ�Í�Î�Ï]Û�Ø�ÊnË'Ö�Ên˵Î�ÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�̼ÊnÚ¹Ù�Ú�Ï]Ì�ÖrÙ9Ô�Ù�Ì�Ó¨Ù�ÒPÓ¼Î�Í�ʹÙ�Ó�Ó�ã�Ù�Ø�Í�̼Î�ÖrÙ�Í�Í�è_ΧÖ�Í�Ù�Ø�ÍrÊ�Û�Ù±ã�ØjÊ�Í�Ê¡ßàÊnÓVõ×�è�Ì�Ö�Ì�ÖrÙ¯þ�Î�Ú_ÎlØjÙ�Ó�Ì���Ù�Í�Ì�ÊnÚ]Ê�ÛGÍ�è_Îr(}�x+�!.��#H�O ��T) %"' ��/0�����k<]˧ʺÏ_Î�Ó � Þ�è_ÎlØ�ΧٯÕ�Ú�Ì�Û¦Ê�ØjË Ï�Ì�Ö�Í�Øj̼ÒPÕ_Í�Ì�ÊnÚ¹Ê9Ô�ÎlØ3ë)�¡Î�Ï_ñÓ�Î�Ú_þ�Í�è©ÒP̼Í5Ö�Í�ØhÌ�Ú_þnÖ5Ì�Ö%Ù�ÖjÖ�Õ�˧Î�Ï�õ

�_Ê�اÙ÷Ï�̨Ö�Í�Øj̼ÒPÕ�Í�̼ÊnÚ�� � Þ:αÖ�Ù9áJø���� Í�Ê�Ï_Î�Ú_Ê�Í�ίÙ�Ú�á»Î�Ó¼Î�˧Î�Ú�Í2Í�èPÙ�Í�è�Ù�Ö�Ú_ÊnÚºñL�lÎlØjÊ÷ã�ØjÊ�ÒPÙ�ÒPÌ�Ó�Ì�Í�á]Ì�Ú � �Ì®õóÎ�õ � Ù�Ú�á�Î�Ó¼Î�˧Î�Ú�Í2̨ڻÍ�è_αÖ�Õ_ã�ãDÊ�Ø�Í�Ê�Û!�¯õ&��ίÖ�Ù9áJø#"$ �'Í�Ê]Ï_Î�Ú_Ê�Í�ίø{̨Ö�ØjÙ�Ú�Ï�Ên˵Ӽá�ßjè_ÊnÖ�Î�ÚÑÙ�ßlßàÊ�ØjÏPÌ�Ú_þLÍ�ÊÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�̼ÊnÚ%�¯õ��_Ê�Ø�Ù«Ö�ÎlÍ'& � Þ:Î3Ö�Ù9á¯ø("$ & Í�Ê«Ï_Î�Ú_Ê�Í�Î�Í�èPÙ�Í�ø(̨Ö%Õ�Ú�̼Û�Ê�Øj˵Ó�áµÏ_ØjÙ�Þ�Ú©Û¦ØjÊnË)&:õ

*,+ 1 -.-./10D13/+2 �B5�2 �4365 �B2 ��7./ 58/87:9<;X/ �=3�1�� �B/ ��9 ×�è_Î:ÕPÚ�̼Ô�ÎlØjÖ�Ù�Ó�ßàÊn˧ãDÊnÖ�Ù�ÒPÌ�Ө̼Í�árÛ�ØjÙ�˧ÎlÞ:Ê�Ø�ý�Þ³Ù�Ö>Ö�Õ_þ�þ�Î�Ö�Í�Î�ÏÒ¡á ɳÙ�Ú�ÎlÍ�Í�Ì�Û�Ê�اÏ_ÎlëPÚ�̨Ú_þ¹Í�è_ΩÖ�Î�ßlÕ_Øh̼Í�áJÙ�ÚPÏÑßàÊn˧ãDÊnÖ�Ì�Í�̼ÊnÚJÊ�Û�ã�Ø�Ê�Í�ʺßàÊnÓ�Ö©ä�å�3�ç®õ � ÚÑÍ�è�Ì�Ö�Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý»ÊnÚ_αë�ØjÖ�ÍÏ�ÎlëPÚ_Î�Ö�Ù�Ú=é�̨Ï_Î�Ù�ÓGÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá¡ìLÊ�Û�Ù÷ã�Ø�Ê�Í�ʺßàÊnÓ � Ù�Ú�Ï Í�è_Î�ÚÑã�Ø�Ê9Ô�Î�Ö�Í�èPÙ�Í�Ù÷ãPÙ�Ø�Í�̨ßlÕ�Ó�Ù�Ø�Ì�˧ãPÓ¼Î�˵Î�Ú�Í�Ù�Í�̼ÊnÚJÊ�ÛÍ�èPÌ�Ö>ã�Ø�Ê�Í�ʺßàÊnÓ�Ê�ãDÎlØjÙ�Í�̨Ú_þ3Ì�ÚµÙ3þn̼Ô�Î�Ú«ßàÊn˵ãPÕ_Í�Ù�Í�̼ÊnÚ�Ù�ÓPÎ�Ú�Ժ̼Ø�ÊnÚP˧Î�Ú�Í�Ö�Î�ßlÕ_Ø�Î�Ó¼á�Ø�Î�Ù�Ó¨Ì��lÎ�Ö?Í�è�̨Ö?Ì�Ï_Î�Ù�Ó_Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá�õ×�è_Î�ÒPÙ�ÖjÌ�ß%Î�Ú�Í�̼Í�̼Î�Ö³Ì�Ú�Ô�ÊnÓ¼Ô�Î�ϱÙ�Ø�Î?>÷ãPÓ�Ù9á�ÎlØjÖA@ ïCBEDEDEDCB @GF � Ù�Ú¯Ù�Ï_Ô�ÎlØjÖjÙ�Ø�áIH � Ù�Ú�ϱÙ�Ú±Î�Ú�Ժ̼Ø�ÊnÚP˧Î�Ú�ÍAJµõ?×�è_Î�Ø�Î�Ù�ÓÎ��ºÎ�ßlÕ_Í�̼ÊnÚJÊ�Û:Ù¹ã�Ø�Ê�Í�ʺßàÊnÓLK � ØjÕPÚ]Ò�á]Í�è_ΧãPÓ¨Ù�á�ÎlØjÖ2Ì�Ú]Í�è_εã�ØjÎ�Ö�Î�Ú�ßàΧÊ�ÛAH Ù�ÚPÏ»Ù�Ú�Î�Ú�ԡ̼ØjÊnÚ�˧Î�Ú�Í�˵Ù�ßjè�̨Ú_ÎMJ �Þ�̼Í�è©Ì�Ú_ãPÕ�ÍAN � Ì�Ö%˵ʡÏ_Î�Ó�Î�Ï©Ù�Ö�Ù«Ö�ÎS!�Õ_Î�ÚPßàÎ�Ê�Û; #�/0���* "/0��%��)�5Ê�ÛNÍ�è_Î�Î�Ú�Í�̼Í�̼Î�Ö�õ�×%è_Î�Î�Ú�Ժ̼Ø�ÊnÚ�˧Î�Ú�Í?J*Ì�Ö%Ù�ßàÍ�Ì�Ô�Ù�Í�Î�ÏëPØjÖ�Í � þ�Î�Ú_ÎlØjÙ�Í�Ì�Ú_þ¯Ì�ÚLãPÙ�Ø�Í�Ì�ßlÕ�Ó¨Ù�Ø�Í�è_Î�Ì�Ú_ãPÕ�Í�Ö5Í�ʯÍ�è_Î2Ê�Í�è�ÎlØ�ã�Ó�Ù�á�ÎlØjÖ�õ�×%è_Î�Ú÷Í�è_Î�ã�Ø�Ê�Í�ʺßàÊnÓ�ãPØ�Ê¡ßàÎlÎ�ÏPÖ�Ò¡áLè�Ù9Ô¡Ì�Ú�þH Î��_ßjè�Ù�Ú�þnÌ�Ú_þ(˵Î�Ö�Ö�Ù�þ�Î�Ö�Þ�̼Í�è©Í�è�Î3ãPÓ�Ù9á�ÎlØjÖ�Ù�ÚPϹÍ�è_Î2Î�Ú�Ժ̼Ø�ÊnÚ�˵Î�Ú�ÍlõW�NÌ�Ú�Ù�Ó�Ó�á � Í�è_Î3Î�Ú�Ժ̼Ø�ÊnÚ�˧Î�Ú�Í%ÊnÕ_Í�ã�Õ_Í�Ö%ÊnÚ_ÎÒ�Ì¼Í � Þ�è�Ì�ßhè©Ì�Ö³Í�è_Î�ÊnÕ�Í�ãPÕ_Í5Ê�Û.Í�è_Î�ã�Ø�Ê�Í�ʺßàÊnÓVõ

×%è_ιÖ�Î�ßlÕ_Øh̼Í�á Ê�Û�Í�è�Ωã�Ø�Ê�Í�ʺßàÊnÓ�ÖµÌ�ÖµÏ�ÎlëPÚ_Î�Ï Ò¡á ßàÊn˧ã�Ù�ØjÌ�Ú_þ]Í�è_ιØjÎ�Ù�Ó³Î��¡Î�ßlÕ_Í�Ì�ÊnÚ Ê�Û�Í�è_ιã�Ø�Ê�Í�ʺßàÊnÓ5Í�ÊJÙ�Ų́Ï_Î�Ù�Ó�ãPØ�Ê¡ßàÎ�ÖjÖ�Ì�ÚLÞ�è�Ì�ßjè]Ù�Ú]Ù�Ï�Ï�Ì�Í�̼ÊnÚ�Ù�ÓÎ�Ú�Í�̼Ívá � Í�è_ΧÌ�Ï_Î�Ù�Ó.Û¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�áJ= � Ì�ÖrÌ�Ú�Í�Ø�ʺÏ�Õ�ßàÎ�Ï�ûDÎ�Ö�Ö�Î�Ú�Í�Ì�Ù�Ó¨Ó¼á � ='̨ÖÙ�Ú�̨Ú�ßàÊ�Ø�ØjÕ_ãPÍ�̼ÒPÓ¼Î�Í�ØjÕPÖ�Í�Î�Ï]ãPÙ�Ø�ÍváLÍ�è�Ù�Í�Ì�Ö�ã�Ø�Ê�þ�ØhÙ�˵˧Î�Ï]Í�Ê©ã�Ø�ʺÏ�Õ�ßàÎ2Í�è_ΧÏ_Î�Ö�̼ØjÎ�Ï÷Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá©Ê�ÛGÍ�è_Χþn̼Ô�Î�ÚÍ�Ù�Ö�ý�õPOAÎlÍ�Q1Ï_Î�Ú_Ê�Í�ίÍ�è_ίÙ�Ï_Ô�ÎlØjÖ�Ù�ØjáJÌ�Ú»Í�èPÌ�Ö2̨Ï_Î�Ù�Ó�Ì��lÎ�ÏJÎ��¡Î�ßlÕ_Í�Ì�ÊnÚ�õ{×�è_Î(ã�Ó�Ù�á�ÎlØjÖ�Ù�Ø�ίØ�ÎlãPÓ¨Ù�ßàÎ�ÏJÒ¡áJÏ�Õ�Ë«Ë3áã�Ó�Ù�á�ÎlØjÖ � Þ�è_ʯÏ_Ê©Ú_Ê�Í�ßàÊn˵Ë2ÕPÚ�Ì�ßlÙ�Í�ΧÞ�̼Í�è÷Î�Ù�ßjè»Ê�Í�è_ÎlØ�ûAÞ�è�Î�Ú_ÎlÔ�ÎlØ�Ù©Ï�Õ�˵Ë2á©ãPÓ�Ù9á�ÎlØ�Ì�ÖrÙ�ßàÍ�̼Ô�Ù�Í�Î�Ï � ̼Í�Ö�Ì�Ú�ãPÕ_Į́Ö5Û¦Ê�ØjÞ:Ù�ØjÏ�Î�ϹÍ�Ê<= Ò¡áRQ � Þ�è_Ê«ßlÙ�Ú{Ö�ÎlÎ3Í�è�ÎLé�ãPÕ_ÒPÓ¨Ì�ßrè_Î�Ù�Ï_ÎlØjì«Ê�Û.Í�è�Î2Ì�Ú_ã�Õ_ÍlõTS ö Ö�Ì�Ú©Í�è�Î�Ø�Î�Ù�Ó�ñ�Ó�Ì�Û¦Î�Î��ºÎ�ßlÕ_Í�̼ÊnÚ �Í�è�Î�ÊnÕ_Í�ã�Õ_ͳÊ�ÛAÍ�è_Îrã�Ø�Ê�Í�ʺßàÊnÓ-Î��¡Î�ßlÕ_Í�Ì�ÊnÚ©Ì�ÖGÍ�è�Î�ÊnÚ�Îàñ®ÒP̼Í:ÊnÕ_Í�ãPÕ_Í:Ê�ÛGJ«õ ��Ê�Þ Ù3ãPØ�Ê�Í�Ê¡ßàÊnÓ�K ���&#����8��!.1H�O�& �!.�VUN���:Ù�Ų́Ï_Î�Ù�Ó�Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á = ̼Û.Û¦Ê�Ø�Ù�Ú�á©Ø�Î�Ù�Ó¼ñ�Ó�̼Û�ÎrÙ�Ï�Ô�ÎlØjÖ�Ù�Ø�áRH Í�è_ÎlØ�Î3Î��ºÌ�Ö�Í�Ö�Ù�ÚL̨Ï_Î�Ù�Ó�ñ®Î��ºÎ�ßlÕ_Í�̼ÊnÚ¹Ù�Ï_Ô�ÎlØhÖ�Ù�Ø�áRQ#Ö�Õ�ßjèÍ�èPÙ�Í�Ú_Ê�Î�Ú�Ժ̼Ø�ÊnÚ�˧Î�Ú�ÍWJ � ÊnÚ±Ù�Ú�á§Ì�Ú�ãPÕ_Í � ßlÙ�Ú«Í�Î�Ó�Ó�Þ�Ì�Í�èµÚ_ÊnÚºñ�Ú_ÎlþnӨ̼þn̼ÒPӼγã�ØjÊ�ÒPÙ�ÒPÌ�Ó�Ì�Í�á3Þ�è_ÎlÍ�è_ÎlØG̼Í�Ì�ÖGÌ�Ú�Í�ÎlØjÙ�ßàÍ�Ì�Ú�þÞ�̼Í�è�H Ù�Ú�Ï ã�Ó�Ù�á�ÎlØjÖ(ØjÕ�ÚPÚ�Ì�Ú_þXK Ì�Ú­Í�è_ιØ�Î�Ù�Ó¼ñ�Ó�̼Û�ιÎ��¡Î�ßlÕ�Í�̼ÊnÚ � Ê�Ø(Þ�̼Í�è�Q Ù�Ú�Ï$=0Ì�ÚaÍ�è_Î{Ì�Ï_Î�Ù�Ó5Î��ºÎ�ßlÕ_Í�̼ÊnÚ�õ�LÊ�Ø�Î�ã�Ø�Î�ßlÌ�Ö�Î�Ó¼á � ̼ÛNÍ�è_Î�Í�Þ:ʱÒPÌ�Ú�Ù�Ø�á©Ï�̨Ö�Í�Øj̼ÒPÕ�Í�̼ÊnÚ¯Î�Ú�Ö�Î�Ë2ÒPÓ¼Î�Ö � ¶?¥ZYW[�\^] _�] ` Ù�ÚPÏ  ba:¥ZY�[�c�] d4] ` � Ï_Î�ÖjßàØj̼ÒPÌ�Ú�þMJI� ÖÊnÕ�Í�ãPÕ_Í>Ù�Û¦Í�ÎlØ�Ì�Ú�Í�ÎlØhÙ�ßàÍ�Ì�Ú_þrÞ�̼Í�è§Ù�Ï_Ô�ÎlØhÖ�Ù�Ø�áeH=Ù�Ú�Ï�ãPÓ�Ù9á�ÎlØjÖ>ØjÕPÚ�Ú�Ì�Ú_þ�ã�Ø�Ê�Í�ʺßàÊnÓ�K�Ü�Ø�Î�Ö�ãAõ � Ù�Ï�Ô�ÎlØjÖ�Ù�Ø�á,QJÙ�Ú�ϧÌ�Ï�Î�Ù�ÓÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá<=(Ý � Ù�Ø�Î3ßàÊn˵ãPÕ_Í�Ù�Í�̼ÊnÚ�Ù�Ó¨Ó¼á(Ì�ÚPÏ�Ì�Ö�Í�Ì�Ú�þnÕ�Ì�Ö�è�Ù�Ò�Ó¼Î3ܸÏ_Î�Ú_Ê�Í�Î�Ï ¶?¥ZYW[�\^] _�] `gfh  ba�¥GY�[=c�] d4] ` ÝhõQ��°ÅP�­W�jiP&STM�P�E*M��F[�GJ��R�GJE�\ ���a] z�P��\V�EN�FÃ|[µKLZ�GlkWUD[\GDE�K)]�SLV�KLV�UDV���R)SLGOP&�\[nmJ[�EN�[µK)�*E*MNGJS=R�V�­WG}R�]"GDUD[µÉ�U}E��N­ÍQ"GJS�V KLZNGDV�SLGJKL[\UxP�R�R��N­W]NKL[\V�ENRD��$VOÃaGD��GJSOz�[µK:ÃaP�RaENV�K}R�Z*VOÃ|EWKLZ�P&K}KLZ*GD[µSxXÄ�NE*UJKL[\V�E*P��\[µK��;UDV��*��M?Q"GY�*R�GOM?[\E;UDV�ENR0K�SL�*UJKL[\EN��]�SLV�KLV�UDV��\R:XÄV&S���GDENGJSTP��"R�GJUD�NSLGY­Í�N�\KL[�Vt]�P8S�K��UDV�­W]N�NKTP&KL[\V�EkzNP�E*MA­WV�SLGR�]"GJUD[µÉ�UOP&���µ��zNV�QN��[\��[\V��NRYK�STP&E*R0XÄGnSO�o �aZ*[\R�XÄGDP&KL�NSLGÅÃaP�RYP�MNMNGOMiKLV�KLZ*Gų ¬ X.STP�­WGnÃaV�SLÈ�[\E \µ»8�^] �

M

Page 5: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

�?Ø�Ê�Í�ʺßàÊnÓ�Ö�Í�á¡ãPÌ�ßlÙ�Ó�Ó�áLÌ�Ú�Ô�Ê�ý�Î�Ê�Í�è�ÎlØrÖjÕ_Ò_ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ölõ � ÚLÍ�èPÌ�Ö�Û�ØjÙ�˧ÎlÞ:Ê�Ø�ý÷Í�è_Î��k1 +�����) 'H%S)k��!DÌ�Ö�Ө̼ý�Î�Ù±Ø�Î�Ù�Ó¼ñӨ̼ۦÎrÎ��ºÎ�ßlÕ_Í�̼ÊnÚ � Î��_ßàÎlã�Í�Í�è�Ù�Í%Ö�Ên˧Î3̨Ú�Ô�ʺßlÙ�Í�̼ÊnÚ�Ö5Ê�ÛNÍ�è_Î�ÖjÕ_Ò_ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ö5Ù�Ø�Î3Ø�ÎlãPÓ�Ù�ßàÎ�Ï©Ò¡á±Í�è_Î�Ì�Ú�Ô�Ê¡ßlÙ�Í�̼ÊnÚ¹Ê�ÛNÙ�Ų́Ú�Ö�Í�Ù�Ú�ßàΧÊ�Û³Ù�Ú»Ì�Ï_Î�Ù�ÓNÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼ÍváN=©ûÍ�èPÌ�Ö�Ì�Ö�ßlÙ�Ó�Ó�Î�Ï]Í�è_ÎJé =�ñ�è�á¡Ò�Øj̨Ï�˧ʡÏ�Î�ÓVõ ì�OAÎlÍ������ c \^] _�] ` Ï_Î�Ú_Ê�Í�ΧÍ�è_ÎÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�̼ÊnÚ�Î�Ú�Ö�Î�Ë3ÒPÓ�Î�Ê�Û�ØhÙ�Ú�Ï_ÊnË=Ô�Ù�ØhÌ�Ù�ÒPÓ¼Î�Ö.Ï_Î�Ö�ßàØjÌ�ÒPÌ�Ú_þ%Í�è�Î:ÊnÕ�Í�ãPÕ_Í.Ê�Û�J � Ù�Û¦Í�ÎlØ?Ì�Ú�Í�ÎlØjÙ�ßàÍ�Ì�Ú�þ�Ì�Ú2Í�è_ÎP=�ñ�è�á¡Ò�Øj̨Ï˵ʡÏ_Î�ÓGÞ�̼Í�èJã�Ø�Ê�Í�ʺßàÊnÓ KNõ OAÎlÍ�K ÒDÎ(Ù÷ã�Ø�Ê�Í�ʺßàÊnÓ³Ì�Ú»Í�è�Î.=�ñ�è�á�Ò�ØhÌ�ϻ˧ʺÏ_Î�Ó � Ù�Ú�Ï��ÙLã�ØjÊ�Í�Ê¡ßàÊnÓ:Í�è�Ù�Í�Ö�Î�ßlÕ_Ø�Î�ÖØjÎ�Ù�Ó�Ì��lÎ�ÖP=©õG×%è_Î�ßàÊn˧ãDÊnÖ�Î�ϯãPØ�Ê�Í�Ê¡ßàÊnÓZK��Ì�Ö5Ú_Ê9Þ ßàÊnÚPÖ�Í�ØjÕ�ßàÍ�Î�Ï©Ò¡á±Ø�ÎlãPÓ�Ù�ßlÌ�Ú�þ2Í�è_Î�ëPØjÖ�Í5˧Î�Ö�Ö�Ù�þ�Î�Í�Ê<= Ì¨Ú K]Ò¡áÙ�ÚÑÌ�Ú�Ô�Ê¡ßlÙ�Í�̼ÊnÚJÊ�Û5ÙLÚ_ÎlÞ�ßàÊ�ã¡á�Ê�Û�� � Þ�̼Í�è�Û�Ø�Î�Ö�è»ØjÙ�Ú�Ï_ÊnË Ì¨Ú_ãPÕ_Í � Í�è�ΫÖ�Ù�˧Π����) � Ù�ÚPÏ»Þ�̼Í�è]Í�è�Î(ßàÊnÚ�Í�Î�Ú�Í�Ö2Ê�ÛÍ�èPÙ�Ír˵Î�Ö�Ö�Ù�þ�ΫÙ�Ö�Ì�Ú�ãPÕ_ÍlûDÎ�Ù�ßjè»ÖjÕ_ÒPÖ�ÎS!¡Õ_Î�Ú�Í�˧Î�ÖjÖ�Ù�þ�ΧÍ�Ê©Í�è�Ù�Í�ßàÊ�ã¡á÷Ê�Û =�Ì�Ö�Ø�Îlã�Ó�Ù�ßàÎ�Ï{Þ�̼Í�è]Ù�Ú�Ù�ßàÍ�̼Ô�Ù�Í�̼ÊnÚ]Ê�ÛÍ�è�Î�ßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�Ï�̨Ú_þ§ßàÊ�ã�á¯Ê�Û � � Þ�Ì�Í�è¯Í�è_Î3ßàÊnÚ�Í�Î�Ú�Í�Ö�Ê�ÛÍ�èPÙ�Í�˧Î�Ö�Ö�Ù�þ�Î�Ù�Ö%Ú_ÎlÞ Ì¨Ú_ãPÕ_ͳÍ�Ê���õ

×%è_Î42�É#ßàÊn˧ãDÊnÖ�̼Í�̼ÊnÚ¯Í�è�ÎlÊ�Ø�Î�Ë ÒPÙ�Ö�̨ßlÙ�Ó�Ó¼á(ÖjÙ�áºÖ%Í�è�Ù�Í%Ì¼Û �(Ö�Î�ßlÕ_Ø�Î�Ø�Î�Ù�Ó�Ì �lÎ�Ö�= Ì�Ú¯Í�è_Î���ñ�è�á�Ò�ØhÌ�ϯ˧ʡÏ�Î�Ó � Û¦Ê�ØÖ�Ên˧ÎrÛ�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá�� � Í�è�Î�Ú±Ù�Ú±Î��ºÎ�ßlÕ_Í�̼ÊnÚ¯Ê�ÛÍ�è_Î�ßàÊn˵ã-ÊnÖ�Î�Ï(ãPØ�Ê�Í�Ê¡ßàÊnÓ�K� � ØjÕPÚ�Ú�Ì�Ú_þ2Ì�Ú«Í�è�Î���ñ�è�á�ÒPØjÌ�ϫ˵ʡÏ_Î�Ó �é�Î�Ë�Õ�Ó�Ù�Í�Î�ÖjìLÙ�Ú�Î��ºÎ�ßlÕ_Í�̼ÊnÚ�Ê�Û³ã�ØjÊ�Í�Ê¡ßàÊnÓLK­Ì�Ú]Í�è_Î =�ñ�è�á¡Ò�Øj̨Ï�˧ʡÏ�Î�ÓVõµ×%è�Ù�Í�Ì�Ö � Ú_Ê©Î�Ú�ԡ̼ØjÊnÚ�˧Î�Ú�Í�˵Ù�ßjè�̨Ú_Î JßlÙ�Ú(Ï�Ì�Ö�Í�̨Ú_þnÕ�Ì�Ö�è�Þ�è�ÎlÍ�è_ÎlØG̼ÍG̨Ö?Ì�Ú�Í�ÎlØjÙ�ßàÍ�Ì�Ú_þ�Þ�̼Í�è H7Ù�Ú�ÏMK��̨ڧÍ�è_Î���ñ�è�á�Ò�ØhÌ�ϵ˧ʺÏ_Î�Ó � Ê�Ø:̼ÍGÌ�ÖGÌ�Ú�Í�ÎlØhÙ�ßàÍ�Ì�Ú_þ�Þ�Ì�Í�èQ+Ù�Ú�Ï K�Ì�Ú¯Í�è�Î =�ñ�è�á¡Ò�ØjÌ�ϩ˵ʡÏ_Î�Ó®õ � Ú©Ê�Í�è_ÎlØ�Þ�Ê�ØjÏPÖ � �������\�� ] _L] ` fh ����� c \^] d4] ` õ��ÎLÙ�Ø�ιÏ_Î�Ö�Ì�þnÚ�Ì�Ú_þ{Ù�Ú�ÏaÙ�Ú�Ù�Ó¼á=��Ì�Ú_þ]ã�Ø�Ê�Í�ʺßàÊnÓ�Ö§Ì�Ú Í�è�ιÉW>A@ ˧ʺÏ_Î�Ó � Ù�ÚPÏaÖ�Ê]Í�è_ÎláÑÞ�Ì�Ó�Ó�Ò-ίÊ�ãDÎlØjÙ�Í�Ì�Ú�þ]̨Ú

Í�è�Î =��?���� ñ�è�á�Ò�ØhÌ�Ï]˧ʡÏ�Î�Ó � Þ�è_ÎlØjÎ =��?���� Ì�Ö�Í�è_Χ۸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼ÍváLÍ�è�Ù�Í3ßhè_Ê¡ÊnÖ�Î�Ö3Ù©Ö�Í�ØhÌ�Ú_þ¯Û¦ØjÊnË Ï�Ì�Ö�Í�Øh̼ÒPÕ_Í�̼ÊnÚ��� Ù�ÚPÏ»è�Ù�Ú�Ï�Ö3̼Í�Í�Ê÷Ù�Ó�ÓNã�Ù�Ø�Í�̼Î�Ölõ7��Õ_Ø�Í�è_ÎlØ � Þ:εÞ�Ì�Ó�ÓNßàÊnÚ�ÖjÌ�Ï_ÎlØ�Í�è_ÎÑé�Ë2Õ�Ó�Í�Ì�ñ�Ö�Î�Ö�Ö�Ì�ÊnÚ�Î��¡Í�Î�ÚPÖ�̼ÊnÚ�Ê�ÛP=µì¹Ê�Û�ɳÙ�Ú_ÎlÍ�Í�ÌÙ�ÚPÏ >�Ù�ÒPÌ�Ú#ä�åS��ç � Ï_Î�Ú�Ê�Í�Î�Ï"!= � Þ�è�Ì�ßjè ØjÕPÚ�Ö2Ë�Õ�Ó¼Í�Ì�ãPӼΫßàÊ�ãPÌ�Î�Ö2Ê�Ûc= Ò¡áÑÌ�Ï�Î�Ú�Í�̼Û�áºÌ�Ú_þ¹Î�Ù�ßhè ßàÊ�ã�áÑÒ�áÑÙ{Ö�ã-Î�ßl̨Ù�Ó���x+ �n����������%�� � )k���x/0� #Í���yõ

×%è_Î�Ï_ÎlëPÚP̼Í�̼ÊnÚ¯Ê�Û !=�$%'& � Í�è_Î3Ë�Õ�Ó¼Í�Ì�ñ�Ö�Î�ÖjÖ�̼ÊnÚ¯Î��ºÍ�Î�Ú�Ö�̼ÊnÚLÊ�Û0=�$%(& � Ì�Ö�Ö�è_Ê9Þ�Ú©ÒDÎ�Ó¼Ê9Þ�õP��Ê�Í�Î2Í�è_Î�ÍvÞ�Ê(Í�á¡ã-Î�Ö�Ê�Ų̂Ú�Ï�Ì�ßàÎ�Ö Í�è_Îr����) � Þ�è�Ì�ßhè«Ï�Ì�DÎlØ�Î�Ú�Í�Ì�Ù�Í�Î�ֳ˧Î�Ö�Ö�Ù�þ�Î�Ö:Í�Ê !=�$%'& Û¦ØjÊnË Ë§Î�ÖjÖ�Ù�þ�Î�Ö5Ö�Î�Ú�Í�Í�Ê�Ê�Í�è_ÎlسÛ�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�̼Î�Ö � Ù�Ú�Ï������) � Í�è_Î2Ö�Õ_Ò_ñ�Ö�Î�ÖjÖ�̼ÊnÚ¯Ì�Ï_Î�Ú�Í�̼ë�ÎlØ � Þ�èPÌ�ßjè©Ì�Ö�Õ�Ú�Ì !¡Õ_Î�ã-ÎlØ%̨Ú_ãPÕ_Í5˧Î�Ö�ÖjÙ�þ�Î(Ü�Ê�Ø�ã�ØjÊ�Ê�ÛhÝhõ

) -.-Z2:7 /���- �T5 /87 9 !=�$%'&!= $%(& ã�ØjÊ¡ßàÎlÎ�Ï�Ö³Ù�Ö³Û�ÊnÓ�Ó¼Ê�Þ�Ö � ØjÕ�Ú�ÚPÌ�Ú_þ�Þ�Ì�Í�è(Ö�Î�ßlÕ�Øj̼Í�á«ãPÙ�ØjÙ�˧ÎlÍ�ÎlØi� � ãPÙ�ØjÍ�̼Î�ÖA@ ïCBEDEDED B @ZF � Ù�ÚPÏ©Ù�Ú±Ù�Ï�Ô�ÎlØ�ñÖ�Ù�Ø�á Q�* 2�ãDÊnÚ Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ Ü,+.-0/21.354�6.7.3 B98':<;=B98(8':<;�B @>= B @�? BA@ZBAB Ý'Û�Ø�ÊnË @C= � ÛEDµÜ @ZBAB Ý Í�è_Î�Ú Ö�Î�Ú�ÏÜGF�HI/9J KILML�N B98O:P;�B98�8O:<;=B @C= B @�? BA@ Ý:Í�ÊM@�?�Ù�Ú�Ï QYÙ�Ú�ϹèPÙ�Ó¼Ílõ�Ð�Í�è_ÎlØ�Þ�Ì�Ö�Î � ̼þnÚ_Ê�ØjÎ�õ

>%ÎlÛ�ÎlØ%Í�Ê{ä�å�3 � å�ò�ç�Û¦Ê�Ø�Û�Õ_ØjÍ�è_ÎlØ%Ï_Î�Ö�ßàØjÌ�ã�Í�̼ÊnÚ±Ê�ÛNÍ�è_Î42rÉ�Û¦ØhÙ�˧ÎlÞ�Ê�Øjý�õ

Q�R<S T�UWVYX[ZI\.Z�]^ZC_P`Ð�Õ_Ø5ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ì�ÊnÚ�Ö:Þ�Ì�Ó�Ó�Õ�Ö�Î�Ù§Ö�ãDÎ�ßlÌ�Ù�Ó�Í�á¡ã-ÎrÊ�ÛF�lÎlØ�Ê�ñ®ý¡Ú�Ê9Þ�Ó¼Î�Ï�þ�Î�ã�Ø�Ê�Í�ʺßàÊnÓ�Ö � Ú�Ù�˧Î�Ó�á ��a � (}�O%"/L%�#�%�!Ç��äóò:G�ç � Þ�è�Ì�ßhèÙ�ØjÎ�Ô�Ù�ØhÌ�Ù�Ú�Í�Ö%Ê�ÛÍ�è_Î2Ö�Ê�ñ�ßlÙ�Ó�Ó¼Î�Ï�b:ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ö3ä�å � � å Mnç®õ��Î3Ø�ÎlԺ̼ÎlÞ Ö�Ên˧Î3Ê�Û.Í�è�Î�Ï_ÎlëPÚ�̼Í�Ì�ÊnÚ�Ö³Ù�Ú�Ï©ã�Ø�Ê�ãDÎlØ�Í�Ì�Î�Ö³Ê�ÛÍ�è_Î�Ö�Î�ã�Ø�Ê�Í�ʺßàÊnÓ�ÖlõOAÎlÍ�Ddcde�Ü @ZBAB Ý9f©Ò-Î(Ù¹ÒPÌ�ÚPÙ�Ø�á÷ØjÎ�Ó�Ù�Í�̼ÊnÚJÙ�ÚPÏÑÙ�Ö�ÖjÕ�˧εÍ�è�Ù�Í�Û�Ê�Ø2Ö�Ên˧Î(þn̼Ô�Î�ÚÑã-ÊnÓ�á¡Ú_ÊnË«Ì�Ù�Óhg.Üji�Ý3̼Í3è_ÊnÓ�ÏPÖ

Í�èPÙ�Í�k B k^lmg.Ü2k @ kóÝGÛ¦Ê�Ø5Ù�Ó�ÓNÜ @ZBAB ÝA�nD�õ���Õ_ØjÍ�è_ÎlØj˧Ê�Ø�Î � Ó¼ÎlÍ�D ÒDÎ�Í�Î�Ö�Í�Ù�ÒPÓ¼ÎrÌ�Ú(ãDÊnÓ¼á¡Ú�Ên˵Ì�Ù�Ó�Í�̨˧Î�õ�OAÎlÍpo $ cqe @ Ü @GBAB Ý �nDrf�ÒDÎ�Í�è_Îr!� ��k< �� �<k�3Ï_ÎlëPÚ_Î�Ï«Ò�á«Í�è_Î�ØjÎ�Ó�Ù�Í�̼ÊnÚ � Ù�Ú�ϱۦÊ�سÙ�Ó�Ó @ �so $ � Ó¼ÎlÍut $ Ü @ Ý�cqe B �Ü @GBAB Ý �sDrfÒDαÍ�è_ÎmvÅ��/0�a���&�7����/3Û�Ê�Ø @ õ ��Ê�Ø«Ù�Ú�á �E�1Ó¨Ù�Ú_þnÕ�Ù�þ�Îno � Ú_Ê�Í�ίÍ�è�Ù�Í�Í�è_ÎlØ�ΩÌ�Ö�Ù]Ú�Ù�Í�Õ_ØjÙ�ÓwvÅ�0/0�:�����7�8��!� "/0��%��xDßàÊnÚ�Í�Ù�Ì�Ú�̨Ú_þ§ãPÙ�̼ØjÖ�Ü @GBAB Ý5Þ�è_ÎlØ�Î B Ì�Ö5Í�è_Î�Þ�Ì�Í�Ú_Î�Ö�Ö5Û�Ê�Ø%Í�è_Î�˧Î�Ë2ÒDÎlØjÖ�è�Ì�ã±Ê�Û @ Ì�Úso � Ù�Ú�ϹÍ�è�Ù�ÍMo $ cyo�õ

ö b:ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ü,z B2{ Ý%Ì�Ö%Ù«Í�è_Ø�ÎlÎ3˧Ê�Ô�Î2Ì�Ú�Í�ÎlØjÙ�ßàÍ�̼Ô�Î2ã�Ø�Ê�Í�ʺßàÊnÓAÒDÎlÍ�Þ:ÎlÎ�Ú{Ùµã�Ø�Ê�Ò�Ù�ÒPÌ�Ó�Ì�Ö�Í�Ì�ß%ãDÊnÓ¼áºÚ_Ên˵Ì�Ù�Ó¼ñ®Í�Ì�˧ÎãPØ�Ê9Ô�ÎlØrz*Ù�ÚPϻ٩ã�Ø�Ê�Ò�Ù�ÒPÌ�Ó�Ì�Ö�Í�Ì�ß�ãDÊnÓ¼áºÚ_Ên˵Ì�Ù�Ó¼ñ®Í�Ì�˧Î�Ô�ÎlØj̼ë�ÎlØ { � Þ�è_ÎlØ�ΧÍ�è_Ϋã�Ø�Ê9Ô�ÎlØ�Ù�ßàÍ�Ö3ëPØjÖ�Ílõ«×%è�ΧÔ�ÎlØj̼ë�ÎlØ3̨ÖÊnÚPÓ¼á÷ØjÎS!�Õ�Ì�Ø�Î�Ï]Í�Ê÷Ö�Î�ÚPÏ�ØjÙ�Ú�Ï_ÊnË ÒP̼Í�Ö�Ù�Ö2Ù¹ßjè�Ù�Ó¨Ó¼Î�Ú_þ�εÍ�Ê÷Í�è_εã�Ø�Ê�Ô�ÎlØ�õw�_Ê�Ø2Ö�Ên˵Î{Ü @ZBAB Ý��|D � Í�è�ΫßàÊn˵˧ÊnŲ́Ú_ãPÕ_Í:Í�ʧÒDÊ�Í�è¯ãPÓ�Ù�á�ÎlØhÖ%Ì�Ö @ Þ�èPÌ�ӼΠB Ì�Ö:ã�ØjÌ�Ô�Ù�Í�Î�Ì�Ú_ã�Õ_Í�Í�ʧÍ�è�Î�ã�Ø�Ê9Ô�ÎlØ9õ��_Ê�Ø�ÖjÕ�ßjè¯þn̼Ô�Î�Ú @ � Ó¼ÎlÍ�ܸø B2} B N¡Ý5Ï_Î�Ú_Ê�Í�ÎÍ�è�Î(ßàÊnÚ�Ô�ÎlØjÖ�Ù�Í�̼ÊnÚ ÒDÎlÍ�Þ:ÎlÎ�Ú Í�è_Ϋã�ØjÊ9Ô�ÎlØ�Ù�Ú�Ï»Í�è_Î(Ô�ÎlØj̼ë�ÎlØ9õ¹×�Ê{ßàÊn˧ãPÕ_Í�Î(Í�è_Î(ëPØjÖ�Í3Ù�Ú�ÏJëPÚ�Ù�Ó�˧Î�Ö�Ö�Ù�þ�Î�Ö � Í�è_ÎãPØ�Ê9Ô�ÎlØ�Ì�Ú�Ô�Ê�ý�Î�ÖrÎ��«ßl̼Î�Ú�Í�Ù�Ó�þ�Ê�Øj̼Í�è�˵Ö�øDÜji�Ý%Ù�Ú�ÏPNPÜji�Ý � Ø�Î�Ö�ã-Î�ßàÍ�Ì�Ô�Î�Ó¼á � Õ�Ö�Ì�Ú_þ{Ü @ZBAB Ý�Ù�Ú�Ï{ØjÙ�ÚPÏ_ÊnË Ò�̼Í�Ö�Ù�ÖrÌ�Ú_ã�Õ_Ílõ2�Ö�Ì�Ú�þ�Ù�Ú�Î��«ßlÌ�Î�Ú�ÍNã�Ø�Î�ÏPÌ�ßlÙ�Í�ÎM~Üji�Ý � Í�è_γÔ�ÎlØj̼ë�ÎlØ>Ï_Î�ßl̨Ï_Î�Ö.Þ�è_ÎlÍ�è_ÎlØ>Í�è_Î5ßàÊnÚ�Ô�ÎlØjÖ�Ù�Í�̼ÊnÚ«Ì�ÖNÙ�ßlßàÎlã�Í�Ì�Ú_þ�Þ�Ì�Í�è2ØjÎ�Ö�ãDÎ�ßàÍ

G

Page 6: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Í�Ê @ õ¯×%è_εØjÎ�Ó�Ù�Í�̼ÊnÚ D � Í�è_ΫÙ�Ó¼þ�Ê�Øh̼Í�è�˵Ö2ø-Üji�Ý � NPÜji�Ý3Ù�Ú�Ï~.Üji�Ý�Ù�ØjεãPÕ_ÒPÓ¨Ì�ß�õ«×%è_Î(Ó¼Î�Ú_þ�Í�è»Ê�Û³Í�è_Î(ßjè�Ù�Ó�Ó�Î�Ú_þ�Î�Ö2̨ÖÏ�Î�Ú_Ê�Í�Î�Ï ��� � Ù�Ú�Ï©Þ:Î3Ù�Ö�Ö�ÕP˧Î�Í�è�Ù�Í ��� ÊnÚ�Ó¼á±Ï_ÎlãDÎ�Ú�Ï�Ö³ÊnÚ©Í�è�Î3Ó¼Î�Ú_þ�Í�è¯Ê�ÛNÍ�è_Î3ßàÊn˫˧ÊnÚ¹Ö�Í�Øj̨Ú_þ @ õ��Î2Þ�̨Ó�Ó�Ú_ÎlÎ�ϹÍ�ʱÒ�Ø�ÊnÙ�Ï�Î�Ú¹Í�è�Ì�Ö�Ï_ÎlëPÚ�̼Í�Ì�ÊnÚ©Ö�Ó�̼þnè�Í�Ó¼á � Í�ʯÏ_Î�Ù�ÓAÞ�̼Í�è¹ßjè�Î�Ù�Í�Ì�Ú_þ±ã�Ø�Ê9Ô�ÎlØhÖlõ���Î2Þ�̨Ó�Ó�Ï_ÎlëPÚ_Î|!o $Í�Ê(ÒDÎ�Í�è_Î3Ì�Ú�ãPÕ_Í%Ó�Ù�Ú_þnÕPÙ�þ�Î � Þ�̼Í�è¹Í�è_Î3ã�ØjÊ�ã-ÎlØjÍ�á¯Í�è�Ù�Í�o $�� !o $ � Ù�ÚPÏL˧Î�Ë2ÒDÎlØjÖ�è�̼ã¹Ì¨Ú !o $ ˵Ù9á©Ò-Î3Í�Î�Ö�Í�Î�Ï÷̨ÚãDÊnÓ¼áºÚ_Ên˵̨Ù�Ó�Í�̨˧Î�õI��Î2Ì�˵ãPÓ�Ì�ßl̼Í�Ó�á�Ù�ÖjÖ�Õ�˧Π{ ÊnÚ�Ó¼á±Î��ºÎ�ßlÕ_Í�Î�Ö%Í�è_Î3ã�Ø�Ê�Í�ʺßàÊnÓA̼Û�Í�è�Î3ßàÊn˵˧ÊnÚ÷Ì�Ú_ãPÕ_Í @ � !o $ õö Ó�ÓCb:ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ö5ã�Ø�Î�Ö�Î�Ú�Í�Î�Ϲè�ÎlØ�Î�Þ�Ì�Ó¨Ó�Ö�Ù�Í�̨Ö�Û¦á±Í�è_Î�Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ§Ö�Î�ßlÕ�Øj̼Í�á±ã�Ø�Ê�ãDÎlØ�Í�Ì�Î�Ö *�� �8 � �L(��8#��� �!$��%"�=�D)��:����� !O�ÎlÍ�ܸø B2} B N�Ý�Ù�Ú�Ï Ü¸ø B2}�8B N � Ý%Ò-Î2Í�Þ:ʱßàÊnÚ�Ô�ÎlØjÖ�Ù�Í�Ì�ÊnÚ�Ö � Í�è�Ù�ÍrÙ�Ø�Î�Ù�ßlßàÎlã�Í�Ì�Ú_þ±Û¦Ê�ØÖ�Ên˧εþnÌ�Ô�Î�Ú @ � !o $ õ � Û }�c } � � Í�è�Î�Ú @ �|o $ õ«×�è_εãPÙ�̼Ø�Ê�Û5Ù�ßlßàÎlã�Í�̨Ú_þ÷ßàÊnÚ�Ô�ÎlØjÖ�Ù�Í�̼ÊnÚPÖ(ܸø B2} B N¡Ý�Ù�Ú�Ïܸø B2} � B N � Ý�Þ�Ì�Í�è }�c } � ̨Ö5ßlÙ�Ó�Ó¼Î�Ϲ٠#�%�!�!Ç�-����%��Lõ*� ()�&#��� �!��=%��a����/Y������� #Í����UN���O% � �}%.vW! � )�<=�5ÜL@@%�ê O`K�Ý �×�è_ÎlØ�Î�̨Ö.Ù�Ü�ã�Ø�Ê�Ò�Ù�ÒPÌ�Ó�Ì�Ö�Í�Ì�ß>ãDÊnÓ¼áºÚ_Ên˵Ì�Ù�Ó�Í�̨˧Î�ÝAÖjÌ�Ë2Õ_ñÓ�Ù�Í�Ê�Ø���Í�è�Ù�ÍNÊnÚ�Ì�Ú_ãPÕ�Í @ �no $ þ�Î�Ú_ÎlØjÙ�Í�Î�Ö?Ù�ßlßàÎlãPÍ�Ì�Ú_þrßàÊnÚ�Ô�ÎlØjÖjÙ�Í�̼ÊnÚ�ÖNÞ�̼Í�è2Í�è_Î:Î��ºÙ�ßàÍ?ÖjÙ�˧Î:ÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�̼ÊnÚÙ�Ö�Þ�è_Î�Únz1Ù�Ú�Ï { Î��¡Î�ßlÕ_Í�Î�Í�è_Î2ã�Ø�Ê�Í�ʺßàÊnÓAÊnÚ{ßàÊn˵˧ÊnÚL̨Ú_ãPÕ_Í @ ܸÙ�Ú�Ï z=̨Ö%þn̼Ô�Î�Ú÷ÙµÞ�̼Í�Ú�Î�Ö�Ö B Û�Ê�Ø @ Ý �Ù�Ú�Ï { Ì�Ú�Ï_ÎlÎ�Ïaè_ÊnÚ�Î�Ö�Í�Ó¼áJßhè_Ê�ÊnÖ�Î�̼֫Í�Ö§ßhè�Ù�Ó�Ó¼Î�Ú_þ�Î�Ö«Õ�Ú�̼Û�Ê�Øj˵Ӽá�Ù�͵ØhÙ�Ú�Ï_ÊnË©õ ×%è_ΩÖ�Ì�Ë�Õ�Ó�Ù�Í�Ê�اÌ�Ö§Ö�ã-Î�ßl̨Ù�ÓÌ�Ú÷Í�è_Î�Ö�Î�Ú�Ö�Î�Í�è�Ù�Í�̼Í�ßlÙ�Ú]Ù�Ï�ÏP̼Í�̼ÊnÚ�Ù�Ó�Ó�á(Í�Ù�ý�εٱØjÙ�Ú�Ï�ÊnË Ö�Í�ØhÌ�Ú_þ } Ù�Ö�Ì�Ú_ãPÕ�Í � Ù�Ú�Ï÷ÊnÕ_Í�ãPÕ_ÍrÙ�Ú]Ù�ßlßàÎlã�Í�Ì�Ú_þßàÊnÚ�Ô�ÎlØhÖ�Ù�Í�̼ÊnÚ±Û¦Ê�Ø @ Þ�è_ÎlØ�Î } Ì�ÖGÍ�è_Î�ßhè�Ù�Ó�Ó¼Î�Ú_þ�Î�õ � Ú§Û¸Ù�ßàÍ � Þ:Î�Þ�Ì�Ó�Ó_Ù�Ö�Ö�ÕP˧Î%Í�è_Î�ÖjÌ�Ë2ÕPÓ�Ù�Í�Ê�Ø�è�Ù�ÖGÍ�è�Ì�Ö?Ö�ã-Î�ßl̨Ù�Óã�Ø�Ê�ãDÎlØ�Ívá(Û�Ê�Ø�Ú_Ê�Í%ÊnÚPÓ¼á @ �no $ � ÒPÕ_Í%Ù�Ó�Ö�ʵÙ�Ú�á @ � !o $ õ

ö Ú a ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓ�Ü,z B2{ Ý�� ���5Û¦Ê�اÙ{Ø�Î�Ó�Ù�Í�̼ÊnÚ D c e�Ü @ZBAB Ý9f{Ù�Ú�Ï É?>A@�� � Ì�Ö�Ù b�ñ®ã�Ø�Ê�Í�ʺßàÊnÓ:Û¦Ê�Ø�Ø�Î�Ó¨Ù�Í�̼ÊnÚ DÎ��_ßàÎlã�Í%Û�Ê�Ø%Í�è_Î3Û¦ÊnÓ�Ó�Ê9Þ�Ì�Ú�þ_õå�õ��_Ê�Ø3Ù±þn̼Ô�Î�Ú»ÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�̼ÊnÚ©Î�Ú�Ö�Î�Ë3ÒPÓ�Îr� � Ù©ßàÊn˵˧ÊnÚ]Ø�ÎlÛ�ÎlØ�Î�Ú�ßàεÖ�Í�ØjÌ�Ú_þ�� Ì�ÖrÏ_ØjÙ�Þ�Ú÷Û�Ø�ÊnË �� (Ù�Ú�Ï{Î�Ù�ßjèÛ�ÕPÚ�ßàÍ�̼ÊnÚ©øDÜji�Ý � NPÜji�Ý � Ù�Ú�Ï ~.Üji�ݳÍ�Ù�ý�Î�Ö���Ù�Ö�Ù�Ú¹Ù�Ï�ÏP̼Í�̼ÊnÚ�Ù�ÓDÌ�Ú_ã�Õ_Ílõ

ò¡õ � Í�̨ÖH#&%�'i(}�)/L "/0��%��} �!i@ %�ê"O`K � ˧Î�Ù�ÚPÌ�Ú_þ¯Í�è�Ù�Í�Í�è_εÖ�̨Ë2Õ�Ó¨Ù�Í�Ê�Ø�ÊnÚ�Ì�Ú_ãPÕ_Í @ � o $ þ�Î�Ú_ÎlØjÙ�Í�Î�Ö3Ù©ßàÊnÚ�Ô�ÎlØ�ñÖ�Ù�Í�̼ÊnÚ­Í�è�Ù�Í(Ì�ÖµÊnÚ�Ó¼á ßàÊn˧ãPÕ_Í�Ù�Í�̼ÊnÚPÙ�Ó�Ó¼á Ì�Ú�Ï�Ì�Ö�Í�Ì�Ú_þnÕ�Ì�Öjè�Ù�ÒPӼεÛ�Ø�ÊnË<ÊnÚ_ιþ�Î�Ú_ÎlØjÙ�Í�Î�Ï+Ì�Ú ÙJØ�Î�Ù�Ó5Î��ºÎ�ßlÕ_Í�̼ÊnÚÒDÎlÍ�Þ:ÎlÎ�Ú z*Ù�Ú�Ï { ÊnÚ»ßàÊn˵˧Ênڻ̨Ú_ãPÕ_Í @�� õ @¡ãDÎ�ßl̼ëPßlÙ�Ó¨Ó¼á � Í�è_ÎlØ�ΫÌ�Ö�Ù©Ú�ÎlþnÓ�̼þn̼ÒPÓ�Î2Û¸Õ�Ú�ßàÍ�̼ÊnÚ��:Ü���Ý�Ù�Ú�Ï�ÙãDÊnÓ¼á¡Ú�Ên˵Ì�Ù�Ó�ñ®Í�Ì�˵Î�Ö�Ì�Ë�Õ�Ó�Ù�Í�Ê�Ø Q­Ö�ÕPßjè¯Í�è�Ù�Í5Û�Ê�Ø5Ù�Ó�ÓDÚ_ÊnÚºñ�Õ�Ú�Ì�Û¦Ê�ØjË ã�Ø�Ê�Ò�Ù�ÒPÌ�Ó�Ì�Ö�Í�Ì�ß³ãDÊnÓ¼á¡Ú�Ên˵Ì�Ù�Ó�ñ®Í�Ì�˵Î�Ù�Ï_Ô�ÎlØ�ñÖ�Ù�Øj̼Î�Ö H c Ü H ïEB H î Ý � Þ:Î(è�Ù9Ô�αÍ�èPÙ�Í k �?Ø�ä��! ^1#" _ Ü���Ý c åyç%$$�?Ø�ä��! ^1#" d_ Ü���Ý c åyçjk�l&��Ü���Ý � Þ�è_ÎlØ�εÍ�è_ÎÎ��¡ãDÎlØj̨˧Î�Ú�Í�Ö'�( ^1#" _ Ü���ݳÙ�Ú�Ï)�( ^1#" d_ Ü���ݳÙ�Ø�Î3Ï�ÎlëPÚ_Î�Ï©Ù�Ö5Û�ÊnÓ�Ó¼Ê�Þ�Ö

*,+.-/10 �32��54 *,+.-�/760 �32n�849;:<>=@? 9;:<A=@?�CBED1F@DHGà� <JILK � 9 � �MB�D1FNDOGà� <PILK � 9 � ¸�>�MB�D1F��RQSUT �vwjtv�n���WV  ¸�>�MBEDXF��YQSUT ��w�t��n���WVZ :<\[ V!D^]#_a` b :<&[ V!D^]#_ ?c < c �MB�D1FND Z D 9 � ¶>wjtv�n��� INd �eG#D1f�� 9 DHbj���b :<\[ V!D�]�_ ?¶>wjtv�n��� I d �3G,Dh� c DHbaDXg¡�MB�D1FND Z DXbaD 9 ���v�

�¡õ�×%è_ÎlØ�Î�Î��_Ì�Ö�Í�ÖrÙ«ãDÊnÓ¼áºÚ_Ên˵̨Ù�Ó�ñ®Í�Ì�˧Î�Î��ºÍ�ØjÙ�ßàÍ�Ê�Øih c Üeh ïEB h î Ý�Ö�Õ�ßjè{Í�è�Ù�ÍrÍ�è_Î�Ø�ÎlÛ�ÎlØ�Î�Ú�ßàΧÖ�Í�Øj̨Ú_þ«ÊnÕ_Í�ã�Õ_Í�Ò¡áh ï Ü�å Ý�Ì�Ö§Ö�Í�Ù�Í�̨Ö�Í�Ì�ßlÙ�Ó�Ó¼á Ì�Ú�ÏPÌ�Ö�Í�Ì�Ú_þnÕPÌ�Ö�è�Ù�ÒPÓ�Î�Û�Ø�ÊnË �� �õ ��Õ_Ø�Í�è�ÎlØj˧Ê�Ø�Î � þn̼Ô�Î�Ú Üj� Blk Ý $ h ï Ü�å Ý � Ì�Û�Í�è_ÎlØ�ÎÎ��ºÌ¨Ö�Í�Ö:Í�Þ:Ê�Ù�ßlßàÎlãPÍ�Ì�Ú_þ�ßàÊnÚ�Ô�ÎlØjÖjÙ�Í�̼ÊnÚ�Ö�ܸø B2} B N¡Ý:Ù�Ú�Ï�ܸø B2} � B N � Ý>Þ�̼Í�è }Uc } � Û¦Ê�سÖ�Ên˧Îrþn̼Ô�Î�Ú @ � !o $ � Í�è_Î�Úh î Ü @GBlk B ܸø B2} B N�Ý�Ý�ÊnÕ_Í�ã�Õ_Í�Ö B ÖjÕ�ßjè©Í�èPÙ�Í2Ü @ZBAB Ý �nD�õnm

o �$V&KL[\UDGKLZ�P8KYKLZNGR�[\­Í�N�ÇP8KLV�SFMNV�GJRYENV�K|��GDE*GJSTP8KLGÅKLZ*GUDV�­W­WV�E;SLGJXÄGnSLGDE*UDG�R0K�SL[\E*���p �$V&KL[\UDGiKLZ*P&KÍKLZN[\RWMNGJÉ*E*[µKL[\V�EHV�E GJI�K�STP�UnKLV�SW[\R?R�[\­W[\��P&S?KLV4KLZ�P8KWV�XYÃaGOP&È6R�]"GDUD[�P��|R�V��*E�M�E*GDR�R V�X8qTV-]�SLV�KLV�UDV��\RD� �$PO��[\EN�4K-ÃaVP�UDUDGJ]NKL[\E*� UDV�E���GnSLRLP&KL[\V�ENR?R�Z�P8SL[�EN� KLZNGiRLP&­WGNr5���*P&STP�E�KLGDGJRÍKLZ�P8KWP Ã|[\KLENGDR�RÍÃ|[\�\�|Q"GiGJI�K�STP&UJKLGOM,z}P&�\KLZNV��N��Z6KLZ*G�GJI�K�STP�UJKLV&SWV�EN�µ�E*GDGOM�R�V�ENGÅUDV�E���GJSLRLP8KL[\V�Ek�

Page 7: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

� Ú�Í�Õ�̼Í�̼Ô�Î�Ó¼á � a ñ®ãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö�Ù�Ø�Î�ã�ØjÊ�Ê�Û.Ê�ÛNý¡Ú_Ê�Þ�Ó¼Î�Ï_þ�Î�ãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö5Þ�̼Í�è©ÙµÖ�Í�ØjÙ�̼þnè�Í�ñ�Ó�Ì�Ú_ÎrÎ��ºÍ�ØjÙ�ßàÍ�Ê�Ø�õ� Ú2ÊnÕ_Ø.Ø�Î�Ö�Õ�Ó¼Í�Ö�Í�ÊrÛ¦ÊnÓ�Ó�Ê9Þ � Þ�γÚ_ÎlÎ�Ï�Ù�ãPÙ�Ø�Í�Ì�ßlÕ�Ó¨Ù�Ø � Ö�̨˧ãPÓ¼ÎG̨Ú�Ö�Í�Ù�Ú�ßàÎ�Ê�Û�Í�è_γ˵Ù�Ì�Ú�Í�è�ÎlÊ�Ø�Î�Ë=Û¦Ø�ÊnË ä�å ��ç®õ�@¡ãDÎ�ßl̼ۦñ

̨ßlÙ�Ó�Ó¼á � Þ:Î2Õ�Ö�Î2Ù«ÖjÓ�̼þnè�Í5þ�Î�Ú_ÎlØjÙ�Ó�Ì���Ù�Í�Ì�ÊnÚLÊ�Û>Ù«ßàÊ�ØjÊnÓ�Ó�Ù�Ø�á¹Ì�ÚÑä�å ��çÞ�èPÌ�ßjè©Î�ÚPÙ�ÒPÓ¼Î�Ö�Ù«ã�Ø�Ê�Ô�ÎlØ � þnÌ�Ô�Î�ÚLÍvÞ�Ê(Ø�Î�Ó�Ù�Í�̼ÊnÚ�ÖÜ,D ïCB D î Ý � Ô�Ù�Ó�Õ_Î�Ö�Ü @AïEBA@ î Ý � !o $ Â�� !o $ Ì � Ù�Ú�ÏLßàÊ�Ø�ØjÎ�Ö�ãDÊnÚ�Ï�Ì�Ú_þ.��ñ�˵Ê9Ô�Î�b�ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ö�Ü�Ü,z ïCB2{2ï Ý B Ü,z î B2{ î Ý�Ý � Í�ÊãPØ�Î�Ö�Î�Ú�Í%Ù.��ñ�˧Ê9Ô�Î�b�ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓGÜ,z���� B2{ ����Ý:Û�Ê�Ø%ã�Ø�Ê�Ô¡Ì�Ú�þ�Í�è�Î�Î��ºÌ¨Ö�Í�Î�Ú�ßàÎ�Ê�ÛNÙ B Ö�Õ�ßjè¹Í�è�Ù�Í%Î�̼Í�è_ÎlØ�Ü @AïCBAB Ý �nD ïÊ�Ø�Ü @ î BAB Ý �sD î õ���Î3ßlÙ�Ó�Ó�Í�èPÌ�Ö³Í�è_ÎLéjÐ_>�ì�ã�Ø�Ê�Í�ʺßàÊnÓAÛ�Ê�Ø2Ü�Ü,z ïCB2{3ï Ý B Ü,z î B2{ î Ý�Ý ���Î�Þ�Ì�Ó�Ó-Ï_Î�Ö�ßàØh̼ÒDÎ�Í�è_Î�ã�ØjÊ�Í�Ê¡ßàÊnÓAÙ�ÖjÖ�Õ�˵Ì�Ú�þ2Í�è_Î�ßhè�Ù�Ó�Ó¼Î�Ú_þ�Î�Ö%Û¦Ø�ÊnË Ü,z ï B2{ ï Ý�Ù�Ú�Ï»Ü,z î B2{ î Ý:Ù�Ø�Î�Ê�ÛÍ�è_Î�Ö�Ù�˧ÎÓ�Î�Ú_þ�Í�è�õ�×�è�Ì�Ö%ßlÙ�ÚLÎ�Ù�Ö�Ì�Ó¼á¯ÒDÎ�þ�Î�Ú_ÎlØjÙ�Ó�Ì �lÎ�Ï � Ù�Ö�Ó¼ÊnÚ_þµÙ�Ö�Í�è_Î3ßjèPÙ�Ó�Ó¼Î�Ú_þ�Î3Ó�Î�Ú_þ�Í�èLÌ�Ú¹Í�è_Î3ßàÊnË2ÒPÌ�Ú_Î�ÏLã�Ø�Ê�Í�ʺßàÊnÓ�Ì�Ö%Ù�ÍÓ�Î�Ù�Ö�Í3Ù�Ö3Ó�ÊnÚ_þ¹Ù�Ö�Í�è_Î(ßjè�Ù�Ó�Ó�Î�Ú_þ�Î�Ö�Û�Ø�ÊnË'Î�̼Í�è�ÎlØ�ã�Ø�Ê�Í�ʺßàÊnÓVõ(×�è_εã�Ø�Ê�Í�ʺßàÊnÓ?ßàÊnÚ�ÖjÌ�Ö�Í�Ö�Ê�Û�Ü,z ï B2{ ï Ý�Ù�Ú�ÏYÜ,z î B2{ î ÝØhÕ�Ú�Ú�Ì�Ú�þµÌ�Ú¹ãPÙ�ØhÙ�Ó�Ó¼Î�Ó � ÒPÕ_Í�Þ�̼Í�è¹Í�è_Î�Ô�ÎlØj̼ëPÎlØC� Ö�ßhè�Ù�Ó�Ó¼Î�Ú�þ�Î } Ö�ãPÓ�̼Í�Ì�Ú�Í�Ê } c }9ï�� } î � Þ�̼Í�è }�ï Ù�Ö�Í�è_Î�ßhè�Ù�Ó�Ó¼Î�Ú�þ�ÎÛ�Ê�Ø2Ü,z ïCB2{2ï Ý � Ù�Ú�Ï } î Ù�Ö5Í�è_Î2ßjè�Ù�Ó�Ó�Î�Ú_þ�Î�Û¦Ê�Ø�Ü,z î B2{ î Ýhõ×%è_Î�ãPØ�Ê�Í�Ê¡ßàÊnÓ�Û�Ê�Ø�z����%Ì�Ö�Ù�Ö�Û�ÊnÓ�Ó¼Ê9Þ�Ö 0�#̼Í�è_ÊnÕ�Í�Ó¼ÊnÖ�Ö�Ê�Û�þ�Î�Ú_ÎlØjÙ�Ө̼Í�á � Ö�Ù9á�z����5ýºÚ_Ê9Þ�Ö B Ö�ÕPßjè(Í�è�Ù�ÍrÜ @�ïEBAB Ý �D ï õ O�ÎlÍ � î ÒDÎ�Í�è_εÖ�Ì�Ë�Õ�Ó�Ù�Í�Ê�ØrÛ�Ê�Ø & î õ2×%è_Î�Úmz����3ØjÕ�ÚPÖ � î Ü @ î Ý�Í�Ê©þ�Î�Ú_ÎlØhÙ�Í�ÎLÜ B�� B N�Ýhõ � Í�Ö�Î�Ú�Ï�Ö�Í�è_Î�ë�ØjÖ�Í˵Î�Ö�Ö�Ù�þ�Î�Ê�Û%Ü,z ï B2{ ï Ý � Ù�Ó�ÊnÚ_þµÞ�̼Í�è Ù�Ö%Í�è_Î�ë�ØjÖ�ͳ˵Î�Ö�Ö�Ù�þ�Î�Ê�Û%Ü,z î B2{ î Ýhõ³ÐrÚ©ßjè�Ù�Ó¨Ó¼Î�Ú_þ�Î } � ̼Í%ßhè_Ê�ÊnÖ�Î�Ö } î c � �Ù�ÚPÏ }�ï c }�� } î õ � Í�Ì�Ö�Ù�ÒPÓ¼Î�Í�ʵã�ØjÊ9ÔºÌ�Ï_Î�Í�è_Î�ë�Ú�Ù�Ó�Ø�Î�Ö�ãDÊnÚ�Ö�Îr̨ÚJÜ,z ïCB2{2ï Ý:ÒDÎ�ßlÙ�Õ�Ö�Î�̼Í%ýºÚ_Ê9Þ�Ö B � Ù�ÚPÏ©Í�è_Î�ëPÚPÙ�ÓØjÎ�Ö�ãDÊnÚ�Ö�Î5Ì�Ú©Ü,z î B2{ î ÝÌ�Ö>Ö�̨˧ãPÓ¼á N_õ>×%è�Î:ë�Ú�Ù�Ó_˧Î�Ö�Ö�Ù�þ�Î5Ê�Û�z����³Ì�Ú�ßlÓ¨Õ�Ï_Î�Ö }9ï Ù�Ó¼ÊnÚ�þ�Þ�̼Í�è�Í�è_Î5ëPÚ�Ù�Ó¡Ø�Î�Ö�ã-ÊnÚPÖ�Î�ÖNÛ¦Ê�ØÜ,z ïCB2{2ï Ý:Ù�Ú�ÏJÜ,z î B2{ î Ýhõ���Î3Ú_Ê�Í�Î�Í�èPÙ�Í%Í�è�Ì�Ö%ßàÊnÚPÖ�Í�ØjÕ�ßàÍ�̼ÊnÚ¹Þ�Ê�Ø�ýºÖ5Û�Ê�Ø a ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓ�Ö�Ù�Ö%Þ:Î�Ó�Ó�Ù�Öwb�ñ®ãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ölõ

� �*Á§È��?�?Ã���� � ����#Ä?���(Ä?�4��� ����+Ä'� � È�ÂP �?Å���� jÈ���È�Ä�Æ��! ÷Ã��%Á��#" �?Ã� Ú Í�è�Ì�Ö�Ö�Î�ßàÍ�̼ÊnÚ Þ:αãPØ�Î�Ö�Î�Ú�Í�Í�è�Î.=#> ? ADC#Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á � Ù�Ú Î��¡Í�Î�ÚPÖ�̼ÊnÚ Ê�Û�É%Ðr×*Þ�è�ÎlØ�α̨ÚÑÙ�Ï�ÏP̼Í�̼ÊnÚ»Í�Ê]Í�è_ÎÊ�Ò�Ó�̼Ժ̼ÊnÕ�ÖNÍ�ØjÙ�Ú�Ö�Û¦ÎlØ � Í�è_Î�Ö�Î�Ú�Ï_ÎlØ:ßlÙ�Ú«ã�ØjÊ9Ô�Î�Í�Ê3Í�è_Î�Ø�Î�ßàÎ�̼Ô�ÎlØ�ܸô:Ê�ÊnÓ¼Î�Ù�Ú-Ý>Ø�Î�Ó¨Ù�Í�̼ÊnÚ�ÖGÙ�˵ÊnÚ_þ3Í�è_Î�ßàÊn˵˵̼Í�Í�Î�Ï(ÒP̼Í�Ö�õ��γÞ�̨Ó�ÓnÓ�Ù�Í�ÎlØ>Õ�Ö�Î�Í�è�Ì�ÖÛ�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�árÍ�Ê�̨˧ãPÓ¼Î�˧Î�Ú�ÍÙ�Ú2Î��«ßl̼Î�Ú�Í ��%"�-�x/ �W /T�P�W�* �!.�� "/0��%��rÛ�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá � Þ�è�Ì�ßhę̀Ú÷Í�Õ�ØjÚ]Þ�Ì�Ó�ÓÎ�Ú�Ù�ÒPÓ�Î�Î��(ßl̼Î�Ú�Í2Ù�Ú�Ï�Õ�Ú�̼Ô�ÎlØhÖ�Ù�Ó�Ó¼á¹ßàÊn˧ãDÊnÖ�Ù�Ò�ӼΧË2Õ�Ó�Í�Ì�ñ®ãPÙ�Ø�Ívá÷ßàÊn˧ãPÕ�Í�Ù�Í�̼ÊnÚ�õ§×%è_Î�Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á= > ? ADC Ì�Ö3Ö�è�Ê9Þ�ÚÑÌ�Ú��N̼þnÕ_Ø�Î÷å�õ � Ú_Û�Ê�Øj˵Ù�Ó¨Ó¼á � ÙLãPÙ�Ø�Í�á�@ = ßàÊn˵˵Ì�Í�Ö�Í�Ê]Ù¹ÒP̼Í�ú�Ò�á»Ö�Î�ÚPÏ�Ì�Ú_þ¹Ù�Ú7%$ 4."</&$ 4(')'+* "˵Î�Ö�Ö�Ù�þ�ιÍ�Ê»Í�è�ιÌ�Ï_Î�Ù�Ó:Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á =#> ? ADC � Ù�Ú�Ï#@>=�ßlÙ�Ú Ó¨Ù�Í�ÎlصÊ�ãDÎ�Ú Í�è�Ì�Ö�ÒP̼Í�Ò¡á Ö�Î�Ú�Ï�Ì�Ú_þ]Ù�Ú 7%$ 4."P/<4^1[7-,˵Î�Ö�Ö�Ù�þ�ΫÞ�̼Í�è»Ù�ã�ã�ØjÊ�ã�ØjÌ�Ù�Í�εßàÊn˵˫̼Í�˧Î�Ú�Í3Ì�Ï_Î�Ú�Í�̼ë�ÎlØ«Ü } :<; Ý�Ô�Ù�Ó�Õ_Î�õ7�_Ê�Ø @ = Í�ʹÊ�ÒPÓ�Ì�ԡ̼ÊnÕPÖ�Ó¼á¹Í�ØjÙ�ÚPÖ�Û¦ÎlØ�Ù©ÒP̼Í�Í�Ê@�? � @>=%Ú_ÎlÎ�Ï�Ö2Í�Ê{ßàÊn˫˵̼Í3ÍvÞ�Ê{ÒP̼Í�Ö�úyùµÙ�Ú�Ï ú ï Ù�Ú�Ï6@�?±Ú_ÎlÎ�Ï�Ö3Í�Ê{ßàÊn˵˫̼Í3Í�Ê{ÊnÚ_ΫÒ�̼Í2ú/.�ûGÙ�Û�Í�ÎlØ�Ö�Î�ÚPÏ�Ì�Ú_þLÙ�Ú7%$ 4."P/7" 3102,4365 7.35Í�Ê�=#> ? ADC � Í�è_αÒP̼Í�úyü87�Ì�Ö3Í�ØhÙ�Ú�Ö�Û�ÎlØ�Ø�Î�Ï»Í�ÊX@�?¯Ù�Ú�Ï Ù�Õ_Í�Ên˵Ù�Í�Ì�ßlÙ�Ó�Ó�áJßàÊn˵˵̼Í�Í�Î�Ï Ò¡á�=#> ? ADCYÊnÚÒDÎ�è�Ù�Ó�ÛAÊ�Û.@�?�õ �LÎ�Ù�Ú�Þ�è�Ì�ӼΠ� @>=NÏ_Ê�Î�Ö5Ú�Ê�Í%Ó¼Î�Ù�ØjÚ©Ù�Ú�á�Í�è�̨Ú_þ � Î��ºßàÎlãPÍ%Í�è�Ù�Í%Ù�Í�ØjÙ�Ú�Ö�Û¦ÎlØ5Í�Ê¡Ê�ý¯ãPÓ�Ù�ßàÎ�õ���Õ_ØjÍ�è_ÎlØj˧Ê�Ø�Î �Í�è�Î2Û¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�á¹Ù�Ó¨Ö�ʱÙ�Ó¨Ó¼Ê9Þ�ÖrÙ(ãPÙ�Ø�Ívá<@>=GÍ�ʯã�Ø�Ê�Ô�Î�Í�ÊR@�?�Í�è�Ù�ÍrÍ�è_Ø�ÎlÎ�ÒP̼Í�Örúyù � ú ï � Ù�Ú�Ï�ú î ̼Í�ßàÊn˵˵̼Í�Í�Î�Ï]Í�ÊÖjÙ�Í�Ì�Ö�Û�á±Ù�ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�Ø5ÒP̨Ú�Ù�Ø�á«ØjÎ�Ó�Ù�Í�̼ÊnÚ¯Ò¡á¯Ö�Î�Ú�Ï�Ì�Ú_þ�Ù�Ú 7%$ 4."P/21.354�6�7�˧Î�Ö�ÖjÙ�þ�Î�Í�Ê�= > ? ADC õ

ö Ö�Ù«ßàÊnÚ�Ô�Î�Ú�Í�̼ÊnÚ � Þ:Î2Õ�Ö�Î�4^129 î�:; Í�Ê(Ï�Î�Ú_Ê�Í�Î2Ù§Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ©ÊnÚ¹ÍvÞ�Ê(ÒP̼Í�Ö � Þ�è_ÎlØ�Î< � e�3 B å�f � Ì�Ö%Í�è_Î3Ö�Í�Øj̨Ú_þ§Ê�ÛÒ�̼Í�Ö:Ê�Û�Í�è_Î�ô:Ê�ÊnÓ�Î�Ù�Ú©Û�ÕPÚ�ßàÍ�̼ÊnÚ�� Ö:Í�ØjÕ_Í�è±Í�Ù�ÒPÓ�ΧÜ�ÊnÕ�Í�ãPÕ_Í5ßàÊnÓ�Õ�˵Ú-ÝhõI��Î�Ù�Ó�Ö�Ê�Ê�Û¦Í�Î�ÚLÌ�Ï_Î�Ú�Í�̼ۦá=�Þ�Ì�Í�è(Í�è�Îr̨Ú�Í�Îlþ�ÎlØÞ�è_ÊnÖ�ΧÒPÌ�ÚPÙ�Ø�áLØ�ÎlãPØ�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ»Ì�Ö<�õ{Ü0�_Ê�Ø3Î��_Ù�˧ãPӼΠ� c åµØ�Îlã�Ø�Î�Ö�Î�Ú�Í�Ö�Í�è_Î ö � � Û�ÕPÚ�ßàÍ�̼ÊnÚ � Þ�è_ÊnÖ�ΧÍ�ØjÕ_Í�èÍ�Ù�Ò�Ó¼Î�Ì�Ö;3"3"3_å�õ�Ý

ö Ö�Ù�Í�Î�ßjè�ÚPÌ�ßlÙ�Ó�Ú_Ê�Í�Î � Þ�Î�Ú_Ê�Í�Î%Í�è�Ù�Í?Í�è�Î�Ð�ãDÎ�Ú�ãPè�Ù�Ö�Î�Ì�Ö>Ú_Ê�ÍGÖ�Í�ØhÌ�ßàÍ�Ó¼á�Ú_Î�ßàÎ�Ö�Ö�Ù�Ø�á§Ö�̨Ú�ßàÎ%̼Í?ßlÙ�ÚµÒDÎ%Ö�̨Ë2Õ�Ó¨Ù�Í�Î�ÏÒ¡áLÍ�è�Î �GØ�Ê9Ô�Χã�è�Ù�Ö�Î�õ2×.Ù�ý�Î�4^129 î�:ù�ù�ù�ù Ù�ÚPÏ 4^1>9 î�:ï�ï�ï�ï � Þ�èPÌ�ßjè{ßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�Ï{Í�Ê©Í�è_ΧÙ�Ó�Ó¼ñL�lÎlØ�ʹÙ�Ú�Ï]Ù�Ó�Ó�ñ®ÊnÚ_Î�Û¸Õ�Ú�ßàÍ�̼ÊnÚPÖlõ×�è_Î�Ú � Ò�á¹ãPØ�Ê9ÔºÌ�Ú_þ(Í�è�Ù�Í�4^1 9 î�:ù�ù�ù�ù ÜVúyù B ú ï Ýpc1ú î Û�Ê�ØrÙ�ØjÒP̼Í�ØjÙ�Ø�á©Ò�̼Í�Ö�úyù�Ù�Ú�Ï]ú ï � ÊnÚ_Î�Î�Ö�Ö�Î�Ú�Í�Ì�Ù�Ó�Ó�á©Ê�ã-Î�ÚPÖ�ÒPÌ�Í�ú î Í�Ê3ºû>Ö�̨˵Ì�Ó�Ù�ØjÓ�á � Ò¡á]ã�Ø�Ê�Ô¡Ì�Ú�þ©Í�è�Ù�Í 4^1>9 î�:ï�ï�ï�ï ÜVúhù B ú ï Ý�c ú î � ÊnÚ_εÊ�ãDÎ�Ú�Ö2ú î Í�Ê å�õ.��αßjè_Ê¡ÊnÖ�εÍ�Ê{Ì�Ú�ßlÓ�Õ�Ï�Î2Í�è_ίÐ�ãDÎ�Úã�è�Ù�Ö�ΧÌ�Ú{Í�è_Î�Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á©Û�Ê�Ø�ßlÓ�Ù�Øj̼Ívá÷Ù�Ú�Ï]Î��«ßlÌ�Î�Ú�ßàá Ü�Í�è_Î(Ðrã-Î�Ú÷ãPè�Ù�Ö�Î�ßlÙ�Ú»Ò-Î�Ø�Î�Ù�Ó�Ì �lÎ�Ï�˧Ê�Ø�Î�Î��«ßl̼Î�Ú�Í�Ó¼áÍ�èPÙ�Ú©Í�è_Î3Ö�̨Ë2Õ�Ó¨Ù�Í�Î�Ï&�?Ø�Ê�Ô�Î�ãPè�Ù�Ö�Î�Ýhõ

ô�ÎlÛ�Ê�Ø�Î�ã�Ø�Î�Ö�Î�Ú�Í�Ì�Ú_þ(Ùµã�Ø�Ê�Í�ʺßàÊnÓÍ�è�Ù�Í�Ö�Î�ßlÕ�Ø�Î�Ó¼á©Ø�Î�Ù�Ó�Ì �lÎ�Ö =#> ? ADC � Þ:Î3ëPØjÖ�Í�Ï�̨Ö�ßlÕ�Ö�Ö%Ö�Ên˧Î2ãPØ�Î�Ó�Ì�˵̨Ú�Ù�Ø�á«ßàÊnÚºñÖ�Í�ØjÕ�ßàÍ�̼ÊnÚ�Ö³Í�èPÙ�Í%Þ�Ì�Ó�Ó-ÒDÎ�Õ�Ö�Î�Ï©Ù�Ö�ÒPÕ�Ì�Ó¨Ï�Ì�Ú_þ�Ò�Ӽʡßjý¡Ö�õ

Page 8: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

) -.-G2:7 / �4- �B58/87:9 =#> ? ADC

= > ? ADC ã�Ø�ʺßàÎlÎ�Ï�Ö5Ù�Ö�Û¦ÊnÓ�Ó�Ê9Þ�Ö � ØhÕ�Ú�Ú�Ì�Ú�þ3Þ�̼Í�è¹ãPÙ�Ø�Í�̼Î�Ö @ ï BEDVDVDVB @ F Ù�Ú�ÏLÙ�Ú¹Ù�Ï_Ô�ÎlØjÖ�Ù�Øjá Q�õ*�� �43 36/87(5 + � 2S1�� �#è_Î�Ú ØjÎ�ßàÎ�̼ԡ̨Ú_þaÛ¦ØjÊnË @>=µÙ­Ë§Î�Ö�ÖjÙ�þ�Î��,7%$ 4."P/&$ 4(' '+* " B98':<;�B2} :<;=B @h? B ú�� �ØjÎ�ßàÊ�ØjÏ�� } :<;=B @ = B @ ? B ú�� � Ö�Î�ÚPÏ Ë§Î�Ö�Ö�Ù�þ�Î�O��IL���/9K5��5�� J�� B98':<;=B2}0:P;�B @ = B @ ? ��Í�Ê @ = � @ ? Ù�Ú�Ï Q �Ù�ÚPÏ Ì¼þnÚ�Ê�Ø�Î Ù�Ó¨Ó�Û¸Õ_Í�Õ_Ø�Π˧Î�Ö�Ö�Ù�þ�Î�Ö]Ê�ÛµÍ�è�ÎÑÛ�Ê�ØjË �,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�B @�? B�� �¹Û�Ø�ÊnË @C=(Ù�Ú�Ï�,7%$ 4."P/7" 3102,4365 7.3 B98':<;�B2} :<;=B�� B�� B�� B @C=��:Û�Ø�ÊnË @�?�õ

*�� /���0T1 5 + � 2S1�� �#è_Î�Ú Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ Û¦Ø�ÊnË @C='٠˧Î�ÖjÖ�Ù�þ�Î �,7%$ 4."P/21.354�6�7 B98O:P;�B98�8O:<;=B} :<; ù B2} :<;_ïCB2} :<; î B @�? B �� � ̼۵Í�è�Î Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ+Í�è_Ø�ÎlÎ Í�Õ_ã�Ó¼Î�Ö � � } :<; ù B @>= B @�? B úyù�� � � } :<;_ï B @C= B @�? B ú ï � �� } :<; î B @>= B @�? B ú î � � Ù�ØjÎ Ù�Ó�Ó7Ø�Î�ßàÊ�ØjÏ_Î�Ï � Ù�Ú�Ï 4^1>9 î�:; ÜVúyù B ú ï Ý c ú î � Í�è_Î�Ú Ö�Î�Ú�Ï Ë§Î�ÖjÖ�Ù�þ�Î�O��CL���/9J>KILML�N B98':<;�B98�8':<;�B2} :<; ù B2} :<;_ï B2}0:P; î B @>= B ��:Í�ÊM@�?�Ù�Ú�ÏRQ�û_Ê�Í�è_ÎlØ�Þ�̨Ö�Î�Ï_ʵÚ_Ê�Í�è�̨Ú_þ_õ

* * / �=-02C;X1 / 5 + � 2S1�� �#è_Î�Ú Ø�Î�ßàÎ�̼ԺÌ�Ú_þ Û�Ø�ÊnË @ = Ù,˧Î�Ö�ÖjÙ�þ�Î��,7%$ 4."P/7"030 , 365,7.3 B98':<;�B2} :<;=B} :<; ù B2} :<;_ïCB � } :<;�B @�?�� � Ì¼Û Í�è�Î'Û¦ÊnÓ�Ó�Ê9Þ�Ì�Ú�þ Í�è�Ø�ÎlÎ'Í�Õ_ãPÓ¼Î�Ö�� } :<; ù B @C= B @�? B úyù�� � � } :<;_ï B @C= B @�? B ú ï � �Ù�ÚPÏ � � } :<;=B @h? B @C= B ú/.�� � Ù�Ø�Î Ù�Ó�Ó+Ø�Î�ßàÊ�ØhÏ_Î�Ï � Ö�Î�Ú�Ï Ë§Î�ÖjÖ�Ù�þ�Î��O��IL���/�� ����� B98':<;=B2}0:P;�B @>= B @h? B} :<; ù B2} :<;_ïCB � } :<;�B úyü 7 � Í�Ê @�? � Ø�Î�ßàÊ�ØjÏ Í�Õ_ãPӼΠ� }0:P;�B @�? B @>= B úyü 7 � � Ù�ÚPÏ Ö�Î�ÚPÏ Ë§Î�ÖjÖ�Ù�þ�Î�O��CL���/9K ��8�� J�� B98':<;�B2} :<;=B @C= B @�? B2} :<; ù B2} :<;_ïCB � } :<; � Í�Ê @>= Ù�Ú�Ï Q � Ù�Ú�Ï Ì�þnÚ_Ê�Ø�ÎÙ�Ó¨Ó Û¸Õ_Í�Õ_Ø�Π˧Î�Ö�Ö�Ù�þ�Î�Ö Ê�Û Í�è_Î Û�Ê�ØjË �,7%$ 4."P/&$ 4(' ' * " B98':<;=B2}0:P;�B @>= B�� � Û�Ø�ÊnË @C= Ù�Ú�Ï�,7%$ 4."P/7" 3102,4365 7.3 B98':<;�B2} :<;=B�� B�� B�� B @h?!�:Û¦ØjÊnË @h?nõ:Ð�Í�è_ÎlØjÞ�Ì�Ö�Î � Ï_ʵÚ_Ê�Í�èPÌ�Ú_þ_õ

* ,I5�1 - 5 + � 2S1�� �#è_Î�Ú Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ{Û�Ø�ÊnË @ = Ù�˧Î�ÖjÖ�Ù�þ�Î"�,7%$ 4."P/<4^1[7-, B98':<;=B2}0:P;�B @ = B @ ? � � ̼Û�Í�è_ÎÍ�Õ�ãPÓ¼Î#� } :<;�B @>= B @�? B ú��³Ì�Ö%Ø�Î�ßàÊ�ØjÏ_Î�Ï � Ö�Î�Ú�Ϲ˧Î�ÖjÖ�Ù�þ�Î$�O��CL���/�� ����� B98O:P;�B2} :<;�B @>= B @�? B ú��:Í�Ê(Ò-Ê�Í�è QÙ�ÚPÏ @h?nû_Ê�Í�è_ÎlØ�Þ�̨Ö�Î � Ï_Ê«Ú_Ê�Í�èPÌ�Ú_þ_õ

�N̼þnÕ_Øjεå &%('�)*)�+-,.)0/�12)�1430576#698:,;,.)�1<52=?>@8:A78B52C�DE,GFIHJ/KD�LM)IFNLOC�/�3�,M8B57/�HJ>@8:,QP

R�R<S S<TVU<_�WXU.YXZ\[ _<Z�]?]�`� Ú ãPÙ�Ø�Í�̨ßlÕ�Ó�Ù�Ø � Þ:ιÖ�è�Ù�Ó�Ó³Ï�̨Ö�ßlÕ�Ö�Ö�Ù�Ú a ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓ^�N_LÛ�Ê�Ø«Ï�Ì�ÖjßàØ�ÎlÍ�ΩӼÊ�þ � Ù�Ú a ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�J%��K5��J­Û�Ê�صãPÙ�Ø�Í�̨Ù�ÓÎS!¡Õ�Ù�Ó�Ì�Í�á±Ê�ÛØ�ÎlãPØ�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ � Ù�Ú�ϹÍ�è_Î�̼Ø%ßàÊn˧ãDÊnÖ�̼Í�Ì�ÊnÚ�Ölõ

`%- aba 50/ � 7 ��2 �T5?; �B/��./ 2C2/+13741�5 �dc � / /87 + 3 9 � Ú÷Í�è_Î�Û�Õ�Ó¨ÓAÔ�ÎlØjÖ�Ì�ÊnÚ÷Ê�ÛGÍ�è_Î�̼Ø�ãPÙ�ãDÎlØ � H�Ù�ØjÙ�á¹��/; �!��?äóò:G�ç.ã�ØjÎàñÖ�Î�Ú�Í�Î�ϵÙ�Ú a ñ®ã�Ø�Ê�Í�ʺßàÊnÓ_Û�Ê�Ø?Í�è_Î%Ï�Ì�ÖjßàØ�ÎlÍ�Î%Ó¼Ê�þnÙ�Øj̼Í�èPË7Ø�Î�Ó�Ù�Í�̼ÊnÚ�õI�LÊ�Ø�Î�ßàÊnÚ�ßàØ�ÎlÍ�Î�Ó�á � Ó¼ÎlÍ�Ü g B�e�BOf Ý>ÒDÎ:ã�Õ_ÒPÓ�Ì�ß�ãPÙ�ØjÙ�˧ñÎlÍ�ÎlØhÖ � Þ�è�ÎlØ�Î e Ù�Ú�Ï g(Ù�Ø�Î�ã�ØjÌ�˧Î�ÖNÖ�Ù�Í�̨Ö�Û¦áºÌ�Ú_þ e k�Ü gL$¹å9Ý?Ù�Ú�Ï f �hg�ij Þ�̼Í�è 4 3lk 7.3�Ü f ÝYc e õ�O�ÎlÍYD#ÒDÎ:Í�è�Î�Ï�Ì�Ö�ßàØjÎlÍ�ÎÓ�Ê�þnÙ�Øj̼Í�è�Ë Ø�Î�Ó�Ù�Í�Ì�ÊnÚ D c e�Ünm BA@ Ý �m<o fqp ˧ʺÏwgIfnõ3×%è_Îlá{ßàÊnÚ�Ö�Í�ØjÕPßàÍ�Î�Ï�Ù�Ú a ñ®ã�Ø�Ê�Í�ʺßàÊnÓNÛ¦Ê�Ø�Í�è_Î�Ø�Î�Ó¨Ù�Í�̼ÊnÚmD�õ��ΧÕ�Ö�Î2Ù±Ö�Ó�̼þnè�Í�Ô�Ù�ØjÌ�Ù�Ú�Í�Ê�Û?Í�èPÌ�Ö%ã�ØjÊ�Í�Ê¡ßàÊnÓ.Û�Ê�Ø�Ù±Ï�Ì�DÎlØ�Î�Ú�Í�Ø�Î�Ó�Ù�Í�Ì�ÊnÚ�D � � Þ�è�Ì�ßjè{Ï�Ì�DÎlØjÖ�Û¦Ø�ÊnËdD Ò¡áL˧Ê�ԡ̨Ú_þ fÛ�Ø�ÊnË Í�è_Î3ã�Õ_ÒPÓ�Ì�ß�ãPÙ�ØhÙ�˧ÎlÍ�ÎlØrÍ�Ê(Í�è_Î�Ì�Ú_ãPÕ�Ílõ � Ú÷Ê�Í�è_ÎlØ�Þ:Ê�ØjÏ�Ö � DsrKt c e�Ü�Ünm BOf Ý BA@ Ý�2m$o fqp Ë§ÊºÏ gIfnõc��Î�ßlÙ�Ó�ÓÊnÕ�Ø%ã�Ø�Ê�Í�ʺßàÊnÓu��__Ünm BOf Ý5Ù�Ú�Ϲã�Ø�Î�Ö�Î�Ú�Í%̼Í%Ì�Ú ö ã�ãDÎ�Ú�Ï�ÌÄ�±ôYÛ�Ê�Ø�ßàÊn˧ã�Ó¼ÎlÍ�Î�Ú_Î�Ö�Ölõv 1 363 �xwR9My �C�)�?(:�D%"/L%�#&%�!u��_ºÜnm BOf Ýr�t� �� a �t(}�O%"/L%�#�%�!��

`%- aba 50/ � 7 ��2 �T5 ; �B/65 �B/47 / �B5"1!z - �B58/87:9 �=;�/+1�50/+13241�-D7 � 7 / �4- 9 ×%è_ΫãPØ�Ê�Í�Ê¡ßàÊnÓE��_LßlÙ�ÚÑÒDÎ(Î��¡Í�Î�Ú�Ï�Î�ÏÑÍ�ÊÊnÚ�Î�Û¦Ê�Ø�ã�Ø�Ê�Ô¡Ì�Ú�þ§ãPÙ�Ø�Í�Ì�Ù�Ó�ÎS!¡Õ�Ù�Ó�Ì�Í�á±Ê�Û>Ø�Îlã�Ø�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ�õ OAÎlÍ�Ü g B�e Ý:ÒDÎ�ãPÕ_ÒPÓ¨Ì�ß�ã�Ù�ØjÙ�˧ÎlÍ�ÎlØjÖ � Þ�è_ÎlØ�Î e Ù�ÚPÏ g÷Ù�Ø�ÎãPØjÌ�˧Î�Ö5Ö�Ù�Í�̨Ö�Û¦áºÌ�Ú_þ e k�Ü g $ å9ÝhõWOAÎlÍMD7ÒDÎ�Í�è_Î�Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ�Ø�Î�Ó�Ù�Í�Ì�ÊnÚ

Ds{�|�}?|~{rcqe�Ü�Ü @ ù BOf ù BOf¡ïCBA@AïCBOf î BOf Ý B Ü �ù B � ïCB � î Ý�Ýuk @ ù�o f!���ù f!� Âï ËµÊ¡Ï g�� @Aï o f!���î f�� Ì Ë§Ê¡Ï�gIf Ü�å9Ý

æ

Page 9: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Þ�è_ÎlØ�Î f ù � f¡ï � f î � Ù�Ú�Ï f Ù�Ó¨Ó�è�Ù9Ô�Î�Ê�ØhÏ_ÎlØ e õ#��ΧßàÊnÚ�Ö�Í�ØhÕ�ßàÍ�Ù�Ú a ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓÛ¦Ê�Ø�Í�è�Ì�Ö�Ø�Î�Ó�Ù�Í�̼ÊnÚ÷Ù�Ú�Ï÷Þ�Î�ßlÙ�Ó�Ó�Ì�ÍJ ��K5�hJNÜ @ ù BOf ù BOf ï BA@ ï BOf î BOf Ýhõ � Ú�Í�Õ�Ì�Í�̼Ô�Î�Ó¼á � Í�è�Ì�Ö�Ì�Ö%Ù§ã�ØjÊ�Ê�ÛNÊ�ÛNý¡Ú�Ê9Þ�Ó¼Î�Ï�þ�Î�Ê�ÛNã�Ø�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ�Ö�Ê�Û @ ù Ê9Ô�ÎlØrÒPÙ�Ö�Î�ÖÜ f ù BOf¡ï Ý:Ù�Ú�Ï @Aï Ê9Ô�ÎlØ�Ò�Ù�Ö�Î�Ö�Ü f î BOf Ý�Ö�Õ�ßhè¯Í�è�Ù�ͳÍ�è_Î�Î��¡ãDÊnÚ_Î�Ú�ͳÊ�Û @ ù�Ê�Ô�ÎlØ f ù�ÎS!¡Õ�Ù�Ó�Ö³Í�è_ÎrÎ��ºãDÊnÚ_Î�Ú�ͳÊ�Û @Aï Ê9Ô�ÎlØf î õ ×%è_Î÷̨Ú�Í�Õ�Ì�Í�̼ÊnÚaÒDÎ�è�Ì�Ú�Ï Í�è�Ì�ÖµãPØ�Ê�Í�Ê¡ßàÊnÓ�Ì�Ö(Ô�ÎlØ�á­Ö�̨˵Ì�Ó�Ù�اÍ�Ê Í�è�Ù�ͱÊ�Û9��__Ünm BOf Ýhõ7×�è_ÎLßàÊn˵˧ÊnÚ+Ø�ÎlÛ�ÎlØ�Î�Ú�ßàÎÖ�Í�ØjÌ�Ú_þ¯ßàÊnÚ�Ö�Ì�Ö�Í�Ö�ÙJ�NÙ�Ì�Ó�Ө̼ÎlØ�ã�Õ_ÒPÓ�Ì�ß�ý�Îlá{Ù�Ú�Ï]Ù�Ï�ÏP̼Í�̼ÊnÚ�Ù�Ó$>A@ ö ˧ʺÏ�Õ�Ó�Õ�Ö�Þ�̼Í�è÷Í�è_Ø�ÎlÎ�þ�Î�Ú_ÎlØjÙ�Í�Ê�ØhÖlõ�×�è_Î2ãPØ�Ê9Ô�ÎlØÖ�Î�Ú�Ï�ÖrÊ9Ô�ÎlØ2Í�è_ΧÍ�è_ΧÎ�Ú�ßàØjá�ã�Í�Ì�ÊnÚ]Ê�ÛY� ������� Ü�Í�è_εØ�ÎlãPØ�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ�ÖhÝ�Õ�ÖjÌ�Ú_þ±Í�è_Î<�NÙ�Ì�Ó�Ó�̼ÎlØrÎ�Ú�ßàØjá�ã�Í�Ì�ÊnÚ]ý�Îlá{Ù�Ú�ÏÍ�è�Î�Ú]ã�Ø�Ê9Ô�Î�Í�è�Ù�Í�Í�è�Î�Ö�Î�Î�Ú�ßàØ�á¡ã�Í�̼ÊnÚ�ÖrÙ�Ø�ΧßàÊ�Ø�Ø�Î�ßàÍlõ�×%è_Î�Ù�Ï�Ï�Ì�Í�̼ÊnÚ�Ù�Ó$>A@ ö ˧ʡÏPÕ�Ó�Õ�Ö�Ù�Ø�ΧÕ�Ö�Î�Ï{Í�Ê©Ø�Î�Ö�ÊnÓ¼Ô�Î�Í�è_ÎÏPÌ�-ÎlØjÎ�Ú�Í�˧ʡÏPÕ�Ó�ÌDã�Ø�Ê�Ò�Ó¼Î�Ë©õ%×%è�̨Ö%Ì�Ö�Ù±Ö�Í�Ù�Ú�ÏPÙ�ØjϹÍ�Î�ßhè�Ú�Ì !¡Õ_αä � � æ � ò:� � �ºå � ò:G�ç®õ#��Î�Ì�Ú�ßlÓ¨Õ�Ï_Î�Í�è_Î�ã�Ø�Ê�Í�ʺßàÊnǪ́Úö ã�ãDÎ�Ú�Ï�Ì��(ô�õv 1 363 �xwR9�� �C�)�?(:�D%"/L%�#&%�!�J ��K ��JNÜ @ ù BOf ù BOf�ïCBA@�ïCBOf î BOf Ý4�t�r �� a �t(}�O%"/L%�#�%�!��

��Î3Ên˵̼Í5Í�è�Î�ã�Ø�Ê¡Ê�ÛÊ�Û.Í�è�Ì�Ö%Ó�Î�˵˵٧Ö�Ì�ÚPßàÎ�Í�è�Î�Í�Î�ßjè�ÚPÌ !�Õ�Î�Ö%Ù�Ø�Î�ØjÙ�Í�è�ÎlØ�Ö�Í�Ù�Ú�Ï�Ù�ØhÏ�õ

� �4365 �B2 /87 / ��- �=; a a 56/��B7 ��2 �D582 �=- � 2 ��-�0T1 /(2 /���- 7 ��� ��� 56/��B7 ��2 �D5829 ö Ö2Þ:αÏ�̨Ö�ßlÕ�Ö�Ö�Î�ÏJÌ�Ú @¡Î�ßyñÍ�Ì�ÊnÚ+ò � Ö�̨Ú�ßàÎ a ñ®ãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö(Ù�Ø�Î{Ö�ãDÎ�ßlÌ�Ù�Ówb:ñ®ã�Ø�Ê�Í�ʺßàÊnÓ�Ö � Í�è_Îlá­Ù�ÏP˵̼͵Î��«ßlÌ�Î�Ú�ͯ˧ÊnÚ_Ê�Í�ÊnÚ_Î]ßàÊn˧ãDÊnÖ�̼Í�Ì�ÊnÚ�Ölõ � Úã�Ù�Ø�Í�Ì�ßlÕ�Ó�Ù�Ø � Í�è_ÎlØ�ΩÎ��_Ì�Ö�ͧÎ��«ßl̼Î�Ú�Í a ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓ�Ö�Û�Ê�صã�ØjÊ9ÔºÌ�Ú_þ�Ù�Ú�á7é ö � � ì»Ù�Ú�Ï1éjÐ >�ì�ßàÊn˧ãDÊnÖ�̼Í�Ì�ÊnÚ�Ö�Ê�Û�Í�è_ÎÏPÌ�Ö�ßàØ�ÎlÍ�Î�Ó¼Ê�þ§ØjÎ�Ó�Ù�Í�̼ÊnÚ�ÖuDsrKt�Ù�Ú�Ï©Í�è�Î�ãPÙ�Ø�Í�Ì�Ù�ÓDÎS!¡Õ�Ù�Ó�̼Ívá(Ê�ÛNØ�ÎlãPØ�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ©Ø�Î�Ó�Ù�Í�Ì�ÊnÚ�ÖuDs{�|�}?|~{_õ

� Ú äóò:G�ç � HrÙ�ØjÙ9á ��/; �!��3Ì�Ú�Í�Ø�Ê¡ÏPÕ�ßàÎ�ÏLÙ±Í�Î�ßjèPÚ�Ì !¡Õ_Î3Í�ʯÍ�ØjÙ�Ú�Ö�Û¦Ê�ØjË�Ù�Ú�á a ñ®ã�ØjÊ�Í�Ê¡ßàÊnÓNÌ�Ú�Í�ʱٱÕ�Ú�Ì�Ô�ÎlØjÖ�Ù�Ó�Ó¼á©ßàÊnË�ñãDÊnÖ�Ù�Ò�Ó¼Î�ã�ØjÊ�Í�Ê¡ßàÊnÓ-Ò¡áµÕ�Ö�Ì�Ú_þ�Ù2ÏP̼þn̼Í�Ù�Ó-Ö�̼þnÚ�Ù�Í�Õ_ØjÎ�Ö�ßhè_Î�˧ÎrÍ�è�Ù�ͳÌ�ÖGÎ��_Ì�Ö�Í�Î�Ú�Í�Ì�Ù�Ó�Ó�áµÕ�Ú_Û�Ê�Ø�þ�Î�Ù�ÒPÓ¼Î�Ù�þnÙ�Ì�Ú�Ö�ͳÙ�Ï�Ù�ã�Í�̼Ô�Îßhè_ÊnÖ�Î�Úºñ�˵Î�Ö�Ö�Ù�þ�ίÙ�Í�Í�Ù�ß�ýºÖlõL×%è_Î�Ì�Ø�Í�ØjÙ�Ú�Ö�Û�Ê�Øj˵Ù�Í�̼ÊnÚÑÌ�Ö�Î��«ßlÌ�Î�Ú�Í � ̼Û:Í�è_ΫÏ�Ì�þn̼Í�Ù�Ó?Ö�̼þnÚPÙ�Í�Õ_Ø�εÖ�ßhè_Î�˧Î(Ù�ÏP˵̼Í�Ö3Ù�ÚÎ��(ßl̼Î�Ú�Í�ã�Ø�Ê¡Ê�Û�Ê�ÛNý¡Ú�Ê9Þ�Ó¼Î�Ï�þ�Î�ã�Ø�Ê�Í�ʺßàÊnÓVõ � Ú©ãPÙ�Ø�Í�Ì�ßlÕPÓ�Ù�Ø � Í�è_Îlá±ã�Ø�Ê9Ô�Î�ÏLÍ�è_Î�Û�ÊnÓ�Ó¼Ê9Þ�Ì�Ú_þ�Ø�Î�Ö�Õ�Ó�Ílõ*,+ 1��T/(1 3 w 9 w ������������� �D),���6/ ���6��/0�O%��k<�� �� *������'i(:/0�0%"� %"�H/ ����� �� *������'i(:/0�0%"� �Y9�%�� ��������1 �8��!� "/0��%��D / �= "/r )�'5��/�� �� a �t(}�D% /L%�#&%�!"!#�5/ �����8� �%$ �t��/�� / �,�8�&� �J�O%����T) (}�D% /L%�#&%�!'&XNä !�ço/ �= "/ ���&#����8��!.1 �O�& �!Ç� U���� / ���!=�$%(& ��),�8 �!9����}#�/0��%��} �!.�0/01¹�-� / ���J=E? ���^�<�k1 +�����) 'H%S)k��! �<k ��-����/ ), �(:/0����� )��"������ ��������(� *������'5�-�k< ���O *���-�k<3�) ����/ ������'H%��8�*� / �)�,+n ) )���/0�����.- %��"���0���8 ) %D9/&u>ä !%ç /L%0! �t� #�%��)��/L "�x/ ����' +&���7%D9 �%$�()%��:���}/0�0 "/0��%��)�r(}!.�=� / ���<=���:���D /0�0%�� %O9r ����<"�} "/0���8�S�

Ü @¡ÎlÎ ö ãPã-Î�ÚPÏ�ÌÄ� ö Û¦Ê�Ø�Ï�Ì�Ö�ßlÕ�ÖjÖ�̼ÊnÚ±ÊnÚ©Í�è_Îo@ºÍ�Ø�ÊnÚ_þH>A@ ö Ù�Ö�ÖjÕ�˧ã�Í�̼ÊnÚAõ�Ý×%è_ÎlØ�ÎlÛ�Ê�Ø�Î � Þ�Î�ßlÙ�Ú±ãPÓ�Õ_þ�Ì�ÚµÍ�è�Î a ñ®ãPØ�Ê�Í�Ê¡ßàÊnÓ¨Ö�Û�Ê�سã�Ø�Ê�Ô¡Ì�Ú�þ2Ï�Ì�ÖjßàØ�ÎlÍ�Î�Ó�Ê�þnÙ�Øj̼Í�è�Ë � ãPÙ�Ø�Í�Ì�Ù�Ó�ÎS!¡Õ�Ù�Ó�̼ÍváµÊ�ÛAØ�Îlã�ØjÎàñ

Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ � Ù�ÚPÏ�Í�è_Î�̼Ø�é ö � � ì -éjÐ_>�ìrßàÊn˧ãDÊnÖ�Ì�Í�̼ÊnÚ�ÖAÌ�Ú�Í�Ê�×�è_ÎlÊ�Ø�Î�Ë �¡õ ��Ù�Ú�Ï3Ê�Ò�Í�Ù�Ì�Ú�Î��(ßl̼Î�Ú�Í$2rÉcO`K ã�ØjÊ�Í�Ê¡ßàÊnÓ�ÖÛ�Ê�Ø%Í�è_Î�Ö�Î�Ø�Î�Ó�Ù�Í�̼ÊnÚPÖlõ � Ú¯ã�Ù�Ø�Í�Ì�ßlÕ�Ó�Ù�Ø � Þ:Î3Ö�èPÙ�Ó�Ó�Õ�Ö�Î42�É#O`K7ã�ØjÊ�Í�Ê¡ßàÊnÓ�Ö%Û¦Ê�Ø%ãPØ�Ê9ÔºÌ�Ú_þ§Í�è_Î3Û¦ÊnÓ�Ó�Ê9Þ�Ì�Ú�þ2Ø�Î�Ó¨Ù�Í�̼ÊnÚ�Ölõå�õ1 , �32 �4;#7 ����� / 2C2/+1 7(1*5 �dcT2��

D'4-} ñ rKt�Ü�Ünmnù BOf ù B m ïEBOf¡ï Ý B Ü @ ù BA@�ï Ý�Ý�cyDsrKt�Ü�Ünmnù BOf ù9Ý BA@ ù9Ý65 DsrKtnÜ�Ünm ïCBOf�ï Ý BA@�ï Ý

ò¡õ1 , �3287 1 `39;: 2 /(1@5 � 7 / �4- �=;E2 /=< �./ 2C2 /(13741 5���cB2��

D'4-} ñ%>�ñ rKt�Ü�Ünmnù B m ïCB m î B m B m � B m S BOf Ý B Ü @ ù BA@AïCBA@ î BA@ BA@ � BA@ S Ý�Ý�cÜ�Ü,DsrKt�Ü�Ünm ù BOf Ý BA@ ù Ý?5 DsrKtnÜ�Ünm ï BOf Ý BA@ ï Ý�Ý � DsrKtnÜ�Ünm î BOf Ý BA@ î Ý�Ý@5Ü,DsrKt�Ü�Ünm BOf Ý BA@ Ý-� DsrKt�Ü�Ünm � BOf Ý BA@ � Ý � DsrKt�Ü�Ünm S BOf Ý BA@ S Ý�Ý

�¡õ � �B/47 / �B5P1�z - �T5 /87 9 �=;E/(1�50/+1 2S1�-D7�� 7 / �4-02��Ds{�|�}q|�{PÜ�Ü @ ù BOf ù BOf�ï BA@AïEBOf î BOf Ý B Ü �ù B � ïCB � î Ý�Ý

Page 10: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

M_õ1 , �32 �4;!5 � /47 /��T5 1�z - �T5 /87 9 �=;E/(1�50/+1 2S1�-D7�� 7 / �4-02��

D 4-} ñ {�|�}?|~{PÜ�Ü @ ù BOf ù BOf�ïCBA@�ïCBOf î BOf B mnù B � ù B � ïCB m ïEB � î B � Ý B Ü ��ù B � ïCB � î B � ù B � ïCB � î Ý�Ý�cD {�|~}?|�{PÜ�Ü @ ù BOf ù BOf¡ïCBA@AïCBOf î BOf Ý B Ü ��ù B � ïCB � î Ý�Ý65D {�|~}?|�{PÜ�Ünmnù B � ù B � ï B m ïEB � î B � Ý B Ü � ù B � ï B � î Ý�Ý

R�R Q ���������� ��EV X[Z \ Z�]^ZC_��λÚ�Ê9Þ ã�ØjÎ�Ö�Î�Ú�Í &X5��CL�� � Ù ã�Ø�Ê�Í�ʺßàÊnÓrÍ�è�Ù�͹Ö�Î�ßlÕ�Ø�Î�Ó¼á­ØjÎ�Ù�Ó�Ì��lÎ�Ö¯Í�è_Î*=#> ? ADC Ì�Ï_Î�Ù�Ó�Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á Ì�Ú+Í�è_ÎÜ8=E? ��� B != $���� ñ����%'& B != $���� ñ���ñ����%(& B != $���� � ���%(& B != $���� ñ ��� � � �%(& Ý�ñ�è�á¡Ò�ØjÌ�Ï ËµÊ¡Ï_Î�Ó � Þ�è_ÎlØ�ÎÑÍ�è_Î É?>_@�ßàÊnÚ�ÖjÌ�Ö�Í�Ö¹Ê�ÛLÜ g B�e^BOf4B � ÝÖjÕ�ßjè�Í�èPÙ�Í e Ù�ÚPÏ g Ù�Ø�λã�ØhÌ�˧Î�Ö¯Ö�Ù�Í�Ì�Ö�Û�áºÌ�Ú_þ e k�Ü g $ å9ݯÙ�Ú�Ï f4B � � g ij Ù�ØjÎ�ØjÙ�Ú�Ï_ÊnË�Î�Ó¼Î�˵Î�Ú�Í�Ö÷Ö�Ù�Í�Ì�Ö�Û�á¡Ì¨Ú_þ4 3lk 7.3�Ü f ÝYcy4[3lk 7.3�Ü � Ý�c e õhg©Ù�Ú�Ï e Þ�̨Ó�Ó_Ù�Ó�Ö�Ê3Ö�ÎlØ�Ô�Î�Ù�ÖGÍ�è_Î�ã�Õ_ÒPÓ�Ì�ß:ãPÙ�ØhÙ�˧ÎlÍ�ÎlØjÖ:Ì�Ú§Í�è_Î�Ø�Î�Ó�Ù�Í�̼ÊnÚ�Ö DsrKt � Ds{�|�}?|~{ �Ù�ÚPÏ©Í�è_Î�̼Ø%ßàÊn˧ãDÊnÖ�Ì�Í�̼ÊnÚ�Ölõ��Î3ë�ØjÖ�ͳÏ�Î�Ö�ßàØj̼ÒDÎrÍ�è_Î�ã�ØjÊ�Í�Ê¡ßàÊnÓVõ

� �43 36/87�5 + � 2S1�� ÐrÚ»ØjÎ�ßàÎ�̼ԡ̨Ú_þLã�Øh̼Ô�Ù�Í�αÌ�Ú_ãPÕ�Í#�,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�B @�? B ú�� � Ù�ÖjÖ�Õ�˵Ì�Ú�þ¹Í�è�Ù�Í }0:P; Ì�Ö3Ú_Ê�ÍÕ�Ö�Î�Ï+ÒDÎlÛ¦Ê�ØjÎ � ãPÙ�Ø�Ívá(@ = ãP̨ß�ýºÖ±Ù ØjÙ�ÚPÏ_ÊnË"! "$ g$# � ßàÊn˧ãPÕ_Í�Î�Ö { $ f � i � ü Ë§ÊºÏ g � Ö�Î�Ú�Ï�ֱ˧Î�Ö�Ö�Ù�þ�ÎÜ�% $ 7%$ 4."P/&$ 4(' ' * " B98':<;�B2} :<;=B2{ Ý�Í�Ê÷ãPÙ�Ø�Ívá�@h? � ˧Î�ÖjÖ�Ù�þ�Î]Ü,+.- /21.354�6.7.3 B98':<;�B2} :<;=B @C= B @h? B Ü { BOf4B2{'& � BOf Ý B Ü(! B !nÝ�ÝÍ�Ê != $ ��� ñ��)�%'& � Ù�Ú�Ï(ÊnÕ_Í�ãPÕ_Í�Ö*�O��IL���/9K5��5�� J�� B98O:<;=B2} :<;�B @>= B @�?~�hõ ö Û¦Í�ÎlØ5Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ3Í�è_Î�˧Î�Ö�ÖjÙ�þ�Î�Ö³Û¦Ø�ÊnË @>=Ù�Ú�Ï != $ ��� ñ����%(& Ø�Î�Ö�ãDÎ�ßàÍ�̼Ô�Î�Ó¼á � @�?�ÊnÕ_Í�ã�Õ_Í�Ö �O��IL���/9K5��5�� J�� B98O:<;=B2} :<;�B @>= B @�?~�hõ�GÖ�Ö�Î�Ú�Í�Ì�Ù�Ó¨Ó¼á#@C=�Ö�Î�Ú�Ï�Ö¯Ù �.Î�Ï_ÎlØjÖ�Î�Ú�ßàÊn˫˵̼Í�˧Î�Ú�Í»ä��:G�ç�Ê�Û3ÒP̼ͯú¹Í�Ê�@�?�Ù�Ú�Ï#Õ�Ö�Î�Ö±Í�è_Î != $���� ñ����%(& Ì�Ï�Î�Ù�ÓÛ�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á�Í�Ê2ãPØ�Ê9Ô�Î�Í�è�Ù�ͳè_Î�Î�̼Í�è�ÎlØ�ýºÚ_Ê9Þ�ÖGÍ�è_ÎrÏ�Ì�ÖjßàØ�ÎlÍ�Î�Ó¼Ê�þ�Ê�Û { ܸÌ�Ú«Þ�è�Ì�ßhè«ßlÙ�Ö�Î @>=�Ì�Ö�ßàÊn˵˵̼Í�Í�Ì�Ú�þÍ�ʵÒP̼Í;3�ݳÊ�Ø%Í�è_Î3Ï�̨Ö�ßàØ�ÎlÍ�Î�Ó¼Ê�þ§Ê�Û {*& � ÒPÙ�Ö�Î f ܸÌ�Ú¯Þ�è�Ì�ßjè©ßlÙ�Ö�Î @>=N̨Ö5ßàÊn˵˵̼Í�Í�Ì�Ú�þ§Í�ʵÒP̼Í�å9Ýhõ

� /���0T1P5 + �B2S1d� @ºÕ�ã�ãDÊnÖ�Î @C=5è�Ù�Ö�ßàÊn˫˵̼Í�Í�Î�Ï»Ò�̼Í�Ö3úyù � ú ï � Ù�Ú�ÏÑú î Í�Ê<@h?«Õ�Ö�Ì�Ú_þ©ßlÌ�ÏPÖ } :<; ù � } :<;_ï � Ù�Ú�Ï } :<; î �Ø�Î�Ö�ãDÎ�ßàÍ�̼Ô�Î�Ó¼á�õ��PÕ_Ø�Í�è_ÎlسÙ�Ö�Ö�Õ�˵Î�Í�è�Ù�Í:Í�è_Î�̼سßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�ÏPÌ�Ú_þ"�.Î�Ï_ÎlØjÖ�Î�Ú¯ßàÊn˵˫̼Í�˧Î�Ú�Í�Ö�Ù�Ø�Î { ùMc f � � i � ü � �{2ï c f �  i � ü  � Ù�Ú�Ï { î c f � Ì i � ü Ì õ0��Ê9Þ � Õ�ã-ÊnÚ�Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ5ã�Øj̼Ô�Ù�Í�Î>Ì�Ú_ãPÕ_Íu�,7%$ 4."P/21.354�6�7 B98O:P;�B98�8O:<;=B2} :<; ù B2}0:P;_ïCB2} :<; î B @h? B �� �@ = Ì�Ö{Í�Ê ã�ØjÊ9Ô�ÎaÍ�Ê @ ? Í�è�Ù�Í 4^1 9 î�:; ÜVú ù B ú ï Ý c ú î � ÕPÖ�Ì�Ú_þ#Ö�Õ_Ò_ñ�Ö�Î�Ö�Ö�̼ÊnÚ7Ì�Ï 8�8O:<; õ ��Î ëPØjÖ�Í�ßàÊnÚ�Ö�Ì�Ï�ÎlØÍ�è_ÎaÖ�̼Í�Õ�Ù�Í�Ì�ÊnÚ Þ�è_ÎlØ�Î c å�å�å�3 � Ì¨Ú Þ�è�Ì�ßjè=ßlÙ�Ö�Î 4^1 9 î�:; ̨Ö÷Í�è�Î � ö � � Ê�ãDÎlØjÙ�Í�̼ÊnÚ�õ � Ú=Í�è�Ì�Ö]Ö�Ì�Í�ñÕ�Ù�Í�̼ÊnÚ � @ = Ö�Î�Ú�Ï�֩˧Î�Ö�ÖjÙ�þ�ÎYÜ�% $ 7%$ 4."P/21.354�6�7 B98O:P;�B98�8O:<;=B2} :<; ù B2}0:P; ï B2} :<; î B ÷ݱÍ�Ê @ ? Ù�Ú�Ï�Ö�Î�Ú�ÏP֩˧Î�Ö�Ö�Ù�þ�ÎÜ,+.- /21�3 4�6�7.3 B98O:<;=B98(8':<;=B @C= B @�? B Ü { ù B2{2ïCB2{ î & � B2{ ù & � B2{2ï�& � B2{ î BOf Ý B Ü(!9ù B ! ïCB ! î B !9ù B ! ïCB ! î Ý�Ý-Í�Êq!= $ ��� ñ���ñ����%'& õ ö Û¦ñÍ�ÎlØ�Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ:Í�è�ÎGßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�ÏPÌ�Ú_þ³Ë§Î�Ö�Ö�Ù�þ�Î?Û�Ø�ÊnË != $ ��� ñ+��ñ����%(& � @h??ÊnÕ_Í�ãPÕ_Í�ÖV�O�dCL���/9J>KIL�L�N B98':<;=B98(8':<;�B2} :<; ù B} :<;_ï B2} :<; î B @C= B ��hõ� Ú�Í�Õ�̼Í�Ì�Ô�Î�Ó¼á � @>=NÌ�ÖGãPØ�Ê9ÔºÌ�Ú_þ�Í�è�Ù�Í�Ü�Ü�ÜVúyù�c 3�Ý 5¹ÜVú ï c 3�Ý�Ý��¹ÜVú î c=å9Ý�Ý 5LÜ�ÜVúyùMc7å9Ý��¹ÜVú ï c7å9Ý��¹ÜVú î c 3�Ý�Ýhõ� Ú÷Í�è_Î�ßlÙ�Ö�Î�Ê�Û?Ù�Ú�á¹Ê�Í�è�ÎlØ�ÒP̨Ú�Ù�Ø�á¯Ê�ãDÎlØjÙ�Í�̼ÊnÚ�Öw4^1 9 î�:; � ̼Í�ßlÙ�Ú{ÒDÎ3Þ%Øh̼Í�Í�Î�ÚLÙ�ÖrÙ«ßàÊn˵ã-ÊnÖj̼Í�̼ÊnÚ¹Ê�Û � ö � � ÖÙ�Ú�ÏÑÍ�è_Î�ÚJãPØ�Ê9Ô�Î�Ï Ö�Í�Îlã Ò�á»Ö�Í�ÎlãAõ<@ = Þ�Ì�Ó�ÓGÚ_ÎlÎ�ÏJÍ�Ê]ßàÊn˵˵Ì�Í3Í�Ê]Ù�Ó�Ó?Í�è_ίÌ�Ú�Í�ÎlØj˧Î�Ï�̨Ù�Í�ΫÒP̼Í�Ö2Ù�Ú�ÏÑã�Ø�Ê9Ô�ÎÎ�Ù�ßjè � ö � � Ê�ãDÎlØjÙ�Í�̼ÊnÚ]Ì�ÖrßàÊ�Ø�Ø�Î�ßàÍlõr�_Ê�Ø�Î��_Ù�˧ãPӼΠ� ßàÊnÚ�Ö�̨Ï_ÎlØrÍ�è_ΧßlÙ�Ö�Î�Þ�è_ÎlØ�Î c 3"3"3_å§Ì¨Ö�Í�è�Î ö � �Ê�ãDÎlØjÙ�Í�̼ÊnÚ�õ���Ê�Í�̨ßàÎ�Í�èPÙ�Í @ �<msc @ �$m*� @ �$m¹×%è�ÎlØ�ÎlÛ¦Ê�ØjÎ � Í�ʯã�ØjÊ9Ô�Î�Í�è�Ù�Í�ú î c1úyùV�÷ú ï � @>=GÚ_ÎlÎ�ÏPÖ�Í�ÊßàÊn˵˵̼Í�Í�Ê(Ù«Ú�ÎlÞ ÒP̼Í�ú c úyùX�©ú ï Õ�Ö�Ì�Ú_þ§Í�è�Î�ã�Ø�Ê�Í�ʺßàÊnÓ�Ì�Ú¹Í�è_Î�É:Ên˫˵̼Í5ãPè�Ù�Ö�Î � Ù�Ú�ϹÍ�è�Î�Ú¹ã�Ø�Ê�Ô�Î3Í�è�Ù�ÍÒDÊ�Í�èLú c úhùX�¹ú ï Ù�ÚPϯÍ�è�Ù�Í�ú î c ú �©ú õ

* / �=-02C;X1 / 5 + � 2S1�� @_Õ_ã�ãDÊnÖ�Î @ = èPÙ�Ö«ßàÊn˵˵Ì�Í�Í�Î�Ï­ÒPÌ�Í�Ö«ú ù Ù�Ú�ÏYú ï � Ù�Ú�Ï @ ? èPÙ�Ö«ßàÊn˵˵Ì�Í�Í�Î�Ï ÒP̼ͫú . � Õ�Ö�̨Ú_þÌ�Ï_Î�Ú�Í�̼ë�ÎlØjÖ } :<; ù � } :<;�ï � Ù�Ú�Ï � } :<; � Ø�Î�Ö�ãDÎ�ßàÍ�̼Ô�Î�Ó¼á�õ ��Õ_ØjÍ�è_ÎlاÙ�Ö�Ö�Õ�˧Î(Í�è�Ù�Í�Í�è_ίßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�Ï�̨Ú_þ��.Î�Ï_ÎlØjÖ�Î�ÚßàÊn˵˵̼Í�˵Î�Ú�Í�Ö±Ù�Ø�Î { ù c f � � i � ü � � { ï c f �  i � ü  � Ù�Ú�Ï { . c f ��7 i � ü 7 õ ��Ê�Þ � Õ_ãDÊnÚ_ΩØ�Î�ßàÎ�Ì�Ô¡Ì�Ú�þã�Øj̼Ô�Ù�Í�Î%Ì�Ú_ã�Õ_Í�7%$ 4."</7" 3102,4365 7.3 B98':<;�B2} :<;=B2} :<; ù B2} :<;_ï B � } :<;�B @�?�� � Ù�Ö�ÖjÕ�˵Ì�Ú_þrÍ�è�Ù�Í } :<; Ì�Ö>Ú_Ê�Í�ÕPÖ�Î�ϧÒDÎlÛ¦Ê�ØjÎ � @C=ĄÖÍ�Ê�Ê�ÒPÓ�̼Ժ̼ÊnÕ�Ö�Ó�á3Í�ØjÙ�Ú�Ö�Û¦ÎlØjÖ�ÒP̼ͳúyü87NÍ�Ê,@�? � Õ�Ö�Ì�Ú�þ2Ö�Î�Ö�Öj̼ÊnÚ«Ì�Ï 8':<; Ù�Ú�Ï(Í�è_Î�ßàÊn˫˵̼Í�˧Î�Ú�Í:Ì¨Ï } :<; Û�Ê�Ø�Í�è�Î�Ú_ÎlÞÒP̼Í5úyü 7 õ � Ú�Í�ÕP̼Í�̼Ô�Î�Ó¼á � @C=NÖ�Î�Ú�ÏPÖ�ÍvÞ�Ê �Î�Ï�ÎlØjÖ�Î�Ú©ßàÊn˵˵Ì�Í�˧Î�Ú�Í�Ö ��, ù�Ù�Ú�Ï , ï � Þ�è_ÎlØjÎ , ùrÌ�Ö³Ù�ßàÊn˵˵Ì�Í�˧Î�Ú�Í

å�3

Page 11: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Í�Ê÷úyù�Õ�Ö�Ì�Ú_þ±ÒPÙ�Ö�Î { . � Ù�ÚPÏ , ï ̨Ö�Ù©ßàÊn˵˵̼Í�˧Î�Ú�Í�Í�Ê÷ú ï Õ�Ö�Ì�Ú�þ±Ò�Ù�Ö�Î { . & � õ � Í�Ù�Ó¨Ö�ʹÖ�Î�Ú�Ï�Ö z�ù§Ù�Ú�Ï z ïþ�Î�Ú_ÎlØjÙ�Í�Î�Ï©Õ�ÖjÌ�Ú_þ3Í�è�Î�Ö�Ù�˵Î�ØhÙ�Ú�Ï_Ên˵Ú_Î�ÖjÖ�Ù�Ö , ù Ù�Ú�Ï , ï õ � Ûú . c 3 � Í�è_Î�Ú%@ ? ý¡Ú_Ê�Þ�Ö:Í�è_ÎrÏ�Ì�Ö�ßàØ�ÎlÍ�ÎrÓ¼Ê�þ�Ê�Û{ .Ù�Ú�ϱßlÙ�Ú(ßhè_Î�ß�ý±Ì¼Û , ù�Ì�Ö�Ù2ßàÊn˫˵̼Í�˧Î�Ú�Í�Í�Êo�lÎlØ�Ê�Ê�Ø:Ú�Ê�Í � Ù�ÚPϫ̼Û�ú/.>c7å � Í�è_Î�Ú @h?�ýºÚ_Ê9Þ�Ö�Í�è_ÎrÏ�Ì�Ö�ßàØjÎlÍ�ÎÓ¼Ê�þµÊ�Û { . & � Ù�Ú�ÏLßlÙ�Ú¹ßjè_Î�ßjý¹Ì¼Û , ï ̨Ö%Ù§ßàÊn˵˵̼Í�˵Î�Ú�Í%Í�Ê �lÎlØ�Ê«Ê�Ø�Ú�Ê�Ílõ��Ê�ÞaÞ�Î:ã�Ø�ʺßàÎlÎ�Ï2Í�Ê�Í�è�Î:Ï_ÎlÍ�Ù�̨Ó�ÖlõG@>=�ØjÙ�ÚPÏ_Ên˵ӼárãPÌ�ßjý¡Ö.ø¡ù B ø ï "$ g # Ù�Ú�Ï2ßàÊn˵ãPÕ_Í�Î�Ö�z�ù $ f�� � � z ï $ f��  �, ù $ { � �. i � ü � � Ù�ÚPÏ , ï $ Ü { . & � Ý �  i � ü  õ @>=�Í�è_Î�Ú Ö�Î�ÚPÏ�Ö2˵Î�Ö�Ö�Ù�þ�λÜ�% $ 7%$ 4."P/7"030 , 365,7.3 B98':<;�B2} :<;=B2}0:P; ù B} :<; ï B � } :<;�B z ù B z ï B , ù B , ï Ý%Í�Ê%@ ? Ù�Ú�Ï÷Ö�Î�Ú�Ï�Ö�Í�è_Î2Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ«Û¦ÊnÕ_Ør˧Î�Ö�ÖjÙ�þ�Î�ÖrÍ�ʱÍ�è�Î�Ì�Ï�Î�Ù�ÓAÛ¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á!= $ ��� � ���%'& õ��

Ü,+.- /21.354�6�7.3 B98':<;�B2} :<;�� 3"3 B @>= B @h? B Ü , ù B � B2{ . B2{ ù B � BOf Ý B ÜVúhù B ø�ù B !�ù9Ý�ÝÜ,+.- /21.354�6�7.3 B98':<;�B2} :<;�� 3_å B @ = B @ ? B Ü , ï B � B2{ . & � B2{ ï B � BOf Ý B ÜVú ï B ø ï B ! ï Ý�ÝÜ,+.- /21.354�6�7.3 B98':<;�B2} :<;�� å�3 B @>= B @h? B Ü,z�ù BOf=B å B , ù B2{ . B � Ý B ܸø¡ù B 3 B úyù9Ý�ÝÜ,+.- /21.354�6�7.3 B98':<;�B2} :<;�� å�å B @>= B @h? B Ü,z ï BOf=B å B , ï B2{ . & � B � Ý B ܸø ï B 3 B ú ï Ý�Ý

ö Û�Í�ÎlØ�Í�èPÌ�Ö � @ = ÎlØjÙ�Ö�Î�Ö�ø ù Ù�Ú�Ϲø ï õö Û�Í�ÎlØ�ØjÎ�ßàÎ�̼ԡ̨Ú_þ§Í�è_Î3˧Î�ÖjÖ�Ù�þ�Î3Û�Ø�ÊnË @ = Ù�Ú�Ï©Û�ÊnÕ_Ø�˧Î�ÖjÖ�Ù�þ�Î�Ö�Û�Ø�ÊnË Í�è�Î2Ì�Ï_Î�Ù�ÓDÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá != $ ��� � ���%(& � @ ?Ï_Ê¡Î�Ö5Í�è_Î�Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ¹Ü�Ê�Í�è_ÎlØ�Þ�Ì�Ö�Î @�?�Ù�ÒDÊ�Ø�Í�ÖhÝhõ� ÛNú/.>c 3 � Í�è_Î�Ú©ßjè�Î�ß�ý©Ì¼ÛCz ��7ù c , ù5Ë§ÊºÏ g � Ù�Ú�Ï©Ö�ÎlÍ%ú $ 3�̼Ûá�Î�Ö%Ù�Ú�Ï©ú $ å�Ê�Í�è_ÎlØ�Þ�̨Ö�Î�û�̼Ûú/.>c7å � Í�è_Î�Úßjè_Î�ßjý(̼ÛIz ��7ï c , ï Ë§ÊºÏ g � Ù�Ú�ϱÖ�ÎlÍ�Ö5ú $ 3�Ì�Û-á�Î�Ö5Ù�ÚPϱú $ å�Ê�Í�è_ÎlØ�Þ�̨Ö�Î�õI��Ê�Þ ú%Ì�ÖGÍ�è_Î�Ò�̼ÍW@�?�Ø�Î�ßàÎ�̼Ô�Î�Ölõ��Î��ºÍ � @h??ãPÌ�ßjý¡Ö�Ù5ØhÙ�Ú�Ï_ÊnË�! "$ g i# Ù�Ú�Ï�Ö�ÎlÍ�Ö { $ f � i � ü ˵ʡÏwg � Ö�Î�Ú�Ï�Ö�˧Î�Ö�Ö�Ù�þ�Î�Ü,7%$ 4."P/&$ 4(' ' * " B98':<;=B2}0:P;�B2{ ÝÍ�Ê�ãPÙ�Ø�Ívá,@>= � Ö�Î�Ú�ÏPÖ>˧Î�Ö�Ö�Ù�þ�Î2Ü,+.- /21.354�6.7.3 B98':<;�B2} :<;=B @h? B @>= B Ü , ù B � B z�ù B2{ B � BOf=B , ï B � B z ï B2{ B � BOf Ý B ÜVú B ! . B ! Bú B ! . B !�Ý�Ý-Í�Ê�Ì�Ï_Î�Ù�Ó�Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá != $���� ñ ��� � ���%(& � Ù�ÚPÏ�ÊnÕ_Í�ãPÕ_Í�ÖV�O�dCL���/�� ����� B98':<;�B2} :<;=B @C= B @h? B2}0:P; ù B2} :<;�ï B � } :<;=Bú��hõ´�NÌ�Ú�Ù�Ó¨Ó¼á � Ù�Û�Í�ÎlØLØjÎ�ßàÎ�̼ԡ̨Ú_þ ˧Î�Ö�Ö�Ù�þ�Î�Ö÷Û�Ø�ÊnË @�?»Ù�Ú�Ï != $ ��� ñ ��� � � �%'& � @C=§ÊnÕ_Í�ã�Õ_Í�Ö �O��IL���/9K5��5�� J�� B8O:P;�B2} :<;�B @>= B @�? B2} :<; ù B2} :<;_ï B � } :<; �hõ

,I5�1 -#5 + � 2S1�� @ºÕ_ã�ãDÊnÖ�Î'@C=Nè�Ù�Ö5ßàÊn˵˵Ì�Í�Í�Î�Ï©Ù�ÒP̼Í%ú%Í�ÊI@�?�Õ�ÖjÌ�Ú_þ�Ö�Î�Ö�Ö�̼ÊnÚ¯Ì�Ï 8':<; � Ù�ÚPÏ©ßàÊn˵˵̼Í�˧Î�Ú�Í5Ì�Ï } :<; õ��Õ_Ø�Í�è�ÎlØ�Ù�Ö�Ö�Õ�˧Î�Í�è�Ù�Í�Í�è�Î�ßàÊn˫˵̼Í�˧Î�Ú�Í�Ì�Ö { c f � i � ü ˧ʺÏwg�õ ��Ê9Þ�Õ�ã-ÊnÚ{Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ(ã�Øh̼Ô�Ù�Í�ΧÌ�Ú�ãPÕ_Í�,7%$ 4."</<4^1[7-, B98':<;�B2} :<;=B @ = B @ ? � � @ = Ê�ãDÎ�Ú�Ö�Í�è_ΧÒP̼Í�ú3Ò�á{Ö�Î�Ú�Ï�Ì�Ú�þ±Ë§Î�ÖjÖ�Ù�þ�ÎLÜ�% $ 7%$ 4."P/<4^1[7-, B98O:P;�B2} :<;�B ú B !�ÝrÍ�Ê@h? � Þ�è_ʧÍ�è_Î�Ú¹Ô�ÎlØj̼ë�Î�Ö³Í�è�Ù�Í { c f � i � ü ˧ʡÏ�g � Ù�Ú�ϯÊnÕ_Í�ãPÕ�Í�Ö3ÜO��IL���/�� ����� B98':<;�B2} :<;=B @C= B @h? B úlÝ:̼Û�Í�è_ÎÔ�ÎlØj̼ëPßlÙ�Í�Ì�ÊnÚ¹Ì�Ö5Ô�Ù�Ó�Ì�Ï�õ×%è�Ì�Ö5ֳ̨Î��ºÙ�ßàÍ�Ó¼á¯Í�è_Î3Ê�ã-Î�ÚPÌ�Ú_þ�Ê�Û.Ù.�Î�Ï�ÎlØjÖ�Î�Ú¹ßàÊn˵˫̼Í�˧Î�Ú�Ílõ

*,+ 1��T/(1 3 w 9�� � �T),���A/ ��� �'�� *�&����'�(a/0�0%����}(:�D%"/L%�#&%�!�&X5��CL�� ���8#��=�8��!.1o�O�& �!Ç� U����A/ ���P=#>@? ADC ��),�8 "!N9����}#E�/0��%��} "!Ç�0/01 �-� / ����Ü8=E? ��� B != $���� � ���%'& B != $���� � �E� ���%'& B != $ ��� � � �%(& B != $���� � ��� � ���%'& Ý:�P�,1,+���� ) 'H%S)k��! �<, ����)��/5 ) �(:/0����� �H'6 "!Ç� �#��0%"�k�r )������&�� ����L��� �? *������'5�-�=< ���O *�����k< �

� /�� �4;�� OAÎlÍLH=ÒDÎ5Ù�Ú«Ù�Ï_Ô�ÎlØjÖjÙ�Ø�á2Í�èPÙ�Í?Ê�ãDÎlØjÙ�Í�Î�ÖGÙ�þnÙ�Ì�ÚPÖ�Í>ã�Ø�Ê�Í�ʺßàÊnÓ�&X5��CL��5õ���Î�ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ù�ÚµÌ�Ï_Î�Ù�Óºã�ØjÊ¡ßàÎ�Ö�ÖÙ�Ï�Ô�ÎlØjÖ�Ù�Ø�á Q»Ö�Õ�ßjè�Í�èPÙ�ÍNÚ�Ê�Î�Ú�Ô¡Ì�Ø�ÊnÚ�˧Î�Ú�ÍLJ�ßlÙ�Ú§Ï�Ì�Ö�Í�̨Ú_þnÕ�Ì�Ö�è�Þ�è_ÎlÍ�è_ÎlØ>̼ÍNÌ�ÖNÌ�Ú�Í�ÎlØjÙ�ßàÍ�Ì�Ú_þ�Þ�̼Í�è,H Ù�Ú�Ï0&X5��CL��̨Ú(Í�è�ΫÜ8=E?���� B != $���� ñ+���%(& B != $���� ñ ��ñ����%(& B != $ ��� � � �%(& B != $���� ñ ��� � � �%'& Ý�ñ�è�á¡Ò�ØjÌ�ϯ˧ʺÏ_Î�Ó � Ê�Ø%Þ�̼Í�è QYÌ�Ú±Í�è_Î�Ì�Ï�Î�Ù�ÓDã�Ø�ʺßàÎ�Ö�Ö³Û¦Ê�Ø=#> ? ADC>õ

�_Ê�صÖ�Ì�˧ã�Ó�Ì�ßl̼Ívá � Þ:αÙ�ÖjÖ�Õ�˧αÍ�è�Ù�Í�ÊnÚ�Ó¼á�ÊnÚ_ΩßàÊ�ã¡á»Ê�Û#=#> ? ADC#Ì�Ö�Ù�ßlßàÎ�Ö�Ö�Ò�á J � ÖjÌ�Ú�ßàΫÊ�Í�è�ÎlØ�Þ�Ì�Ö�αÞ:αßlÙ�ÚÎ�Ù�ÖjÌ�Ó¼á(ÏPÕ_ãPÓ�Ì�ßlÙ�Í�ÎrÍ�è_Î2Ù�ßàÍ�̼ÊnÚ�Ö5Û�Ê�Ø!QYÛ¦Ê�Ø%Î�Ù�ßhèLßàÊ�ã¡á©Ê�Û;=#> ? ADC>õ

ö Í�Í�è_Î�Ò-Îlþn̨Ú�Ú�Ì�Ú_þrÊ�Û�Í�è_Î�Ì�Ï_Î�Ù�Ó_ãPØ�Ê¡ßàÎ�ÖjÖ � Í�è�Î�Ì�Ï_Î�Ù�Ó�Ù�Ï_Ô�ÎlØjÖ�Ù�Ø�á Q Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�ÖGÍ�è_Î�ßàÊn˫˧ÊnÚ«ØjÎlÛ¦ÎlØ�Î�ÚPßàÎ�Ö�Í�Øj̨Ú_þ=E? ����õ��N̼ØjÖ�Í � Q þ�Î�Ú_ÎlØhÙ�Í�Î�Í�Þ:ʵã�Øj̨˧Î�Ö>gLÙ�Ú�Ï e Ê�ÛÙ�ã�ã�ØjÊ�ã�ØjÌ�Ù�Í�Î�Ö�Ì��lÎ�Ö�ÕPßjè¯Í�è�Ù�Í e k�Ü g $ å9Ýhõ ��Î��¡Í Q­þ�Î�Ú_ÎlØjÙ�Í�Î�Ö

$# GYP&R�R��*­WGaKLZ�P8KaP&�\�NKLZ*G|[�M�� R}P&SLG|Q*[\E*P&S��?R0K�SL[\EN��RDz�P�E*M ÃaG|�NR�G� 3r������ÅKLVÅ[�E*MN[\UOP8KLG|KLZNG|UDV�ENUOP&KLGDE*P&KL[\V�E;V&X R0K�SL[�EN��r�Ã|[µKLZWR0K�SL[\E*����

å�å

Page 12: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

ØhÙ�Ú�Ï_ÊnË f � g�ij Ö�ÕPßjè{Í�è�Ù�Í�4[3lk 7.3�Ü f ÝMc e � Ù±ØjÙ�Ú�Ï�ÊnË � "$ g$# � Ù�Ú�Ï � $ f . õ�×%è_Î�Ú Q�ÊnÕ_Í�ãPÕ�Í�Ö�Ü g B�e^BOf4B � ÝrÙ�ÖÍ�è�Î�ßàÊn˫˧ÊnÚ�Ø�ÎlÛ�ÎlØ�Î�Ú�ßàΧÖ�Í�Øj̨Ú_þ¯Û¦Ê�Ø�=E?��h�µÙ�Ú�Ï�Ö�Í�Ê�Ø�Î�Ö � õ ��Ê�Í�Ì�ßàΧÍ�è�Ù�Í�Ö�Ì�ÚPßàÎ Q ýºÚ_Ê�Þ�ÖrÍ�è_εÏ�Ì�ÖjßàØ�ÎlÍ�ΧӼÊ�þ¯Ê�Û �Ò�Ù�Ö�Î f � Ì�Í%ßlÙ�Ú©ÎS!�ÕP̼Ô�Ê¡ßlÙ�Í�Î2Í�è_Î"�.Î�Ï_ÎlØjÖ�Î�Ú÷ßàÊn˵˵̼Í�˵Î�Ú�Ílõ� Õ�ØjÌ�Ú_þ§Í�è_Î2Ì�Ï_Î�Ù�ÓDã�Ø�ʺßàÎ�Ö�Ö � QYØjÕPÚ�Ö5ÙµÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏLßàÊ�ã�á¯Ê�Û H(õ��¹Î�ÖjÖ�Ù�þ�Î�Ö�ØjÎ�ßàÎ�̼Ô�Î�ÏLÛ�Ø�ÊnË J*Ù�Ø�Î3Û¦Ê�Ø�Þ³Ù�ØjÏ_Î�Ï

Í�Ê«Í�è_Î3Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï H � Ù�ÚPϹ˧Î�Ö�Ö�Ù�þ�Î�Ö�Ö�Î�Ú�Í%Ò¡á±Í�è�Î3Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï%H Í�ʫ̼Í�Ö%Î�Ú�Ժ̼Ø�ÊnÚ�˵Î�Ú�Í�Ù�ØjÎ�Û¦Ê�ØjÞ:Ù�ØjÏ�Î�Ï©Í�Ê J«õ�QÙ�Ó¨Ö�Ê«Ö�ÎlÎ�Í�è_Î_(}�x+�!Ç��#r���& ),���{ܸÖ�Îlίä�å�ò�ç�ݳÊ�Û?Ù�Ó�Ó�Í�è�Î2˧Î�Ö�ÖjÙ�þ�Î�Ö�Û�Ø�ÊnË Õ�Ú�ßàÊ�Ø�ØjÕ�ã�Í�Î�Ï©ãPÙ�Ø�Í�Ì�Î�Ö%Í�Ê<= > ? ADC Ù�Ú�ÏL˵Ù9áÏ�Î�ßlÌ�Ï_Î�Þ�è_ÎlÍ�è_ÎlØ�Ê�Ø�Ú_Ê�Í�Í�ʯÛ�Ê�Ø�Þ³Ù�ØjÏ]Í�è_Î�Ö�ε˧Î�ÖjÖ�Ù�þ�Î�Ölõ � Ú{Í�è_εßlÙ�Ö�ΧÊ�Û =#>@?BADC � Ù�Ó�ÓN˧Î�Ö�Ö�Ù�þ�Î�Ö�Í�ÊJ=#>@? ADCaÙ�Ø�Į̂Ú(Í�è�Î�ãPÕ_ÒPÓ�̨ß�è_Î�Ù�Ï�ÎlØ3ܸ˧Î�Ù�Ú�Ì�Ú_þ§Í�èPÙ�Í QYßlÙ�Ú¹Ö�ÎlÎ�Ù�Ó¨ÓDÍ�è_Î�Ö�Î�˧Î�Ö�ÖjÙ�þ�Î�Ö%Û�Ø�ÊnË Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�ϱãPÙ�Ø�Í�̼Î�Ö³Í�Ê<=#>@? ADCÝÎ��_ßàÎlã�Í2Í�è_ΫÒPÌ�Í3ú�Ì�Ú�Í�è_Î 7%$ 4."</&$ 4(')'+* "�˧Î�Ö�Ö�Ù�þ�Î�õJ��ίÕ�Ö�ÎJéa$�ì¹Í�Ê÷Ï�Î�Ú_Ê�Í�ΫÍ�è�̨Ö�ã�Øj̼Ô�Ù�Í�εÒPÌ�Ílõ ��Õ_ØjÍ�è_ÎlØj˧Ê�Ø�Î �Q7Ù�Ó�Ö�Ê÷ãPÓ�Ù9á¡Ö3Í�è_ΫØ�ÊnÓ�Î�Ö�Ê�Û5Í�è_Î(Û¦ÊnÕ_Øo2rÉcO`K Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�̼Î�Ö != $���� ñ����%(& � != $���� ñ ��ñ����%(& � != $���� � ���%(& � Ù�Ú�Ï != $���� ñ ��� � � �%'& õ�PÕ_Ø�Í�è_ÎlØj˵Ê�Ø�Î � QY˵Ù�̨Ú�Í�Ù�Ì�ÚPÖ5Ù§Ø�Î�ßàÊ�ØjÏ©Ê�Û>ßàÊn˵˫̼Í�Í�Î�Ï©ÒP̼Í�Ö³ÎlØhÙ�Ö�Î�ÏLÌ�Ú = > ? ADC õG×�è_ÎrØ�Î�ßàÊ�ØjÏLßàÊnÚPÖ�Ì�Ö�Í�Ö³Ê�ÛNÍ�Õ_ãPÓ�Î�ÖÊ�Û:Í�è_Î�Û�Ê�ØjË<Ü } :<;�B @>= B @�? B2{ B ! B úlÝ � Ì�Ú�Ï�̨ßlÙ�Í�Ì�Ú_þ«Í�èPÙ�Í�Í�è_εßàÊn˵˵̼Í�Í�Î�Ï»ÒP̼Í�Ê�Û:Ì�Ï } :<; Ì�ÖrÛ�Ø�ÊnË @C=:Í�Ê @�? � Í�è_ΧÒ�̼Į́Örú�Ù�Ú�Ï{Í�è_Î<�.Î�Ï_ÎlØhÖ�Î�Ú�ßàÊn˵˵̼Í�˵Î�Ú�Í�Ì�Ö { c � ü i f � õ � Û�@ = Ì�Ö�Õ�ÚPßàÊ�Ø�ØjÕ_ã�Í�Î�Ï � Í�è�Î�Ú]Í�è_Î�ÒP̼Í�ú3̨Ö�Ö�ÎlÍrÍ�Ê­é�� � į̀Ú�Ï�Ì�ßlÙ�Í�Ì�Ú�þ2Í�è�Ù�Í!QYßlÙ�Ú¹ÎS!¡Õ�̼Ô�ʺßlÙ�Í�Î�ÊnÚLÍ�è�Ì�Ö³ÊnÚ_Î3Ù�ÚPÏ©Ì�Ú¯Í�è�Ì�Ö%ßlÙ�Ö�Î � { c f � õ��Î��ºÍ � Þ�ÎrÏ_Î�Ö�ßàØh̼ÒDÎ%Í�è_ÎrÒDÎ�è�Ù�Ժ̼Ê�Ø:Ê�Û�Q Ì�Ú«Ï�ÎlÍ�Ù�Ì�ÓVõ Q Ì�Ö�é�ÎlÔ�Î�Ú�Í�ñ�Ï_Øh̼Ô�Î�Ú � ì§Ì�Ú(Í�è�Ù�Í5̼Í�Ö:Ù�ßàÍ�̼ÊnÚ�Ö³Ù�Ø�ÎrÍ�Øj̼þ�þ�ÎlØ�Î�Ï

Ò¡á±Í�è_Î3˧Î�Ö�ÖjÙ�þ�Î�Ölõ���Î2Ì�Í�Î�˵Ì��lÎ�Í�è_Î�ÒDÎ�è�Ù9Ô¡Ì�Ê�Ø5Ê�Û.Q+Ù�ßlßàÊ�ØjÏ�Ì�Ú�þ§Í�ʵÍ�è_Î3˧Î�ÖjÖ�Ù�þ�Î�Ö�̼Í%Ö�ÎlÎ�Ölõå�õ � Û Q Ö�ÎlÎ�Ö©ÙaßàÊ�Ø�ØjÕ�ã�Í�Î�Ï+ãPÙ�Ø�Í�á @C=�Ö�Î�Ú�Ï#Ùa˧Î�Ö�ÖjÙ�þ�έÜ�% $ 7%$ 4."P/&$ 4(' '+* " B98':<;�B2} :<;=B2{ ݯÍ�Ê ã�Ù�Ø�Í�á @h? � Í�è�̨Ö˧Î�Ù�Ú�Ö�Í�è�Ù�Í!@ = Ü�Þ�è�Ì�ßhè¹Ì�Ö%ßàÊnÚ�Í�Ø�ÊnÓ�Ó¼Î�ϹÒ�á%HµÝ:Ì�Ö%ßàÊn˵˵Ì�Í�Í�Ì�Ú_þ«Ù§Ò�̼Í5Í�ÊM@ ? õ�×�è_Î�ÚRQ+Î��¡ãDÎ�ßàÍ�Ö�٫˧Î�Ö�Ö�Ù�þ�ÎÜ,+.- /21�3 4�6�7.3 B98O:<;=B2} :<;�B @>= B @�? B Ü { BOf=B2{*& � BOf Ý B Ü(!�ù B ! ï Ý�Ý�Û�Ø�ÊnË @>=�Í�Ê != $ ��� ñ����%'& õ ö Û¦Í�ÎlØ�Ø�Î�ßàÎ�̼ԺÌ�Ú_þ{Í�è�Ì�Ö�˧Î�Ö�ñÖ�Ù�þ�Î � Q=ßjè_Î�ßjý¡Ö2Ì¼Û { c f � � Ù�ÚPÏ�Ì¼Û {*& � c f �  õ � Û�Ú�Î�̼Í�è_ÎlØ�è_ÊnÓ�ÏPÖ � Q ̼þnÚ_Ê�ØjÎ�ÖrÍ�è�Ì�Ör˧Î�Ö�Ö�Ù�þ�Î � Ö�Ì�Ú�ßàÎ @ =Û�Ù�̨Ó�ÖGÍ�è�Î�ãPØ�Ê�Ê�Û�õ � Û�Ò-Ê�Í�è¯è_ÊnÓ¨Ï � Í�è�Î�Ú H ˵Ù�Ú�Ù�þ�Î�Ö5Í�Ê�ëPÚ�Ï(ÊnÕ_ͳÍ�è_Î�Ï�Ì�ÖjßàØ�ÎlÍ�Î�Ó�Ê�þ�Ê�Û � Ü�Þ�è�Ì�ßhè¯Ì�Ö !�ù�$ ! ï Ý �Ù�Ú�ÏRQ+Ì�Ú¯Í�è�Ì�Ö5ßlÙ�Ö�Î�Ù�ÒDÊ�Ø�Í�Ö%Ù�Ú�Ï©ÊnÕ�Í�ãPÕ_Í�Ö%Ù�Ú©ÎlØjØ�Ê�Ø�˧Î�Ö�ÖjÙ�þ�Î���������� ��������� � c f � ��� � Â������� Û { c f � � ÒPÕ_Í {*& � c f �  � Í�è�Î�ÚX@C=:Ì�Ö�ßàÊn˵˵̼Í�Í�̨Ú_þ(Í�Ê©Í�è�ΧÒP̼Í�ú c 3ºû.Ì¼Û { c f � � ÒPÕ_Í {*& � c f �  �Í�è_Î�Ú @C=?Ì�Ö�ßàÊn˵˵̼Í�Í�̨Ú_þ«Í�Ê(Í�è_Î3ÒPÌ�Í�ú�c å�õ � Ú¹Î�̼Í�è�ÎlØ�ßlÙ�Ö�Î � Q#Ø�Î�ßàÊ�ØjÏ�Ö�Ü } :<;�B @�? B @C= B2{ B !9ü B úlÝ5Ù�Ú�Ï{Ö�Î�Ú�ÏPÖ˧Î�Ö�Ö�Ù�þ�Î �,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�B @�? B ú��:Í�ʵÊnÚ©ÒDÎ�è�Ù�Ó¼ÛÊ�Û.@>=NÍ�Ê�=#> ? ADC?õ

ò¡õ � ÛWQ Ö�ÎlÎ�Ö�ٯ˧Î�ÖjÖ�Ù�þ�Î��,7%$ 4."P/&$ 4(' '+* " B98O:P;�B2} :<;�B @ ? B $ ��Û�Ø�ÊnË'Ù�Ú�Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï{ãPÙ�Ø�Ívá<@ = Í�ÊJ= > ? ADC � Í�è�̨Ö˧Î�Ù�Ú�Ö.Í�è�Ù�Í�@>=�Ì�ÖNßàÊn˵˵Ì�Í�Í�Ì�Ú_þrÙ�ÒPÌ�ÍÍ�Ê @�?�õ���Ê�Í�Ì�ßàγÍ�è�Ù�Í QJßlÙ�ÚPÚ_Ê�Í>Ö�ÎlγÍ�è_Î:Ù�ßàÍ�ÕPÙ�ÓºÒP̼Í.ú�Ì�Ú2Í�è_γ˧Î�Ö�Ö�Ù�þ�Î �Ù�Ú�Ï»Í�è�ΫãPÓ�Ù�ßàΫÊ�Û�ú�Ì�Ö�ëPÓ¨Ó¼Î�Ï]Ò�á éa$�õ ì Q7Í�è_Î�ÚJÛ¦Ê�ØjÞ:Ù�ØjÏPÖ�Í�è�Ì�Ö3˧Î�Ö�Ö�Ù�þ�αÍ�ÊLÍ�è_Î.=#>@?BADC � Ù�Ú�ÏJÊ�Ò�Í�Ù�Ì�Ú�ÖÙ«ØjÎ�ßàÎ�̼ã�Ís�O��CL���/9K5��8�� J�� B98O:P;�B2} :<;�B @>= B @�?~�%Ì�Ú¹Ø�ÎlÍ�Õ�ØjÚ�õ#��Î��¡Í � Q#Û�Ù�ý�Î�ÖrÙ.�.Î�Ï_ÎlØhÖ�Î�Ú÷ßàÊn˫˵̼Í�˧Î�Ú�Í�Ò¡áãPÌ�ßjý¡Ì�Ú�þ¹Ù¹ØjÙ�Ú�Ï�ÊnË !6"$ g # � Ö�ÎlÍ�Í�Ì�Ú�þ { $ f � � Ù�ÚPÏJÖ�Î�Ú�ÏPÌ�Ú_þ©Ë§Î�Ö�Ö�Ù�þ�Î]Ü�% $ 7%$ 4."P/&$ 4(' ' * " B98':<;�B2} :<;=B2{ Ý�Í�ÊãPÙ�Ø�Ívá<@�?�õ3Ù�Ú�Ï{Ø�Î�ßàÊ�ØhÏ�Ö§Ü }0:P;�B @>= B @h? B2{ B ! B ��ÝhõeQ Ù�Ó¨Ö�Ê(Û�Ê�Ø�þ�Î�Ö�٩˧Î�ÖjÖ�Ù�þ�ιÜGF�HI/9J KILML�N B98':<;�B2} :<;=B @C= B @�? BÜ { BOf=B2{*& � BOf Ý�Ý�Û�Ø�ÊnË != $ ��� ñ����%'& Í�Ê%@�?�õ���Ê�Í�̨ßàÎ2Í�è�Ù�Í�̼Û�@�?�Ì�Ö�ßàÊ�ØjØjÕ_ã�Í�Î�Ï � Í�è_Î�Ú÷Í�è_Î�˧Î�Ö�Ö�Ù�þ�Î�Ö�Í�ÊR@h?�Ù�Ø�ÎÌ�Ú¯Û¸Ù�ßàÍ�Ï�̼ØjÎ�ßàÍ�Î�ϯÍ�ÊIH(õ

�¡õ � ÛAQ7Ö�ÎlÎ�Ö3Ù÷ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï�ãPÙ�Ø�ÍváP@>=3ܸßàÊnÚ�Í�Ø�ÊnÓ�Ó¼Î�ÏJÒ�áPHµÝrÖ�Î�Ú�Ï»Ù÷˧Î�Ö�Ö�Ù�þ�Î]Ü�% $ 7%$ 4."P/7"030 , 365,7.3 B98':<;�B2} :<;=B2}0:P; ù B} :<;_ï B � } :<;�B z�ù B z ïCB , ù B , ï Ý�Í�Ê]Ù�ÚÑÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï�ãPÙ�Ø�Ívá @h? � Í�è�Ì�Ö3˵Î�Ù�Ú�Ö3Í�èPÙ�Í,@>=3ܸßàÊnÚ�Í�ØjÊnÓ�Ó¼Î�ÏÑÒ�áXHµÝ�̨ÖãDÎlØ�Û¦Ê�Øh˵Ì�Ú_þ3Ù�Ú±Ê�ÒPӨ̼ԡÌ�ÊnÕ�Ö>Í�ØjÙ�Ú�Ö�Û¦ÎlسÍ�Ê�@ ? õ � Ú(Í�è�Ì�Ö:ßlÙ�Ö�Î � Qaë�ØjÖ�ÍGÔ�ÎlØj̼ëPÎ�ÖGÍ�è_Îr̼ͳè�Ù�Ö³Ö�Í�Ê�Ø�Î�ϱÍ�è_Ø�ÎlÎ�Í�Õ_ãPÓ�Î�ÖÜ } :<; ù B @>= B @h? B2{ ù B !9ù B úyù9Ý � Ü } :<;_ï B @C= B @�? B2{2ïCB ! ïCB ú ï Ý � Ù�Ú�Ï1Ü � } :<;�B @�? B @>= B2{ . B ! . B ��Ý � Ù�Ú�Ï#Ï_Ê¡Î�Ö¯Ú_Ê�Í�è�Ì�Ú_þ ̼ÛÚ_Ê�Ílõ ��Î��¡Í � Q ãPÓ�Ù9á¡Ö³Í�è_Î�̨Ï_Î�Ù�Ó-Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá != $ ��� � � �%(& Í�ʵØjÎ�ßàÎ�̼Ô�Î�Û¦ÊnÕ�Ø%˧Î�Ö�Ö�Ù�þ�Î�Ö�Û¦Ø�ÊnËg@>= � Ù�Ú�ϯÔ�ÎlØj̼ë�Î�ÖÍ�è�Ù�Í

, ùuc � ���M� i { ��� Â. � { ùMc � ���M� i f ��� Ì, ï c � �  � i�Ü { . & � Ý � Â� � {2ï c � �  � i f � Â-Ì

z�ùuc f�� �M� � , ùMc { � �M�. i � � � Âz ï c f��  � � , ï c Ü { . & � Ý �  � i � � Â�Â

å�ò

Page 13: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

� Û8�ù�ù c�úyù � QYÙ�ÒDÊ�Ø�Í�Ö%Ù�Ú�ϹÊnÕ_Í�ãPÕ_Í�Ö � ��� � ��� ��� � � � c f 9 ��� Ì � � � : � 9 ü ���d���M� : ������ Û8� ï ù c�ú ï � QYÙ�ÒDÊ�Ø�Í�Ö%Ù�Ú�ϹÊnÕ_Í�ãPÕ_Í�Ö � ��� � ��� ��� � � � c f 9 � Â-Ì � �  : � 9 ü  �d�  � : ������ Û � ù ï c�úyù � Q#Ù�ÒDÊ�Ø�Í�Ö%Ù�Ú�ϯÊnÕ_Í�ã�Õ_Í�Ö � � ����� � � ����� � c f � Ì�� 9 ���  � � �M� : � 9 � �  � ü � : � � �� Û � ï�ï c�ú ï � Q#Ù�ÒDÊ�Ø�Í�Ö%Ù�Ú�ϯÊnÕ_Í�ã�Õ_Í�Ö � � ����� � � ����� � c f � Ì�� 9 � Â� � �  � : � 9 � Â�Â�� � Â� � �  ��� ü ï : � � �Ð�Í�è_ÎlØ�Þ�Ì�Ö�Î � Þ�Î{è�Ù�Ô�Î�� ù�ù c � ù ï c ú ù � � ï ù c � ï�ï c ú ï � � ù î c ! ù � � ï î c ! ï � � ù�ù c � ù ï � Ù�Ú�Ï� ï ùwcA� ï�ï õ�@¡ÎlÍ�Í�̨Ú_þ«ø¡ù�c �ù ï Ù�Ú�ÏLø ï c � ï�ï � Þ:Î3Ö�ÎlÎ3Í�èPÙ�Í�Ü,z�ù B z ïCB , ù B , ï ݳ̨Ö�Ì�Ú�Ï�ÎlÎ�Ï©Ê�ÛNÍ�è_Î3Û�Ê�ØjË Ù�Öã�Ø�Î�Ö�ßàØh̼ÒDÎ�ϱ̨گã�Ø�Ê�Í�ʺßàÊnÓ &u8��IL��%õö Û�Í�ÎlØ�Í�è�Ì�Ö � Q Ö�Î�ÚPÏ�Ö2˵Î�Ö�Ö�Ù�þ�Î �,7%$ 4."</7" 3102,4365 7.3 B98O:<;=B2} :<;�B2} :<; ù B2} :<; ï B � } :<;=B @ ? ��Í�ÊF= > ? ADC � Ù�Ú�Ï Þ³Ù�̼Í�Ö�Û¦Ê�ØÍ�è_Îa˧Î�Ö�Ö�Ù�þ�Î �O��CL���/9K ��8�� J�� B98O:P;�B2} :<;�B @>= B @�? B2} :<; ù B2} :<;_ï B � } :<; �÷Ò�Ù�ß�ý=Û¦Ø�ÊnË Ì¼Ílõ ��Î��ºÍ � Q Û�Ù�ý�Î�Ö�ÙßàÊn˵˵̼Í�˵Î�Ú�Í2Ò�á�Ö�ÎlÍ�Í�Ì�Ú�þ !6"$ g # � { $ f � � Ù�ÚPÏJÖ�Î�Ú�ÏPÌ�Ú_þ¯Í�è_Î(˵Î�Ö�Ö�Ù�þ�Î{Ü,7%$ 4."P/&$ 4(' ' * " B98':<;�B2} :<;=B2{ Ý�Í�Ê@ = � Ù�Ú�Ï»Û�Ê�Ø�þ�Î�Ö3ÙL˧Î�Ö�Ö�Ù�þ�Î]ÜGF�HI/9J KILML�N B98':<;�B2} :<;=B @ ? B @ = B Ü , ù B � B z ù B2{ B � BOf=B , ï B � B z ï B2{ B � BOf Ý�Ý�Û¦Ø�ÊnË!= $���� ñ ��� � � �%'& Í�ÊM@>=�õ

M_õ � Û Q Ö�ÎlÎ�Ö�Ù{˧Î�Ö�ÖjÙ�þ�Î �,7%$ 4."P/7"030 , 365,7.3 B98':<;�B2} :<;=B2}0:P; ù B2} :<;�ï B � }0:P;�B @�?J��Û�Ø�ÊnË Ù�Ú Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�ÏJãPÙ�Ø�Í�á6@C=%Í�Ê=#>@?BADC � Í�è�Ì�Ö�˧Î�Ù�Ú�Ö�Í�è�Ù�Í�@C=�Ì�Ö2Í�ØjÙ�Ú�Ö�Û�ÎlØ�ØjÌ�Ú�þLÙ{ÒP̼Í3Í�ÊX@�?±Ê�ÒPÓ�̼Ժ̼ÊnÕ�ÖjÓ¼á�õ � Û @h?¯Ì�Ö�Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï � Í�è_Î�ÚQ=ßlÙ�Ú Ö�Ì�Ë�Õ�Ó�Ù�Í�ΫÍ�è_Ϋã�ØjÊ�Í�Ê¡ßàÊnÓGÎ�Ù�Ö�Ì�Ó¼á�õ � Ú»ãPÙ�Ø�Í�̨ßlÕ�Ó�Ù�Ø � Ì�Í3Û�Ê�Ø�Þ:Ù�ØhÏ�Ö�Í�è�Ì�Ö2˧Î�Ö�Ö�Ù�þ�Î(Í�Ê�=#> ? ADC � Ê�Ò�Í�Ù�Ì�Ú�Öٵ˧Î�Ö�ÖjÙ�þ�Î$�O��IL���/9K5��5�� J�� B98':<;=B2}0:P;�B @>= B @h? B2} :<; ù B2}0:P;_ïCB � }0:P; ��Ì�Ú¯Ø�ÎlÍ�Õ_ØjÚ¹Û¦Ø�ÊnË =#>@? ADC � Û¦Ê�Ø�þ�Î�Ö�ٵ˧Î�Ö�ñÖ�Ù�þ�ÎÑÜ�% $ 7%$ 4."</7" 3102,4365 7.3 B98':<;=B2}0:P;�B2} :<; ù B2} :<; ï B � } :<;=B z ù B z ï B , ù B , ï Ý�Û�Ø�ÊnË @ = Í�Ê @ ? Ù�Ú�ÏaÍ�è�Î�ÚaÙ�˧Î�Ö�Ö�Ù�þ�ÎÜ�% $ 7%$ 4."P/&$ 4(' ' * " B98':<;�B2} :<;=B2{ ݳÛ�Ø�ÊnË @�?rÍ�ÊM@>= � Þ�è�ÎlØ�Î�z�ù B z ïCB , ù B , ï Ù�Ø�Î�þ�Î�Ú_ÎlØjÙ�Í�Î�Ï÷Ù�Ö%Ì�Ú¯Í�è_Î�ãPØ�Ê�Í�Ê¡ßàÊnÓܸÕ�Ö�Ì�Ú�þ3úhùMc ú ï c 3�Ý?Ù�Ú�Ï { $ f � Û�Ê�Ø�Ù�ØjÙ�Ú�Ï_ÊnË ! "$ g # õ Q Ù�Ó�Ö�Ê3Ö�̨Ë2Õ�Ó¨Ù�Í�Î�Ö>Í�è_Î�Ù�ãPã�Ø�Ê�ã�Øj̨Ù�Í�Î%˧Î�Ö�Ö�Ù�þ�Î�ÖÍ�Ê«Ù�Ú�Ï©Û�Ø�ÊnË Í�è�Î 2rÉcO`K1Ì�Ï_Î�Ù�ÓDÛ¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�̼Î�Ö != $���� � ���%(& Ù�Ú�Ï != $ ��� ñ ��� � � �%'& õ��Ê�Þ � ßàÊnÚ�Ö�Ì�Ï_ÎlØ{Í�è_Î ßlÙ�Ö�Î Í�è�Ù�Í @ ? Ì�Ö÷ßàÊ�Ø�ØjÕ_ã�Í�Î�ÏAõ ö Ö�Ö�Õ�˧ÎÑÍ�è�Ù�ÍPQ'è�Ù�Ö÷̨Ú�Í�ÎlØjÚPÙ�Ó�Ó¼á�Ö�Í�Ê�Ø�Î�Ï=Í�Õ_ãPÓ�Î�ÖÜ } :<; ù B @>= B @h? B2{ ù B !9ù B ��Ý � Ü } :<;_ï B @C= B @�? B2{2ïCB ! ïCB ��Ý«Ù�ÚPÏ7Ü � } :<;�B @�? B @C= B2{ . B ! . B ú/.vÝhõ � Û2̼ͯÌ�Ö¯Ú_Ê�ͯÍ�è_Î�ßlÙ�Ö�Î �Q Ï_Ê¡Î�Ö�Ú_Ê�Í�è�̨Ú_þ_õ�Ð�Í�è�ÎlØ�Þ�Ì�Ö�Î,Q Ï_Ê¡Î�ÖrÍ�è_ΧÛ�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ_õ � Ú�Í�Õ�̼Í�̼Ô�Î�Ó¼á � Q ßlÙ�Ú»Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Í�è_Î�ã�Ø�Ê�Í�ʺßàÊnÓ?ßàÊ�Ø�ñØ�Î�ßàÍ�Ó¼á»Ö�̨Ú�ßàΫ̼Í�ýºÚ_Ê�Þ�Ö�Í�è_Î(ÒP̼Í2ú/.�ÕPÖ�Î�ϻҡá�Í�è_Î(ßàÊ�ØjØjÕ_ã�Í�Î�Ï�ØjÎ�ßàÎ�̼Ô�ÎlØ � Ù�Ú�ÏJÌ�Í2Ó¼Î�Ù�ØjÚPÖ�Û¦ØjÊnË =#> ? ADC­Í�è_ÎÍ�ØjÙ�Ú�Ö�Û�ÎlØ�Ø�Î�Ï«ÒP̼Ílõ � Í�Ö�ÎlÍ�Ö�Í�è_Î�ÏPÙ�Í�Ù3Û�Ê�ØGÍ�è_Î�Ê�ã�ãDÊnÖ�̼Í�Î�ßlÙ�Ö�Î�Ü�å $©ú�.�ÝNØjÙ�Ú�Ï�Ên˵Ӽá � ÖjÌ�Ú�ßàÎ%Í�è_Î�Ø�Î�ßàÎ�̼Ô�ÎlØ5Ö�è_ÊnÕPÓ�ÏÚ_Ê�Í�Ó¼Î�Ù�ØhÚ©Ù�Ú�á¡Í�è�Ì�Ú_þµÙ�ÒDÊnÕ_Í5Í�è�Î3Ï�Ù�Í�ÙµßàÊ�Ø�ØjÎ�Ö�ãDÊnÚ�Ï�Ì�Ú_þ�Í�ÊLåN$Ñú�.�õ?ô:Î�Ó¼Ê�Þ Ì�Ö%Í�è_Î�Ï_ÎlÍ�Ù�Ì�Ó¼Î�ÏLÏ_Î�Ö�ßàØj̼ãPÍ�̼ÊnÚ�õܸÙ�Ýw�_Ê�Ø�Þ³Ù�ØjÏ(Í�è_Î�˧Î�ÖjÖ�Ù�þ�Î�Í�Ê�=#>@? ADC{Ù�Ú�ϵÊ�ÒPÍ�Ù�Ì�Ú�Ö>Í�è_Î�˧Î�Ö�Ö�Ù�þ�Î �O��IL���/�� ����� B98':<;=B2}0:P;�B @>= B @h? B2} :<; ù B

} :<;_ï B � } :<;�B ú��%Ì�Ú¯Ø�ÎlÍ�Õ_ØjÚAõÜ�ÒDÝ @¡ÎlÍ 8 ù B98�ïEB98 î "$ g # � z�ü 7 $ f�� � ��, ü 7 $ Ü { . & � ü87 Ý � � i � ü � z ï � ü 7 $ f��  � Ù�Ú�Ï , ï � ü 7 $ f�� Ì õܸß9Ý @¡Î�Ú�Ï�˧Î�Ö�ÖjÙ�þ�Î2Ü�% $ 7%$ 4."P/7"030 , 365,7.3 B98O:P;�B2} :<;�B2} :<; ù B2} :<;�ï B � }0:P;�B z�ù B z ïCB , ù B , ï Ý�Í�Ê @�?5Ù�Ú�ϧÖ�̨Ë2Õ�Ó¨Ù�Í�Î�Í�è_Î

Û¦ÊnÕ�Ø�˧Î�Ö�Ö�Ù�þ�Î�Ö�Û¦ØjÊnË != $���� � ���%'& Í�ÊM@�?�õܸÏ-Ýw>%Î�ßàÎ�̼Ô�ί˧Î�Ö�ÖjÙ�þ�Î{Ü,7%$ 4."P/&$ 4(' '+* " B98':<;�B2} :<;=B2{ Ý3Û�Ø�ÊnË @�?(Ù�Ú�ÏJ˧Î�ÖjÖ�Ù�þ�Î{Ü,+.- /21.354�6�7.3 B98':<;�B2} :<;=B @h? B @>= B

Ü , ù B � B z�ù B2{ B � BOf=B , ï B � B z ï B2{ B � BOf Ý B Ü �ù B � ïCB � î B � ù B � ïCB � î Ý�Ý:Û�Ø�ÊnË @�?�Í�Ê != $���� ñ ��� � � �%'& õÜ�Î�ݹɳè_Î�ßjý2Ì¼Û , ùuc � � � i z � Âù � { c � � � i f � Ì Ü�Þ:γßlÙ�Ó¨Ó¡Ì�Í�é�ÎlÔ�Î�Ú�Í�3�ì�Ý � Ê�Ø , ï c � � � i z � Âï � { c � � � i f � Ì

Ü�Þ�Î�ßlÙ�Ó¨Ó-Ì�Í�é�ÎlÔ�Î�Ú�Í3ålì�Ýhõ � ÛÚ�ÊnÚ_ÎrÊ�Û�Í�è_ÎrÎlÔ�Î�Ú�Í�Ö�è�Ù�ã�ãDÎ�Ú�Ö � Í�è_Î�Ú©Ö�Í�Ê�ã¹Ù�ÚPϯÛ�Õ_ØjÍ�è_ÎlØj˧Ê�Ø�Î � ̼þnÚ_Ê�ØjÎrÙ�Ó¨Ó% $ 7%$ 4."P/&$ 4(' ' * "�Ù�Ú�Ï % $ 7%$ 4."P/7"03102, 365,7.3N˧Î�Ö�Ö�Ù�þ�Î�Ö�Û¦ØjÊnË @h?�Þ�̼Í�è } :<; Ù�Ö�ßàÊn˵˫̼Í�˧Î�Ú�Í�Ì�Ï�õ

Ü�ÛhÝ � ÛGÎlÔ�Î�Ú�͵Ü�å�$Ñú/.vÝ%èPÙ�ã�ãDÎ�Ú�Ö � Ù�ÒDÊ�Ø�ÍrÙ�Ú�Ï÷ÊnÕ_Í�ãPÕ_ÍrßàÎlØ�Í�Ù�Ì�Ú{ÎlØ�ØjÊ�Ø�˧Î�ÖjÖ�Ù�þ�Î�Ölõ � Ú{ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�Ø � Ù�Ö�Ö�Õ�˵ÎÍ�è�Ù�Í2ú/.uc 3¹Ù�ÚPÏ]ÎlÔ�Î�Ú�ͱå§è�Ù�ã�ãDÎ�Ú�Ö � Í�è_Î�Ú»Þ�εè�Ù9Ô�Î z ï c f��  � , ï c f�� Ì � Ù�Ú�Ï , ï c � � � i z � Âï õ� Û � ù�c 3 � Í�è�Î�ÚaÊnÕ_Í�ãPÕ_Í � ��� � ��� � ��� � , ï c z � Âï � � � Î�Ó�Ö�ΩÊnÕ�Í�ãPÕ_Í � � ����� � � ����� � cf 9 � Ì � � Â�� �  : � � � ��� � õ?×%è�Î�Ö�̼Í�Õ�Ù�Í�̼ÊnÚ¯Û�Ê�Ø�ú î c7å�̨Ö5Ö�Ì�˵Ì�Ó¨Ù�Ø�õ

Ü�þ�ÝJ��Ê�Þ*Ù�ÖjÖ�Õ�˧εÍ�èPÙ�Í3ÎlÔ�Î�Ú�Í�ú/.�è�Ù�ãPã-Î�ÚPÖlõ �_Ê�Ø�Ö�Ì�˵ãPÓ�Ì�ßl̼ÍváLÞ:Î(Ù�Ö�ÖjÕ�˧εÍ�è�Ù�Í�ú/.�c 3ÑÜ�Í�è_αÖ�̼Í�Õ�Ù�Í�Ì�ÊnÚÛ¦Ê�ØLú/.sc å]Ì�Ö¯Ö�̨˵Ì�Ó�Ù�ØyÝhõ � Ú+Í�è�Ì�Ö¯ßlÙ�Ö�λßhè_Î�ß�ý#Ì¼Û �ù c ú�õ � Û3Ú_Ê�Í � Þ�λèPÙ�Ô�Î�Í�è�Ù�Ísz�ù c f�� � �

åS�

Page 14: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

, ù�c { � �. i � ü c � ü i f � Ì � Ù�Ú�Ï , ù�c � ��� i z � Âù c � ��� i f � � � �  õµ×%è_Î�Ú � Q Ù�ÒDÊ�Ø�Í�Ö�Ù�ÚPÏ�ÊnÕ_Í�ãPÕ_Í�Ö��������� ��������� � c f 9 � � � �  � ��7 : � 9 ü �d��� : � � �

ܸè-ݹÐ�Í�è_ÎlØ�Þ�Ì�Ö�Î � ̼Û_ú/.>c 3 � Í�è�Î�Ú�Ö�Í�Ê�Ø�Î?Í�Õ�ãPÓ¼Î%Ü } :<;�B @�? B @C= B2{ B � î B úlÝhû�Ê�Í�è_ÎlØjÞ�Ì�Ö�Î>Ö�Í�Ê�Ø�Î�Ü }0:P;�B @�? B @>= B2{ B � î B úlÝhõG¡õ � ÛMQ Ö�ÎlÎ�Ö]Ù+˵Î�Ö�Ö�Ù�þ�Î �,7%$ 4."</21.354�6.7 B98':<;=B2}0:P; ù B2} :<;�ï B2} :<; î B @�? B ��©Û�Ø�ÊnË Ù�Ú7Õ�Ú�ßàÊ�Ø�ØjÕ�ã�Í�Î�Ï ã�Ù�Ø�Í�á @>=«Í�Ê=#>@?BADC � Í�è�Ì�Ö�˧Î�Ù�ÚPÖrÍ�è�Ù�Íe@>=:Ì�Ö�ã�ØjÊ9ÔºÌ�Ú_þ¯Í�Ê @h?µÍ�è�Ù�Í�4^1 9 î�:; ÜVúyù B ú ï ÝMc ú î õ Q Í�è_Î�Ú]Û�Ê�Ø�Þ:Ù�ØhÏ�ÖrÍ�è�Ì�Ör˧Î�Ö�ñÖ�Ù�þ�Î:Í�Ê�Í�è_Î =#>@?BADC>õ6��Î��¡Í � Ù�Û�Í�ÎlØNØ�Î�ßàÎ�̼ԺÌ�Ú_þ�˧Î�ÖjÖ�Ù�þ�ÎE�O��CL���/9J>KILML�N B98':<;�B98�8':<;�B2} :<; ù B2} :<;_ï B2} :<; î B @>= B ��Û¦ØjÊnË =#> ? ADC � Q#Û�Ê�Ø�þ�Î�Ö�Í�è�Î�Ô�Ù�Øh̼ÊnÕ�Ö�˵Î�Ö�Ö�Ù�þ�Î�Ö�Û¦Ø�ÊnË Í�è_Î ORK1Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á != $ ��� ñ���ñ����%'& Í�Ê @�?�õW�_Ê�Ø�Î���ñÙ�˧ãPӼΠ� Ì�Û c å�å�å�3JÜ�Í�è_Î<� ö � � Û�Õ�ÚPßàÍ�̼ÊnÚ�Ý � Í�è_Î�ÚPQ Ö�Î�Ú�Ï�Ör˧Î�Ö�Ö�Ù�þ�ÎLÜGF�HI/9J KILML�N B98O:P;�B98�8O:<;=B @C= B @�? BÜ { ù B2{3ïCB2{ î & � B2{ ù & � B2{2ï & � B2{ î BOf Ý�Í�Ê!@h? � Ù�Ö�ÖjÕ�˵Ì�Ú_þ³Í�è�Ù�ÍZQ÷è�Ù�Ö�ØjÎ�ßàÊ�ØjÏ_Î�Ï�Í�Õ_ã�Ó¼Î�Ö?Ü } :<; ù B @>= B @h? B2{ ù B !9ù B ��Ý �Ü } :<;_ï B @>= B @h? B2{2ïCB ! ïCB ��Ý5Ù�Ú�ÏJÜ } :<; î B @C= B @�? B2{ î B ! î B ��Ýhõ �_Ê�Ø�Ê�Í�è_ÎlØ� � Ë�Õ�Ó¼Í�Ì�ãPÓ¼Îr˧Î�Ö�Ö�Ù�þ�Î�Ö�Þ�ÊnÕPÓ�ϯÒDÎ�Ö�Î�Ú�ÍÙ�ßlßàÊ�ØjÏ�Ì�Ú�þnÓ¼á � Ì�Ú�Ê�ØjÏ�ÎlØ?Í�Ê3Ö�Ì�Ë�Õ�Ó�Ù�Í�γÍ�è_Î5ãPØ�Ê9Ô�Î%ã�è�Ù�Ö�Î%Ì�Ú�ã�ØjÊ�Í�Ê¡ßàÊnÓ�&X8�dCL���õ ö þnÙ�Ì�Ú � ̼Û4@ ? ̨Ö>ßàÊ�Ø�ØjÕ_ã�Í�Î�Ï �Í�è_Î�Ú©Í�è_Î2˧Î�Ö�Ö�Ù�þ�Î�Ö�Í�ÊM@h?3Ù�Ø�Î3Ì�Ú¯Û¸Ù�ßàÍ�ÏP̼Ø�Î�ßàÍ�Î�ϯÍ�ÊIH±õ

�¡õ � Û Q Ö�ÎlÎ�ÖLÙaßàÊ�Ø�ØjÕ_ã�Í�Î�Ï#ã�Ù�Ø�Í�á�@ = Ö�Î�ÚPÏ+٭˧Î�Ö�Ö�Ù�þ�έÜ�% $ 7%$ 4."</21.354�6.7 B98':<;=B98(8':<;�BEB2} :<; ù B2} :<; ï B2} :<; î B ÷Ý(Í�Ê@h? � Í�è�Ì�ֵ˧Î�Ù�ÚPÖµÍ�è�Ù�ÍM@>=«Ü¸ßàÊnÚ�Í�Ø�ÊnÓ�Ó¼Î�Ï­Ò�á HµÝ�Ì�Ö§ã�ØjÊ9ÔºÌ�Ú_þ�Í�ÊJã�Ù�Ø�Í�á�@�?LÙ�ÒDÊnÕ_Í«Ù»ÒPÌ�Ú�Ù�Ø�á Ø�Î�Ó�Ù�Í�Ì�ÊnÚaÛ¦Ê�ØÍ�è_Ø�ÎlÎ�ÒP̼Í�Ölõ � ÚµÍ�è�Ì�Ö�ßlÙ�Ö�Î � Qaßjè_Î�ßjý¡Ö5̼ÛDÍ�è_Î�Í�Õ_ãPÓ¼Î�Ö�Ü } :<; ù B @>= B @h? B2{ ù B !�ù B úhù9Ý � Ü } :<;_ï B @C= B @�? B2{2ïCB ! ïCB ú ï Ý>Ù�Ú�ÏÜ } :<; î B @>= B @h? B2{ î B ! î B ú î ÝDÙ�Ø�Î�Ù�Ó¨Ó�Ö�Í�Ê�Ø�Î�Ï � Ù�Ú�Ï2Ö�Í�Ê�ãPÖ�Ê�Í�è_ÎlØ�Þ�Ì�Ö�Î�õ6��Î��¡Í � Q{Ø�Î�ßàÎ�̼Ô�Î�ÖÛ�Õ�Ø�Í�è_ÎlØ˵Î�Ö�Ö�Ù�þ�Î�ÖÛ¦Ø�ÊnË@C=_Í�Ê�Í�è_ÎGÌ�Ï�Î�Ù�Ó�Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á != $ ��� ñ���ñ����%'& � Ô�ÎlØj̼ëPÎ�ÖAÍ�è_Î�Ë©õ � Û¡Ù�Ó�Ó�Í�è_Î?ã�ØjÊ�Ê�Û¸Ö�Í�Ê != $ ��� ñ+��ñ����%(& Ù�Ø�Î?Ô�Ù�Ó�Ì�Ï � Þ�è�Ì�ßhèÌ�˧ãPӨ̼Î�Ö:Í�è�Ù�ÍM4^1 9 î�:; ÜVúyù B ú ï Ý�c�ú î � Q+Í�è_Î�Ú¹Ö�Î�ÚPÏ�Ö%˧Î�Ö�ÖjÙ�þ�Î �,7%$ 4."</21.354�6.7 B98':<;=B98(8':<;�B2} :<; ù B2} :<;�ï B2} :<; î B @�? B ��Í�Ê�=#> ? ADC?û_Ê�Í�è_ÎlØ�Þ�Ì�Ö�Î�Ï_ʵÚ_Ê�Í�èPÌ�Ú_þ_õ

��õ � Û'Q Ö�ÎlÎ�֫ٻ˵Î�Ö�Ö�Ù�þ�Î �,7%$ 4."P/<4^1[7-, B98O:P;�B2} :<;�B @>= B @�?7��Û¦Ø�ÊnË<Ù�Ú­Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï ãPÙ�ØjÍ�á�@C=rÍ�Ê�=#>@? ADC � Í�è�̨Ö˧Î�Ù�Ú�Ö>Í�èPÙ�Í�@ = Ì�Ö>Ê�ãDÎ�Ú�̨Ú_þ�Ù3ßàÊn˵˵̼Í�Í�Î�ϧÒ�̼ÍNÍ�Êe@ ? õ QÑßjè�Î�ß�ýºÖ?Û�Ê�Ø?Í�è_Î%Ö�Í�Ê�Ø�Î�Ï«Í�Õ_ãPÓ¼Î�Ü } :<;=B @ = B @ ? B2{ B ! B ��ÝÙ�Ú�ÏaÏ_Ê¡Î�Ö�Ú_Ê�Í�è�Ì�Ú_þ�Ú_Ê�Ö�ÕPßjè Í�Õ_ãPӼίÌ�Ö�Í�Ê�Ø�Î�ÏAõÑ×%è_Î�Ú Q Û�Ê�Ø�Þ³Ù�ØjÏ�Ö�Í�è�Ì�Ö�˧Î�ÖjÖ�Ù�þ�ιÍ�Ê =#> ? ADC�Ù�Ú�Ï Þ:Ù�̼ÍÍ�ÊÑØjÎ�ßàÎ�̼Ô�ÎLÍ�è_Î]˧Î�Ö�ÖjÙ�þ�Î��O��CL���/�� ����� B98O:<;=B2} :<;�B @>= B @�? B ú���Ò�Ù�ß�ý Û�Ø�ÊnË<Ì�Ílõ ��Î��ºÍ � Q Ö�Î�Ú�Ï�֫˧Î�Ö�Ö�Ù�þ�ÎÜ�% $ 7%$ 4."P/<4^1[7-, B98O:P;�B2} :<;�B ú B ! $ ú�i � ÝrÍ�ÊP@h?nõ�%�ÎlØ�Î � ̨Ö�Í�è_ίÏ�Ì�Ö�ßàØjÎlÍ�Î(Ó¼Ê�þ÷Ê�Û � ÒPÙ�Ö�Î f õ ö þnÙ�Ì�Ú � Ì¼Û @h?¯Ì¨ÖßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï � Í�è�Ì�Ö5˧Î�ÖjÖ�Ù�þ�Î2Ì�Ú¯Û¸Ù�ßàÍ%þ�Ê¡Î�Ö%Í�ÊIH±õ

æ¡õ � ÛZQ­Ö�ÎlÎ�Ö³Í�è�Ù�Í5Ù�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï(ã�Ù�Ø�Í�á @C=NÖ�Î�Ú�ϯÙ�˧Î�Ö�Ö�Ù�þ�εÜ�% $ 7%$ 4."</<4^1[7-, B98':<;�B2} :<;=B ú B !nݳÍ�ʵÙ�Ú±ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�ÏãPÙ�Ø�ÍváM@h? � Í�èPÌ�ÖG˧Î�Ù�ÚPÖGÍ�è�Ù�ÍA@>=�ܸßàÊnÚ�Í�ØjÊnÓ�Ó¼Î�Ï(Ò�á HµÝ>Ì�Ö�Ê�ãDÎ�Ú�Ì�Ú_þ2Ù�ÒP̼Í�Í�Ê,@�?�õ � Ú«Í�èPÌ�ÖGÖ�Ì�Í�Õ�Ù�Í�̼ÊnÚ � Q ßhè_Î�ß�ýºÖÍ�è_Î2Ö�Í�Ê�Ø�Î�ÏLÍ�Õ�ãPÓ¼Î(Ü } :<;=B @ = B @ ? B2{ B ! � B ú � Ýhõ � Û { c � ü i f � � Í�è�Î�ÚLÍ�è�̨Ö%Ì�Ö�Ù�Ú{Ì�Ú�Ô�Ù�Ó¨Ì�ϯÊ�ãDÎ�Ú�Ì�Ú_þµÙ�Ú�Ï Q�Ï_Ê�Î�ÖÚ_Ê�Í�è�Ì�Ú�þ_õNÐ�Í�è_ÎlØjÞ�Ì�Ö�ÎWQJßjè_Î�ßjý¡Ö>Ì�Û�úpc�ú � õ � Û�̼Í.̨Ö.Ú�Ê�ÍÍ�è_Î5ßlÙ�Ö�Î � Q»Ù�Ò-Ê�ØjÍ�Ö.Ù�ÚPÏ2ÊnÕ�Í�ãPÕ_Í�Ö.Ù�Ú�ÎlØ�Ø�Ê�ØN˧Î�Ö�Ö�Ù�þ�Î��������� ��������� � c f 9 ��� � � : � 9 ü � ü � : � � � � Û>úpc�ú � � QYÍ�è_Î�Ú¹Ö�Î�Ú�ÏPÖ%˧Î�Ö�Ö�Ù�þ�Î$�,7%$ 4."</<4^1[7-, B98':<;�B2} :<;=B @C= B @�?~�Í�Ê�=#> ? ADC?õ

�¡õ � Û'H ßàÊ�Ø�ØjÕ_ã�Í�Ö«Ù]ãPÙ�Ø�Í�á�@>= � Q*Ú_ÎlÎ�Ï�Ö§Í�ÊJÖjÌ�Ë2ÕPÓ�Ù�Í�ι̼Í�ÖµÌ�Ú�Í�ÎlØjÚ�Ù�Ó5Ö�Í�Ù�Í�Î � Þ�è�Ì�ßjèaßàÊnÚPÖ�Ì�Ö�Í�ÖµÖ�ÊnÓ¼Î�Ó¼á Ê�Û�Í�è_ÎÌ�Ú_Û�Ê�Øj˵Ù�Í�̼ÊnÚ±Í�Ê«Ù�Ó¨Ó�Í�è�Î�Õ�Ú_Ê�ãDÎ�Ú_Î�ϯßàÊn˵˵̼Í�˵Î�Ú�Í�Ölõ �+è_Î�ÚR@>=NÌ�Ö5ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï � Í�è_Î�Ö�Î�ßàÊn˵˫̼Í�Í�Î�Ï©ÒP̼Í�Ö³Ù�Ø�Îþn̼Ô�Î�Ú�Í�Ê?Q�õ6��Ê9Þ � Ö�Ì�Ú�ßàÎLQ]ýºÚ_Ê9Þ�ÖÍ�è�Î�Ï�Ì�ÖjßàØ�ÎlÍ�Î�Ó¼Ê�þ�Ê�Û � � Ì�ÍßlÙ�Ú�Ê�ãDÎ�Ú�Í�è_Î�Ö�γßàÊn˵˵Ì�Í�˧Î�Ú�Í�Ö.Í�ÊrÙ�Ú�á3Ô�Ù�Ó�Õ�Î�õ@¡Ê Q�Ö�Ì�˧ã�Ó¼á÷þ�Î�Ú�ÎlØjÙ�Í�Î�Ö§Í�è_Î�Ö�ίÊ�ãDÎ�Ú�Ì�Ú�þnÖ3Ò¡á»ëPÚ�ÏPÌ�Ú_þLÙ�Ó�Ó�Í�è_ίÍ�Õ_ãPÓ�Î�Ö3Ê�Û%Í�è�αÛ�Ê�ØjË Ü } :<;�B @ = B�� B2{ B ! B ��Ý �Û¦Ê�Ø�Î�Ù�ßhèaÖ�Õ�ßjè Í�Õ_ã�ӼΫëPÚPÏ�Ì�Ú_þ¹ÊnÕ�Í2Í�è_ΩßàÊ�Ø�ØjÎ�Ö�ãDÊnÚ�Ï�Ì�Ú_þLÒP̼Í�úµÛ¦ØjÊnË Í�è�αßàÊ�Ø�ØhÕ_ã�Í�̼ÊnÚ Ì�Ú_Û�Ê�Øj˵Ù�Í�̼ÊnÚ � Ù�Ú�Ïã�Ø�ʺÏ�Õ�ßlÌ�Ú�þ ! � c !N${ú i � Ù�Ö:Í�è_ÎrÊ�ãDÎ�Ú�Ì�Ú�þ2Ì�Ú_Û�Ê�Øj˵Ù�Í�Ì�ÊnÚ�õGQ Ù�Ó¨Ö�ʧßjè�Ù�Ú_þ�ÎrÍ�è_ÎrÍ�Õ_ã�Ó¼Î�Ü } :<;=B @C= B�� B2{ B ! B ��Ý:Í�ÊÜ } :<;�B @>= B�� B2{ B ! �8B úlÝhõ

×�è�Ì�Ö:ëPÚ�Ì�Öjè_Î�Ö³Í�è_Î�Ï_Î�Ö�ßàØh̼ã�Í�̼ÊnÚ¯Ê�Û.Q�õ��Ê�Þ � Þ:Î3Ö�è_Ê�Þ Í�è�Ù�Í

����� c ����� ]��c " ��� ���]��c " ��� ����

]��c " ��� � � ��]��c " ��� ñ ��� � � �� ��� | � 4�� ] _�] ` fh  ba�¥GY�[ c�� ����� ] d4] ` B

å M

Page 15: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Þ�è�Ì�ßjè]Ì�˵ãPÓ�̼Î�Ö�ÊnÕ_Ø�Í�è_ÎlÊ�Ø�Î�Ë©õ ��εë�ØhÖ�Í�Ö�ý�ÎlÍ�ßjè»Í�è�Î�Ö�Í�ØjÕ�ßàÍ�Õ_Ø�Î�Ê�Û:Í�è_Î�ã�Ø�Ê¡Ê�Ûvõ ��εßàÊnÚ�Ö�Í�ØjÕPßàÍ�Ö�Ù©Ö�ÎS!¡Õ_Î�Ú�ßàÎ�Ê�Ûé�Ë«ÌÄ�ºì3Î��¡ãDÎlØj̨˧Î�Ú�Í�ÖGÒDÎlÍ�Þ:ÎlÎ�Ú(Í�è�Î�Î��ºãDÎlØjÌ�˧Î�Ú�ÍY�����»Ù�Ú�Ï(Í�è_Î�Î��ºãDÎlØjÌ�˧Î�Ú�Í  ba:¥ZYW[ � Ï_Î�Ö�ßàØjÌ�Ò-Î�Ï«Ì�ګ˧Ê�ØjÎ�Ï_ÎlÍ�Ù�Ì�ÓÒDÎ�Ó¼Ê�Þ � ÖjÕ�ßjè(Í�è�Ù�ÍGÍ�è�Î�Î��ºãDÎlØjÌ�˧Î�Ú�Í�Ö?ÊnÚ«Í�è_Î�ÍvÞ�Ê�Î��ºÍ�Ø�Î�˧Î�Ö³Ù�Ø�Î������»Ù�ÚPÏ  ba:¥ZYW[ ×�è_Î�Ú � Þ:Î�ãPØ�Ê9Ô�Î�Í�è�Ù�Í:ÎlÔ�ÎlØ�áÍvÞ�ʱÙ�Ï+'�Ù�ßàÎ�Ú�Í%Î��ºãDÎlØjÌ�˧Î�Ú�Í�Ö%Ù�Ø�Î�̨Ú�Ï�Ì�Ö�Í�̨Ú_þnÕ�Ì�Ö�èPÙ�ÒPÓ¼Î�õ� * �ù � ö Ó¨Ó5ãPÙ�Ø�Í�̼Î�Ö«ØjÕ�Ú Í�è_Î÷Ø�Î�Ù�Ó%ãPØ�Ê�Í�Ê¡ßàÊnÓ � Ù�ÚPÏ Í�è_ÎL̨Ï_Î�Ù�Ó5Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�̼Î�Ö§Ù�Ø�Î÷ØjÕ�Ú Ù�Ö(Ö�ãDÎ�ßl̼ë�Î�Ï�õ+×%è�Ì�ֵ̨Ö

Ì�Ï_Î�Ú�Í�Ì�ßlÙ�ÓDÍ�ʵÎ��ºãDÎlØjÌ�˧Î�Ú�Íu����� c ��� � ] �c " ��� ñ��)�� ] �c " ��� ñ���ñ��)��

] �c " ��� � ���� ] �c " ��� ñ ��� � ���� ��� | � 4 � ] _�] ` õ

� * ï � ×%è�Ì�Ö3Î��¡ãDÎlØj̨˧Î�Ú�Í�Ì�Ö�Í�è�αÖ�Ù�˵αÙ�Ö � * �ù � Î��ºßàÎlã�Í�Í�è�Ù�Í�Þ�è_Î�ÚÑÙ]ßàÊ�Ø�ØjÕ_ã�Í�Î�Ï Ö�Î�Ú�Ï�ÎlØ3ãDÎlØ�Û�Ê�Øj˵Ö�Ù÷ßàÊnË�ñ˵̼Í�˧Î�Ú�Í�Ê�اÊ�ã-Î�ÚPÌ�Ú_þLÍ�Ê�Ù�ÚaÕ�ÚPßàÊ�Ø�ØjÕ_ã�Í�Î�ÏJØ�Î�ßàÎ�̼Ô�ÎlØ � Í�è_αÍ�Î�Ö�Í�Ö§Û�Ø�ÊnË Q Ù�Ø�ίØjÕ�Ú�õ � Ú ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�Ø � Í�è_ÎÎ��¡ãDÎlØj̨˧Î�Ú�Í�Ï_Ê¡Î�Ö2Í�è�Î(Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ ¹Ü�å9Ý2Õ�Ö�Ì�Ú�þ¹Í�è_αÔ�Ù�Ó�Õ�Î�Ö2Ö�Î�Ú�ͧÌ�Ú»Í�è�Î&O`K ã�Ø�Ê¡Ê�Û5Ê�Û%Í�è_ίßàÊn˵˵Ì�Í�˧Î�Ú�ÍãPè�Ù�Ö�Î:Í�ÊrÎ�̼Í�è_ÎlØNÎ��ºÍ�ØjÙ�ßàÍ>Í�è_Î5ßàÊn˵˵̼Í�Í�Î�Ï�ÒPÌ�ÍÊ�ØNÎ�Ó�Ö�γÙ�ÒDÊ�Ø�Í.Þ�̼Í�è�Ù � ��� � ��� ��� � ��� ÎlØ�ØjÊ�ØN˵Î�Ö�Ö�Ù�þ�Î � Ù�Ú�ÏÜVònÝ5Í�ʱÔ�ÎlØj̼Û�á¯Í�è�Ù�Í�Í�è_Î2Ö�Ù�˵Î3ÒP̼Í�Ì�Ö�Õ�Ö�Î�ÏLÌ�ÚLÍ�è_Î3Ê�ãDÎ�Ú¹ã�è�Ù�Ö�Î � Ê�Ø�Î�Ó�Ö�Î�Ù�ÒDÊ�Ø�Í�Ö%Þ�Ì�Í�èLÙ � ��� � ��� ��� � ���ÎlØ�Ø�Ê�Ø�˵Î�Ö�Ö�Ù�þ�Î�õ

� * î � ×%è�Ì�Ö:Î��ºãDÎlØjÌ�˧Î�Ú�Í%Ì�Ö³Í�è_Î�Ö�Ù�˵Î3Ù�Ö � * ï � Î��ºßàÎlã�Í�Í�è�Ù�Í%Þ�è_Î�Ú©Ù«ßàÊ�Ø�ØjÕ_ã�Í�Î�Ï©Ö�Î�Ú�Ï_ÎlØ%ãDÎlØ�Û�Ê�Øj˵Ö5Ù�Í�ØjÙ�ÚPÖ�Û¦ÎlØÍ�Ê�Ù�ÚÑÕ�ÚPßàÊ�Ø�ØjÕ_ã�Í�Î�ÏJØ�Î�ßàÎ�̼Ô�ÎlØ � Í�è�Î(Í�Î�Ö�Í�Ö�Û�Ø�ÊnË Q Ù�Ø�Î(ØhÕ�Ú�õ � ÚJã�Ù�Ø�Í�Ì�ßlÕ�Ó�Ù�Ø � Í�è_Î(Î��¡ãDÎlØj̨˧Î�Ú�Í�Ï_Ê¡Î�Ö3Í�è_ÎÛ¦ÊnÓ¨Ó¼Ê9Þ�̨Ú_þ �Õ�Ö�̨Ú_þ�Í�è_γÔ�Ù�Ó�Õ_Î�Ö>Ö�Î�Ú�Í>Ì�Ú2Í�è_ÎcO`K+ã�Ø�Ê¡Ê�Û�Ê�ÛPÍ�è_γÍ�ØjÙ�ÚPÖ�Û¦ÎlØ>ã�è�Ù�Ö�Î:Í�Ê�Î�̼Í�è_ÎlØNÔ�ÎlØj̼Û�á�Í�è�Ù�Í>Í�è_γÒPÌ�Í�ÖÍ�ØjÙ�Ú�Ö�Û�ÎlØ�Ø�Î�ÏJßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�Ï�Í�Ê÷Í�è_εã�ØjÎlԡ̼ÊnÕPÖ�Ó¼á÷ßàÊn˫˵̼Í�Í�Î�Ï»Ò�̼Í�Ö � Ê�Ø2Î�Ó�Ö�ΫÙ�ÒDÊ�Ø�Í3Þ�̼Í�è»Ù � ��� � ��� ��� � ���ÎlØ�Ø�Ê�Ø�˵Î�Ö�Ö�Ù�þ�Î�õ

� * � ×%è�Ì�Ö«Î��¡ãDÎlØj̨˧Î�Ú�ͯÌ�Ö(Í�è_Î]Ö�Ù�˧Î�Ù�Ö � * î � Î��_ßàÎlã�Í©Í�è�Ù�ͱÞ�è_Î�Ú#Ù�Ú#Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�ÏYÖ�Î�Ú�Ï_ÎlدãDÎlØ�Û�Ê�Øj˵֫ÙÍ�ØjÙ�Ú�Ö�Û�ÎlØ%Í�Ê«Ù«ßàÊ�Ø�ØjÕ_ã�Í�Î�ϹØ�Î�ßàÎ�̼Ô�ÎlØ � Ì�Í%Ì�Ö5Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï©Ù�Ö�Ì�Ú%Q � Þ�è�ÎlØ�Î�Í�è_Î�ÒPÌ�Í5ú�ÒDÎ�Ì�Ú�þ�Í�ØhÙ�Ú�Ö�Û�ÎlØ�Ø�Î�Ï©Ì�Ö5Ú_Ê�ÍÍ�Ù�ý�Î�ÚLÛ�Ø�ÊnË = > ? ADC � Ò�Õ_Í%Ì�Ú�Ö�Í�Î�Ù�ÏLÌ�Ö%ßàÊn˧ãPÕ�Í�Î�ϹÙ�Ö�úyü87 � Þ�è_ÎlØ�Î3ú . Ì�Ö%Í�è_Î(Ü�ã�Ø�ÎlԺ̼ÊnÕ�ÖjÓ¼áµÎ��ºÍ�ØjÙ�ßàÍ�Î�Ï�Ý%ÒP̼ͳÊ�ÛÍ�è_Î5Ø�Î�ßàÎ�̼Ô�ÎlØ � Ù�Ú�ϵúyù�Ù�ÚPϵú ï Ù�Ø�γÍ�è_Î5Ò�̼Í�Ö.Ê�Û�Í�è_Î5Ö�Î�ÚPÏ_ÎlØ�õ ö Ó�Ö�Ê � Ú_Ê�Í�Î5Í�è�Ù�Í>Ì�Ú�Í�è�Ì�Ö>ÖjÌ�Ë2ÕPÓ�Ù�Í�̼ÊnÚ � Õ�Ö�̨Ú_þ�Í�è_ÎÔ�Ù�Ó¨Õ_Î�Ö%Ö�Î�Ú�Í�Ì�Ú¯Í�è�Î O`K1ã�ØjÊ�Ê�Û.Ê�ÛNÍ�è_Î�Ø�Î�ßàÎ�̼Ô�ÎlØ � Ö�ßàÊn˵˵Ì�Í�˧Î�Ú�Í�Í�ʱú � ̼Í5Î�̼Í�è�ÎlØ%Î��¡Í�ØhÙ�ßàÍ�Ö�Í�è_Î2ßàÊn˵˵̼Í�Í�Î�ÏÒP̼Í5Ê�Ø%Î�Ó¨Ö�Î�Ù�ÒDÊ�Ø�Í�Ö5Þ�̼Í�è©Ù ����� � � ����� � � � ÎlØ�Ø�Ê�Ø�˵Î�Ö�Ö�Ù�þ�Î�õ

� * � � ×%è�Ì�Ö3Î��¡ãDÎlØj̨˧Î�Ú�Í�Ì�Ö2Í�è_ίÖ�Ù�˧ΩÙ�Ö � * � Î��_ßàÎlã�Í�Í�è�Ù�͹Ü�å9Ý3Þ�è_Î�Ú Ù�Ú Õ�Ú�ßàÊ�Ø�ØjÕ�ã�Í�Î�ÏÑÖ�Î�Ú�Ï_ÎlØ�ã-ÎlØjÛ¦Ê�ØjË«ÖÙ]ã�Ø�Ê¡Ê�Û�Í�Ê»ÙJßàÊ�Ø�ØjÕ_ã�Í�Î�Ï ØjÎ�ßàÎ�̼Ô�ÎlØ � Í�è_ιÕ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï Ö�Î�Ú�Ï_ÎlصÏ�Ê�Î�ÖµÚ�Ê�ͧã�Ø�Ê9ÔºÌ�Ï_ίÍ�è_ΩÔ�ÎlØh̼ëPßlÙ�Í�̼ÊnÚ Ì¨ÚºñÛ¦Ê�Øh˵Ù�Í�̼ÊnÚ�Í�Ê©Í�è_ε̨Ï_Î�Ù�Ó O`K Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á � Ù�Ú�ÏPQ Ì�Ö�Õ�Ö�Î�Ï]Í�ʹÛ�Ê�Ø�þ�ΧÍ�è_Î F�HI/9J KILML�NL˧Î�Ö�Ö�Ù�þ�Î�Ö3Û¦Ø�ÊnËÍ�è_Î<O`K�Û¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá � Ù�Ú�Ï ÜVònÝ�Þ�è_Î�Ú{Ù(ßàÊ�ØjØjÕ_ã�Í�Î�Ï{Ö�Î�Ú�Ï_ÎlØ�ãDÎlØ�Û�Ê�Øj˵Ö�Ù«ã�ØjÊ�Ê�Û>Í�ʯÙ�Ú]Õ�Ú�ßàÊ�Ø�ØjÕ�ã�Í�Î�ϹØ�ÎàñßàÎ�̼Ô�ÎlØ � ̼Í?Ô�ÎlØh̼ë�Î�Ö?Í�èPÙ�Í�Í�è_Î�Ô�Ù�Ó¨Õ_Î�ÖGÖ�Õ�ÒP˵̼Í�Í�Î�Ï(ßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�ϧÍ�Ê�Í�è_Î�Ô�Ù�Ó�Õ�Î�ÖGã�ØjÎlԡ̼ÊnÕPÖ�Ó¼á3Î��ºÍ�ØjÙ�ßàÍ�Î�ϯÛ�Ê�ØGÍ�è_ÎßàÊn˵˵̼Í�˵Î�Ú�Í�Ölõ

� * S � ×%è�Ì�Ö�Î��ºã-ÎlØhÌ�˧Î�Ú�Ír̨Ö�Í�è�Î�ÖjÙ�˧εÙ�Ö � * � � Î��_ßàÎlã�Í�Í�èPÙ�ͧÜ�å9Ý�=E?����«Ì�Ö�Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï]Ù�Ö�Ì�Ú Q � Ö�Ê©Í�è_Î�Ô�Ù�Ó�Õ�ÎÊ�Û�Ó�Ê�þ�� � ̨Ö(ýºÚ_Ê�Þ�Ú � Ù�Ú�Ï ÜVònݱÞ�è_Î�Ú�Ù�Ú�ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï+Ö�Î�Ú�Ï_Îlع˵Ù�ý�Î�Ö¹ÙaßàÊn˫˵̼Í�˧Î�Ú�ÍJÜ�Î�̼Í�è_ÎlØL̨Ú+ÙßàÊn˵˵̼Í�˵Î�Ú�ͯãPè�Ù�Ö�Î÷Ê�Ø©Ù�Ö©Ù Ø�Î�ßàÎ�̼Ô�ÎlعÌ�ÚYÍ�è_Î{Í�ØjÙ�Ú�Ö�Û�ÎlدãPè�Ù�Ö�Î�Ý«Ê�دÊ�ãDÎ�Ú�Ì�Ú_þ � Ê�د̨Ö(ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï � ̼ͱ̨ÖÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï]Ù�Ö�Ì�Ú Q ܸÌVõóÎ�õ � ˵Ù�ý¡Ì¨Ú_þ±ßàÊn˫˵̼Í�˧Î�Ú�Í�Ö�Í�Êw�lÎlØ�Ê � Ù�ÚPÏ÷Í�è�Î�Ú]ÎS!�Õ�Ì�Ô�Ê¡ßlÙ�Í�Ì�Ú�þ±ÊnÚ�Ù�Ú]Ê�ãDÎ�Ú�̨Ú_þ(Ê�ØÙ§ßàÊ�Ø�ØjÕ_ãPÍ�̼ÊnÚ�ÝhõGÐrÚ_ÎrßlÙ�Ú©Ô�ÎlØj̼Û�áµÍ�è�Ù�Í5Í�èPÌ�Ö³Ì�Ö³Ú_Ê9Þ�ÎS!¡Õ�̼Ô�Ù�Ó¼Î�Ú�Í�Í�ʧÍ�è_Î�Î��ºãDÎlØjÌ�˧Î�Ú�Í  ba�¥GY�[ c�� ����� ] d4] ` � Ù�ÍÓ¼Î�Ù�Ö�Í�Û¦Ø�ÊnË Í�è_Î2ԡ̼ÎlÞ Ê�Û>Í�è_Î2Ù�Ï�Ô�ÎlØjÖ�Ù�Ø�áRH±õ#��Ê�Í�̨ßàÎ2Í�è�Ù�Í�ÊnÚ�Ó¼á©Ì�Ú¹Í�è�Ì�Ö%ë�Ú�Ù�ÓAÎ��ºãDÎlØjÌ�˧Î�Ú�Í�Ï_Ê¡Î�Ö Q�Ú_ÎlÎ�Ï�ÖÍ�ʵýºÚ_Ê9Þ=Ó¼Ê�þ�� � Ì�Ú¯Ê�ØjÏ_ÎlØ%Í�Ê«ÎS!�Õ�Ì�Ô�Ê¡ßlÙ�Í�Î2Í�è_Î"�.Î�Ï_ÎlØjÖ�Î�Ú¹ßàÊn˫˵̼Í�˧Î�Ú�Ílõ

��Î3ã�Ø�Ê�Ô�Î3ÙµÖ�ÎS!¡Õ_Î�Ú�ßàÎ�Ê�Û>Ó¼Î�˫˵Ù�ÖlõI��Î3ÊnÚPÓ¼á(Ö�ý�ÎlÍ�ßhè÷Ö�Ên˧Î�Ê�Û.Í�è_Î�ã�Ø�Ê¡Ê�Û¸Ö5̼ÛÍ�è�Îlá±Ù�Ø�Î2Ö�Í�Ù�Ú�Ï�Ù�ØjÏAõ

v 1 363 �xwR9�� � �T),���4/ ��� )��-��#��O��/T�o!Ä%&< *�&���='i(:/0��%�� � � * �ù fh � * ï �

� /�� �4;�� ×%è_Î�ÏPÌ�-ÎlØjÎ�Ú�ßàÎ�ÒDÎlÍ�Þ:ÎlÎ�Ú � * �ù�Ù�Ú�Ï � * ï Ì�Ö5Í�è�Ù�Í�Ì¨Ú � * ï � Q+Î��¡Í�ØhÙ�ßàÍ�Ö%Í�è_Î3ßàÊn˫˵̼Í�Í�Î�ϹÒ�̼Í�Ölõ @¡Ê§Í�è_ÎÊnÚPÓ¼á(ãDÊnÖ�Öj̼ÒPÌ�Ó�Ì�Í�á§Í�Ê«Ï�̨Ö�Í�Ì�Ú_þnÕ�̨Ö�è � * �ùrÛ�Ø�ÊnË � * ï Ì�Ö³Þ�è_Î�Ú¹Ù«ßàÊ�Ø�ØjÕ_ã�Í�Î�ϹãPÙ�Ø�Í�á±Ê�ãDÎ�Ú�Ö%ÙµßàÊn˫˵̼Í�˧Î�Ú�Í�Ì�Ú¯ÍvÞ�ÊÏPÌ�-ÎlØjÎ�Ú�Í?Þ³Ù�áºÖlõ � ÚµÍ�è�̨ÖGßlÙ�Ö�Î � * ï Ù�ÒDÊ�Ø�Í�Ö:Ù�Ú�Ï«ÊnÕ_Í�ãPÕ_Í�ÖGÍ�è_Î�˧Î�ÖjÖ�Ù�þ�Î � � ��� ��� � ��� ��� Ù�Ó¼ÊnÚ_þ3Þ�̼Í�è§Í�è_Î�Ô�Ù�Ó�Õ�Î

åSG

Page 16: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Ê�Û%Ó�Ê�þ�� � õ � Û³Í�è�Ì�Ö2Ï_Ê�Î�Ö2Ú_Ê�Í�è�Ù�ã�ãDÎ�Ú�Ö�Þ�Ì�Í�è»Ú_ÎlþnÓ�̼þnÌ�ÒPӼΧã�Ø�Ê�Ò�Ù�ÒPÌ�Ó�̼Ívá � Þ:Î(ßlÙ�Ú ßàÊnÚ�Ô�ÎlØjÍ2Í�è�̨Ö�Î��ºã-ÎlØhÌ�˧Î�Ú�Í3Ì�Ú�Í�ÊÙ�Ú�Ù�Ó�þ�Ê�Øj̼Í�è�Ë�� Í�è�Ù�Í�Ò�Ø�Î�Ù�ýºÖrÍ�è_ΧÏ�Ì�Ö�ßàØjÎlÍ�εӼÊ�þ©Ù�Ö�Ö�Õ�˧ãPÍ�̼ÊnÚ�õ2×�è_ΧßàÊnÚ�Ô�ÎlØjÖj̼ÊnÚ�Ì�Ö�Ö�Í�Ù�Ú�ÏPÙ�ØjÏ �Í�è_εÙ�Ó¼þ�Ê�Øj̼Í�èPË� Þ�̼Í�è{Ì�Ú_ãPÕ_Í�Ü�� B�� Ý%ØjÕ�Ú�Ö�Í�è_Î�Î�Ú�Í�Ì�Ø�Î2Î��ºãDÎlØjÌ�˧Î�Ú�Í � Ø�ÎlãPÓ�Ù�ßl̨Ú_þ«Í�è_Î¹Ü f=B � Ý�Ì�Ú÷Í�è_ΧßàÊn˵˧ÊnÚ]Ø�ÎlÛ�ÎlØ�Î�Ú�ßàΧÖ�Í�Øj̨Ú_þÒ¡áÑÜ�� B�� Ýhõ���Ê�Í�Ì�ßàÎ�Í�è�Ù�Í�̨ÚLÎ��ºãDÎlØjÌ�˧Î�Ú�Í � * ï � Í�è�Î2ýºÚ_Ê9Þ�Ó¼Î�Ï_þ�Î�Ê�ÛGÓ¼Ê�þ�� � ̨Ö�Ú_Ê�ÍrÚ_ÎlÎ�Ï_Î�ÏAõ�×%è�ÎlØ�ÎlÛ¦Ê�ØjÎ � ��ßlÙ�ÚßlÙ�ØjØ�á(ÊnÕ�Í%Í�è_Î�Î��ºã-ÎlØhÌ�˧Î�Ú�Í � * ï � Ù�Ú�ϹÞ�è_Î�Ú � * ï Ù�Ò-Ê�ØjÍ�Ö � �­ÒPØ�Î�Ù�ý¡Ö%Í�è_Î3Ï�Ì�ÖjßàØ�ÎlÍ�Î�Ó¼Ê�þµÙ�Ö�Ö�ÕP˧ã�Í�̼ÊnÚ�õ

v 1 363 �xwR9�� � �T),���4/ ��� )��-��#��O��/T�o!Ä%&< *�&���='i(:/0��%�� � � * ï fh � * î �

� /�� �4;�� ×�è_ΫÏ�Ì�DÎlØ�Î�Ú�ßàΫÒ-ÎlÍvÞ�ÎlÎ�Ú � * ï Ù�Ú�Ï � * î Ì�Ö3Í�è�Ù�Í3Þ�è�Î�Ú»ÙLßàÊ�Ø�ØhÕ_ã�Í�Î�Ï»ã�Ù�Ø�Í�á�@>=5Í�ØjÙ�Ú�Ö�Û�ÎlØjÖ2Ù©ÒP̼Í�Í�ÊÙ�Ú�ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï÷ÊnÚ�Î�@�? � Í�èPÌ�Ö�è�Ù�ã�ãDÎ�Ú�Ö�Ì�Ú{Í�è_Î�Ø�Î�Ù�Ó>Ó�̼Û�Î�˧ʡÏ�Î�Ó.Ì¨Ú � * ï � Þ�è�̨ӼÎ2Ì�Ú � * î Q Ì�Ö�ØjÕ�Ú{Í�Ê©Ô�ÎlØj̼Û�áÌ�Û5Í�è_Î(Ò�̼Í�Ö3Í�ØhÙ�Ú�Ö�Û�ÎlØ�Î�ÏÑßàÊ�ØjØ�Î�Ö�ãDÊnÚ�ÏÑÍ2Í�è_ίã�ØjÎlԡ̼ÊnÕPÖ3ßàÊn˵˵Ì�Í�Í�Î�ÏÑÊnÚ�Î�Ölõ�×%è_ÎlØ�ÎlÛ�Ê�Ø�Î � Í�è_αÊnÚ�Ó¼á�ãDÊnÖ�Ö�Ì�ÒPÌ�Ó�̼Ívá÷Í�ÊÏPÌ�Ö�Í�Ì�Ú_þnÕPÌ�Ö�è2Í�è_Î�Ö�Î%ÍvÞ�Ê2Î��¡ãDÎlØjÌ�˵Î�Ú�Í�Ö>Ì�Ö?Þ�è_Î�Ú � * î Ù�ÒDÊ�Ø�Í�Ölõ ö ÖGÌ�Ú§Í�è_Î%ãPØ�ÎlÔ¡Ì�ÊnÕ�ÖNã�Ø�Ê¡Ê�Û � Ù�Ú�á�Í�Ì�˧ΠQ Ù�ÒDÊ�Ø�Í�Ö � Ì�Íë�Ú�Ï�Ö.ÊnÕ_ÍGÓ¼Ê�þ � � � Ù�ÚPÏ�Í�è�Õ�ÖGÌ¼Û � * î Ù�ÒDÊ�Ø�Í�ÖNÞ�̼Í�è§Ú_ÊnÚ_ñ�Ú_ÎlþnÓ�̼þn̼Ò�Ó¼Î�ãPØ�Ê�ÒPÙ�ÒPÌ�Ө̼Í�á � Þ:Î�ßlÙ�ÚµßàÊnÚ�Ö�Í�ØhÕ�ßàÍ?Ù�ÚµÙ�Ï_Ô�ÎlØjÖjÙ�Ø�á�­Í�è�Ù�Í%Ò�ØjÎ�Ù�ý¡Ö5Í�è�Î�Ï�Ì�Ö�ßàØ�ÎlÍ�Î�Ó�Ê�þµÙ�Ö�Ö�Õ�˧ãPÍ�̼ÊnÚ¯Ò�á±ØjÕ�ÚPÚ�Ì�Ú_þ � * î õ ö Ó�Ö�ʵÙ�Ö%̨گÍ�è_Î�ã�Ø�ÎlԺ̼ÊnÕ�Ö³ã�ØjÊ�Ê�Û � ý¡Ú�Ê9Þ�Ó¼Î�Ï�þ�ÎÊ�ÛNÓ¼Ê�þ � � Ì�Ö5Ú�Ê�Í�Ú_ÎlÎ�Ï_Î�Ï©Ò¡á � * î õ

v 1 363 �xwR9�� � �T),���4/ ��� �'�� *�&���='i(:/0��%�� � � * î fh � * �

� /�� �4;�� �N̼ØjÖ�Í � Þ�̼Í�è¹Í�è_Î2Ö�Ù�˵Î3Ø�Î�Ù�Ö�ÊnÚPÌ�Ú_þ(Ù�Ö�Ì�Ú©Í�è�Î3ã�Ø�ÎlԺ̼ÊnÕ�Ö%ã�Ø�Ê¡Ê�Û � Þ�Î�˵Ù�á¹Ù�Ö�ÖjÕ�˧Î3Í�èPÙ�Í�Í�è_Î2ã�Ø�Ê�ÒPÙ�ÒP̨Ó�̼Í�áÍ�èPÙ�Í QYÙ�ÒDÊ�Ø�Í�Ö%Ì¨Ú � * Ì�Ö5Ú�ÎlþnÓ�̼þn̼ÒPÓ�Î � ÕPÚ�Ï_ÎlØ5Í�è_Î�Ï�̨Ö�ßàØ�ÎlÍ�Î�Ó¼Ê�þ«Ù�ÖjÖ�Õ�˧ã�Í�Ì�ÊnÚ � Þ�è�Ì�ßhè©Ì�Ö5Ì�˧ãPӨ̼Î�ϱÒ�á � � %2õ��Ê�Þ � Ù�Ö�ÖjÕ�˵Ì�Ú_þ±Í�è�Ù�Í Q Ú�ÎlÔ�ÎlØ3Ù�ÒDÊ�Ø�Í�Ö � Í�è_Î�ÊnÚ�Ó�á÷Ï�Ì�DÎlØ�Î�Ú�ßàΧÌ�ÖrÍ�èPÙ�Í�Í�è_Î�Í�Õ_ãPӼιÜ,z ù B , ù B z ï B , ï Ý�Ì�Örþ�Î�ÚºñÎlØhÙ�Í�Î�Ï­Ï�Ì�DÎlØ�Î�Ú�Í�Ó¼áÑÌ¨Ú Í�Þ:ÊÑßlÙ�Ö�Î�Ölõ %�Ê�Þ�ÎlÔ�ÎlØ � Þ:ÎLÖ�èPÙ�Ó�Ó³ã�Ø�Ê�Ô�ιÍ�è�Ù�Í«Ù�Ö�ÖjÕ�˵Ì�Ú_þ � � % � Í�è_Î�Ö�ιÍvÞ�Ê ßlÙ�Ö�Î�Ö«Ù�Ø�Į̂Ú�Ï�Ì�Ö�Í�̨Ú_þnÕ�Ì�Ö�èPÙ�ÒPÓ¼Î�õÅ�_Ø�ÊnË Ú_Ê9Þ=ÊnÚ � Þ�Î�Ù�Ö�Ö�Õ�˧Î3Í�è�Ù�Í�ú . c 3µÛ�Ê�Ø�ÖjÌ�˧ãPÓ�̨ßl̼Í�á]Ü�Í�è_Î2ßlÙ�Ö�Î�Û�Ê�Ø�ú . c�å3Ì�Ö�Ö�Ì�˵̨Ó�Ù�ØyÝhõ×�è_Î�Ú¯Þ�Î2è�Ù�Ô�Î { .>c f � 7 Ù�Ú�ÏLúyùMc�ú�õ

� Ú � * î � Í�è_ΫÜ,z�ù B , ù B z ïCB , ï Ý5Ù�Ø�Î�þ�Î�Ú_ÎlØjÙ�Í�Î�Ï{Ù�Öø ù B ø ï $ g$# z ù $ f � � B , ù $ { � �. i � ü � c1Ü { . Ý � � i � ü B z ï $ f �  B , ï $ Ü { . & � Ý �  i � ü  c1Ü { . Ý �  i � ü  � �  B� Ú � * � Í�è_Îlá¯Ù�Ø�Î�þ�Î�Ú_ÎlØjÙ�Í�Î�Ï÷Ù�Ö

8 ù B98nïCB98 î $ g # �z�ù $ f � � B , ù $ Ü { .vÝ � � i � ü B z ï $ f � Â B , ï $ f � Ì D

� ͧÌ�Ö�Î�Ù�Ö�áJÍ�Ê»Ö�ÎlαÍ�èPÙ�͵Ì�Ú Ò-Ê�Í�è ßlÙ�Ö�Î�Ö � Ü,z�ù B , ù9Ý3Ù�ØjίÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�Î�ÏJÌ�Ï�Î�Ú�Í�Ì�ßlÙ�Ó¨Ó¼á � Ù�Ú�Ï Ù�Ø�ΩÌ�Ú�Ï_ÎlãDÎ�Ú�Ï�Î�Ú�Í2Û¦Ø�ÊnËÜ,z ïCB , ï Ýhõ_@ºÊ(Þ:Î�Ú�Ê9Þ7ۦʺßlÕ�Ö�ÊnÚ Ü,z ïCB , ï Ýhõ � Ú � * î � Í�è_Î�Í�Õ_ãPÓ¼Î¯Ü f=B � B z ï B Ü { .vÝ �  i �

ü  & , ï Ýpc*Ü f=B � BOf �  B � �  Ý̨Ö5Ù�ØjÙ�Ú�Ï_ÊnË � %�ñ®Í�Õ_ãPÓ¼Î�õ � Ú � * � Í�è_Î�Í�Õ_ãPÓ¼Î§Ü f=B � B z ï B Ü { .�Ý �  i �

ü  & , ï Ý�c1Ü f=B � BOf��  BOf � � Ì i�Ü { .vÝ �  i � ü  Ý:Ì�Ö5ÙØhÙ�Ú�Ï_ÊnË M�ñ®Í�Õ_ãPÓ�Î�õ

×%è�ÕPÖ � ̼ÛNÙ�Ú¹Î�Ú�Ô¡Ì�Ø�ÊnÚ�˧Î�Ú�Í?J ßlÙ�ÚLÏPÌ�Ö�Í�Ì�Ú_þnÕPÌ�Ö�è � * î Û¦Ø�ÊnË � * � Þ�Î�ßlÙ�ÚLÕ�Ö�Î2̼Í%Í�Ê«Ò�Õ�Ì�Ó�ϯÙ�Ú_Ê�Í�è_ÎlØrÙ�Ï_Ô�ÎlØ�ñÖjÙ�Ø�á�=Í�è�Ù�Í3Ò�ØjÎ�Ù�ý¡Ö2Í�è_Î � � % Ù�Ö�ÖjÕ�˧ã�Í�̼ÊnÚAõ¯×�è�Ì�Ö�Ì�Ö2Ù¹ØjÙ�Í�è�ÎlØ�Ö�Í�Ù�ÚPÏ�Ù�ØjÏJßàÊnÚ�Ô�ÎlØjÖ�Ì�ÊnÚ �7Ø�Î�ßàÎ�̼Ô�Î�Ö�ÙLÍ�Õ_ã�Ó¼ÎÜ @ ù BA@AïEBA@ î BA@ ÝNÙ�Ö?Ì�Ú�ãPÕ_Ílõ � Í>þ�Î�Ú_ÎlØjÙ�Í�Î�Ö:Ù�ØjÙ�ÚPÏ_ÊnË � �hg # � Õ�Ö�Î3Ü @ ù BA@�ï ÝNÌ�Ú�Í�è_Î5ãPÓ¨Ù�ßàÎ5Ê�Û>Ü f4B � Ý>̨Ú�Í�è_Î%ßàÊn˵˧ÊnÚØjÎlÛ¦ÎlØ�Î�ÚPßàÎ�Ö�Í�ØjÌ�Ú_þ � Ù�Ú�Ï�ßlÙ�Ø�Øh̼Î�Ö�ÊnÕ_ÍrÍ�è_Î�Î��¡ãDÎlØjÌ�˵Î�Ú�Í � * õ � Ú]ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�Ø � ��ØjÕPÚ�Ö�Í�è_Î�Î�Ú�Ժ̼Ø�ÊnÚ�˵Î�Ú�Í J Ù�Ú�ÏÍ�è¡Õ�Ö�ßlÙ�Ú÷Ö�ÎlÎ2Ù�Ó�Ó�Í�è_Î2Ò�̼Í�Ö%ßàÊn˵˵Ì�Í�Í�Î�Ï÷Í�Ê_û���Ù�Ó�Ö�Ê(ØjÕ�Ú�Ö Q#Ù�ÚPϹÍ�è¡Õ�Ö�ßlÙ�Ú{Ö�ÎlÎ�Ù�Ó�ÓAÍ�è�Î2˧Î�Ö�ÖjÙ�þ�Î�Ö�Í�ØhÙ�Ú�Ö�˵̼Í�Í�Î�Ï �Ù�Ö�Þ�Î�Ó¨Ó�Ù�Ö5Í�è_Î3Ì�Ú�Í�ÎlØjÚ�Ù�ÓDØhÙ�Ú�Ï_ÊnË ÒPÌ�Í�Ö5Õ�Ö�Î�Ï©Ò¡á Q�õ

×%è_Î�Ú��]ãPÌ�ß�ýºÖNÙ�ØhÙ�Ú�Ï_ÊnË=Ö�Î�Ö�Ö�̼ÊnÚ2Þ�è_ÎlØjÎ�Ù�Ú�Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï�ã�Ù�Ø�Í�áe@>=PÍ�ØhÙ�Ú�Ö�Û�ÎlØjÖÙ�ÒP̼ÍÍ�Ê�Ù�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï2ÊnÚ_Î @h?�õ��Î2Õ�Ö�Î�Í�è_Î�Ú�Ê�Í�Ù�Í�̼ÊnÚ�Ö�̨Ú%Q � Ù�Ú�ϹÖ�ʵÍ�è_Î�ÒP̼Í%ßàÊn˫˵̼Í�Í�Î�Ï©Ò¡áR@ ? Ì�Ö%ú . Þ�̼Í�èN�.Î�Ï_ÎlØjÖ�Î�Ú¹ßàÊn˵˵̼Í�˵Î�Ú�Í { . õG×%è�Î�#ØjÎlãPÓ�Ù�ßàÎ�Ö�Í�è_Î3Í�Õ_ã�ӼαÜ,z�ù B z ïCB , ù B , ï Ý�Ö�Î�Ú�Í�Ò¡á©Í�è_Î,Q+Í�Ê%@�?3Ò¡áJÜ,z�ù BA@ î B , ù B Ü { .vÝ �  i �

ü  &(@ Ýhõ � Û>Í�è�Î2Í�Õ_ã�Ó¼ÎÜ @ ù BA@AïEBA@ î BA@ Ý � ̨Ö�ÙµØhÙ�Ú�Ï_ÊnË � %7Í�Õ_ã�ӼΠ� Í�è_Î�Ú÷Í�è_Î2Î��¡ãDÎlØjÌ�˵Î�Ú�Í�̨Ö%Ì�Ï_Î�Ú�Í�Ì�ßlÙ�ÓAÍ�Ê � * î ûD̼Û%Ü @ ù BA@�ïCBA@ î BA@ Ý%Ì�Ö�ÙØhÙ�Ú�Ï_ÊnË M�ñ®Í�Õ_ã�ӼΠ� Í�è_Î�Ú©Í�è_Î3Î��¡ãDÎlØjÌ�˵Î�Ú�Í%Ì�Ö5̨Ï_Î�Ú�Í�̨ßlÙ�ÓDÍ�Ê � * õ�×%è�ÕPÖ%̼Û�Í�è�Î�Î�Ú�Ժ̼Ø�ÊnÚP˧Î�Ú�Í!J*ßlÙ�Ú¹ÏPÌ�Ö�Í�Ì�Ú_þnÕPÌ�Ö�è� * î Û�Ø�ÊnË � * � �YßlÙ�Ú©ÒPØ�Î�Ù�ý¯Í�è_Î � � %1Ù�ÖjÖ�Õ�˧ã�Í�Ì�ÊnÚ�õ

å �

Page 17: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

v 1 363 �xwR9�� � * fh � * � �

� /�� �4;�� ×�è�Ì�Ö?Ì�Ö:Ö�Í�ØjÙ�Ì�þnè�Í�Û�Ê�Ø�Þ³Ù�ØjÏ�õ � ÚµÛ¸Ù�ßàÍ:Í�è�Î�ÊnÚ�Ó�áµÏ�Ì�DÎlØ�Î�Ú�ßàÎ�ÒDÎlÍvÞ�ÎlÎ�Ú � * Ù�Ú�Ï � * � Ì�Ö�Í�è�Ù�Í:Ì¨Ú � * � Í�è_Î=#> ? ADC Ô�ÎlØj̼ëPÎ�Ö�Í�è_Î�ã�ØjÊ�Ê�Û¸Ö � Þ�è�Ì�Ó¼Î�Ì�Ú � * � � ̼ÍrÌ�Ö�Í�è_Î�Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Í�Ì�Î�Ö != $���� � ���%(& Ù�Ú�Ï != $ ��� ñ ��� � � �%'& Í�èPÙ�ÍrÏ�Ê(Í�è_ÎÔ�ÎlØh̼ëPßlÙ�Í�̼ÊnÚ�õ?×�è�Õ�Ö³Í�è�Î�Ö�Î�Í�Þ:Ê(ÏPÌ�Ö�Í�Øj̼Ò�Õ_Í�̼ÊnÚ�Ö:Ù�Ø�Î�Ì�Ú¯Û¸Ù�ßàÍ ��),���}/0�0#� �!�õ

v 1 363 �xwR9�� � * � fh � * S �

� /�� �4;�� � Ú � * � � Í�è�Î{ßàÊn˵˫̼Í�˧Î�Ú�Í�Ö«Ò¡á­Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ïaã�Ù�Ø�Í�̼Î�Ö±Ù�Ø�Î{ßlÙ�ØjØj̼Î�Ï Ò¡áaÍ�è_Î�˵Ö�Î�Ó¼Ô�Î�Ölõ � Ú � * S � QÖjÌ�Ë2ÕPÓ�Ù�Í�Î�Ö>Í�è_Î�Ö�Î�ßàÊn˵˵̼Í�˧Î�Ú�Í�Ö?Þ�̼Í�è_ÊnÕ_ÍNÍ�è_Î%ýºÚ_Ê�Þ�Ó¼Î�Ï_þ�Î%Ê�Û-Í�è_Î�Ù�ßàÍ�ÕPÙ�Ó�ÒP̼Í�ÖGßàÊn˵˵̼Í�Í�Î�Ï«Í�Ê_õ � Ú§Û¸Ù�ßàÍ � Q Ó¼Î�Ù�ØjÚPÖÍ�è�Î�Ö�Î5ÒP̼Í�Ö>ÊnÚPÓ¼á2Ó�Ù�Í�ÎlØ � Ù�Í?Í�è_Î5Ê�ãDÎ�Ú�Ì�Ú�þ�Ê�ØGÞ�è_Î�Ú§Ù�ãPÙ�ØjÍ�á�Ì�Ö>ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï�õI%�Ê�Þ�ÎlÔ�ÎlØ � Ö�Ì�ÚPßàÎ QJýºÚ_Ê9Þ�Ö?Í�è_Î%Ï�Ì�Ö�ßàØjÎlÍ�ÎÓ�Ê�þ � ̼Í�ßlÙ�Ú«ÎS!¡Õ�̼Ô�ʺßlÙ�Í�Î�Ù�Ú�Ï«Ê�ãDÎ�ÚµÍ�è_Î�ßàÊn˵˵̼Í�˧Î�Ú�ÍGÍ�Ê�Ù�Ø�ÒP̼Í�ØhÙ�Ø�á�Ô�Ù�Ó�Õ�Î�Ölõ>×%è�ÕPÖ � Í�è_Î�Ó¼Î�˵˵Ù�Û�ÊnÓ�Ó¼Ê�Þ�ÖGÏP̼Ø�Î�ßàÍ�Ó¼áÍ�è�Î�Û�Ù�ßàÍ�Í�è�Ù�Í%Í�è_Î"�.Î�Ï_ÎlØjÖ�Î�ÚLßàÊn˵˵Ì�Í�˧Î�Ú�Í%̨Ö5ãDÎlØ�Û¦Î�ßàÍ%èPÌ�Ï�Ì�Ú_þ_õ

@ºÕ�˵˵̨Ú_þ�ÎlÔ�ÎlØ�á¡Í�è�Ì�Ú_þ Õ_ã � Þ:Î÷ýºÚ_Ê�Þ Í�èPÙ�Í Q Ì�Ö(ÙJÔ�Ù�Ó�Ì�ÏYÌ�Ï_Î�Ù�Ó�ã�Ø�ʺßàÎ�Ö�Ö±Ù�Ï_Ô�ÎlØjÖ�Ù�Ø�áYÙ�Ú�Ï­Í�è�ÎlØ�ÎlÛ¦Ê�ØjÎ÷Í�è_Î&X8�dCL�� ã�Ø�Ê�Í�ʺßàÊnÓ%Ö�Î�ßlÕ_Ø�Î�Ó�áJØ�Î�Ù�Ó�Ì �lÎ�Ö§Í�è_ÎJ=#> ? ADC Ì�Ï_Î�Ù�Ó:Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á Ì�Ú Í�è_ÎJÜ8=E?���� B != $���� ñ �)�%(& B != $���� ñ���ñ����%'& B!= $ ��� � � �%(& B != $ ��� ñ ��� � � �%'& Ý�ñ�è�á¡Ò�ØjÌ�ϩ˵ʡÏ_Î�Ó®õ

� � Ä�ÈjÁrÂ�� �: ��� �A� hÆ��:ÂDÈjÄ�Á� Ú Í�èPÌ�Ö«Ö�Î�ßàÍ�Ì�ÊnÚYÞ:Î÷Öjè_Ê9Þ�è_Ê9Þ Í�Ê Ö�Î�ßlÕ_Ø�Î�Ó¼á Ø�Î�Ù�Ó¨Ì��lÎ÷ÙÑÍ�Þ:Ê�ñ®ãPÙ�Ø�Ívá­Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�á Í�è�Ù�ͱÞ�Î]ßlÙ�Ó�Ó �)%����x/��W "/T��W�* �!.�� "/0��%��+Ü8=���D>AÝ�Ì�Ú Í�è_ÎJ=#> ? ADCDñ�è�á�ÒPØjÌ�Ï Ë§ÊºÏ_Î�Ó5Ì�Ú Í�è_Ωã�ØjÎ�Ö�Î�Ú�ßàΩÊ�ÛrÙ�˵Ù�Ó�Ì�ßl̼ÊnÕPÖ � Ù�Ï�Ù�ã�Í�̼Ô�Î÷Ù�Ï_Ô�ÎlØjÖ�Ù�Ø�á�õ� Ú_Û�Ê�Øj˵Ù�Ó�Ó�á � =���D>LÙ�Ó¨Ó¼Ê9Þ�Ö�Í�Þ:Ê�ãPÙ�Ø�Í�̼Î�Ö:Í�ÊE'�ÊnÌ�Ú�Í�Ó�á�ÎlÔ�Ù�Ó�Õ�Ù�Í�Î�Ù�Ú�áµÒPÌ�ÚPÙ�Ø�á§Ê�ãDÎlØjÙ�Í�̼ÊnÚ(ÊnÚ(ÍvÞ�Ê�ÒPÌ�Í�Ö � Ù�ÚPÏ«Í�è�̨ÖGÞ�̨Ó�ÓÙ�Ó¨Ó¼Ê9Þ1Õ�Ö�Í�Ê©ßàÊnÚ�Ö�Í�ØjÕ�ßàÍrþ�Î�Ú_ÎlØhÙ�ÓÍvÞ�Ê�ñ®ã�Ù�Ø�Í�á�ßàÊn˧ãPÕ�Í�Ù�Í�̼ÊnÚ{ã�Ø�Ê�Í�ʺßàÊnÓ�Ö�ÊnÚ÷Í�Ê�ã�Ê�Û�=���D>õ ��Î�ë�ØhÖ�Í�ãPØ�Î�Ö�Î�Ú�Í�Í�è_ÎÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá � Öjè_Ê9Þ�ÚLÌ�Ú �N̼þnÕ_Ø�Î3ò¡õ

) -G-G2:7 / ��- �B58/87 9 =���D>

=���D>{ã�Ø�ʺßàÎlÎ�Ï�Ö%Ù�Ö�Û¦ÊnÓ�Ó�Ê9Þ�Ö � ØjÕPÚ�Ú�Ì�Ú_þ�Þ�̼Í�è¯ã�Ù�Ø�Í�̼Î�Ö @ ïCBEDVDVDVB @ZF � Ù�Ú�ÏLÙ�Ï_Ô�ÎlØjÖ�Ù�Ø�á%Q�õ*�� �43 36/87<5 + � 2S1�� �+è�Î�ÚJØ�Î�ßàÎ�Ì�Ô¡Ì�Ú�þLÛ�Ø�ÊnË @C=�Ù{˧Î�Ö�ÖjÙ�þ�Îx� $ 4(')'+* " B98':<;=B2}0:P;�B @�? B ú�� � ØjÎ�ßàÊ�ØjÏ� } :<;=B e @C= B @h? f B ú�� � Ö�Î�Ú�ϯ˧Î�Ö�Ö�Ù�þ�Î#� K5��8�� J�� B98O:<;=B2} :<;�B @>= B @�?7�GÍ�ÊI@>= � @�?�Ù�Ú�Ï Q � Ù�ÚPϱÌ�þnÚ_Ê�Ø�Î�Ù�Ó�ÓÛ¸Õ_Ø�Í�è�ÎlØr˵Î�Ö�Ö�Ù�þ�Î�Ö�Ê�Û�Í�è_ΧÛ�Ê�ØjË � $ 4(' ' * " B98O:P;�B2} :<;�BA@GB�� ��Ù�ÚPÏ\�,7 6-0 B98':<;�B2} :<;=B�� B�� BA@ZB�� �2Û�Ø�ÊnË @�?Ê�Ø?@h? � Þ�è_ÎlØ�Î @ �me @>= B @�?.fnõ

*�� 0 �B5 - � 741 5 + � 2S1�� �#è_Î�Ú]Ø�Î�ßàÎ�̼ԺÌ�Ú_þ¯Û�Ø�ÊnË @ = ٩˧Î�ÖjÖ�Ù�þ�Î��,7 6-0 B98O:<;=B2} :<;�B2} :<; ù B2} :<; ï B @ ? B �� �Ì�Û ÒDÊ�Í�è � } :<; ù B e @C= B @�? f B úyù�� Ù�Ú�Ï � } :<;�ï B e @>= B @h? f B ú ï � Ù�Ø�Î Ø�Î�ßàÊ�ØjÏ�Î�Ï � Í�è_Î�ÚßàÊn˵ãPÕ_Í�Î ú c 4^1 9 î�:; ÜVú ù B ú ï Ý � Ø�Î�ßàÊ�ØjÏ � } :<;�B e @ = B @ ? f B ú�� � Ö�Î�Ú�Ï Ë§Î�ÖjÖ�Ù�þ�Î�O� -��_(/9K5�d8�� J�� B98':<;�B2} :<;=B2}0:P; ù B2}0:P;_ï B @>= B @h? B ��+Í�Ê @>= � @h?�Ù�Ú�Ï Q � Ù�Ú�Ï Ì�þnÚ_Ê�Ø�Î Ù�Ó�Ó]Û¸Õ_Ø�ñÍ�è�ÎlØ3˧Î�Ö�Ö�Ù�þ�Î�Ö3Ê�Û�Í�è�εۦÊ�ØhË � $ 4(' '+* " B98O:P;�B2} :<;�BA@GB�� ��Ù�ÚPÏ �,7 6-0 B98':<;=B2}0:P;�B�� B�� BA@ZB�� �2Û�Ø�ÊnË @>=³Ê�Ø@�? � Þ�è_ÎlØ�Î @ � e @C= B @�? fnõ:ÐrÍ�è_ÎlØ�Þ�Ì�Ö�Î � Ï�ʵÚ_Ê�Í�è�Ì�Ú_þ_õ

* ,I5�1 - 5 + � 2S1�� �#è_Î�Ú#Ø�Î�ßàÎ�̼ԺÌ�Ú_þ Û�Ø�ÊnË @C=�٠˧Î�Ö�Ö�Ù�þ�Î��,4^1[7-, B98':<;�B2} :<;=B @h?�� � ̼Û�Í�è�Î�Í�Õ_ãPÓ¼Î� } :<;=B e @ = B @ ? f B ú��Ì�ÖØ�Î�ßàÊ�ØjÏ�Î�Ï � Í�è_Î�Ú�Ö�Î�Ú�Ï�˧Î�Ö�Ö�Ù�þ�Î^�O� ����� B98':<;�B2} :<;=B @ = B @ ? B ú���Í�Ê @ ? ûnÊ�Í�è_ÎlØ�Þ�Ì�Ö�Î �Ï�ʵÚ_Ê�Í�è�Ì�Ú_þ_õ

�N̼þnÕ_Ø�Î3ò &% '�)���578l/�,��!H2,.)s)7A0HJ>.CqH2,M8B57/4LOC�/�3�,M8B57/dHJ>l8:,QP�

å4�

Page 18: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

ö Í:Ù�è�̼þnèµÓ�ÎlÔ�Î�Ó � Í�è�Î�Ù�ã�ã�ØjÊnÙ�ßjè±Þ�Î�Þ�Ì�Ó�ÓPÕ�Ö�Î�Í�Ê2Ø�Î�Ù�Ó�Ì��lÎ�Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á =���D>¹Ì�Ö�Ö�Ì�˵Ì�Ó¨Ù�Ø>Í�Ê2Í�è�Ù�ͳÌ�Ú{äóò � � å�ò�ç®õ� Ú»ã�Ù�Ø�Í�Ì�ßlÕ�Ó�Ù�Ø � Î�Ù�ßhèÑÒ�̼Í3Ö�Í�Ê�Ø�Î�Ï Ì�Ú = ��D> Þ�Ì�Ó¨ÓNÒ-Î���Ð_>5ñ�Öjè�Ù�Ø�Î�ÏÑÒ�á�@ = Ù�Ú�Ï6@ ? � Ù�ÚPÏJÎ�Ù�ßhèÑþnÙ�Í�ίÎlÔ�Ù�Ó�Õ�Ù�Í�̼ÊnÚÞ�Ì�Ó�Ó_ÒDÎrÏ_ÊnÚ_Î�Ò�á«Ù]í � ïvðlñ®Ê�ÒPÓ�̼Ժ̼ÊnÕ�ÖGÍ�ØjÙ�Ú�Ö�Û�ÎlØ�õI%�Ê9Þ:ÎlÔ�ÎlØ � ÊnÕ_Ø:Ø�Î�Ö�Õ�Ó�Í�Ì�Ú_þ2ßàÊnÚPÖ�Í�ØjÕ�ßàÍ�̼ÊnÚ±Ì�Ö:Ï�̼Ø�Î�ßàÍ�Ó�áµÖ�Î�ßlÕ_Ø�ÎrÙ�þnÙ�Ì�Ú�Ö�Íٯ˵Ù�Ó¨Ì�ßl̼ÊnÕ�Ö � Ù�Ï�Ù�ã�Í�Ì�Ô�Î�Ù�Ï�Ô�ÎlØjÖ�Ù�Ø�á � Ù�ÚPÏ÷Í�è�ÎlØ�ÎlÛ¦Ê�ØjÎ�Þ:ΧÏ_Ê©Ú_Ê�Í�Ú_ÎlÎ�Ï]Í�è_Î�é�ßàÊn˧ãP̨ӼÎlØjì±Õ�Ö�Î�Ï]Ì�Ú äóò � � å�ò�ç®õ3×%è�̨Öé�ÏP̼Ø�Î�ßàÍ�ìLܸÙ�Ö�Ê�ã�ãDÊnÖ�Î�ϯÍ�ʵÍ�è_ÎLé�ÍvÞ�Ê�ñ®ã�è�Ù�Ö�Î�ì�Ý�Ù�ã�ã�Ø�ÊnÙ�ßhè¹ËµÙ�ý�Î�Ö�ÊnÕ_Ø%ã�Ø�Ê�Í�ʺßàÊnÓAË�Õ�ßjèL˧Ê�Ø�Î�Î��«ßlÌ�Î�Ú�Ílõ

� Ú]ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�Ø � Þ:ΧÞ�Ì�Ó¨Ó�Ø�Î�Ù�Ó¨Ì��lΧÍ�è_Î =���D>ÑÛ�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá÷Õ�ÖjÌ�Ú_þ¯Ù¯Û�Õ_ØjÍ�è_ÎlØ�þ�Î�Ú_ÎlØjÙ�Ó�Ì ��Ù�Í�̼ÊnÚ]Ê�Û�Í�è�Î =#>@? ADCÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá � Þ�èPÌ�ßjè«Þ�Î�ßlÙ�Ó�ÓB= �>@?BADC õ?×%è_Î�É:Ên˵˵̼Í:Ù�Ú�ϹÐ�ãDÎ�ÚµãPèPÙ�Ö�Î�Ö:Ê�ÛR= �>@?BADC Ù�ØjÎ�Ì�Ï�Î�Ú�Í�Ì�ßlÙ�Ó�Í�Ê�Í�è_ÊnÖ�Î�Ê�Û=#> ? ADC � ÒPÕ�Í:Í�è�Î3×�ØhÙ�Ú�Ö�Û�ÎlØ5ãPè�Ù�Ö�ÎrãDÎlØ�Û�Ê�Øj˵ֳ٠í � ï�ð ñ®Í�ØjÙ�Ú�Ö�Û�ÎlØ2ܸÌ�Ú�Ö�Í�Î�Ù�ϯÊ�Û í î ïvð Ý � Þ�è�̨ӼÎ�Í�è�Î"�?Ø�Ê�Ô�Î�ãPè�Ù�Ö�Î�ãPØ�Ê9Ô�Î�ÖØjÎ�Ó�Ù�Í�̼ÊnÚ�Ö5ßàÊnÚ�ÖjÌ�Ö�Í�Ì�Ú_þ§Ê�ÛNô:Ê�ÊnÓ�Î�Ù�Ú¹Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ö³Ê�Û?/ �,�8�&�H+���/��(ܸÙ�Ö�Ê�ã�ãDÊnÖ�Î�ϯÍ�Ê«ÍvÞ�Ê�Ýhõ��ιÚ_Ê9Þ ã�Ø�Î�Ö�Î�Ú�Í = �> ? ADC Ì�Ú �N̼þnÕ�Ø�ÎJ�¡õ���ιÕPÖ�Î 4^129 :; Í�Ê{Î��ºã�Ø�Î�ÖjÖ�Ù]ô:Ê¡ÊnÓ¼Î�Ù�Ú Û�ÕPÚ�ßàÍ�̼ÊnÚÑÊ�Û�Í�è_Ø�ÎlαÒP̼Í�Ö �

Þ�è_ÎlØ�Î< �me�3 B å�f � Ï_Î�Ú_Ê�Í�Î�Ö5Í�è_Î3ÊnÕ_Í�ãPÕ_Í5ÒPÌ�Í�Ö³Ê�Û.Í�è�Î�ô�Ê¡ÊnÓ¼Î�Ù�ÚLÛ�Õ�ÚPßàÍ�̼ÊnÚ�� Ö³Í�ØjÕ_Í�è¹Í�Ù�ÒPÓ¼Î�õ

) -.-Z27 / �4- �B58/ 7 9 = �> ? ADC

= �> ? ADC ã�Ø�ʺßàÎlÎ�Ï�Ö%Ù�Ö�Û¦ÊnÓ�Ó�Ê9Þ�Ö � ØhÕ�Ú�Ú�Ì�Ú�þ3Þ�̼Í�è¯ã�Ù�Ø�Í�̼Î�Ö @ ïCBEDVDVDVB @ZF©Ù�Ú�ÏLÙ�Ú¹Ù�Ï_Ô�ÎlØjÖ�Ù�Øjá Q *�� �43 36/87 5 + �B2S1d� �#è_Î�Ú­Ø�Î�ßàÎ�̼ԺÌ�Ú_þ�Û�Ø�ÊnË @ = ÙJ˧Î�Ö�ÖjÙ�þ�Î �,7%$ 4."P/&$ 4(' '+* " B98O:P;�B2} :<;�B @ ? B ú�� �ØjÎ�ßàÊ�ØjÏ � } :<;�B @>= B @�? B ú�� � Ö�Î�Ú�Ï�˧Î�Ö�Ö�Ù�þ�Î �O�dCL���/9K5�d8�� J�� B98':<;�B2} :<;=B @C= B @�?J��Í�Ê @C= � @�?µÙ�Ú�Ï�Q �Ù�ÚPÏY̼þnÚ�Ê�Ø�Î÷Ù�Ó�Ó�Û�Õ�Í�Õ_Ø�ÎL˧Î�ÖjÖ�Ù�þ�Î�Ö¯Ê�Û�Í�è_ÎLÛ�Ê�ØjË �,7%$ 4."</&$ 4(')'+* " B98O:P;�B2} :<;�B @�? B�� �«Û¦Ø�ÊnË�@>=�Ù�Ú�Ï�,7%$ 4."P/7" 3102,4365 7.3 B98':<;�B2} :<;=B�� B�� B�� B�� B��jB @ =���Û�Ø�ÊnË @�?�õ

*�� /���0T1 5 + � 2S1�� �#è_Î�ÚYØ�Î�ßàÎ�̼ԺÌ�Ú_þ�Û�Ø�ÊnË @>=�ÙJ˧Î�ÖjÖ�Ù�þ�Î �,7%$ 4."P/21.354�6�7 B98O:P;�B2} :<; ù B2} :<;_ï B2} :<; î B} :<; B @h? B �� � ̼Û(Í�è_ÎaÛ�ÊnÓ�Ó¼Ê9Þ�Ì�Ú_þ#Û¦ÊnÕ�Ø�Í�Õ_ãPÓ¼Î�Ö\� } :<; ù B @C= B @�? B úyù�� � � } :<;_ï B @C= B @�? B ú ï � � � } :<; î B @C= B@�? B ú î � � Ù�Ú�Ï � } :<; B @C= B @�? B ú � � Ù�Ø�Î�Ù�Ó�Ó�Ø�Î�ßàÊ�ØhÏ_Î�Ï � Ù�Ú�Ï�4^1 9 :; ÜVúyù B ú ïCB ú î Ý c ú � Í�è_Î�Ú÷Ö�Î�Ú�Ϲ˵Î�Ö�ñÖjÙ�þ�Î �O��CL���/9J KILML�N B98':<;�B2} :<; ù B2} :<; ï B2} :<; î B2} :<; B @ = B ��NÍ�Ê,@ ? Ù�Ú�ÏMQ�û¡Ê�Í�è_ÎlØ�Þ�Ì�Ö�Î�Ï_Ê3Ú_Ê�Í�èPÌ�Ú_þ_õ* * / �=-02C;X1 / 5 + � 241d� �#è_Î�Ú Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ Û�Ø�ÊnË @>=0٠˵Î�Ö�Ö�Ù�þ�Î �,7%$ 4."P/7"030 , 365,7.3 B8':<;=B2}0:P;�B2} :<; ù B2} :<; ï B2} :<; î B2} :<; B � } :<; ù B � } :<; ï B @ ? � � Ì¼Û Í�è�Î Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ Ö�ÌÄ� Í�Õ�ãPÓ¼Î�Ö� } :<; ù B @>= B @�? B úyù�� � � }0:P;_ï B @>= B @h? B ú ï � � � } :<; î B @C= B @h? B ú î � � � } :<; B @C= B @�? B ú � � � � } :<; ù B @h? B @C= B ú � � �� � } :<;_ï B @h? B @C= B ú S � � Ù�ØjÎ Ù�Ó�Ó ØjÎ�ßàÊ�ØjÏ_Î�Ï � Í�è_Î�Ú Ø�Î�ßàÊ�ØjÏ � } :<;=B @h? B @>= B úyü87;� � � �e�3 B å B ò B � f � Þ�è_ÎlØ�Î � � Ö ÒPÌ�Ú�Ù�Ø�á ØjÎlã�Ø�Î�Ö�Î�Ú�Í�Ù�Í�̼ÊnÚ ÎS!¡Õ�Ù�Ó�Ö ú � ú S � Ö�Î�Ú�Ï Ë§Î�Ö�ÖjÙ�þ�Î�Ö�O��CL���/�� ����� B98':<;�B2} :<;=B @C= B @�? B2} :<; ù B2} :<;_ï B2}0:P; î B2} :<; B � } :<; ù B � } :<;�ï B úyü 7 � Í�Ê @�? Ù�Ú�Ï�O��CL���/9K ��8�� J�� B98O:P;�B2} :<;�B @ = B @ ? B2} :<; ù B2} :<; ï B2} :<; î B2} :<; B � } :<; ù B � } :<; ï �=Í�Ê @ = Ù�Ú�Ï Q � Ù�Ú�ÏÌ�þnÚ_Ê�Ø�Î�Ù�Ó�Ó»Û�Õ_Í�Õ�Ø�Π˧Î�Ö�Ö�Ù�þ�Î�Ö Ê�ÛaÍ�è_Î Û�Ê�ØjË �,7%$ 4."</&$ 4(')'+* " B98':<;=B2}0:P;�B @>= B�� ��Û�Ø�ÊnË @�? Ù�Ú�Ï�,7%$ 4."P/7" 3102,4365 7.3 B98':<;�B2} :<;=B�� B�� B�� B�� B��jB��jB @�?��:Û�Ø�ÊnËg@>=�õ:Ð�Í�è_ÎlØ�Þ�Ì�Ö�Î � Ï_ʵÚ_Ê�Í�è�̨Ú_þ_õ

* ,I5�1 - 5 + � 2S1�� �+è_Î�Ú»Ø�Î�ßàÎ�̼ԺÌ�Ú_þ¯Û¦ØjÊnË @C=5٩˧Î�Ö�ÖjÙ�þ�Î��,7%$ 4."</<4^1[7-, B98':<;=B2}0:P;�B @>= B @h?7� � ̼Û�Í�è_ÎÍ�Õ�ãPÓ¼Î4� } :<;�B @>= B @h? B ú��rÌ�ÖrØ�Î�ßàÊ�ØjÏ�Î�Ï � Ö�Î�Ú�Ï�˧Î�ÖjÖ�Ù�þ�Î��O��IL���/�� ����� B98O:<;=B2} :<;�B @>= B @�? B ú���Í�Ê©ÒDÊ�Í�è@�?�Ù�ÚPÏRQ�û_Ê�Í�è_ÎlØjÞ�Ì�Ö�Î � Ï_Ê(Ú_Ê�Í�è�Ì�Ú_þ_õ

�N̼þnÕ�Ø�Î�� �� L CJF~,M'�)IF ��)0/�)IFIHJ>@8���H2,M8B57/�57Lb,M'�)^)�+-,.)0/�12)�1430576#698:,;,.)�1 52=�>l8:A~8n52C�DE,GFIHJ/KD�LM)IFNLOC�/�3�,M8B57/dHJ>l8:,QP

%�ÎlØ�ΩÞ:ιÖ�ý�ÎlÍ�ßhè­è_Ê�Þ Í�è_ÎJ= �>@?BADC Û�Õ�ÚPßàÍ�̼ÊnÚ�Ù�Ó�̼Ívá»ßlÙ�ÚaÒDΩÌ�˧ãPÓ¼Î�˵Î�Ú�Í�Î�Ï Ì�Ú Í�è_ÎJ= > ? ADC ñ�è�á�Ò�ØhÌ�ÏÑ˵ʡÏ_Î�Ó®õ� ˧ãPÓ�Î�˧Î�Ú�Í�Ù�Í�Ì�ÊnÚ(Ê�Û�Í�è_Î3É:Ên˵˵̼ͳÙ�Ú�ϹÐ�ãDÎ�Ú(ãPè�Ù�Ö�Î�Ö³Ì�Ö³Ö�Í�ØjÙ�Ì�þnè�Í�Û�Ê�Ø�Þ³Ù�ØjÏ�õ?×%è_Î��?ØjÊ9Ô�Î�ãPè�Ù�Ö�Î�Ì�Ö:ÒPÙ�Ö�Î�Ï(ÊnÚ±Í�è_ÎÛ�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ(ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ì�ÊnÚ÷Ê�Û�Ù(Í�è�Ø�ÎlÎàñ®ÒP̼Í�ô�Ê¡ÊnÓ¼Î�Ù�Ú]þnÙ�Í�Î f Õ�Ö�̨Ú_þ«ë�Ô�Î�Í�Þ:Ê�ñ®ÒP̼Í�ô:Ê¡ÊnÓ¼Î�Ù�Ú{þnÙ�Í�Î�Ölõ���OAÎlÍ @ZB m B N¯ÒDÎÍ�è�Î�Ì�Ú�ãPÕ_Í�Ö5Í�Ê f � Ù�Ú�Ï{Ó¼ÎlÍ f p� ü%Ï�Î�Ú_Ê�Í�Î�Í�è_Î2ÍvÞ�Ê�ñ®Ò�̼Í�ô:Ê¡ÊnÓ¼Î�Ù�Ú{þnÙ�Í�ΧßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�Ï�̨Ú_þµÍ�Ê f Þ�è_Î�Ú @ ̨Ö�Ö�ÎlÍ�Í�Ê©ú9õ

�$V&KLGKLZ�P8K�ÃaGÅ�*R�G�P ��P&KLGKLZ*P&K|KTP�È�GDRaK�Ã:V?[\E*]*��KLR r�� �ÅP�E*MiUDV�­W]*��KLGDR� r�� ���

å�æ

Page 19: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

×�è_Î�Úf Ü @ZB m B N�ÝYc Ü��@ � f p� ù Ünm B N¡Ý�Ý�5]Ü @ � f p� ï Ünm B N�Ý�Ý D

2�Ö�Ì�Ú�þ�Í�è�Ì�Ö:ßàÊnÚ�Ö�Í�ØjÕPßàÍ�̼ÊnÚ � ÊnÚ�Î�ßlÙ�Ú¯Ì�˧ã�Ó¼Î�˧Î�Ú�Í?Í�è_ÎE�GØ�Ê9Ô�ÎrãPèPÙ�Ö�Î�Ê�Û`= �> ? ADC Þ�̼Í�èµëPÔ�Îc=#> ? ADC÷̨Ú�Ô�ʺßlÙ�Í�̼ÊnÚ�Ö�Ê�Û�GØ�Ê9Ô�ÎrÙ�Ú�Ï(Û¦ÊnÕ�Ø�Ê�ÛÉ:Ên˫˵̼Í�Ü�Í�ʧßàÊn˵˵Ì�ÍGÍ�Ê�Í�è_Î�̨Ú�Í�ÎlØj˵Î�Ï�Ì�Ù�Í�Î%ÒPÌ�Í�ÖhÝhõ>×%è_Î�×ØjÙ�Ú�Ö�Û¦ÎlØ:ãPè�Ù�Ö�Î�Ê�Û`= �>@?BADC ßlÙ�Ú(ÒDÎØjÎ�Ù�Ó�Ì��lÎ�Ï�Õ�Ö�Ì�Ú_þ¯í î ïvðlñ®Í�ØjÙ�Ú�Ö�Û�ÎlØjÖ.Í�Ê�þ�ÎlÍ�è_ÎlØ>Þ�̼Í�è�Ù�Õ��_Ì�Ó�Ì�Ù�ØjárßàÊn˫˵̼Í�˧Î�Ú�Í�Ölõ ö ÖjÖ�Õ�˧Î�Í�è_Î5Ö�Î�Ú�Ï_ÎlØ.Þ:Ù�Ú�Í�ÖNÍ�ÊrãDÎlØ�Û¦Ê�ØhËÍ�è�Î÷í � ï�ðyñ®Í�ØhÙ�Ú�Ö�Û�ÎlØ�Ê�Û.ÒP̼Í�Ö�ú0= � : �me�3 B å B ò B � fnõ%×�è_Î�Ö�Î�Ú�Ï_ÎlØ�ßàÊn˵˵̼Í�Ö5Í�Ê(Ù�Õ��ºÌ¨Ó�Ì�Ù�Ø�á(ÒP̼Í�Ö } = � : �me�3 B å B ò B � f � Ù�Ú�Ï; ù B2;_ï � Ù�Ú�Ϲã�Ø�Ê9Ô�Î�Ö�Í�ʵÍ�è_Î�ØjÎ�ßàÎ�̼Ô�ÎlØ�Í�è�Ù�Í

úyù c } ù � ; ù Bú ï c }�ï � ; ù Bú î c } î � ;_ïEBú c } � ;_ïED

×�è_εí � ïvðàñ®Í�ØhÙ�Ú�Ö�Û�ÎlØ>Ê�ÛNÜVúyù B ú ïCB ú î B ú Ý.ßlÙ�Ú«Ú_Ê9Þ ÒDγØ�Î�Ù�Ó�Ì �lÎ�Ï�Ò�á�Í�è_Ø�ÎlΫí î ï�ðyñ®Í�ØjÙ�Ú�Ö�Û�ÎlØjÖ>Ê�Û-Í�è_Î5ãPÙ�Ì�ØjÖ5Ü ; ù B2;_ï Ý � Ü } ù B2}9ï ÝÙ�ÚPÏJÜ } î B2} ÝhõWOAÎ�Ù�ÔºÌ�Ú_þ«Û�Õ_ØjÍ�è_ÎlØ%Ï_ÎlÍ�Ù�Ì�Ó�Ö³Û�Ê�Ø%Í�è_Î2Ì�Ú�Í�ÎlØjÎ�Ö�Í�Î�Ï©Ø�Î�Ù�Ï_ÎlØ � Þ�Î2Ö�Ì�˧ãPÓ�áµÖ�Í�Ù�Í�Î3Í�è_Î3Û¦ÊnÓ�Ó�Ê9Þ�Ì�Ú�þ�Ó�Î�˵˵ٺõv 1 363 � � 9My �C�)���8� �%$ �t��/��7 �� ���·#��L���x/�(}�O%"/L%�#�%�!;/ �= "/ ���8#����O��!Ç1 �8�8 �!.�VUN��� / ��� = �> ? ADC 9��=�:#�/0��%��} �!.��/01 ���¹/ ���=#> ? ADC��P�,1,+���� ) 'H%S),��! �<, ����)��/ 'H �!.�0#��0%��=� �? ) �(:/0�-�"�6 )��"������ ����L��� ���Î%Ú_Ê9Þ Í�Õ�ØjÚ�ÊnÕ_Ø?Ù�Í�Í�Î�Ú�Í�Ì�ÊnÚ§Í�Ê�Í�è_γØ�Î�Ù�Ó�Ì���Ù�Í�Ì�ÊnÚ�Ê�ÛPÍ�è_Î�=���D>(Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á3Ì�Ú2Í�è_Î�= �> ? ADC ñ�è�á�Ò�ØhÌ�Ï2˵ʡÏ_Î�Ó®õ

ö Ö?Þ:Î�˧Î�Ú�Í�̼ÊnÚ_Î�Ï«Ù�Ò-Ê�Ô�Î � Î�Ù�ßhè«Ò�̼ÍGúrܸÙ�Ö�Ö�ʺßlÌ�Ù�Í�Î�Ï«Þ�̼Í�è§Ù�Ú«Ì�Ï�Î�Ú�Í�̼ëPÎlØ } :<; Ý>Ö�Í�Ê�ØjÎ�Ï«Ì�Ú = � �D> Ì�ÖGÖ�è�Ù�Ø�Î�ϧÒDÎlÍvÞ�ÎlÎ�Ú@>=AÙ�Ú�Ï @h?nõ �¹Ê�Ø�Î�ã�Ø�Î�ßlÌ�Ö�Î�Ó¼á � @>=Aè�Ù�Ö:Ù�ÒP̼ÍGú ï Ù�Ú�ÏM@�?�èPÙ�Ö�Ù3ÒP̼Í�ú î Ö�ÕPßjèµÍ�èPÙ�ͳúpc�ú ï � ú î õ���Õ_Ø�Í�è�ÎlØj˧Ê�Ø�Î � Î�Ù�ßjèÊ�Û³ú ï Ù�Ú�Ï�ú î Ì�ÖrÙ(ØhÙ�Ú�Ï_ÊnË�ÒP̼Í�Ò�á{̼Í�Ö�Î�Ó¼Û�õ�ô:Ê�Í�èX@C=³Ù�Ú�ÏP@�?�Þ�Ì�Ó�Ó.ßàÊn˵˵̼Í�Í�Ê©Í�è_Î�̼ØrÒPÌ�Í�Ö�Í�Ê©Î�Ù�ßhè�Ê�Í�è_ÎlØ�Õ�Ö�̨Ú_þ̨Ï_Î�Ú�Í�Ì�ë�ÎlØ }0:P; õ�×ʵÊ�ãDÎ�Ú©Í�è�ֳ̨ÒP̼ͳÍ�ÊM@>= � @h?�Ê�ã-Î�ÚPÖ5̼Í�Ö%Ö�è�Ù�ØjÎ � ú î � Í�Ê @C= � Þ�è�ʧÍ�è_Î�Ú¹ßàÊn˧ãPÕ�Í�Î�Ö�úpc�ú ï � ú î õ

� Ú�Ê�ØjÏ_ÎlØNÍ�ÊrÎlÔ�Ù�Ó�Õ�Ù�Í�Î } cy4^1 9 î�:; ܸø B úlÝ � Ö�Õ_ãPã-ÊnÖ�Î @>=�è_ÊnÓ¨Ï�Öø ï Ù�Ú�ϧú ï ܸÖ�è�Ù�ØjÎ�ÖNÊ�ÛPø3Ù�Ú�ϧú � Ø�Î�Ö�ãDÎ�ßàÍ�̼Ô�Î�Ó�á_Ý � Ù�Ú�Ï@�??è�ÊnÓ�Ï�Ö�ø î Ù�Ú�Ï3ú î õ×%è�Î�Ú � @>=ºþ�Î�Ú_ÎlØjÙ�Í�Î�Ö.Ù5ØjÙ�Ú�Ï_ÊnË ÒPÌ¼Í }�ï "$ e�3 B å�f³Ù�ÚPÏ�ßàÊn˧ãPÕ_Í�Î�Ö�Û¦ÊnÕ�ØAÒP̼Í�Ö � ù�ù B � ù ïCB � ï ù B � ï�ï �Þ�è�Ì�ßjè{Ù�Ø�Î�Í�è_Î�é�ßlÙ�Ú�Ï�Ì�ÏPÙ�Í�Î2ÒPÌ�Í�Ö�ì«Û�Ê�Ø } î � @ ? � ÖrÖ�è�Ù�ØjÎ2Ê�Û�ÒPÌ¼Í } õ��#è�Ì�ßjè{ÒPÌ�Í�Ì�Ö } î Ï_ÎlãDÎ�Ú�ÏPÖ%ÊnÚP@ ? � ÖrÖ�è�Ù�Ø�Î�Örø îÙ�ÚPÏLú î õ�×%è_Î�Ù�ßàÍ�Õ�Ù�Ó�Ò�̼Í�Ö%Ù�Ø�Î�ßàÊn˧ã�Õ_Í�Î�ÏLÙ�Ö%̨گÍ�è_Î�Í�Ù�ÒPÓ¼Î�ÒDÎ�Ó¼Ê�Þ�õ

� c d D�� d � ���9¾ �Np���tv����t b d�CV!DHV�� �� ��b K���� -�� d��� � c K D�� K ��CV!D�]y� � K ��b K���� - � d��� � c K Dh��� K�� ]y�v��7]#DHV�� � K ��b K ��� - � d��� �v� c K � ]y�lD�� K ��7]#D�]y� � KHK ��b K ��� - � d��� �v� c K � ]y�lDh��� K � ]à���

@>=GÍ�è�Î�Ú�ßàÊn˵˵̼Í�ÖrÍ�Ê©Í�è_Î�ÒPÌ�Í�Ö }�ïCB � ù�ù B � ù ïCB � ï ù B � ï�ï Ù�ÚPÏ÷ãPØ�Ê9Ô�Î�Ö�Í�Ê @�?�Í�è_ΧØ�Î�Ó¨Ù�Í�̼ÊnÚ�ÖrÌ�Ú{Í�è_Î�Í�Ù�ÒPӼΧÕ�Ö�̨Ú_þ«Í�è_Î= �> ? ADC �?ØjÊ9Ô�Î�ãPèPÙ�Ö�Î�õI��Î�Õ�Ö�Î ù B ï B î B Í�ʵÏ�Î�Ú_Ê�Í�Î�Í�è_Î�Î�Ú�ßàʺÏ�Ì�Ú_þnÖ:Ê�ÛÍ�è_ÎrÛ�Õ�ÚPßàÍ�̼ÊnÚ�Ö³Ù�Ö�Ö�ʺßlÌ�Ù�Í�Î�Ï©Þ�Ì�Í�èÍ�è�Î�Ö�Î�Ø�Î�Ó�Ù�Í�̼ÊnÚPÖlõI��Î��ºÍ � @>=.Ù�Ú�ÏR@�?rÎ�Ú_þnÙ�þ�Î2Ì�Ú¯Ù»í � ï�ðyñ®Ê�ÒPӨ̼ԡÌ�ÊnÕ�Ö:Í�ØjÙ�Ú�Ö�Û�ÎlØ5Ö�ʧÍ�è�Ù�Í @�?rØ�Î�ßàÎ�̼Ô�Î�Ö�ÒPÌ¼Í � � Ì ü Ì � Þ�è�Ì�ßhèÌ¨Ö @h? � Ö�Ö�è�Ù�ØjÎ�Ê�ÛÒPÌ�Í } õ

×%è_Î�Û¸Õ�Ó�Ó�ã�Ø�Ê�Í�ʺßàÊnÓ &u���� �+Û¦Ê�ØLÖ�Î�ßlÕ_Ø�Î�Ó¼áYØ�Î�Ù�Ó�Ì���Ì�Ú�þ =�� �D>=Ì�Ú#Í�è�Î = �>@? ADC ñ�è�á¡Ò�ØjÌ�Ï#˧ʺÏ_Î�Ó�Ì�Ö©Ö�è�Ê9Þ�Ú Ì¨Ú�N̼þnÕ�Ø�Î M � Þ�̼Í�è¯Í�è�Î�ßjè_Î�ßjý¡Ö�Û¦Ê�Ø�Ø�Î�Õ�Ö�Î�Ï } :<; � Ö%Ên˵Ì�Í�Í�Î�Ï�õ*,+ 1��T/(1 3 �;9 ��� �O%"/L%�#�%�! &X���� �·���&#����O��!Ç1 �8�8 "!Ç� U����w/ ����=���D> 9����}#�/0�0%"�} �!.��/01 ���·/ ��� = �>@? ADC �P�,1,+�����) 'H%S),��! �<, ����)��/Í'H �!.�0#��0%"�k� �Í ) �(:/0�-�"�6 )��"������ ����L��� �

� /�� �4;�� OAÎlÍ'H ÒDÎ2Ù�Ú]Ù�Ï�Ù�ã�Í�̼Ô�ΧÙ�Ï_Ô�ÎlØhÖ�Ù�Ø�á¹Í�è�Ù�ÍrÊ�ãDÎlØjÙ�Í�Î�ÖrÙ�þnÙ�Ì�Ú�Ö�Í�ãPØ�Ê�Í�Ê¡ßàÊnÓ &X���� �¹Ì¨ÚLÍ�è_Î = �> ? ADC ñ�è�á�Ò�ØhÌ�Ï˵ʡÏ_Î�Ó®õ���ΫßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ù�Ú»Ì�Ï_Î�Ù�ÓNÙ�Ï_Ô�ÎlØhÖ�Ù�Ø�á Q Ö�Õ�ßhè]Í�è�Ù�Í�Ú_Ê©Î�Ú�ԡ̼ØjÊnÚ�˧Î�Ú�Í J ßlÙ�Ú�Í�Î�Ó�ÓNÞ�Ì�Í�è{Ú�ÊnÚºñ�Ú_ÎlþnÓ�̼þnÌ�ÒPÓ¼Î

�

Page 20: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

� /��B7 ��2 �D5 &u�� � �

*�� �43 36/87 5 + � 2S1��å�õ ÐrÚ¯Ì�Ú_ãPÕ_ÍE� $ 4(' ' * " B98O:P;�B2} :<;�B @�? B ú�� � @C=.ßjè_Ê¡ÊnÖ�Î�Ö�ú ï "$ e�3 B å�f � Ö�ÎlÍ�Ö�ú î c�ú ï � ú � Ù�Ú�Ï©Ö�Î�Ú�Ï�Ö

˧Î�Ö�Ö�Ù�þ�Î�Ö �,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�B @�? B ú ï �³Í�Ê�= �>@?BADC � Ù�Ú�ÏJÜ ��� 7 / 3�� 0 357 B98':<;�B2} :<;=B ú î Ý�Í�ÊM@�?�õò¡õ 2�ãDÊnÚ¯Ø�Î�ßàÎ�̼ԺÌ�Ú_þ©Ü ��� 7 / 3�� 0.3 7 B98':<;�B2} :<;=B ú î Ý�Û¦ØjÊnËg@C=NÙ�Ú�Ï �O��IL���/9K5��5�� J�� B98':<;=B2}0:P;�B @>= B @h?7�Û¦Ø�ÊnË = �> ? ADC � @�? Ö�Î�Ú�Ï�Ö �,7%$ 4."</&$ 4(')'+* " B98O:P;�B2} :<;�B @>= B ú î �©Ù�ÚPÏ �,4^1[7-, B98O:P;�B2} :<;�B @�? B @C=.�¹Í�Ê

= �>@? ADC � Ù�ÚPÏ{Õ�ã-ÊnÚ÷Ø�Î�ßàÎ�̼ԺÌ�Ú_þ �O��IL���/9K5��5�� J�� B98':<;=B2}0:P;�B @�? B @>=.��Û¦Ø�ÊnË = �> ? ADC � @�?�ÊnÕ_Í�ñãPÕ_Í�Ö � K5��5�� J�� B98':<;�B2} :<;=B @ ? B @ = �

�¡õ 2�ãDÊnÚ Ø�Î�ßàÎ�̼ԺÌ�Ú_þ�O��IL���/9K5��5�� J�� B98':<;�B2} :<;=B @ = B @ ? � � �O��IL���/9K5��5�� J�� B98':<;=B2}0:P;�B @ ? B @ = �Ù�Ú�Ï��O��CL���/�� ����� B98O:P;�B2} :<;�B @�? B @C= B ú��#Û�Ø�ÊnË = �>@? ADC � Þ�è_ÎlØ�Î ú c ú î � @C=aÊnÕ_Í�ã�Õ_Í�Ö� K ��8�� J�� B98O:P;�B2} :<;�B @ = B @ ? �hõ

*�� 0 �B5 - � 741 5 + � 2S1��å�õ ÐrÚ¹Ì�Ú_ã�Õ_ÍE�,7 6 0 B98':<;�B2} :<;=B2}0:P; ù B2}0:P;_ï B @�? B �� � @>=>ßjè_Ê¡ÊnÖ�Î�Ö }9ï "$ e�3 B å�f � Ö�ÎlÍ�Ö

� ù�ù c } ï � 4^1 9 î�:; ܸø ï B ú ï Ý � � ù ï c } ï � 4^1 9 î�:; ܸø ï B ÜVú ï � å9Ý�Ý �� ï ùMc }�ï � 4^1 9 î�:; Ü�ܸø ï � å9Ý B ú ï Ý � � ï�ï c }�ï � 4^1 9 î�:; Ü�ܸø ï � å9Ý B ÜVú ï � å9Ý�Ý �

Þ�è_ÎlØ�Î5ø ï B ú ï Ù�Ø�Î @>= � ÖGÖ�è�Ù�Ø�Î�Ö.Ê�Û-ÒP̼Í�Ö.Ê�Û�Ì�Ï�Î�Ú�Í�̼ëPÎlØjÖ } :<; ù5Ù�Ú�Ï } :<;_ï � Ø�Î�Ö�ãDÎ�ßàÍ�̼Ô�Î�Ó¼á � Ù�Ú�ϧÖ�Î�Ú�Ï�ÖÍ�è_Î�Û�ÊnÓ�Ó¼Ê9Þ�Ì�Ú_þ�˧Î�Ö�Ö�Ù�þ�Î�Ö�Í�Ê�= �> ? ADC �,7%$ 4."P/&$ 4(' '+* " B98O:P;�B2} :<;�B @ ? B2} ï �

�,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�� 3"3 B @h? B � ù�ù�� � �,7%$ 4."P/&$ 4(' '+* " B98O:P;�B2} :<; � 3_å B @�? B � ù ï ��,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�� å�3 B @h? B � ï ù�� � �,7%$ 4."P/&$ 4(' '+* " B98O:P;�B2} :<; � å�å B @�? B � ï�ï �

Ù�Ú�ÏJÜ ��� 7 /<7 6-0 B98O:<;=B2} :<;�B2} :<; ù B2} :<;�ï B ÷Ý:Í�ÊM@�?�õò¡õ @C=>Ö�Î�Ú�ÏPÖ³Í�è_Î�Û�ÊnÓ�Ó¼Ê9Þ�Ì�Ú_þ§Ë§Î�Ö�Ö�Ù�þ�Î�Ö�Í�Ê�= �> ? ADC

�,7%$ 4."</21.354�6.7 B98':<;�B2} :<; ù B2} :<;_ï B2}0:P;�B2} :<; � 3"3 B @�? B ¹ù�� ��,7%$ 4."</21.354�6.7 B98':<;�B2} :<; ù B2} :<;_ï B2}0:P;�B2} :<; � 3_å B @�? B ï � ��,7%$ 4."</21.354�6.7 B98':<;�B2} :<; ù B2} :<;_ï B2}0:P;�B2} :<; � å�3 B @�? B î � � Ù�Ú�Ï�,7%$ 4."</21.354�6.7 B98':<;�B2} :<; ù B2} :<;_ï B2}0:P;�B2} :<; � å�å B @�? B �hõ

�¡õ @C=�Ö�Î�ÚPÏ�Öh�,7%$ 4."P/7"03102, 365,7.3 B98O:<;=B2} :<;�B2} :<; � 3"3 B2} :<; � 3_å B2} :<; � å�3 B2} :<; � å�å B2} :<; ù B2} :<;_ï B @�?~�§Í�Ê= �>@? ADC � Ù�Ú�Ï«Õ�ã-ÊnÚµØ�Î�ßàÎ�̼ԺÌ�Ú_þ#�O��IL���/9K5��5�� J�� B98':<;=B2}0:P;�B @>= B @h? } :<; � 3"3 B2} :<; � 3_å B2} :<; � å�3 B}0:P;� å�å B2} :<; ù B2} :<; ï � � ÊnÕ_Í�ãPÕ_Í�Ös�O� -��_(/9K5��5�� J�� B98':<;=B2}0:P;�B2} :<; ù B2} :<; ï B @ = B @ ? B ��hõ

M_õ 2�ãDÊnÚ(Ø�Î�ßàÎ�̼ԺÌ�Ú_þ3ØjÎ�ßàÎ�̼ã�Í�Ö:Û¦Ê�Ø5Ù�Ó�Ó�Í�è_ÎrßàÊn˵˵̼Í�˵Î�Ú�Í�Ö³Ù�Ú�ϱã�Ø�Ê¡Ê�Û�Ô�ÎlØj̼ë�ßlÙ�Í�̼Ênگ˧Î�Ö�Ö�Ù�þ�Î�Ö³Û�Ê�ØÙ�Ó�Ó_Í�è_Î%ãPØ�Ê�Ê�Û¸Ö � Ù�Ó¼ÊnÚ_þ�Þ�Ì�Í�èh�O��IL���/�� ����� B98':<;=B2}0:P;�B @>= B @h? B2} :<; � 3"3 B2} :<; � 3_å B2} :<; � å�3 B9} :<; �å�å B2}0:P; ù B2} :<; ï B � � Ì ü Ì �AÛ¦Ø�ÊnË = �> ? ADC � Þ�è_ÎlØ�Î:ø î B ú î Ù�Ø�Î @ ? � Ö.Ö�è�Ù�ØjÎ�ÖÊ�Û_Ò�̼Í�Ö�Ê�Û�Ì�Ï_Î�Ú�Í�̼ë�ÎlØjÖ } :<; ùÙ�Ú�Ï } :<;_ï � Ø�Î�Ö�ãDÎ�ßàÍ�̼Ô�Î�Ó¼á � @�?�ÊnÕ_Í�ã�Õ_Í�Ö �O� ��N_'/9K5��8�� J�� B98O:P;�B2} :<;�B2} :<; ù B2} :<;�ï B @C= B @�? B ��hõ

* ,I5�1 -#5 + � 2S1��å�õ ÐrÚ¹Ì�Ú_ã�Õ_ÍE�,4^1[7-, B98':<;�B2} :<;=B @ ? � � @ = Ö�Î�Ú�Ï�Ö^�,4^1[7-, B98O:P;�B2} :<;�B @ = B @ ? �:Í�Ê<= �>@? ADC õò¡õ 2�ãDÊnÚ§Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ �O��IL���/�� ����� B98':<;=B2}0:P;�B @>= B @h? B ú ï �.Û¦ØjÊnË = �> ? ADC � @�?�ßàÊn˧ãPÕ_Í�Î�Ö:úuc�ú ï(�

ú î Ù�Ú�Ï«ÊnÕ�Í�ãPÕ_Í�Ö �O� ����� B98O:<;=B2} :<;�B @>= B @�? B ú��hõ�%�ÎlØ�Î�ú î ̨Ö�@�? � Ö:Ö�è�Ù�Ø�Î%Ê�Û�Í�è�Î�ÒP̼ÍGÊ�Û�Ì�Ï_Î�Ú�Í�̼ë�ÎlØ}0:P; õ

�.Ì�þnÕ_Ø�Î M ��XF05�,.523057> &u�� � �

ò�3

Page 21: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Ù�Ï�Ô�Ù�Ú�Í�Ù�þ�Î�Þ�è_ÎlÍ�è�ÎlØ%̼Í:̨Ö5Ì�Ú�Í�ÎlØhÙ�ßàÍ�Ì�Ú_þ�Þ�̼Í�è H�Ù�Ú�ϯã�Ù�Ø�Í�̼Î�Ö³ØjÕ�ÚPÚ�Ì�Ú_þ�ãPØ�Ê�Í�Ê¡ßàÊnÓ�&X�� � �±Ì�Ú±Í�è_Î�= �> ? ADC ñ�è�á�Ò�ØhÌ�Ï˵ʡÏ_Î�Ó � Ê�Ø�Þ�̼Í�è%Q+Ì�Ú¯Í�è_Î3Ì�Ï�Î�Ù�ÓDã�Ø�ʺßàÎ�Ö�Ö5Þ�̼Í�è = � �D> õQ=ØhÕ�Ú�Ö3Ù÷ßàÊ�ã�áJÊ�Û H � Ö�Ì�Ë�Õ�Ó�Ù�Í�̨Ú_þLÕ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï�ã�Ù�Ø�Í�̼Î�Ö�Ù�Ú�Ï�= �> ? ADC Û¦Ê�Ø,H(õ*�¹Î�Ö�ÖjÙ�þ�Î�Ö2ØjÎ�ßàÎ�̼Ô�Î�Ï Û¦Ø�ÊnË

J Ù�Ø�ΫۦÊ�Ø�Þ³Ù�ØjÏ_Î�Ï»Í�ÊLÍ�è_ΫÖjÌ�Ë2ÕPÓ�Ù�Í�Î�ÏXH � Ù�Ú�ϻ˵Î�Ö�Ö�Ù�þ�Î�Ö�Ö�Î�Ú�Í2Ò�á{Í�è_αÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏPH�Í�ÊL̼Í�Ö3Î�Ú�Ժ̼Ø�ÊnÚ�˵Î�Ú�Í�Ù�Ø�ÎÛ�Ê�Ø�Þ³Ù�ØjÏ_Î�ÏJÍ�ÊPJ«õRQ1Ù�Ó�Ö�ÊLÖ�ÎlÎ�Ö�Í�è_Î(ãPÕ_ÒPÓ¨Ì�ß�è_Î�Ù�Ï_ÎlØ3Ê�Û%Ù�Ó¨Ó?Í�è_α˧Î�Ö�Ö�Ù�þ�Î�Ö�Û�Ø�ÊnË Í�è_αÕ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï»ã�Ù�Ø�Í�̼Î�Ö�Í�Ê=�� �D> � Ù�Ú�Ï©Ë«Ù�á¯Ï_Î�ßlÌ�Ï_ÎrÞ�è_Î�Ú � Ì�Û�ÎlÔ�ÎlØ � Í�ʧÛ�Ê�Ø�Þ³Ù�ØjÏ©Í�è_Î�Ö�Î�˧Î�Ö�Ö�Ù�þ�Î�Ölõ ��Õ_Ø�Í�è_ÎlØ � Q­ËµÙ�Ì�Ú�Í�Ù�Ì�Ú�Ö5Ù�Ø�Î�ßàÊ�ØhÏ©Ê�Û�Í�è_ÎÖjè�Ù�Ø�Î�Ö%Ê�Û?ßàÊn˫˵̼Í�Í�Î�Ï÷ÒP̼Í�Ö%Í�Ê = ��D> è_Î�Ó�ÏLÒ�á¹ßàÊ�ØjØjÕ_ã�Í�Î�ÏLãPÙ�Ø�Í�̼Î�ÖlõW�_Ê�ØrÎ�Ù�ßhè{Öjè�Ù�Ø�Î�Ê�Û?Ù(ÒP̼Í5Ê�Û?Ì�Ï�Î�Ú�Í�̼ëPÎlØ } :<; �Í�è�ΫØ�Î�ßàÊ�ØjÏ ßàÊnÚ�Í�Ù�Ì�Ú�Ö�ÙLÍ�Õ�ãPÓ¼Î�Ö�Ê�Û�Í�è_ΫÛ�Ê�ØjË�Ü }0:P;�B @>= B @h? B úlÝhõ÷×%è�̨Ö3Ì�Ú�ÏPÌ�ßlÙ�Í�Î�Ö�Í�è�Ù�ͧú§Ì�Ö @C= � Ö�Ö�è�Ù�Ø�Î(Ê�Û%ÒP̼Í�ú �Þ�è�Ì�ßjè©Ì¨Ö5Ö�è�Ù�Ø�Î�Ï©Þ�̼Í�èR@ ? õQ­ã�Ø�ʺßàÎlÎ�Ï�Ö5Ò¡á±Ø�Î�Ù�ßàÍ�Ì�Ú_þµÍ�Ê«Í�è_Î3˧Î�Ö�ÖjÙ�þ�Î�Ö�̼Í%Ö�ÎlÎ�Ö�õ

��/83�-05 � 7 / - c�7 + 1 � �43 36/87�5 + � 2S1�� � Ú§Í�è�Ì�ÖGãPè�Ù�Ö�Î = �> ? ADC ̨ÖGÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï«Ù�ÖGÚ_Ê�ØjË«Ù�ÓVõ���Î�Ò�Ø�Î�Ù�ý§Í�èPÌ�Ö>ãPè�Ù�Ö�ÎÌ�Ú�Í�Ê«ßlÙ�Ö�Î�Ö å�õ @ºÙ9áMQ Ö�ÎlÎ�Ö5˧Î�ÖjÖ�Ù�þ�Î#� $ 4(' ' * " B98O:P;�B2} :<;�B @ ? B $ �GÊnÚ±Í�è_ÎrÊnÕ_Í�þ�ÊnÌ�Ú_þ§ßàÊn˵Ë�Õ�Ú�Ì�ßlÙ�Í�Ì�ÊnÚ«Í�Ù�ãDÎrÊ�ÛG@ = Ï_Î�Ö�ñ

Í�Ì�Ú_Î�Ï(Û�Ê�Ø = ��D> õ?×%è_Î�ÚMQ ßjè_Ê¡ÊnÖ�Î�Ö³Ù2ÒP̼Í:ú î "$ e�3 B å�f3Ù�Ú�ϱÖ�Î�Ú�ÏPÖ�˧Î�ÖjÖ�Ù�þ�Î�Ö�Ü � � 7 / 3��-0.357 B98O:<;=B2} :<;�B ú î ÝÛ�Ø�ÊnË Ü¸Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï�ÝM@C=3Í�Ê @�?]Ù�Ú�Ï��,7%$ 4."P/&$ 4(' '+* " B98':<;�B2} :<;=B @h? B 3q�«Û¦ØjÊnË�@>=3Í�Ê = �> ? ADC õ � Û @h?�̨ÖßàÊ�Ø�ØjÕ�ã�Í�Î�Ï#Q Ö�Í�Ê�Ø�Î�Ö]Ü } :<;�B @ ? B @ = B ú î ݧÙ�Ú�Ï­Û¦Ê�Ø�Þ³Ù�ØjÏ�Ö«Í�è_Î $ 4(')'+* "µËµÎ�Ö�Ö�Ù�þ�Î{Í�Ê = ��D> õ Ü � Û @ ? ̨ÖÕ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï � QYÞ�Ì�Ó�ÓDÖ�Í�Ê�Ø�Î3Í�è_Î�Í�Õ_ãPÓ¼Î�Ù�ÚPÏ©Û¦Ê�ØjÞ:Ù�ØjϯÍ�è�Î3˧Î�Ö�Ö�Ù�þ�Î2Ì�Ú©Í�è_Î�Ö�̨Ë2Õ�Ó¨Ù�Í�̼ÊnÚ¯Ê�Û.@�?�õ�Ý

2�ãDÊnÚ @C=�Ø�Î�ßàÎ�Ì�Ô¡Ì�Ú�þ �O�dCL���/9K5�d8�� J�� B98':<;�B2} :<;=B @C= B @�?J� � �O�dCL���/9K5�d8�� J�� B98':<;�B2} :<;=B @h? B @>=:�Ù�Ú�Ï#�O��CL���/�� ����� B98':<;�B2} :<;=B @h? B @>= B ú��-Û�Ø�ÊnË = �> ? ADC � Þ�è_ÎlØ�ÎGúuc#ú î � Q÷Û¦Ê�ØjÞ:Ù�ØjÏPÖX� K5�d8�� J�� B98':<;�B2} :<;=B @C= B @�?~�Û�Ø�ÊnË =���D>�Í�Ê @h?nõ�Ü ��Ê�Í�Î�Í�è�Ù�Í�̼Û.@�?�Ì�Ö�ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï � Í�è_Î)$ 4(')'+* "%˧Î�Ö�Ö�Ù�þ�Î�Þ�Ì�Ó�Ó�èPÙ�Ô�Î�Ò-ÎlÎ�Ú¹Û¦Ê�Ø�ñÞ³Ù�ØjÏ_Î�Ï©Í�Ê�= � �D> Ù�ÚPÏ»Ü }0:P;�B @ ? B @ = B ú î Ý�Þ�Ì�Ó�ÓDè�Ù9Ô�Î3ÒDÎlÎ�Ú©Ö�Í�Ê�Ø�Î�ÏLÒ�á Q�õ�Ý

ò¡õ @ºÙ9á Q�Ö�ÎlÎ�Ö{Ù�Ú=Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï @�?ÑØ�Î�ßàÎ�̼Ô�Π˧Î�Ö�Ö�Ù�þ�Î�Ö�O��CL���/9K5��8�� J�� B98O:<;=B2} :<;�B @>= B @�? B �¹Û¦Ø�ÊnË= �> ? ADC Ù�Ú�Ï Ü ��� 7 / 3�� 0.3 7 B98':<;�B2} :<;=B ú î Ý÷Û¦ØjÊnË ãPÙ�Ø�Í�á @C=�õ � Û±Ù�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï�@C=¯Ö�Î�Ú�Í�Í�è_Îa˧Î�Ö�Ö�Ù�þ�Î�,7%$ 4."P/&$ 4(' '+* " B98':<;�B2} :<;=B @ ? B ú ï �3Í�ÊF= �> ? ADC � Q�Ö�Î�Ú�Ï�Ö � $ 4(' '+* " B98O:<;=B2} :<;�B @ ? B ú��2Û�Ø�ÊnË @ = Í�Ê*= ��D> �Þ�è_ÎlØjÎ2úMc ú ï � ú î � Ù�Ú�ÏLÖ�Í�Ê�ØjÎ�Ö%Í�Õ_ãPÓ�Î«Ü }0:P;�B @>= B @h? B ú ï Ýhõ � ÛNÚ_Ê�Í � Q+Û¦Ê�ØjÞ:Ù�ØjÏPÖ%Í�è_Î $ 4(')'+* "%˧Î�Ö�Ö�Ù�þ�ÎÛ�Ø�ÊnË Í�è_Î�Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï%@>=.Í�Ê =�� �D> ܸÖ�ÎlÎ�Ù�ÒDÊ�Ô�Î�Ýhõ � Ú¯Î�̼Í�è_ÎlØ�ßlÙ�Ö�Î�Ì�Í%Ö�Í�Ê�Ø�Î�Ö5Í�Õ_ã�Ó¼ÎµÜ }0:P;�B @�? B @>= B ú î Ýhõ×%è_Î�Ú%Q#Ö�Î�Ú�ÏPÖ^�,7%$ 4."</&$ 4(')'+* " B98O:<;=B2} :<;�B @�? B @C= B ú î �:Ù�Ú�Ï �,4^1[7-, B98O:P;�B2} :<;�B @�? B @C=.�:Í�Ê<= �>@? ADC õ2�ãDÊnÚR@h?�Ø�Î�ßàÎ�̼ԺÌ�Ú_þ4�O��IL���/9K5��5�� J�� B98':<;�B2} :<;=B @h? B @>=.��Û¦ØjÊnË = �>@? ADC � Q Û¦Ê�ØjÞ:Ù�ØjÏPÖ³Í�è_Î�˧Î�Ö�Ö�ñÖ�Ù�þ�Î � K ��8�� J�� B98O:P;�B2} :<;�B @ ? B @ = ��Û�Ø�ÊnË = ��D> Í�ÊM@ ? õ

��/83�-05 � 7 / - c�7 + 1 � 0 �B5 - � 741<5 + � 2S1�� ö þnÙ�Ì�ÚJÌ�Ú{Í�è�Ì�Ö�ãPè�Ù�Ö�Î = �> ? ADC Ì�Ö�Ö�̨Ë2Õ�Ó¨Ù�Í�Î�Ï{Ù�Ö3Ú_Ê�Øj˵Ù�Ó � Î��_ßàÎlã�Í�Í�è�Ù�ÍÞ�è_Î�Ú÷̼Í�Ø�Î�ßàÎ�Ì�Ô�Î�Öw7%$ 4."P/21.354�6�7§ËµÎ�Ö�Ö�Ù�þ�Î�Ö�Û¦ØjÊnË Õ�ÚPßàÊ�Ø�ØjÕ_ã�Í�Î�ÏLãPÙ�Ø�Í�̼Î�Ö�Í�è_Î �dCL���/9J>KIL�L�N¯Ë§Î�Ö�Ö�Ù�þ�Î�Ö�þ�ÎlÍÖ�Î�Ú�ͧÞ�̼Í�è_ÊnÕ_Í2Ô�ÎlØj̼Û�á¡Ì¨Ú_þ÷Í�è�αØjÎ�Ó�Ù�Í�̼ÊnÚ � Ù�ÚPÏ Ù{Í�ØjÙ�ÚPÖ�Û¦ÎlصßlÙ�ÕPÖ�Î�Ö§Ù��lÎlØ�Ê�ÒP̼Í�Í�Ê]Ò-ΩßàÊn˵˫̼Í�Í�Î�Ï Û¦Ê�Ø�Í�è_ÎØ�Î�ßàÎ�̼Ô�ÎlØ � Þ�̼Í�è_ÊnÕ_Í%Ù�ßàÍ�ÕPÙ�Ó�Ó¼á±ã-ÎlØjÛ¦Ê�ØjË«Ì�Ú_þ�Ù�Ú�á±Í�ØjÙ�Ú�Ö�Û�ÎlØ�õ��Î3Ò�ØjÎ�Ù�ý(Í�è�ֳ̨ãPè�Ù�Ö�Î�Ì�Ú�Í�Ê«ßlÙ�Ö�Î�Ö å�õ @ºÙ9á�Q Ö�ÎlÎ�Ö�@h?±ØjÎ�ßàÎ�̼Ô�ίë�Ô�Î �O��CL���/9K ��8�� J�� B i i iQ�3˧Î�Ö�Ö�Ù�þ�Î�Ö�Û�Ø�ÊnË = �>@?BADC � ßàÊn˵˵̼Í�Í�̨Ú_þ � Ö�Ù�á �

Í�Ê©ÒP̼Í�Ö }�ïCB � ù�ù B � ù ïCB � ï ù B � ï�ï Û�Ø�ÊnË @C= � ٩˧Î�ÖjÖ�Ù�þ�ÎLÜ ��� 7 /<7 6-0 B98':<;=B2}0:P;�B2} :<; ù B2} :<;_ï B ÷Ý�Û¦Ø�ÊnË @>= � Û¦ÊnÕ_Ø�O��IL���/9J KILML�N B i i i:@>= B ��0��˧Î�Ö�Ö�Ù�þ�Î�Ö�Û¦Ø�ÊnË = �> ? ADC Ü�Þ�̼Í�è{Î�Ù�ßjè ���ßàÊ�Ø�Ø�Î�Ö�ãDÊnÚ�Ï�̨Ú_þµÍ�Ê Ù�Ö�̨ÚÍ�è_Î�ãPØ�Ê�Í�Ê¡ßàÊnÓ�ݳÙ�Ú�Ϲٵ˵Î�Ö�Ö�Ù�þ�Î �O��CL���/�� ����� B98O:P;�B2} :<;�B @>= B @�? } :<;�� 3"3 B2} :<; � 3_å B2} :<;�� å�3 B2} :<; � å�å B} :<; ù B2}0:P; ï B2} î ��Û�Ø�ÊnË = �> ? ADC õ�Ü ö Ö�Ö�Õ�˧Î�Í�è�Î } :<; � Ö�Ù�Ø�Î3ßàÊnÚ�Ö�̨Ö�Í�Î�Ú�Í�Ì�Ú©Ù�Ó�ÓDÍ�è_Î�Ö�Î2˧Î�Ö�Ö�Ù�þ�Î�Ölõ�Ý � Û @ = ̨ÖßàÊ�Ø�ØjÕ�ã�Í�Î�Ï � Q Ö�Í�Ê�Ø�Î�Ö?Í�è�Î5Í�Õ_ãPÓ¼Î�Ü } :<;�B @>= B @�? B2}�ï Ýhõ � ÛPÍ�è_Î�˵Î�Ö�Ö�Ù�þ�Îs�,7 6 0 B98O:P;�B2} :<;�B2} :<; ù B2} :<;�ï B @�?~�Nè�Ù�ÖÚ_Ê�ÍrÒDÎlÎ�Ú]Ö�Î�Ú�Í�Í�ÊJ=�� �D> � Q Ö�Î�ÚPÏ�Ö�Í�è�Ì�Ör˧Î�Ö�Ö�Ù�þ�ÎLÜ�Ê�Ø�ÖjÌ�˧ãPÓ¼á©Û�Ê�Ø�Þ³Ù�ØjÏ�Ör̼Ír̼Û�@>=:̨Ö�ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï�Ýhõ×%è_Î�˧Î�Ö�ÖjÙ�þ�Îh�O� -��_(/9K5��5�� J�� B98':<;�B2} :<;=B2} :<; ù B2} :<;_ï @>= B @h? B ���Ì�Ö�Í�è_Î�Ú{Û¦Ê�Ø�Þ³Ù�ØjÏ_Î�Ï{Û�Ø�ÊnË =���D>JÍ�Ê@�?�õ

ò¡õ @ºÙ9á Q�Ö�ÎlÎ�Ö�˵Î�Ö�Ö�Ù�þ�Î<�,7 6-0 B98':<;=B2}0:P;�B2} :<; ù B2} :<;_ï B @�? B ��%Û�Ø�ÊnË Ù�Ú{Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï©ã�Ù�Ø�Í�á @>=>Í�Ê =���D>õQ Ö�Î�Ú�Ï�Örë�Ô�Îh�,7%$ 4."P/&$ 4(' ' * " B i i i ��˧Î�Ö�Ö�Ù�þ�Î�Ö3Í�Ê&= �>@?BADC Þ�̼Í�è]Ô�Ù�Ó�Õ_Î�ÖrÎS!¡Õ�Ù�Ó>Í�Ê �lÎlØ�Ê � Ù�Ú�Ï]Í�è_εÙ�ã_ñ

òºå

Page 22: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

ã�Ø�Ê�ãPØjÌ�Ù�Í�Î } :<; � ܸ֯ßàÊ�ØjØ�Î�Ö�ãDÊnÚ�Ï�Ì�Ú�þ©Í�ÊLÍ�è_αßàÊn˵˵̼Í�˧Î�Ú�Í�Ö2Í�Ê }�ïCB � ù�ù B � ù ïCB � ï ù B � ï�ï Ý � Ö�Î�Ú�ÏPÖ�Í�è_Î(Û¦ÊnÕ_Ø�,7%$ 4."P/21�3 4�6�7 B i i iQ�³Ë§Î�ÖjÖ�Ù�þ�Î�Ö�Í�Ê.= �>@?BADC Ü�Þ�è�Ì�ßhè¹Ù�Ø�Î3Ú�Ê�Í�ßjè_Î�ßjý�Î�Ï{Ò�á¯Í�è_Î3ÖjÌ�Ë2ÕPÓ�Ù�Í�Î�ÏJ= �> ? ADC Ý � Ù�Ú�ÏÖ�Î�Ú�ÏPÖÍ�è�γ˵Î�Ö�Ö�Ù�þ�Î �,7%$ 4."P/7"03102, 365,7.3 B98O:P;�B2} :<;�B i i i B @�?~��Í�Ê�= �> ? ADC õ��+è_Î�Ú�@>=�ØjÎ�ßàÎ�̼Ô�Î�Ö?Ù�Ó�Ó¡Í�è_γØ�Î�ßàÎ�̼ãPÍ˧Î�Ö�ÖjÙ�þ�Î�Ö � Q»Û�Ê�Ø�Þ³Ù�ØjÏ�ÖNÍ�è_γÊ�ØjÌ�þnÌ�Ú�Ù�Ӻ˧Î�Ö�Ö�Ù�þ�Î5Í�Ê�=�� �D>]ܸ̼Û�̼Í>è�Ù�Ö>Ú_Ê�Í>ÒDÎlÎ�Ú�Û¦Ê�Ø�Þ³Ù�ØjÏ_Î�ϧÙ�Ó�Ø�Î�Ù�Ï_á � Ö�ÎlÎÙ�ÒDÊ9Ô�Î�Ý2Ù�Ú�ÏÑÛ¦Ê�ØjÞ:Ù�ØjÏPÖ3Í�è_ί˧Î�Ö�ÖjÙ�þ�Îx�O� -��_(/9K5�d8�� J�� B98':<;�B2} :<;=B2} :<; ù B2}0:P;_ï @>= B @h? B ���Û�Ø�ÊnË =���D>Í�ÊM@>=�õ

��/83�-05 � 7 / - c�7 + 1�, 5 1�- 5 + � 2S1�� ��Î�Ò�Ø�Î�Ù�ý±Í�è�Ì�Ö�Ì�Ú�Í�ʵßlÙ�Ö�Î�Ö å�õ @ºÙ9á Q»Ö�ÎlÎ�ÖNÙ�Ú�Õ�Ú�ßàÊ�Ø�ØjÕ�ã�Í�Î�Ïe@�?:Ø�Î�ßàÎ�̼Ô�γ˧Î�Ö�Ö�Ù�þ�Î^�O��CL���/�� ����� B98O:P;�B2} :<;�B @>= B @�? B $ ��Û�Ø�ÊnË = �>@? ADC õ

� Û�Í�è_ÎlØ�Î�̨Ö:Ú�Ê�Í%Ù�Ó¼Ø�Î�Ù�Ï_á±Ù�˵Î�Ö�Ö�Ù�þ�Î �O� ����� B98':<;=B2}0:P;�B @>= B @h? B $ �GÊnÚ©Í�è_ÎrÊnÕ_Í�þ�Ên̨Ú_þµßàÊn˵Ë�Õ�Ú�Ì�ßlÙ�Í�̼ÊnÚÍ�Ù�ãDÎ�Ê�Û�= ��D> � Í�è_Î�ÚP@ = Ë2Õ�Ö�Í�ÒDÎ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï � Ö�Ê Q Ö�Î�Ú�Ï�Ö �,4^1[7-, B98':<;=B2}0:P;�B @ ? ��Û�Ø�ÊnË @ = Í�Ê = ��D> õ� Ú©Î�̼Í�è_ÎlØ�ßlÙ�Ö�Î � Q­Í�è_Î�Ú¹Û¦Ê�Ø�Þ³Ù�ØjÏ�Ö%Í�è_Î3˧Î�Ö�ÖjÙ�þ�Î �O� ����� B98':<;�B2} :<;=B @C= B @h? B $ �:Û�Ø�ÊnË =���D>{Í�ÊM@�?�õ

ò¡õ @ºÙ9á�Q1Ö�ÎlÎ�Ö�˧Î�Ö�Ö�Ù�þ�Î �,4^1[7-, B98':<;=B2}0:P;�B @�??�rÊnÚ Í�è_ΫÊnÕ�Í�þ�ÊnÌ�Ú_þ]ßàÊn˵Ë2ÕPÚ�Ì�ßlÙ�Í�̼ÊnÚÑÍ�Ù�ãDÎ(Ê�Û�Ù�Ú Õ�Ú�ßàÊ�Ø�ñØjÕ_ãPÍ�Î�Ï @ = Ï�Î�Ö�Í�Ì�Ú_Î�Ï+Û�Ê�ØJ= ��D> õ ×%è_Î�Ú Q Ö�Î�Ú�ÏPֱ٠˧Î�ÖjÖ�Ù�þ�Î �,7%$ 4."</<4^1[7-, B98':<;=B2}0:P;�B @ = B @ ? �«Û¦Ø�ÊnËܸÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï-ÝA@C=NÍ�Ê.= �>@? ADC õ%ÐrÚ�ßàÎ3Í�è�Ù�Í�˧Î�Ö�ÖjÙ�þ�Î2Ì�Ö5Û�Ê�Ø�Þ³Ù�ØjÏ_Î�ÏLÍ�Ê�= �>@?BADC � Q+Û¦Ê�Ø�Þ³Ù�ØjÏ�Ö%Í�è_Î3˧Î�Ö�ñÖ�Ù�þ�Î4�,4^1[7-, B98O:<;=B2} :<;�B @�?��5Í�Ê =���D>.õ���Ê9Þ Ö�Ù9áLٱ˧Î�Ö�Ö�Ù�þ�Î4�O� ����� B98':<;�B2} :<;=B @C= B @h? B $ ��Ù�ã�ãDÎ�Ù�ØjÖ�ÊnÚÍ�è_Î�ÊnÕ_Í�þ�ÊnÌ�Ú�þ¹ßàÊn˵Ë2ÕPÚ�Ì�ßlÙ�Í�̼ÊnÚ{Í�Ù�ãDΧÊ�Û�=���D>.õ � ÛW@�?§Ì�ÖrßàÊ�Ø�ØhÕ_ã�Í�Î�Ï � Q Þ�̨Ó�ÓÖ�ÎlÎ�Í�è�ΧÒP̼Í�ú3Ì�Ú{Í�è�̨Ö˧Î�Ö�ÖjÙ�þ�Î � Ù�Ú�Ï]Þ�̨Ó�ÓèPÙ�Ô�ΫÖ�Í�Ê�ØjÎ�ϻٯÍ�Õ_ãPÓ¼Î¹Ü } :<;=B @h? B @>= B ú î Ýhõ§×%è_Î�Ú�̼Í�ßlÙ�Ú»ßàÊn˵ãPÕ_Í�Ϋú ï ܸÙ�Ú�Ï�Ö�Í�Ê�Ø�ÎÜ } :<;�B @>= B @�? B ú ï Ý�Ý5Ì�Û.̼Í�Ì�Ö%Ú_Ê�Í�Ù�Ó¼Ø�Î�Ù�Ï_á¹Ö�Í�Ê�Ø�Î�Ï � Ù�ÚPÏLÖ�Î�Ú�ÏLÍ�è_Î2˵Î�Ö�Ö�Ù�þ�Î<�O�dCL���/�� ����� B98':<;�B2} :<;=B @C= B@�? B ú ï ��Û¦ØjÊnË = �>@?BADC Í�Ê!@�?�õ � Û @h?GÌ�Ö�Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï � Q{Ö�Ì�˧ã�Ó¼á�Ö�Î�Ú�ÏPÖDÍ�è_ÎG˧Î�Ö�ÖjÙ�þ�Î �O��IL���/�� ����� B98':<;=B} :<;�B @ = B @ ? B 3q�³Û�Ø�ÊnË = �> ? ADC Í�ÊM@ ? õ

��/83�-05 � 7 / - c�2 �T/(/ -G5;7 / ��-;2�� � Û�H ßàÊ�Ø�ØhÕ_ã�Í�ÖGٯܸÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï-ÝNãPÙ�Ø�ÍváM@>= � QaßàÊ�Ø�ØjÕ_ãPÍ�Ö�@>=�Ù�Ú�ϱÚ_ÎlÎ�Ï�Ö�Í�Ê�Ö�̨Ë2Õ�Ó¨Ù�Í�Î̼Í�Ö�Ì�Ú�Í�ÎlØhÚ�Ù�ÓGÖ�Í�Ù�Í�Î � Þ�è�Ì�ßjèÑßàÊnÚ�Ö�Ì�Ö�Í�Ö3Ê�Û5Í�è_αÖ�è�Ù�Ø�Î�Ö2Ê�Û�Ù�Ó�Ó�Õ�Ú_Ê�ãDÎ�Ú_Î�Ï»ßàÊn˫˵̼Í�Í�Î�Ï Ù�Ú�ÏJÎlÔ�Ù�Ó�Õ�Ù�Í�Î�ÏÑÒP̼Í�Ö�õ�_Ê�Ø�ÎlÔ�ÎlØjá±Ò�̼Í�ú�Ê�ÛN̨Ï_Î�Ú�Í�Ì�ë�ÎlØ }0:P; Öjè�Ù�Ø�Î�ϯҡá @>=>Ù�Ú�Ï @�? � Q+Ï_Ê�Î�Ö%Í�è_Î�Û¦ÊnÓ¨Ó¼Ê9Þ�̨Ú_þ_õ* � Û5Ü } :<;�B @>= B @�? B ú ï ݳÌ�Ö%Ö�Í�Ê�Ø�Î�Ï � QYÕPÖ�Î�Ö5Í�è_Î3Öjè�Ù�Ø�Î3ú ï õ* � Û?Ü } :<;=B @h? B @C= B ú î Ý>̨Ö?Ö�Í�Ê�Ø�Î�Ï � Ù�Ú�ÏIQ Ó¼Î�Ù�ØjÚPÖGú³Õ_ãDÊnÚ§ßàÊ�Ø�ØjÕ_ãPÍ�̼ÊnÚLÜ�ÒDÎ�ßlÙ�Õ�Ö�Î�Í�è�Ì�Ö>ÒPÌ�Í>Þ:Ù�Ö�ßàÊn˵˵̼Í�Í�Î�ÏÒ�áP@ = Ê�Ø�Ê�ãDÎ�Ú_Î�Ï]Ò¡áP@ ? � Ê�Ø�ÒDÎ�ßlÙ�Õ�Ö�ÎI@ ? Ì�Ö�Ù�Ó�Ö�ʹßàÊ�Ø�ØjÕ_ã�Í�Î�Ï � Ù�Ú�Ï�Í�è¡Õ�Ö Q ßlÙ�Ú»ßàÊn˵ãPÕ_Í�εÙ�Ó�ÓNÒPÌ�Í�ÖÖ�è�Ù�Ø�Î�ϯҡá%@C=NÙ�Ú�ÏR@�?�Ý � Q­ßàÊn˧ã�Õ_Í�Î�Ö�ú ï c�ú � ú î � Ö�Í�Ê�Ø�Î�Ö2Ü }0:P;�B @>= B @h? B ú ï Ý:Ù�Ú�Ï©Õ�Ö�Î�Ö³Í�è_Î�Öjè�Ù�Ø�Î�ú ï õ

* � Û�Ú_Î�̼Í�è_ÎlØ�Í�Õ_ãPÓ�Î(Ì�Ö�Ö�Í�Ê�Ø�Î�Ï Ü�ÒDÎ�ßlÙ�Õ�Ö�αÍ�èPÌ�Ö2̨Ö�Ù�Ú Õ�Ú�Ê�ã-Î�Ú�Î�ÏJÒPÌ�Í2Û�Ø�ÊnË Ù�Ú ÎlÔ�Ù�Ó�ÕPÙ�Í�̼ÊnÚ Ù�Ú�Ï @h?©Ì¨ÖÕ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï�Ý � Q+þ�Î�Ú_ÎlØjÙ�Í�Î�Ö�ú ï "$ e�3 B å�f � Ö�Í�Ê�Ø�Î�Ö�Ü } :<;�B @>= B @h? B ú ï Ý � Ù�Ú�ϹÕ�Ö�Î�Ö5Í�è_Î�Ö�è�Ù�ØjÎ3ú ï õ

×�è�Ì�Ö�ßàÊnÚ�ßlÓ¨Õ�Ï_Î�Ö�Í�è�ΫÏ_Î�Ö�ßàØjÌ�ã�Í�̼ÊnÚ�Ê�Û Q�õ � Í2̨Ö�Ö�Í�ØjÙ�̼þnè�Í�Û�Ê�Ø�Þ:Ù�ØhÏJÍ�Ê{Ö�ÎlΫÍ�è�Ù�Í�Í�è_ΫÖjÌ�Ë2ÕPÓ�Ù�Í�̼ÊnÚ»Ì�Ö ()����9��8#�/ � ÌVõóÎ�õ ������ c "� ������������ | ] _�] ` Ù�Ú�Ï  ba�¥GY�[�c���� � ] d4] ` Ù�Ø�Î3Ì�Ï�Î�Ú�Í�Ì�ßlÙ�Ó¨Ó¼á«Ï�Ì�Ö�Í�Øj̼ÒPÕ_Í�Î�ÏAõ

� � Ç?È��GÁr �%Á�Å � Á§È��?�GÃ��4� : �� �+Ä?���±Ä!����� �� ��#Ä� ^ �5Ã- ���+Ä'� �§Æ2Â��:Â�È�Ä�Á� Ú©Í�è�̨Ö%Ö�Î�ßàÍ�̼ÊnÚ©Þ:Î2Öjè_Ê9Þ è�Ê9Þ Í�Ê(Ö�Î�ßlÕ_Ø�Î�Ó¼á¯Ø�Î�Ù�Ó¨Ì��lÎ�Ù�Ú�á©Ù�Ï�Ù�ãPÍ�̼Ô�Î�Ó¼á±Þ�Î�Ó�Ó¼ñ®Û¦Ê�Øj˵Î�Ï©Í�Þ:Ê�ñ®ãPÙ�Ø�Ívá¹Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á(̨ÚÍ�è�Î�ã�Ø�Î�Ö�Î�Ú�ßàÎ�Ê�ÛN˵Ù�Ó�̨ßl̼ÊnÕ�Ö³Ù�Ï�Ù�ã�Í�̼Ô�Î�Ù�Ï�Ô�ÎlØjÖ�Ù�Øj̼Î�Ö5̨گÍ�è_Î�=���D>�ñ�è�á¡Ò�ØjÌ�ϯ˧ʺÏ_Î�ÓVõGÐrÕ�Ø%ßàÊnÚ�Ö�Í�ØjÕ�ßàÍ�Ì�ÊnÚ©Ì�Ö5Ö�Ì�˵̨Ó�Ù�ØÍ�Ê�Í�è_ÎrßàÊnÚ�Ö�Í�ØjÕPßàÍ�̼ÊnÚ�Ö³Ì�Ú÷äóò� � ò � � å�ò�çDÛ¦Ê�سÖ�Î�˵̼ñ�è_ÊnÚ_Î�Ö�ͳÙ�Ï_Ô�ÎlØjÖ�Ù�Øh̼Î�ÖlõI%�Ê�Þ�ÎlÔ�ÎlØ � Ö�Ì�Ú�ßàÎ�ÊnÕ_Ø =���D>¹Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�ą́Ö�Ö�Î�ßlÕ_Ø�ΧÌ�Ú{Í�è_Χã�ØjÎ�Ö�Î�Ú�ßàΧÊ�۳˵Ù�Ó�Ì�ßl̼ÊnÕPÖ�Ù�Ï_Ô�ÎlØhÖ�Ù�Øj̼Î�Ö � Þ�εÙ�Ø�εÙ�Ò�Ó¼Î�Í�Ê©Ê�Ò�Í�Ù�Ì�Ú�Ù¯Í�Þ:Ê�ñ®ãPÙ�Ø�Ívá�ã�Ø�Ê�Í�ʺßàÊnÓ>Ö�Î�ßlÕ_Ø�ÎÙ�þnÙ�̨Ú�Ö�Í�˵Ù�Ó¨Ì�ßl̼ÊnÕ�Ö³Ù�Ï_Ô�ÎlØjÖjÙ�Øj̼Î�Ö%Ï�Ì�Ø�Î�ßàÍ�Ó¼á�õ��Î{ëPØjÖ�͵Ø�ÎlԺ̼ÎlÞ�Ö�Ên˧Î{Ê�Û�Í�è_Î]Ù�Ö�Ö�Õ�˧ãPÍ�̼ÊnÚ�ÖµÙ�ÒDÊnÕ_Í(Í�Þ:Ê�ñ®ãPÙ�Ø�Í�á+Û¸Õ�Ú�ßàÍ�̼ÊnÚ�Ù�Ө̼Í�̼Î�Ö§Þ:Î÷Õ�Ö�Î÷Ì�ÚYÊnÕ_Ø«ã�Ù�ã-ÎlØ �

Þ�è�Ì�ßjè±Ù�Ø�ÎrÙ�Ó�Ö�Ê�Õ�Ö�Î�ϱÌ�Ú]ä�å�ò�ç®õI��Î�Ó¼ÎlÍ =*ÒDÎ�Ù�Ú±Ì�Ï_Î�Ù�Ó�Í�Þ:Ê�ñ®ãPÙ�Ø�Ívá(Û¸Õ�Ú�ßàÍ�̼ÊnÚPÙ�Ó�̼Í�á � Ù�Ú�Ï(Þ�ÎrÓ¼ÎlÍA@ ï Ù�Ú�Ï @ î ÒDÎ�Í�è_Îã�Ù�Ø�Í�Ì�ßl̼ãPÙ�Í�̨Ú_þ2ã�Ù�Ø�Í�̼Î�Ölõ���Î2Ù�Ö�Ö�Õ�˧Î�Í�èPÙ�Í = ˵Ù�á¯ÒDÎrØ�Îlã�Ø�Î�Ö�Î�Ú�Í�Î�Ï©ÔºÌ�٧٧۸Ù�˵Ì�Ó�á�� c Ê�Û>ô�Ê¡ÊnÓ¼Î�Ù�Ú¹ßlÌ�ØjßlÕ�̼Í�Ö � Í�è_Î

ò�ò

Page 23: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

�¡Í�è÷ßl̼ØjßlÕ�Ì�Í5Ø�Îlã�Ø�Î�Ö�Î�Ú�Í�Ì�Ú_þ(Ù�ÚLÙ�ßàÍ�̼Ô�Ù�Í�̼ÊnÚ÷Ê�Û0= Þ�̼Í�è¹Ö�Î�ßlÕ�Øj̼Í�á¯ãPÙ�ØhÙ�˧ÎlÍ�ÎlØ_�-õP�#̼Í�è_ÊnÕ_Í�Ó¼ÊnÖ�Ö5Ê�Û>þ�Î�Ú_ÎlØhÙ�Ó�̼Í�á � Þ�ÎÙ�ÖjÖ�Õ�˧Î�Í�è_Î2ßl̼ØjßlÕ�̼Í�Ö5Ù�ØjÎ�ßàÊn˧ãDÊnÖ�Î�Ï©Î�Ú�Í�̼Ø�Î�Ó¼á±Ê�ÛI� ö � � þnÙ�Í�Î�Ö�õ ï ù

�_Ê�Ø3Ö�Ì�˵ãPÓ�Ì�ßl̼Ívá � Þ:ΧÙ�Ö�Ö�Õ�˧Î�Í�è�Ù�Í�Ì�Ú{Î�Ù�ßhè»Ù�ßàÍ�̼Ô�Ù�Í�̼ÊnÚ � Ü�å9Ý�Ù�Í�˧ÊnÖ�ÍrÊnÚ_Î�ãPÙ�ØjÍ�á÷èPÙ�Ö�Ù�Ú{̨Ú_ãPÕ_Í�Í�ÊJ= Þ�Ì�Í�èÙ�Í�˧ÊnÖ�Í_�©ÒP̼Í�Ö � ÜVònݳÎ�Ù�ßhè{ãPÙ�Ø�Í�á¹Ë«Ù�á¹Ø�Î�ßàÎ�Ì�Ô�Î2Ù�Ír˧ÊnÖ�ÍA�¹ÒP̼Í�Ö�Ù�Ö�ÊnÕ_Í�ã�Õ_Í%Û�Ø�ÊnË = � Ü��nÝ�= ̨Ö�Ù«Ï�ÎlÍ�ÎlØj˵Ì�Ú�̨Ö�Í�Ì�ßÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ � Ù�ÚPÏ Ü M�Ý�Í�è_ΧӼʺßlÙ�ÓNÖ�Í�Ù�Í�ΧÊ�ÛI='Ù�Û¦Í�ÎlØ�Î�Ù�ßhè»Ù�ßàÍ�̼Ô�Ù�Í�̼ÊnÚ]ßlÙ�Ú]ÒDÎ�Ï_Î�ÖjßàØj̼ÒDÎ�Ϲҡá÷Ù�Í�˧ÊnÖ�Í4�LÒP̼Í�Ölõ�×%è�Į̂Ú�̼Í�Ì�Ù�Ó�Ö�Í�Ù�Í�Î�Ê�Û = Ì�Ö³Ï_Î�Ö�ßàØjÌ�Ò-Î�Ï(Ò¡á7� �lÎlØ�Ê�ÒP̼Í�Ölõ���Î�Ù�Ö�ÖjÕ�˧ÎrÍ�è�Ù�Í5˧Î�Ö�Ö�Ù�þ�Î�Ö�Ö�Î�Ú�Í5Û�Ø�ÊnË�H Í�Ê = Ù�Ø�Î�̼þnÚ_Ê�Ø�Î�Ï �Ù�ÚPÏ©Í�è_ÎlØ�Î�Ù�Ø�Î2Ú_ʵ˧Î�Ö�Ö�Ù�þ�Î�Ö�Û�Ø�ÊnË = Í�ÊIH±õ��Î3Ú_Ê�Í�Î�Í�è�Ù�Í5Í�è_ιé�Ï�ÎlÍ�ÎlØj˵Ì�Ú�̨Ö�Í�Ì�ß�Û�Õ�ÚPßàÍ�̼ÊnÚ�ì�Ù�Ö�Ö�Õ�˧ãPÍ�̼ÊnÚ¯Ù�Ò-ÊnÕ�Í = Ì�Ö³Þ�̼Í�è_ÊnÕ_ͳӼÊnÖ�Ö³Ê�Û.þ�Î�Ú_ÎlØjÙ�Ó�Ì�Í�á � Ö�Ì�ÚPßàÎ

Þ:Î%ßlÙ�ÚµÙ�Ó¼Þ³Ù�áºÖ?ØjÎ�Ù�Ó�Ì��lÎ5Ùrã�Ø�Ê�ÒPÙ�Ò�Ì�Ó�Ì�Ö�Í�̨ß>Û�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á"= Õ�ÖjÌ�Ú_þrÙ�Ï_ÎlÍ�ÎlØj˵Ì�ÚPÌ�Ö�Í�Ì�ß�ÊnÚ_Î�= � Ù�Ö?Û¦ÊnÓ�Ó�Ê9Þ�Ölõ ö Ö�Ö�Õ�˵̨Ú_þÍ�èPÙ�Í = Ú_ÎlÎ�Ï�ÖW�«ØjÙ�Ú�Ï_ÊnË ÒP̼Í�Ö � Í�è_Î�Ú = � Ø�Î�ßàÎ�̼Ô�Î�Ö5Ù6��ñ®ÒPÌ�Í�Ö�Í�Øj̨Ú_þ2Ù�Ö5Ù�Õ��_Ì�Ó�̨Ù�Ø�á�̨Ú_ãPÕ_Í?Û�Ø�ÊnË*Î�Ù�ßjè¯ã�Ù�Ø�Í�Ì�ßl̼ãPÙ�Í�̨Ú_þã�Ù�Ø�Í�á]Õ_ãDÊnÚ]Í�è_εë�ØhÖ�Í�Ù�ßàÍ�̼Ô�Ù�Í�Ì�ÊnÚ � Ù�Ú�Ï»Í�è_Î�Ú�ØjÕ�ÚPÖ�='ÕPÖ�Ì�Ú_þ¯Í�è_Î ��Ð_> Ê�Û:Í�è_Î�Ö�ΫÖ�Í�ØhÌ�Ú_þnÖ�Ù�Ö�Í�è�ΧØjÙ�Ú�Ï_ÊnË'Ò�̼ÍÖ�Í�ØjÌ�Ú_þ_õ � Í%Ì�Ö³Î�Ù�Ö�á¯Í�Ê(Ö�ÎlÎ�Í�è�Ù�Í5Í�è_Î3ÖjÌ�˧ãPÓ¼Î�ã�Ø�Ê�Í�ʺßàÊnÓAÞ�è_ÎlØ�Î�Î�Ù�ßjèLãPÙ�Ø�Ívá¯Ö�Î�Ú�Ï�Ö5Ù§ØjÙ�ÚPÏ_ÊnË ��ñ®ÒP̼Í5Ö�Í�ØhÌ�Ú_þµÙ�Ö5Í�è_ÎÙ�Õ)�ºÌ�Ó¨Ì�Ù�Ø�á«Ì�Ú�ãPÕ_Í:Í�Ê�= � Ö�Î�ßlÕ_Ø�Î�Ó¼á±Ø�Î�Ù�Ó�Ì��lÎ�Ö%Í�è_Î3Ì�Ï_Î�Ù�ÓDÛ¸Õ�Ú�ßàÍ�Ì�ÊnÚ�Ù�Ó�̼Ívá<= Ì�Ú¯Í�è_Î = � ñ�è�á�Ò�ØhÌ�ϯ˧ʡÏ�Î�ÓVõ ï�ï

×%è_Î3Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ«ã�Ø�Ê�Í�ʺßàÊnÓ8! c Ø�Î�Ù�Ó�Ì��lÎ�Ö�Ù�Ú]Ù�ßàÍ�̼Ô�Ù�Í�̼ÊnÚ{Ê�Û�= Þ�è_Î�Ú @ ï Ö�Î�Ú�ÏPÖ�ٱ˧Î�Ö�Ö�Ù�þ�Î�Í�Ê =©õ�ܸ×%è�Î�ßlÙ�Ö�ÎÛ�Ê�Ø @ î ̨Ö3Ù�Ú�Ù�Ó¼Ê�þ�ÊnÕPÖlõ�Ý ��ίÙ�Ö�Ö�Õ�˧ΫÍ�è�Ù�Í2ÒDÊ�Í�è6@ ï Ù�Ú�Ï @ î è_ÊnÓ�ÏÑÙ�Ú 8':<; Ù�Ö�Ù�Õ��ºÌ¨Ó�Ì�Ù�Ø�á{Ì�Ú_ãPÕ_Ílõ �#è_Î�Ú6@ ï ̨ÖÙ�ßàÍ�Ì�Ô�Ù�Í�Î�ÏLÞ�̼Í�è©Ì�Ú_ã�Õ_Í�Ü 8O:<;=B�� Ý � ̼Í%Ì�Ú�̼Í�̨Ù�Í�ÎrÙ�ãPØ�Ê�Í�Ê¡ßàÊnÓAÞ�̼Í�èR@ î Í�ʵãDÎlØ�Û¦Ê�ØhË Ù�'�Ên̨Ú�Í5þnÙ�Í�Îàñ®Ò¡á�ñ®þnÙ�Í�Î�ÎlÔ�Ù�Ó�Õ�Ù�Í�̼ÊnÚÊ�Û.Í�è_Î3Ù�ãPã�Ø�Ê�ã�Øj̨Ù�Í�ÎrßlÌ�ØjßlÕ�̼Í5Ì�Ú � c õ

�_Ê�Øj˵Ù�Ó�Ó�á � Í�è_Îlá¯ßlÙ�Ø�Ø�á±ÊnÕ_Í%Í�è_Î�Û�ÊnÓ�Ó¼Ê�Þ�Ì�Ú_þ�ã�Ø�Ê�Í�ʺßàÊnÓVõ� -./87 / �B58/�� � 7 / �4-X� �+è�Î�Ú @ ï Ø�Î�ßàÎ�̼Ô�Î�Ö�Ü 8':<;=B�� Ý � ̼Í�ßhè_Î�ß�ýºÖ%̼Û�Í�èPÌ�Ö�̨Ö�Í�è�Î�ë�ØhÖ�Í:Ù�ßàÍ�Ì�Ô�Ù�Í�̼ÊnÚ±Ê�Û;= � Ù�Ú�ϯ̼ÛAÖ�Ê�̼ͳÖ�ÎlÍ�Ö

Õ_ã©Í�è_Î3̨Ú�Í�ÎlØjÚPÙ�Ó�Ö�Í�Ù�Í�Î�õ�×%è�Î�ڹ̼Í%ßàÊn˵˵Ì�Í�Ö5Í�ʫ̼Í�Ö5ãPØj̼Ô�Ù�Í�Îr̨Ú_ãPÕ_Ílõ�`137S7 /8- c -G5 7 + 1 / -D7(13/ - �T5 2S7 � 741�� �_Ê�Ø : c å B ò BEDVDVDVB � � @ ï Ö�Î�Ú�Ï�Ö�˧Î�Ö�Ö�Ù�þ�Î�Ö � $ 4(' ' * " B98':<;�B2} :<; = B @ î B 3q�Ù�Ú�ÏaÍ�è_Î�Ú��,4^1[7-, B98':<;�B2} :<; = B @ î ��Í�Ê�=���D>.õ @ ï Þ:Ù�̼Í�ÖµÍ�Ê»Ø�Î�ßàÎ�̼Ô�Î÷Í�è_ιÙ�ã�ã�Ø�Ê�ãPØjÌ�Ù�Í�ΩØ�Î�ßàÎ�̼ãPÍ�Ölõ @ îÙ�ÒDÊ�Ø�Í�Ör̼ÛGÙ�Ú�áLÊ�ÛGÍ�è�Î�Ò�̼Í�Ö�Ù�ØjÎ�Ú�Ê�ÍA�lÎlØjÊ_õ��IDÎ�ßàÍ�̼Ô�Î�Ó¼á � @ ï ßàÊn˵˵̼Í�ÖrÍ�ʯÍ�è_ΧÌ�Ú�Ì�Í�Ì�Ù�Ó�Ì�Ú�Í�ÎlØjÚ�Ù�Ó.Ö�Í�Ù�Í�Î

Ê�Û =,Ü�Þ�è�̨ßjè¯Ì�Ö³Ù�Ó�Ó:�lÎlØ�ÊnÖhÝ � Ù�ÚPÏ(Ò¡á«Ê�ãDÎ�Ú�̨Ú_þ2Í�è�Î�Ë Ì�˵˧Î�Ï�̨Ù�Í�Î�Ó¼á � Ì�Í:ãPØ�Ê9Ô�Î�Ö5Í�ÊI@ î Í�è�Ù�Í5Í�è_Î�Ö�Î�ÒPÌ�Í�ÖÙ�Ø�Î3Ì�ÚPÏ_ÎlÎ�Ï©Ù�Ó�Ó�ñL�lÎlØ�Ê_õ ï î� �43 3 / 747 /8- c 7 ��7 + 1 56/ / 0 � 741P/8-Z5�- 7q� ��Ê�Ø : c1å B ò BEDVDVDVB � � @ ï Ö�Î�ÚPÏ�ÖA˧Î�Ö�ÖjÙ�þ�Î�ÖV� $ 4(' ' * " B98':<;�B2} :<; = B @ ï B�� =:�

Í�Ê#= ��D> Ù�Ú�Ï3Þ:Ù�̼Í�ÖAÍ�Ê�Ø�Î�ßàÎ�̼Ô�Î�Í�è_ÎGÙ�ã�ãPØ�Ê�ã�ØjÌ�Ù�Í�ÎNØ�Î�ßàÎ�Ì�ã�Í�Ölõ�%�ÎlØ�ÎGÞ�ÎGÙ�ÖjÖ�Õ�˧Î?Í�èPÙ�Í � c � ï � î i i i � �õï

@ î ÖjÌ�˧ãPÓ¼á«Ø�Î�ßàÊ�ØjÏ�Ö5Í�è�Î�Ø�Î�ßàÎ�̼ã�Í�Ö%Ø�Î�ßàÎ�̼Ô�Î�ϹÛ�Ø�ÊnË =�� �D>õ��� 741 a 7 9 a c � 7(1�1^0 �T5 - � 7 / ��-u� �_Ê�ØGÎ�Ù�ßjè.� ö � � þnÙ�Í�Î�Ì�Ú�Í�è�γßlÌ�ØjßlÕ�Ì¼Í � @ ï Ï_ÎlÍ�ÎlØj˵Ì�Ú�Î�ÖNÍ�è_Î%ßàÊn˵˵̼Í�˵Î�Ú�Í?Ì�Ï�Î�Úºñ

Í�̼ë�ÎlØjÖrÙ�Ö�Ö�Ê¡ßlÌ�Ù�Í�Î�Ï»Þ�̼Í�è÷Í�è_ΧÌ�Ú_ãPÕ_Í�Ö�Í�Ê©Í�è�Ù�Í"� ö � � þnÙ�Í�Î � Ö�Ù9á } :<; ù�Ù�ÚPÏ }0:P;_ï � ßàØ�Î�Ù�Í�Î�Ö2Ù±Ú�ÎlÞ7ÕPÚ�Ì !¡Õ_ÎßàÊn˵˵̼Í�˵Î�Ú�Í�Ì�Ï_Î�Ú�Í�̼ë�ÎlØ } :<; � Ö�Î�Ú�Ï�Ör˧Î�Ö�Ö�Ù�þ�Îh�,7 6 0 B98':<;�B2} :<;=B2}0:P; ù B2}0:P;_ïCB @ î B å�å�å�3q��Í�Ê =���D> � Ù�Ú�Ï{Þ:Ù�̼Í�ÖÛ¦Ê�Ø�Í�è_Î3Ù�ã�ãPØ�Ê�ã�ØjÌ�Ù�Í�Î3Ø�Î�ßàÎ�̼ã�ÍlõP%�ÎlØ�Î)dc�å�å�å�3«Ì�Ö�Í�è_Î�Î�Ú�ßàʺÏ�Ì�Ú_þ§Ê�ÛNÍ�è_Î � ö � � Ê�ãDÎlØjÙ�Í�̼ÊnÚ�õ @ î ÖjÌ�˧ãPÓ¼áØ�Î�ßàÊ�ØjÏ�Ö%Í�è_Î�Ø�Î�ßàÎ�̼ãPÍ�Ö5Ø�Î�ßàÎ�̼Ô�Î�ÏLÛ¦Ø�ÊnË =���D>õ

,.-;7 5�- 7?� @ î Ô�ÎlØh̼ë�Î�Ö�Û¦Ø�ÊnË�Ù�Ó�ÓN̼Í�Ö�Ø�Î�ßàÎ�̼ãPÍ�Ö�Í�èPÙ�Í @ ï è�Ù�Ï�= ��D> ãDÎlØ�Û�Ê�ØjË Í�è_Î�ßàÊ�ØjØ�Î�ßàÍ�ßàÊn˧ãPÕ_Í�Ù�Í�Ì�ÊnÚ÷ÊnÚ{Í�è_ÎÙ�ã�ã�Ø�Ê�ãPØjÌ�Ù�Í�ιÒPÌ�Í�Ölõ ×%è_Î�Ú­Û¦Ê�رÎ�Ù�ßjè#ÊnÕ_Í�ã�Õ_͵ÒP̼Í(Ê�Û�= � ̼ͫÌ�Ö«Î�̼Í�è_ÎlدÙ�ÚY̨Ú�Í�ÎlØjÚPÙ�Ó�Ö�Í�Ù�Í�Î{ÒPÌ¼Í � Ê�دÙJÒ�̼ÍÙ�Ï�Ï_Ø�Î�ÖjÖ�Î�ÏLÍ�ʯÎ�̼Í�è�ÎlØ @ ï Ê�Ø @ î Ü�Þ:Χè�Ù�Ô�ΧÙ�Ö�ÖjÕ�˧Î�Ï{Í�è�Ù�ÍE=�Ï_Ê¡Î�ÖrÚ_Ê�ÍrßàÊn˵Ë2ÕPÚ�Ì�ßlÙ�Í�Î�Þ�̼Í�è H§Ýhõ � ÚLÍ�è_ÎÛ¦Ê�Øh˧ÎlØ�ßlÙ�Ö�Î � @ ï Ù�Ú�Ï @ î Ï_ʵÚ_Ê�Í%Ú_ÎlÎ�Ï©Í�ʵÏ�ʵÙ�Ú�á¡Í�è�Ì�Ú_þ_õ?×�è_Îlá(Ö�̨˧ãPÓ¼á«Ö�Í�Ê�Ø�Î�Í�è_Î�Ì�Ï_Î�Ú�Í�̼ë�ÎlسÊ�ÛNÍ�è�Ì�Ö:ÒPÌ�Í �

 � �aZ*[\R|[\RFGJE�KL[µSLGD�µ�;XÄV&S$R�[\­W]N��[\UD[µK-���)�$V&KLGKLZ�P&K�KLZNG�� ��� � X.�*E*UnKL[�V�E�P&��[µK-�AUOP&E_Q"GÅ�*R�GDM�KLV?GD��P��\�*P&KLG�P�E�����P8KLGÅV�X)XÄP�E(Vt[\EiK-ÃaV��Â� �$V&KLG}KLZ*P&K�[µX�P�M*P&]NKL[\��G}UDV�S�SL�N]NKL[\V�ENR)P&SLG:P��\�\V8Ã:GOM,z�KLZ*GDEÅKLZN[�R=[�R�P�UJKL�*P��\�µ� V�E*�µ�ÅK�SL�*G�XÄV&SxP�MNP�]NKL[\��GJ�\�$ÃaGD�\��V XÄV�SL­WGDM�X.�*E*UnKL[�V�E�P&��[µKL[\GDRD�

®�GDG \\»O�a]"XÄV&S$P MN[\R�UD�NR�R�[�V�EAV�EiKLZ*[\Ra]"V�[\E�KOzNP&E�M�KLZNG$­WV�M�[\É*UOP&KL[\V�E*R�E*GJUDGDR�RLP&S��;X.V�SYP�Ei[�MNGOP&�kP�MN��GnSLRLP&S���[\E�KLZNG$UOP&R�GÅV�X�]�SLV�Q*P�Q*[\�\[\R0KL[\UXÄ�*ENUJKL[\V�ENRD�Â-Ì �$V&KLG$KLZ�P8KaKLZ*G�� � � R��*R�GOM;Z*GnSLGÅP�E*M�GD�\R�GJÃ|ZNGJSLGÅ[\E�KLZNG$]�SLV�KLV�UDV��k­ �*R0K�P&�\�,Q"G$�*E*[�Ë��*GYQ*[µK�R0K�SL[\EN��RaKLZ�P8K|[\E�MN[\UOP8KLG$KLZ*GFQ*[µK � R��*R�G

[�E�KLZ*GÅUD[µSLUD�N[µKO� � V�S|[\E*R0KTP&E*UDG�zNKLZ*G��� ���xZNGJSLGÅUDV��N��MAQ"G$KLZ*GÅQN[µK|GDE*UDV�M�[\E*�?V&X������������ ����n� U �=V_[\E*MN[\UOP&KLGWKLZNG?�*R�GWV&X|GOP�UTZ5V�X:KLZ*GDR�GWQN[\KLR�[\EoKLZNG?UD[µSLUD�*[µKOz)V�E*GWUDV��*��M,z=XÄV�S�[\E*R0KTP&E*UDG�z)R�GJK��� ���YKLVAQ"GÍKLZ*GWQN[\K�GJE*UDV�M�[�EN� V�X

�����! !"�� �$#  � r ����nz�Ã|Z*GJSLG8r;[\R�KLZ*GÅP&UJKL[\��P8KL[�V�E E��*­ÍQ"GnSO�

ò:�

Page 24: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

Ö�Ê©Í�è�Ù�Í�Í�è_Îlá]ßlÙ�Ú»Õ�Ö�ε̼Í�Ì�Ú{Í�è_εÚ_Î��ºÍ�Ù�ßàÍ�̼Ô�Ù�Í�Ì�ÊnÚ�õ � Ú]Í�è_εӨÙ�Í�Í�ÎlØ3ßlÙ�Ö�Î � Ù�ÖjÖ�Õ�˵Ì�Ú�þ(Í�è�Ù�Í�Í�èPÌ�ÖrÒPÌ¼Í � Þ�Ì�Í�èÌ�Ï_Î�Ú�Í�̼ë�ÎlØ } :<; � Ì�ÖGÙ�Ï�Ï�Ø�Î�Ö�Ö�Î�Ï(Í�Ê @ î � @ ï Ö�Î�Ú�Ï�ÖGÙ�˧Î�Ö�ÖjÙ�þ�Î �,4^1[7-, B98O:<;=B2} :<;�B @ î �>Í�Ê�= � �D> Ù�Ú�ÏM@ î Î��ºÍ�ØjÙ�ßàÍ�ÖÍ�è_Î�Ò�̼Í�ú:Û�Ø�ÊnË�Í�è�Î�˧Î�Ö�Ö�Ù�þ�Î �O� ����� B98':<;=B2}0:P;�B e @ ï B @ î f B ú��NØ�Î�ßàÎ�̼Ô�Î�Ï(Û�Ø�ÊnË =�� �D>õ>×%è_Î�ã�Ø�Ê�Í�ʺßàÊnÓPÛ�Ê�ØGÍ�è_ÎßlÙ�Ö�Î3Û�Ê�Ø�Ù�ÒPÌ�Í%Ù�Ï�Ï_Ø�Î�Ö�Ö�Î�ϯÍ�ÊM@ ï Ì�Ö³Í�è_Î3Ö�Ù�˵Π� ÒPÕ_ͳÞ�̼Í�èR@ ï Ù�Ú�Ï @ î Ö�Þ�Ì�Í�ßjè_Î�Ï�õ

�¹Î�Ö�ÖjÙ�þ�Î�Ö�Í�è�Ù�Í�Ù�Ø�Î�ÊnÕ_Í5Ê�ÛNÊ�ØjÏ_ÎlØ�Ù�Ø�Î�Ï_Î�Ù�Ó¼Í%Þ�̼Í�è©Õ�Ö�Ì�Ú�þ�Í�Ù�þ�þn̨Ú_þ � Ù�Ö�̨ڻä�å�ò�ç®õ*,+ 1��T/(1 3 �R9Gy�� ��/6= +&�H /,v�% � () ���/01� ) �(:/0������!Ç1 v ��!�! �.9�%"��' � )49����}#�/0�0%"�} �!.��/01�F�C�)���@! c ���8#����8��!.1w�O�& �!Ç� U����= �-� / ���c=���D>.�P�,1,+���� )7'H%S),��! �Í�-� / ���;(}�8�������}#��6%D94'H �!.��#���%��=�r ) �(:/0�����H )������&�� �������� �

� /�� �4;�� ×%è_Î�ãPØ�Ê�Ê�ÛÛ¦ÊnÓ¨Ó¼Ê9Þ�Ö%Í�è_Î�ã�Ø�Ê¡Ê�ÛÊ�Û?ɳÓ�Ù�Ì¨Ë M_õ�M(Ì�Ú»ä�å�ò�ç®õOAÎlÍ�H,ÒDίÙ{Ë«Ù�Ó�Ì�ßl̼ÊnÕ�Ö � Ù�ÏPÙ�ã�Í�̼Ô�ΩÙ�Ï_Ô�ÎlØjÖ�Ù�ØjáJÍ�è�Ù�͵Ì�Ú�Í�ÎlØjÙ�ßàÍ�Ö�Þ�Ì�Í�èÑã�Ù�Ø�Í�̼Î�Ö�ØjÕ�Ú�ÚPÌ�Ú_þ@! c Ì�ÚÑÍ�è_Î =���D>Pñ

è�á¡Ò�ØjÌ�ϯ˧ʺÏ_Î�ÓVõI��Î2Þ�Ì�Ó�Ó-ßàÊnÚ�Ö�Í�ØjÕPßàÍ�Ù�Ú¹Ù�Ï_Ô�ÎlØjÖjÙ�Ø�á%QYÌ�Ú©Í�è�Î�Ì�Ï_Î�Ù�ÓDã�Ø�ʺßàÎ�Ö�Ö5Û�Ê�Ø#= Ö�ÕPßjè¹Í�èPÙ�Í�Ú_ʧÎ�Ú�ԡ̼ØjÊnÚ�˧Î�Ú�ÍJ*ßlÙ�Ú÷Ï�Ì�Ö�Í�Ì�Ú�þnÕ�Ì�Ö�è(Þ�è_ÎlÍ�è_ÎlØ�̼Í%Ì�Ö%Ì�Ú�Í�ÎlØjÙ�ßàÍ�Ì�Ú�þµÞ�̼Í�è%H Ù�Ú�Ï6! c ̨گÍ�è_Î = ��D> ñ�è�á¡Ò�Øj̨ϩ˧ʡÏ�Î�Ó � Ê�Ø�Þ�Ì�Í�èRQYÙ�Ú�Ï= ̨گÍ�è_Î3Ì�Ï_Î�Ù�ÓAã�Ø�ʺßàÎ�Ö�ÖlõLQ+ØjÕ�Ú�ÖAH'ܸÖ�Ì�Ë�Õ�Ó�Ù�Í�̨Ú_þ�Í�è_Î3Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�ϯãPÙ�ØjÍ�̼Î�Ö%Ù�Ú�ÏN=���D>÷Û�Ê�Ø�Ì�ÍhÝ � Ù�Ú�Ï©ã�ØjÊ¡ßàÎlÎ�Ï�ÖÙ�Ö�Û¦ÊnÓ¨Ó¼Ê9Þ�Ö�õ��/83�-05 � 7 / - c�7 + 1 2 �43 3�-G-./ 2 � 7 / ��- � /87 + J � Q+Ï�̼Ø�Î�ßàÍ�Ó¼á(Û¦Ê�ØjÞ:Ù�ØjÏPÖ%Ù�Ú�á¯Ë§Î�Ö�ÖjÙ�þ�Î�Ö�ÒDÎlÍ�Þ:ÎlÎ�Ú<J Ù�Ú�ÏRH±õ��/83�-05 � 7 / - c�7 + 1P/ -G5�- 7<2L7��KcB1�� � Ú Í�èPÌ�Ö§Ö�Í�Ù�þ�Î � =���D>#Ì�Ö§Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏaÚ�Ê�Øj˵Ù�Ó�Ó¼á�õ·@ºÙ9á�@ ï Ì�Ö§ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï�õ

×%è_Î�Ú±Þ�è_Î�Ú%@ ï ØjÎ�ßàÎ�̼Ô�Î�Ö%Ù�Ú¯Ì�Ú_ãPÕ�ÍGÛ�Ø�ÊnËgJ � Q Ö�ÎlÎ�Ö5٧˵Î�Ö�Ö�Ù�þ�ÎrÊnÚ¯Í�è_ÎrÊnÕ_Í�þ�ÊnÌ�Ú_þ§ßàÊn˵Ë�Õ�Ú�̨ßlÙ�Í�̼ÊnÚ±Í�Ù�ã-ÎÊ�Û @ ï Û�Ê�Ø = � ÒPÕ_Í�ßlÙ�ÚJÊnÚ�Ó¼áLԺ̼ÎlÞ1Í�è_Î(è_Î�Ù�Ï_ÎlØ�õ�Q Ï�Ê�Î�Ö�Ú�Ê�Í�Û¦Ê�Ø�Þ³Ù�ØjÏ�Í�èPÌ�Ö�˧Î�Ö�Ö�Ù�þ�Î(Ï�̼Ø�Î�ßàÍ�Ó¼á � Ö�Ì�Ú�ßàÎ�H˵Ù�á§ßàÊ�Ø�ØhÕ_ã�Í�@ ï Ù�Í?Ù�Ú�á�Í�Ì�˧Î�õ QÑÖ�̨Ë2Õ�Ó¨Ù�Í�Î�Ö�@ ï Û�Ê�ØLH Ò¡á2ã�Ó�Ù�ßlÌ�Ú_þ4��˧Î�Ö�ÖjÙ�þ�Î�Ö � $ 4(' ' * " B98':<;=B2} :<; = B @ î B 3q�ÊnÚ]̼Í�Ö�ܸÖjÌ�Ë2ÕPÓ�Ù�Í�Î�Ï�ݳÊnÕ�Í�þ�ÊnÌ�Ú_þ¯ßàÊn˵Ë2ÕPÚ�Ì�ßlÙ�Í�̼ÊnÚ÷Í�Ù�ãDÎ2Û�Ê�Ø�=���D>õ � Û�H Û¦Ê�Ø�Þ³Ù�ØjÏ�Ö�Ù�Ó�ÓÍ�è_Î�Ö�Î�˧Î�Ö�Ö�Ù�þ�Î�ÖrÍ�Ê=���D>¹ÒDÎlÛ�Ê�Ø�Î�ßàÊ�ØjØjÕ_ã�Í�Ì�Ú�þ @ ï � QaÛ�Ê�Ø�Þ³Ù�ØjÏ�Ö�Í�è_Î�Ê�Øj̼þnÌ�Ú�Ù�Ó-˧Î�Ö�ÖjÙ�þ�Î�Û�Ø�ÊnË @ ï Í�Ê =©õ�Q Ù�Ó�Ö�Ê�Þ:Ù�Ì�Í�Ö:ÕPÚ�Í�Ì�Ó�@ ïØ�Î�ßàÎ�̼Ô�Î�Ö�Í�è_Î3Ù�ã�ã�ØjÊ�ã�ØjÌ�Ù�Í�ÎrØ�Î�ßàÎ�̼ãPÍ�Ölõ� Û?@ ï Ì�Ö�ßàÊ�Ø�ØjÕ_ã�Í�Î�Ï � Q ÖjÌ�˧ãPÓ¼á]Þ³Ù�̼Í�Ö�Û¦Ê�Ø�H Í�Ê�Ï_Î�Ө̼Ô�ÎlØ3Í�è_Ω˧Î�ÖjÖ�Ù�þ�Î�Ö4� $ 4(' ' * " B98O:P;=B2} :<; = B @ î B�� =:� � Û¦Ê�Ø: c å B ò BEDVDVDVB � � Û�Ø�ÊnË @ ï Í�Ê =�� �D> � Ù�Ú�Ï+Í�è_Î�Ú�Ï_Î�Ó�Ì�Ô�ÎlØjÖ(Ùa˵Î�Ö�Ö�Ù�þ�Î­Ü 8':<;�B�� Ý(Û¦Ø�ÊnË @ ï Í�Ê = � Þ�è_ÎlØjÎ� c � ï � î i i i � �õ� Û @ î Ì�Ö%Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï � Q#Ö�Ì�˧ãPÓ�áµØ�Î�ßàÊ�ØjÏ�Ö%Í�è_Î�Ø�Î�ßàÎ�̼ã�Í�Ö%Í�è_Î3Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏR@ î Ø�Î�ßàÎ�̼Ô�Î�ÏLÛ¦ØjÊnË Û¦ØjÊnË = ��D> õ� ÛNÍ�è�̨Ö%Ì�Ö5Í�è_Î2ë�ØjÖ�Í%Ù�ßàÍ�Ì�Ô�Ù�Í�̼ÊnÚ � Í�è_Î3̨Ú�Í�ÎlØjÚPÙ�ÓAÖ�Í�Ù�Í�Î�Û¦Ê�Øc= Ë2Õ�Ö�Í�Ù�Ó�Ö�Ê«Ò-Î�Ö�ÎlÍ�Õ_ãAõ � Û�@ ï Ì�Ö%ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï �Q7ØjÕ�Ú�Ö3Í�è_αÙ�ßàÍ�ÕPÙ�Ó�ãPØ�Ê�Í�Ê¡ßàÊnÓ�Û�Ê�Ø2Í�è_ίÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï6@ ï õ � Û @ ï ̨Ö3ßàÊ�Ø�ØjÕ�ã�Í�Î�Ï � Q7Þ:Ù�̼Í�Ö2Û�Ê�Ø,H'Í�Ê]Ï_Î�Ó�Ì�Ô�ÎlØÍ�è_Î $ 4(')'+* "5˧Î�ÖjÖ�Ù�þ�Î�Ö�Û�Ø�ÊnË @ ï Í�Ê�=�� �D>õ � Û @ î ̨Ö%Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï � Q­ØhÕ�Ú�Ö5Í�è_Î�Ù�ßàÍ�ÕPÙ�ÓAã�Ø�Ê�Í�ʺßàÊnÓAÛ�Ê�Ø%Í�è_ÎÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏR@ î õ

��/83�-05 � 7 / - c�7 + 1 2j/ / 2 -./87.1^0 �T5 - � 7 / ��- 2L7��KcB1�� � ÚµÍ�è�Ì�ÖGÖ�Í�Ù�þ�Î � =�� �D>¯Ì¨ÖGÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï«Ú_Ê�Øj˵Ù�Ó�Ó�á�õ6��Ù�ßjèJ� ö � �þnÙ�Í�Î2Ì�Ö�ßàÊn˧ãPÕ_Í�Î�Ï©Ò¡á%@ ï Ö�Î�ÚPÏ�Ì�Ú_þ§Ù�Ún7 6 0 ¡Ë§Î�Ö�Ö�Ù�þ�Î2Í�Ê�=���D>.õ� ÛW@ ï Ì�ÖrÕ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï � Q Ö�Î�Ú�Ï�Ö�Í�è_Î�7 6-0�˧Î�ÖjÖ�Ù�þ�εÖ�ãDÎ�ßl̼ë�Î�Ï{Ì�Ú÷Í�è_Î�ã�Ø�Ê�Í�ʺßàÊnÓÛ�Ø�ÊnË Í�è_ΧÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï @ ïÍ�ÊN=���D> � Ù�ÚPÏ�Þ:Ù�Ì�Í�Ö�Û�Ê�Ø3Í�è_ΫÙ�ãPã�Ø�Ê�ã�Øj̨Ù�Í�Î�ØjÎ�ßàÎ�̼ã�Ílõ � Û @ ï ̨Ö�ßàÊ�Ø�ØjÕ_ãPÍ�Î�Ï � Q Þ:Ù�̼Í�Ö3Û¦Ê�Ø H Í�Ê{Ï_Î�Ó�̼Ô�ÎlØ�Ù˧Î�Ö�Ö�Ù�þ�Î �,7 6-0 B98':<;=B2}0:P;�B2} :<; ù B2} :<;_ïCB @ ï B ���Û�Ø�ÊnË @ ï Í�Ê<=���D>.õ� Û @ î Ì�Ö%Õ�Ú�ßàÊ�ØjØjÕ_ã�Í�Î�Ï � Q#Ö�Ì�˧ãPÓ�áµØ�Î�ßàÊ�ØjÏ�Ö%Í�è_Î�Ø�Î�ßàÎ�̼ã�Í�Ö%Í�è_Î3Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏR@ î Ø�Î�ßàÎ�̼Ô�Î�ÏLÛ¦ØjÊnË Û¦ØjÊnË =���D>.õ

��/83�-05 � 7 / - c�7 + 1 �T-;7 56-;7<2L7 �dcB1�� ö Í?Í�è�Ì�Ö>ãDÊnÌ�Ú�Í � = Öjè_ÊnÕ�Ó�ϧè�Ù9Ô�Î�Ö�Î�Ú�ÍN� �����Ñ˧Î�Ö�Ö�Ù�þ�Î�Ö�Í�Ê @ ï Ù�Ú�ÏI@ î � ÒPÕ�ÍQYè�Ù�Ö�Ú_Ê�Í�Ï_Î�Ó�Ì�Ô�ÎlØ�Î�Ï©Ù�Ú�áx� ����� ˵Î�Ö�Ö�Ù�þ�Î3Í�Ê(Ù�Ú¹Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�ϯãPÙ�Ø�Ívá(á�ÎlÍlõ� Û?@ î Ì�Ö2ÕPÚ�ßàÊ�Ø�ØjÕ_ãPÍ�Î�ÏJÙ�Ú�Ï è�Ù�Ö�Ú_Ê�Í�Ø�Î�ßàÎ�̼Ô�Î�Ï Í�è_ίÙ�ã�ã�Ø�Ê�ã�ØhÌ�Ù�Í�ΫØ�Î�ßàÎ�Ì�ã�Í�Ö�Ì�Ú»Í�è_αã�Ø�ÎlÔ�Î�Ì�ÊnÕ�Ö2Ö�Í�Ù�þ�Î�Ö � QÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ö @ î Ù�ÒDÊ�Ø�Í�Ì�Ú_þ_õ��Ê�Þ Þ:Î�Ò�Ø�Î�Ù�ý±Í�è_Î3ÖjÌ�Ë2ÕPÓ�Ù�Í�̼ÊnÚ¯Ì�Ú�Í�Ê«ßlÙ�Ö�Î�Ölõ

ò+M

Page 25: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

å�õ!@ ï Ù�ÚPÏR@ î Ù�Ø�Î�Õ�ÚPßàÊ�Ø�ØjÕ_ã�Í�Î�Ï .QYÖ�Î�Ú�Ï�Ö³Í�è_Î�Ù�ãPã�Ø�Ê�ã�Øj̨Ù�Í�Î 4^1[7-,�˧Î�Ö�Ö�Ù�þ�Î�Ö%Û¦Ø�ÊnË Í�è_Î�Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�ÏR@ ïÙ�Ú�Ï @ î Í�Ê"= ��D> � Ù�Ú�Ï�ÊnÚ�ßàÎ%Ù�Ú 4^1[7-,�˵Î�Ö�Ö�Ù�þ�Î�Ì�Ö.Ø�Î�ßàÎ�Ì�Ô�Î�ϧÒ�á©Ü�Í�è_Î%Ö�̨Ë2Õ�Ó¨Ù�Í�Î�Ï�Ý`= � �D> � Q Ö�Î�Ú�Ï�Ö.Í�è_ÎÙ�ã�ã�Ø�Ê�ãPØjÌ�Ù�Í�Î^� �����J˵Î�Ö�Ö�Ù�þ�Î�Û�Ø�ÊnË�=�� �D> � ÒPÕ_Í>Þ�̼Í�è«Ù�Ï�Ù�Í�Ù�Ô�Ù�Ó�Õ_Î5Ê�Ûa�lÎlØ�Ê_õ�Ü ��Ê�Í�Î�Í�è�Ù�Í?Í�è_Î%Ô�Ù�Ó�Õ_Î�ÖÌ�Ú©Í�è_Î�Ö�Î3˧Î�Ö�Ö�Ù�þ�Î�Ö�Ù�Ø�Î3Ú�Ê�Í�Ö�ÎlÎ�Ú¹Ò¡á%H*Ù�ÚPÏL˵Ù9á±ÒDÎ�Ö�ÎlÍ�Ù�Ø�ÒP̼Í�ØhÙ�ØjÌ�Ó¼á�õ�ݧÐrÚPßàÎ3Ù�Ó�Óu� �����Y˧Î�Ö�Ö�Ù�þ�Î�Öè�Ù�Ô�ΧÒDÎlÎ�Ú÷Ø�Î�ßàÎ�̼Ô�Î�Ï]Ò¡áÑÜ�Í�è_Î�Ö�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï�Ý!@ ï Ü�Ø�Î�Ö�ãAõ � @ î Ý � Q�Û�Ê�Ø�Þ:Ù�ØhÏ�Ö�Í�è�Î � �����#˧Î�ÖjÖ�Ù�þ�Î�Û¦Ø�ÊnË= Í�ÊM@ ï Ü�Ø�Î�Ö�ãAõ � @ î Ýhõò¡õ!@ ï ̨Ö�ßàÊ�Ø�ØjÕ_ã�Í�Î�Ï � ÒPÕ_Í @ î ̨Ö�Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï 'Q Ö�Î�Ú�ÏPÖ�Í�è_εÙ�ã�ã�ØjÊ�ã�ØjÌ�Ù�Í�Î 4^1[7-,(˵Î�Ö�Ö�Ù�þ�Î�Ö�Û�Ø�ÊnË�Í�è_ÎÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï @ î õ;@ºÌ�Ú�ßàÎ @ ï Ì�Ö%ßàÊ�ØjØjÕ_ã�Í�Î�Ï � Q+è�Ù�Ö�Ø�Î�ßàÎ�̼Ô�Î�Ï÷Í�è_Î#� �����Y˧Î�Ö�Ö�Ù�þ�Î�Û¦ØjÊnË = Í�Ê @ ï � Ù�Ú�ÏÍ�è�ÕPÖ2ýºÚ_Ê�Þ�Ö�Í�è_Î(ÊnÕ�Í�ãPÕ_Í3Ò�̼Í�Ö3Û�Ê�Ø @ ï õ�@¡Ê Q Ö�Î�Ú�Ï�Ö�Ù{˧Î�Ö�Ö�Ù�þ�Î �O� ����� B98':<;=B2}0:P;�B e @ ï B @ î f B ú���Í�Ê@ ï Û�Ê�Ø�Î�Ù�ßhè�ÊnÕ_Í�ãPÕ�Í�ÒPÌ�Ílõ ö Ó�Ö�ʱÛ�Ê�Ø�Î�Ù�ßhè 4^1[7-,«Ë§Î�Ö�ÖjÙ�þ�Î�Ï�Î�Ó�̼Ô�ÎlØ�Í�ÊJ=���D>»Û�Ø�ÊnË @ ï � Q Ö�Î�Ú�ÏPÖ�Í�è_ÎÙ�ã�ã�Ø�Ê�ãPØjÌ�Ù�Í�Î4� �����=˧Î�Ö�Ö�Ù�þ�Î(Û¦Ø�ÊnË =���D> Í�Ê<@ î ÒPÕ_Í�Þ�̼Í�è»ÙLÏPÙ�Í�Ù¹Ô�Ù�Ó�Õ_ΧÊ�ÛW�lÎlØ�Ê_õ»Ü ö þnÙ�Ì�ÚJÚ_Ê�Í�ÎÍ�è�Ù�Í�Í�è_Î3Ô�Ù�Ó�Õ_Î�Ö�Ì�Ú¹Í�è_Î�Ö�Î�˧Î�ÖjÖ�Ù�þ�Î�ÖrÙ�Ø�Î2Ú_Ê�Í�Ö�ÎlÎ�Ú÷Ò�áRH Ù�Ú�Ï÷˵Ù�áLÒ-Î2Ö�ÎlÍ�Ù�ØjÒP̼Í�ØjÙ�Øj̨Ӽá�õ�ݯÐrÚ�ßàÎ2Ù�Ó¨Ó� ����� ˧Î�ÖjÖ�Ù�þ�Î�Ö�è�Ù�Ô�Î(Ò-ÎlÎ�ÚJØ�Î�ßàÎ�̼Ô�Î�ÏJÒ�á Ü�Í�è_αÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï�Ý'@ î Q=Û¦Ê�Ø�Þ³Ù�ØjÏ�Ö3Í�è_Îh� ����� ˧Î�Ö�Ö�Ù�þ�ÎÛ¦ØjÊnË = Í�ÊM@ î õ

�¡õ!@ ï ̨Ö%Õ�Ú�ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï � ÒPÕ_Í @ î Ì�Ö%ßàÊ�Ø�ØhÕ_ã�Í�Î�Ï ö Ú�Ù�Ó¼Ê�þ�ÊnÕ�Ö%Í�ʵÍ�è_Î�ã�ØjÎlԡ̼ÊnÕPÖ5ßlÙ�Ö�Î�õM_õ�ô:Ê�Í�è @ ï Ù�Ú�Ï @ î Ù�Ø�Î3ßàÊ�Ø�ØjÕ�ã�Í�Î�Ï �QYÖjÌ�Ë2ÕPÓ�Ù�Í�Î�Ö�= ��D> Ú_Ê�Øj˵Ù�Ó¨Ó¼á�õ

��/83�-05 � 7 /���- �4;!2 �T/(/ -Z507 / �4-02�� � Û>Ù{ܸÖjÌ�Ë2ÕPÓ�Ù�Í�Î�Ï�Ý:ãPÙ�ØjÍ�á @ ï Ì�Ö�ßàÊ�Ø�ØjÕ_ã�Í�Î�ÏLÒ�áRH � Q�ßàÊ�Ø�ØhÕ_ã�Í�Ö!@ ï Ù�ÚPÏLÚ_ÎlÎ�Ï�ÖÍ�Ê{ÖjÌ�Ë2ÕPÓ�Ù�Í�εÍ�è_αÌ�Ú�Í�ÎlØhÚ�Ù�ÓGÖ�Í�Ù�Í�αÊ�Û3Ü�Í�è_ίÖ�Ì�Ë�Õ�Ó�Ù�Í�Î�Ï�Ý @ ï Û�Ê�Ø,H'Í�Ê÷˵Ù�Í�ßhèÑÍ�è�Î(˧Î�Ö�ÖjÙ�þ�Î�Ö�Ö�Î�Ú�Í�Í�Ê÷Ù�Ú�ÏØ�Î�ßàÎ�̼Ô�Î�Ï�Û�Ø�ÊnË = � Ù�Ú�Ï]Ì�Ú{ãPÙ�ØjÍ�Ì�ßlÕ�Ó�Ù�Ø � Í�è_ε̨Ú_ãPÕ_Í�Ù�Ú�Ï]ÊnÕ_Í�ãPÕ_ÍrÒPÌ�Í�Ö�Ê�Û�Í�è_εßàÊn˧ãPÕ�Í�Ù�Í�̼ÊnÚ]Û¦Ê�Ø @ ï õ2ô³Õ_ÍÍ�è�Ì�ÖrÌ�ÖrÖ�Í�ØjÙ�Ì�þnè�Í�Û�Ê�Ø�Þ³Ù�ØjÏ � Ö�Ì�ÚPßàÎ�Õ�ã-ÊnÚ]ßàÊ�Ø�ØhÕ_ã�Í�Ì�Ú_þ @ ï � Q Ê�Ò�Í�Ù�Ì�Ú�ÖrÍ�è_Χ˧Î�ÖjÖ�Ù�þ�Î�Ö�Ö�Î�Ú�Í�Í�ʹÙ�Ú�Ï{Ø�Î�ßàÎ�̼Ô�Î�ÏÛ¦ØjÊnË = � Ù�Ú�ϹßlÙ�Ú©ã�Ù�Í�ßjè¹Í�è�Î3Ö�Í�Ù�Í�Î3Ê�Û @ ï Ù�ã�ã�Ø�Ê�ã�ØhÌ�Ù�Í�Î�Ó¼á�õ?×%è_Î�ÖjÙ�˧Î�Ì�Ö5Í�ØjÕ�Î�Û¦Ê�Ø!@ î õ

� ͧÌ�Ö�Ö�Í�ØjÙ�̼þnè�Í�Û�Ê�Ø�Þ:Ù�ØhÏ Í�Ê]Ö�ÎlαÍ�èPÙ�Í�Í�è�ίÖjÌ�Ë2ÕPÓ�Ù�Í�̼ÊnÚ Ì�Ö ()����9��&#�/ � Ì®õóÎ�õ � Í�è�Ù�ͧÎlÔ�Î�ÚaÙ]ßàÊn˧ãPÕ_Í�Ù�Í�Ì�ÊnÚ�Ù�Ó�Ó¼á»Õ�Ú_ñÒDÊnÕ�ÚPÏ_Î�ϱÎ�Ú�Ժ̼Ø�ÊnÚ�˵Î�Ú�Í%ßlÙ�Ú�Ú�Ê�Í�Ï�Ì�Ö�Í�̨Ú_þnÕ�Ì�Ö�è(Í�è_Î�Ì�Ï�Î�Ù�ÓDã�Ø�ʺßàÎ�Ö�Ö5Þ�̼Í�è©Ì¨Ï_Î�Ù�ÓDÛ�ÕPÚ�ßàÍ�̼ÊnÚ�Ù�Ó�Ì�Í�á<= Û�Ø�ÊnË Í�è�Î�= ��D> ñè�á¡Ò�ØjÌ�ϯ˧ʺÏ_Î�ÓDÞ�̼Í�è¯Í�è�Î�ã�Ø�Ê�Í�ʺßàÊnÓ�! c õ

� � Ç?È��GÁr �%Á�Å � Á§È��?�GÃ��4� : �� �+Ä?���±Ä!����� ���� ������� �����������������������������

!�"�# è%$'&(&*),+ # $'- "/. )0&�è%- . è1- .2# -3)54 # ) "%6/# è%)87�),&�9%: # &<;=7�-?>A@%7�)CBD$E-?9%&<&*),+ # $E- " & # -F&*),+C917�),:'GF7�),HI:'$EJC)8H " G. ),:':LKM;=-I7N>�) 6 >O9%: # $EKM@PHQ7 # G<;R9 " + # $E- " HI:S$ # GT$ "T# è1)U@P7�),&*) " +5)U-I;V>WHI:S$'+C$E-?9%&�H 6 HQ@ # $EBI)XH 6 BI)C7N&NHQ7N$E),&Y+5-I7�7Z91@ # $ "1[ H "HQ7N\P$ # 7NHQ7�G " 9P>X\])C7^-I;_@PHQ7 # $E),&,`bac917U+5- " & # 7Z9%+ # $E- " $'&^&N$'>W$':'HQ7 # - # èPH # $ "FdLe,fhg ;=-I7�&*),>i$LK�è1- " ),& # H 6 BI)C7N&�HQ7N$E),&,`j 9 #Ck H [ HI$ " HI&<$ "F# è1) #l. -QKM@PHQ7 # G�+CHI&*) kY. )mHQ7�)nHQ\P:E) # -�+5- " & # 7N9P+ # \P9%$S: 6 $ "1[ \P:E-D+�oD& # èPH # +CH "p. $ # è%& # H "P6>iHI:'$'+C$E-?9%&qH 6 BI)C7N&�HQ7Z$E),& k H "%6b# è1)C7�)C;=-I7�)c-?917�+5- " & # 7N9%+ # $E- " $'&�&*),+C917N)�H [ HI$ " & # >iHI:'$'+C$E-?9%& k H 6 HQ@ # $'BI)cH 6 BI)C7N&�HQ7N$E),&6 $E7�),+ # :EGI`

!�" @PHQ7 # $'+C9%:SHQ7 kq. )mHI&N&�9%>�) # èPH #i# è1)C7�)nHQ7�)srt@PHQ7 # $E),&i@PHQ7 # $'+C$E@uH # $ "1[ $ "3# è%)m+5-?>�@P9 # H # $E- "k�. è%$S+jèFHQ7�)6 ) " - # ) 6 \�G�vxw,y�v{zQyC|E|E|Ey�v{}]`�~8)�HI&�&N9%>�) # è%H #�# è%)0;=9 " + # $E- " HI:'$ # G3��$'&T7N)C@%7�),&*) " # ) 6 \�G�H�;RHI>W$':'GF����-I;j -�-?:E),H " +C$'7N+C9%$ # & kP# è1)�� # ès+C$'7N+C9%$ # 7�)C@P7�),&*) " # $ "1[ H " HI+ # $EBhH # $'- " -I;�� . $ # ès&*),+C917N$ # GT@PHQ7NHI>�) # )C7c��`�~/$ # è1-?9 #:'-?&�&�-I; [ ) " )C7NHI:S$ # G k. )�HI&�&�9%>�) # è1)�+C$E7Z+C9%$ # &�HQ7�)�+5-?>�@]-?&*) 6 ) "�# $'7�),:EG(-I;q������H "P6m� a^� [ H # ),&C`���&c\])C;�-I7N) k. )�HI&N&�9%>�) # è%H # $ " ),HI+jè�HI+ # $EBQH # $E- "kV��e�� H # >�-?& # - " )�@PHQ7 # G¹è%HI&�H " $ " @u9 #�# -i� . $ # è�H # >�-?& # �(\u$ # & k{��f?�),HI+hè0@PHQ7 # G�>WH,G�7�),+5),$EBI)WH # >�-?& # �n\P$ # &^HI&U-?9 # @P9 # ;=7�-?>�� k����?� ��$'&^H 6 ) # )C7N>W$ " $S& # $'+�;=9 " + # $E- "k H "%6F�=� �# è%)U:E-�+CHI:& # H # )U-I;���HQ; # )C7�),HI+jèsHI+ # $EBhH # $'- " +CH " \]) 6 ),&�+57N$E\]) 6 \�GbH # >�-?& # �i\u$ # &C`��%è1)^$ " $ # $'HI:�& # H # )U-I;{��$S&6 ),&�+57N$E\]) 6 \�Gs�(JC)C7�-<\P$ # &,`�~8)OHI&�&N9%>�) # èPH # >�),&N&�H [ ),&c&*) "�# ;=7�-?> # è%)X� # -W� HQ7�)�$ [?" -I7�) 6k H "%6s# è%)C7�)�HQ7�)" -W>�),&�&�H [ ),&�;�7N-?>�� # -W�b`

!�" -I7 6 )C7 # -8&*),+C917N),:EG07�),HI:'$'JC)i� k�. )T\PHI&�$'+CHI:':'G8;�-?:':'- .¡# è1)T&�HI>�)(HQ@%@%7�-?HI+hè3HI&�$ "¢# è1) #l. -QKM@PHQ7 # G�+CHI&*)I`£ - . )CBI)C7 k�. )�¤%7N& #V" )C) 6�# -X)54 # ) "%6 &*-?>�)Y-I;�-?917x+5- " & # 7N9%+ # $E- " &�;=7�-?>¥@P7�)CB�$'-?9%&�&*),+ # $E- " & # -�&�9%$ #�# è1)�>O9%: # $'@P:E)5K@uHQ7 # G(+CHI&*)I`

fI¦

Page 26: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

����� ����� ���� ���� ��������������������� �"!# �%$'&)(+*-,.���/��01���� 2�03��0��4�~8)bHI&�&�9%>�)WH "6587:9<;�=?>49A@3BC5�9�=EDGF?H-IJ5�DBC5LK?9 +jèPH "%" ),:�H�BhHI$':SHQ\P:E) # -sHI:':�@PHQ7 # $'+C$E@PH # $ "%[ @PHQ7 # $E),&C`��%è1)W+hè%H "%" ),:�$S&>W- 6 ),:') 6 \�G # è1)_\%7�-?H 6 +CHI& # ;=9 " + # $E- " HI:'$ # G��NMPOb$ "RQ $ [ 917N) ¦ `V�%è1)q;=9 " + # $E- " HI:'$ # G [ 9%HQ7NH " # )C),& # è1)_HI9 # è1) " # $'+C$ # G-I;�Hm>�),&N&�H [ ) k $�` )I` kV# è%H # " -n@PHQ7 # G0v%S�+CH " ;RHQoI)THn>�),&�&NH [ )<;=7�-?> vUT?`m�%èP$'&X$'&�HI:'&*- # è1)bHI&N&�9%>�@ # $'- " 9%&*) 6$ "0dLe,fhgMk H "%6(. )U7�)C;=)C7 # è1)^7N),H 6 )C7N& # - dLe,f%kPf�Vhg ;=-I7�>�-I7�)U$ " K 6 )C@ # è 6 $S&�+C9%&�&�$'- " &C`

WYX%Z%[�\�]<^_Z%`4aA]b\�c �NMPO

�NMPOn@%7�-D+5)C) 6 &YHI&�;�-?:S:E- .Uk 7Z9 "%" $ "%[�. $ # èT@PHQ7 # $E),&�vxw,yC|E|E|Ey�v }(H "%6 H " H 6 BI)C7N&�HQ7NGed�`f�g @]- " 7�),+5),$EBD$ "1[ H >�),&N&�H [ ) �ih8jlk�mLnoJm�p-q yCrts�uPy-v�y-w � ;=7�-?>¡v S kD. è1)C7N)xv¥$'&_HO&*) # -I; @PHQ7 # $'),& k &�) "%6�-y{z |�}P~�����}U� yCrts�u%y�v S y-v�y-w ��# -iHI:':]@PHQ7 # $'),&Y$ " v H "%6 d k H "%6�� HI: # `

Q $ [ 917�) ¦/���{�U� �NMPO��8���8���P���8�t���i��� �t�3�b�L�4�8�������

~8)nHI:'&�- " )C) 6 H " )54 # ) " &�$'- " -I; #E� )�� ��;R9 " + # $'- " HI:'$ # G k_" HI>�),:EG #E� )s- " )5K # -QKl>WH " G � ��;R9 " + # $'- " HI:'$ # G k6 ) " - # ) 6 \ Gi�¢¡�£t¤q` !�"�# 9%$ # $EBI),:EG kD#E� $'&�;R9 " + # $E- " HI:S$ # GWHI:':'- . &�H &�$ "%[ :E)�@P7�-�BI)C7 # -�@%7N-�BI)UH #E� )C-I7�),> # -W>O9%: # $'@P:E)BI)C7Z$E¤%)C7N&�&�$S>�9%: # H " )C-?9%&�:EGI`¦¥D)C) Q $ [ 917�)¨§ ;=-I7�H�;=-I7N>WHI: 6 )C¤ " $ # $E- " `

W%XYZ�[�\L]b^UZ�`_a<]b\�c �ª©¡�£�¤

� ©¡�£�¤ @%7N-�+5)C) 6 &VHI&�;�-?:':'- .Uk 7N9 "%" $ "1[�. $ #E� @PHQ7 # $'),&Vv�w,yC|E|E|Ey�v{}OH "P6 H " H 6 BI)C7N&�HQ7�G«d k H "%6 @PHQ7NHI>�) # 7Z$EJC) 6. $ #E� HW7�),:'H # $E- "­¬®�

f�g @]- " 7�),+5),$'B�$ "%[ Hm>W),&�&�H [ ) �A¯8°��E±8jlkL²8³8j yCrts�uPy-v�y-w y-´ � ;=7�-?> v�S k .N� )C7�)µv�$S&XHs&�) # -I;�@PHQ7 # $E),& k$'; ¬W� w y-´ �N¶�eQk#E� ) " &*) "%63�1·{¸��C¹%º�»N»½¼ yCrts�u%y�v S y-v�y-w ��# -(HI:': @PHQ7 # $E),&c$ " v�H "P6 d H "%6¾� HI: #"¿- #E� )C7 . $'&*) � HI: # `

Q $ [ 917�)¨§ �À�{�_� � ©¡�£�¤ �����8���P���8�J���Á��� �t�3�<�L�4�8���1�Â�

~8)n-I\u&*)C7�BI) #E� H #<#E� ) gÄà � � +5- " & # 7Z9%+ # $E- " \ GÆÅcHQ7NH,G ="9�58ÇÂÈUd fI¦Qg +CH " \]) " H # 917ZHI:':EG3)54 # ) "%6 ) 6�# -�H- " )5K # -QKl>WH " G gÄà � � @%7�- # -D+5-?: . $ #E��#E� )�H 6%6 $ # $E- " HI: \%7�-?H 6 +CHI& # ;R9 " + # $E- " HI:S$ # GI`c�Y-?9 [�� :EG�&*@]),HQo�$ "1[1k v S �=#E� )@P7�-�BI)C7 � 7N9 " &UH " $ "%6 )C@]) "%6 ) " # +5-I@ G�-I; #E� ) # . -QKM@uHQ7 # G gÀà � ��@%7�- # -D+5-?: . $ #E� )CBI)C7NG0@PHQ7 # G�v T�É v 9%&�$ "1[H�9 " $�Ê 9%)µrJsËu k H "%6 HI:':_>�),&�&NH [ ),&�HQ7�)T\%7�-?H 6 +CHI& # `GÌqHI+ � v T HI+C+5)C@ # &�$E;�H "P6 - " :'G�$E;YHI:': #E� )(+5- " BI)C7Z&�H # $E- " &HQ7N) HI+C+5)C@ # $ "1[ ` ! # $'&c& # 7NHI$ [���# ;=-I7 . HQ7 6�# -�+5- " & # 7N9P+ # H " $ 6 ),HI:�H 6 BI)C7N&�HQ7�G�d�` ! ; #E� )O@P7�-�BI)C7X$S&�9 " +5-I7�7N91@ # ) 6kd &N$'>�@P:EGT7N9 " &�H<>O9%: # $LKM@PHQ7 # G gÄà � ��&�$'>O9%:'H # -I7�;�-I7c)CBI)C7�G�+5-I@�Gs-I; #E� ) gÄà � � @%7�- # -D+5-?:�` ! ; #E� ) @%7�-�BI)C7U$S&+5-I7N7N91@ # ) 6 H "%6m#E� )C7�) $'&�H # :E),HI& # - " ) 9 " +5-I7N7N91@ # ) 6 BI)C7N$E¤%)C7 k d +CH " )54 # 7ZHI+ #�#E� ) . $ #�" ),&�&C` ! ;�HI:':{@PHQ7 # $'),&�HQ7�)+5-I7N7N91@ # ) 6kD#E� )^&�$S>�9%:SH # $E- " $'&_& # 7NHI$ [�� # ;�-I7 . HQ7 6 `�� � )^+5- " BI)C7Z&�$E- " 7�),>WHI$ " &�)"Íi+C$E) " # `x� � )C7�)C;=-I7�) . ) � H,BI) #E� );=-?:':E- . $ "1[ #E� )C-I7N),>(`

ΫÏ%Ð�^_Ñ�Ð�ÒÔÓ�ÕAÖØ× >_D=?H®9<;�=®K�9AH-I8>�ÙµÚÀÛ/ÜÝ58KCK�7�ÞxßU9A@3I8>àI�H®9<;:=¨á¨Û/Üâ5LK�K�7�ÞxßU9A@3I8>:ã�ä"I8H�="å8="H�æ�HE=?ÇÂ5�9A@AI�>�¬ç9<;/5�95D8Þ®@<9<K�58>6èÀélß_H-I�91IJBCI�Çëê¨ãª9<;�="H-=G=Cì�@bK�9<Kí5�9<;H-=�=?éiHÁI�7/>UDíß_H-I�91IJB�I8ǽî�z�d ê�g�9<;�5�9«K"=CB"7�H-="Çïæ�HE=E58Çð@�ñt=�K¾9<;�= � ©¡�£�¤@3D=C58Ç_ä�7�>_BJ9A@3I8>_5�Çï@A9Aæ¾@b>ò9<;�=s� �xO:ó_ô y��NMPO ��éË;æF?H�@AD¾ÞõIJD�=?Ç)5JÙ58@<>:K?9½5�D�5�ßU9A@<å8=ö5�D8å�=?H�K?58H�@3=?KCãR5LK�K�7�Þ®@b>/Ù÷=?H-5LK�@b>�Ù�Èø 7�H�9<;�="H�ÞõI8HE=�ãY9<;�=¢B�I8Þxß_7:915�9A@AI�>eBCI�ÞNßUÇÂ=Cì�@A9Aæ¨IÁä.î�zVd ê�g�@bK.9<;/5�9�IÁäYêòßUÇï7/KùB�I8>:K�915�>49 >P7�ÞªFC="HëIÁä)=Eì?ß�I8>U=">49A@A5�9A@AI8>PK5�>_De9<;�=ÀÙ�=">U=?H-5�9A@3I8>úIÁäR5ªK�@ÂÙ�>U5�9A7�H-=?ãù9A@<Þõ=?K½9<;�=«>P7�ÞªFC="HRIÁäÄHE=EB�="@bå�@b>/Ù¨ß�58H?9A@3=?K?È

f §

Page 27: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

����� �Ý,�23��03���� ���t$������~8)^HI:'&�-�)54 # ) "%6<#E� )��� �O�����;R9 " + # $E- " HI:'$ # G # - #E� )c>�9P: # $LKM@PHQ7 # Gi+CHI&*) kD.N� )C7N) #E� )�@%7�-�-I;@ � HI&�)�$'&q7�)C@P:SHI+5) 6 \�GH�- " )5K # -QKl>WH " G<@%7�-�-I;�H "%6i#E� )�7�),+5),$E@ # &�HQ7N)�&*) " #�# - HI:S:%@PHQ7 # $S+C$E@PH # $ "1[ @PHQ7 # $E),&C`M� � )�;R9 " + # $E- " HI:'$ # G�$'& 6 ) " - # ) 6\�Gb�¢¡ �O���� H "P6 $'& [ $EBI) " $ "­Q $ [ 917�)�� `

W%XYZ%[�\�]<^_Z%`4aA]b\�c � ¡ :O����

�¢¡ :O���� @%7�-D+5)C) 6 &YHI&Y;=-?:':E- . & k 7N9 "%" $ "1[�. $ #E� @PHQ7 # $E),&Yv w yC|E|E|Ey�v } H "%6 H " H 6 BI)C7N&�HQ7�G�d�`

f�� ^_ÒàÒ÷]b\��{Ï�`��tÐ�� ~ � ) " 7�),+5),$EBD$ "1[ ;�7�-?>�v S H�>W),&�&�H [ )�� ³"o"k8q���o"k����� q yCrJs�uPy"!�s�u%y-v�y$#&% k 7N),+5-I7 6�'!�s�uPy�v S y-v�y$#&% k &�) "%6 >�),&N&�H [ )(�*) z�»)~��Cº ) z ),+ ¹_~ yCrJsËu%y"!�s�u%y�v S y-v�% # -nHI:S:V@PHQ7 # $E),&O$ " v�H "%6 d kH "P6 $ [?" -I7�)<HI:S:�;R9 # 917�)<>�),&�&�H [ ),& -I; #E� )b;=-I7N>-� ³"o"k8q���o"k����� q yCrJs�uPy"!�s�u%y-v�y/.0%�;�7�-?> v S H "%6 HI:':;R917 # 9%7�)^>�),&�&�H [ ),&�-I; #E� )U;�-I7Z>1� ³"o"k8q��1q�jbm32tp*43³8j yCrts�uPy"!�sËu%y/.?y/.?y/.?y�v S y-v�%�;�7�-?>�H " GT@PHQ7 # G<$ " v�`

f65 Ñ�^87_Ð9�YÏ%`��JÐ:� ~ � ) " 7�),+5),$EBD$ "1[ ;�7N-?> v S H >�),&�&NH [ );� ³"o"k8q��E±8jlkL²8³ yCrts�u%yCrLrts�u%y"!�s�u3<?y!�s�u%w,y"!�s�u�zIy-v�y>=?% k $E; #E� )t;=-?:':E- . $ "1[ #E� 7N)C) # 91@P:'),& k �'!�s�u3<Iy�v S y-v�y$#�<@% k �'!�s�u1w,y�v S y-v�y$#�w/% k H "%6�'!�s�u zIy�v S y-v�y$# z@% HQ7N) HI:':¡7�),+5-I7 6 ) 6�k H "%6 k�±BA z"CD � #E<hy$#�w � ¶ # z k�#E� ) " &*) "%6 >�),&N&�H [ )�*) z�»)~��C¹%º�»N»½¼ yCrts�u%yCrLrts�u%y"!�s�u3<?y"!�s�u1w y"!�sËu�zIy�v S y-v�y>=?% # -�HI:':Y@PHQ7 # $E),&W$ " v�H "%6 d ¿ - #E� )C7 . $'&*)6 - " - #E� $ "1[ `

f ÎùÑL`4ZF�HG�Ð/ÑI�YÏ%`��JÐ:� ~ � ) " 7�),+5),$EBD$ "1[ ;=7�-?> v S H >�),&�&NH [ )J� ³"o"k8qË�1q�j<mB2�p*4A³8j yCrJsËu%y"!�s�u%y!�s�u3<Qy"!�s�u1w�y>K*!�s�u%y�v T y-v�% k $E; #E� ) ;�-?:':'- . $ "%[ #E� 7�)C) # 91@P:E),& k �'!�s�uB<?y�v S y-v�y$#�<@% k �'!�s�u1w�y�v S y-v�y$#�w/% kH "P6 �LK*!�sËu%y�v T?y-v�y$#EMN%�HQ7�) HI:': 7�),+5-I7 6 ) 6k &*) "%6 >�),&�&�H [ )O�*) z�»¢~��$PY|_~ | yCrts�uPy"!�sËu%y�v%S�y�vUT�y!�s�u3<Qy"!�s�u1w�y>K*!�s�u%y$#�QSR"% # - v T k 7�),+5-I7 6 # 91@u:E) �'!�sËu%y�v T y-v�y$#�QSR"% k H "%6 &*) "%6 >�),&N&�H [ )�*) z�»)~��Cº ) z ),+ ¹_~ yCrJsËu%y"!�s�u%y�v S y�v T y"!�s�u3<Iy"!�s�u1w,y>K*!�s�u%y-v�% # -�HI:':�@uHQ7 # $E),&¥$ " v k v S k H "%6 d ¿H "P6 $ [?" -I7�)iHI:':x;=9 # 9%7�)i>�),&�&NH [ ),& -I; #E� )i;=-I7N>-� ³"o"k8q���o"k����� q yCrts�uPy"!�sËu%y-v�y/.0% ;=7�-?> v T H "%6 HI:':;R9 # 917N)m>�),&�&NH [ ),&T-I; #E� )n;�-I7Z>O� ³"o"k8qË�1q�jbm32tp*43³8j yCrts�u%y"!�s�uPy/.?y/.?y/.?y�v T y-v�%W;=7�-?>AH " G @PHQ7 # G $ " v�`a #E� )C7 . $'&�) kD6 - " - #E� $ "%[ `

f�T ��Ð�ZU�{Ï�`��tÐ�� ~ � ) " 7N),+5),$EB�$ "1[ ;=7�-?> v S H�>�),&�&NH [ )V� ³"o"k8qË��k�±�³W2 yCrts�uPy"!�sËu%y�v S y-v�% k $E; #E� )# 9%@P:E)��'!�s�uPy�v�S*y-v�y$#&%x$'&�7�),+5-I7 6 ) 6k &*) "%6 >W),&�&�H [ )X�*) z�»)~��$P%|U~ | yCrts�u%y"!�s�uPy�v�S�y-v�y$#&% # -¨d3H "%6 HI:':@uHQ7 # $E),&Y$ " v ¿ - #E� )C7 . $'&�) k%6 - " - #E� $ "%[ `

Q $ [ 917�)�� �À�{�_�ZY ���1�3�\[��L����� �$] � � � � � ����� Y^Y �1�-� � �­�������`_L�b�����À�A���8�P�C� � �N�Á��� �t�3�<�L�4�8���1�Â�

! # $'&<& # 7NHI$ [�� # ;�-I7 . HQ7 6p# -�)54 # ) "P6p#E� ) î�z ) z�»)~ @%7�- # -D+5-?: # - #E� )n>O9%: # $E@u:E)5KM@PHQ7 # GF+CHI&*)I` a " )m&N$'>�@P:EG7N)C@P:'HI+5),& #E� )�� £�¤ ;R9 " + # $E- " HI:'$ # $E),&x\ G #E� )�� ¡�£t¤ ;=9 " + # $E- " HI:'$ # $'),&xH "%6 7�)C@u:'HI+5),& #E� )c@�-?$ "�# K # -QKM@]-?$ " # >�),&�&�H [ ),&\�Gs\P7�-?H 6 +CHI& # >�),&�&NH [ ),&C`�~8) 6 ) " - # ) #E� ) )54 # ) "P6 ) 6 @%7�- # -D+5-?:�\ G î�za� ) z�»)~�k H "%6n. ) � H�BI) #E� ) ;�-?:S:E- . $ "1[#E� )C-I7�),>(`

ΫÏ%Ð�^_Ñ�Ð�ÒÔÓ�Õ � × >_D=?He9<;�=�áÀácb 5LKCK�7�ÞNß 9A@AI8>Pãëß_HÁI�91IJBCI8Ç)î�zd� ) z�»¢~ K"=CB"7�H-="ÇïæíHE=E58Çð@Âñ�=�K�9<;�= �¢¡ :O���� @3D=E5�Çä�7�>_BJ9A@3I8>_5�Çï@A9Aæ½@<>µ9<;�=�� �xO:óUô�y��NMPOVyfe� ©:gHh é`i�j¡�£t¤ yfe� ©:gHh élk"éSi�j¡�£�¤ yfe� ©,mNn h nlm¡�£�¤ yfe� ©:gHh é m*n h nom¡�£�¤ ��éË;æF?H�@AD¨ÞõIJD=?ÇU5JÙ58@<>:K?9%5�D�5�ßU9A@bå�=5D8å8="HCK"58H�@1=�K�ãë5LK�K�7�Þ®@b>/Ù�=?H-5LK�@<>�ÙÈ

����p �Ý,�23��03���� ���t$rq��%0��%�s�U�����utU 2Ë,�U��01���~8)()54 # ) "%63#E� )wv*-?$ " # [ H # )�)CBhHI:S9%H # $E- " ;R9 " + # $E- " HI:S$ # G # - #E� )T>O9%: # $EKM@PHQ7 # G�+CHI&*)I` Q 9 " + # $'- " HI:'$ # G8� ¡yx$z p$S&& � - .�" $ "eQ $ [ 917�)|{�`x� � )�- " :'Gi+ � H "%[ ),& . $ #E� 7�),&*@]),+ #�# - #E� ) # . -QKM@uHQ7 # G(+CHI&*)UHQ7�) #E� H #_#E� )c7�),+5),$'@ # &_HQ7�)^&*) "�#

f �

Page 28: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

# -bHI:':]@PHQ7 # $'+C$'@PH # $ "1[ @PHQ7 # $'),& k H "%6(#E� H # HI:':]@PHQ7 # $E),& � H,BI) # -bH [ 7�)C)X;=-I7 #E� )U-I@]) " $ "1[ # - # HQoI)X@P:SHI+5)I`

W%XYZ�[�\L]b^UZ�`_a<]b\�c � ¡yx$z

� x$z n@%7�-D+5)C) 6 &YHI&�;�-?:':'- . & k 7N9 "%" $ "1[O. $ #E� @uHQ7 # $E),&Yvxw,yC|E|E|Ey�v } k H "%6 H 6 BI)C7N&�HQ7�G�d�`

f�� ^_ÒàÒ÷]b\��{Ï%`��JÐ:� ~ � ) " 7�),+5),$EBD$ "1[ ;=7�-?>�v S H >�),&�&�H [ ) � o"k��w�� q yCrJs�uPy"!�s�u%y-v�y$#&% k $E; v S�É v k#E� ) " 7N),+5-I7 6 �'!�sËu%y-v�y$#&% k &�) "%6 >W),&�&�H [ )� º ) z ),+ ¹U~ yCrts�u%y"!�s�u�% # -(HI:': @PHQ7 # $E),&�$ " v¡H "P6 d k H "%6$ [?" -I7�)_HI:':�;=917 #E� )C7 >W),&�&�H [ ),&V-I; #E� )_;�-I7N> � o"k����� q yCrts�u%y"!�s�uPy-v�y/.0%�H "P6 � ³�²8m � yCrts�u%y"!�s�uPy/.?y/.?y-v�y/.0%;=7�-?>�H " GT@PHQ7 # G<$ " v�`

f�� 7:`4aAX%`:\tÐ �{Ï�`��tÐ�� ~ � ) " 7�),+5),$EBD$ "1[ ;=7�-?> v S Hm>�),&�&�H [ ) � ³�²8m � yCrts�uPy"!�sËu%y"!�s�uB< y"!�s�u1w y-v�y>=?% k$';pv%S É v H "%6 \]- #E� �'!�s�u < y-v�y$# < % H "%6 �'!�s�u w y-v�y$# w % HQ7�)�7�),+5-I7 6 ) 6k/#E� ) " +5-?>�@u9 # )# ¶ k�± A z"CD � #�< y$#�w �Zk 7�),+5-I7 6 �'!�s�uPy-v�y$#/% k &*) "%6 >�),&�&�H [ )��*)�� |��t�Cº ) z )�+ ¹U~ yCrJsËu%y"!�s�u%y"!�s�u3<�y"!�s�u%w�yv S y-v�y>=?% # - HI:':<@PHQ7 # $E),&�$ " v H "%6 d k H "%6 $ [?" -I7�)/HI:':<;=9%7 #E� )C73>�),&�&�H [ ),&F-I; #E� )/;=-I7N>� o"k����� q yCrts�uPy"!�sËu%y-v�y/.0%bH "%6 � ³�²8m � yCrJs�uPy"!�s�u%y/.?y/.?y-v�y/.0%b;=7�-?> H " G¢@PHQ7 # G�$ " v ¿ - #E� )C7 . $'&*) k�6 -" - #E� $ "1[ `

f�T ��Ð�Z �{Ï�`��JÐ�� ~ � ) " 7�),+5),$'B�$ "%[ ;=7�-?> v%SXH�>�),&�&�H [ ) � k�±�³W2 yCrJs�uPy"!�s�u%y�v T % k $E;Uv%S É v kv T÷É v k H "P6�#E� ) # 91@P:E) �'!�s�uPy-v�y$#/%i$'& 7�),+5-I7 6 ) 6kx#E� ) " 7�),+5-I7 63# 91@u:E) � k�±�³W2jl³� yCrJsËu%y"!�s�u%y�v T %Z`~ � ) " H # 9%@P:E) � k�±�³W2�jl³� yCrts�uPy"!�sËu%y�v S y�v T %�$S&�7�),+5-I7 6 ) 6 ;=-I7�)CBI)C7�Gsv T�É v k�#E� ) " &*) "P6 >�),&N&�H [ )� PY|U~U| yCrJs�uPy"!�s�u%y$#&% # -iv S `

Q $ [ 9%7�)w{ ���{�_�ZY ���1�3�� [��L�t��� �?�L�l�_������� �R� _��8������3�b�L���Á��� �t�3�b�L�4�8�l�1���!�" ;RHI+ #Ckq. )(- " :EG " )C) 6 H�� . ),HQoI) " ) 6�� BI)C7N&�$E- " -I; #E� )<� ¡yx$z ;R9 " + # $'- " HI:'$ # G0;�-I7 [ ) " )C7NHI:�>O9%: # $LKM@PHQ7 # G

+5-?>W@P9 # H # $E- " `�� � ) . ),HQoI) " ) 6 BI)C7Z&�$E- "kI6 ) " - # ) 6 \�Gc���4¡yx$z k�� HI& #E� )_H 6%6 $ # $'- " HI:?+5- " & # 7ZHI$ "�#{#E� H # - " :EG � ac�H "P6 ����� -I@])C7NH # $E- " &cHQ7�)�HI:':'- . ) 6 $ "s#E� )O)CBQHI:'9%H # $E- " @ � HI&�)I` !�# $'&�-I\�BD$E-?9%& #E� H # &�$ " +5)�� � ac��y��������W$'&�H+5-?>W@P:E) # )x&*) # -I; j -�-?:E),H " -I@�)C7ZH # $E- " & k ���4¡yx$z $S&@]- . )C7�;=9P:I) " -?9 [��^# -�7�),HI:'$EJC)�H " G^>�9P: # $LKM@PHQ7 # G�;R9 " + # $'- " HI:'$ # GI`

��&<$ "3#E� ) #l. -QKM@PHQ7 # Gp+CHI&*) kY. )mHI:S&*- " )C) 6 H " )54 # ) " &�$E- " -I; #E� )s� ¡ �O���� ;R9 " + # $'- " HI:'$ # G k_6 ) " - # ) 6 \�G���¡ :O���� k�#E� H # @])C7�;=-I7N>W&�� � w�� KM-I\P:S$EB�$'-?9%& # 7NH " &�;�)C7_H "%6 @P7�-�BI),&�7�),:'H # $'- " &xHI>�- "%[ ;=-?917x\P$ # &C`�~8)�- " :'G & # H # ) #E� );=-?:':E- . $ "1[ #E� )C-I7N),>�H "%6 -?>W$ #�#E� ) 6 ) # HI$':S&C`

ΫÏ%Ð�^_Ñ�Ð�ÒÔÓ�Õ �"! ;�="H-=½=Cì�@bK�9<Kx58>�=$# B"@1=?>49Pß_H-I�91IJB�I8Ç�9<;/5�9%K"=CB"7�HE=?ÇðæRHE=E58Çð@�ñt=�KÀ9<;�= ���¡ :O���� ä�7�>_BJ9A@3I8>_5�Çï@A9AæR@<>­9<;�=�¢¡ :O���� éË;æF?H�@ADµÞõIJD="ÇY5JÙ�58@b>PK�9ùÞõ58Çï@3B"@3I87/KCãë5�D�5�ßU9A@<å8=ª5�D8å�=?H�K?58H�@3=?K?È

��)54 #Ck]. )O\%7N$E)%PGT&�oI) # + � Hb@%7�- # -D+5-?: î�zd�'&�� ) #E� H # &*),+C917�),:EG(7N),HI:'$EJC),&�� �4¡yx$z 8$ "m#E� )U���¡ :O���� K � G \%7Z$ 6>W- 6 ),:M`�� � $'&q@%7�- # -D+5-?:�$'&�),&�&*) " # $'HI:':EGbH�>�9%: # $LKM@PHQ7 # GW)54 # ) " &�$E- "T# - #E� ) î�z(&�� )b@%7�- # -D+5-?:�` !�"�î�za�)&�� ) k H \u$ ##c$S& " - . & � HQ7�) 6 HI>�- "1[ HI:':@PHQ7 # $S+C$E@PH # $ "1[ @PHQ7 # $E),& � @uHQ7 # G(v S � HI&Y\P$ # # S &�9P+ ��#E� H #'* }S,+ w # S ¶ #3>�- 6if ` !�"#E� )�;=-?:':E- . $ "1[U6 ),&�+57Z$E@ # $E- "k�. )�-?>W$ # &*-?>�) 6 ) # HI$':'&x$ "W#E� )�@%7�- # -D+5-?:u&�9P+ � HI& #E� )�;�-I7Z>WH # -I; #E� )�>�),&�&�H [ ),&_H "%6#E� )U$ 6 ) " # $E¤%)C7N&_-I; #E� )U\P$ # &C`�� � ),&*) 6 ) # HI$':'&�& � -?9P: 6 \])c+C:'),HQ7�;=7�-?> #E� )U+5- " # )54 # `

� ^_ÒàÒ÷]b\ �YÏ%`��JÐ:� Q -I7�@PHQ7 # G(v S # -i+5-?>W>W$ #Y# -bHi\P$ # # k $ #Y[ ) " )C7NH # ),&�7ZH "%6 -?>�\P$ # & #�w,y$#ZzhyC|E|E|Ey$#Z}�-w/.0 �21Dy e �H "%6 #Z} 0 #43r#�w5376663 #Z}8-wC`�� � ) " v S +5-?>i>W$ # & # -�# S #E� 7N-?9 [�� � �¡ �O���� k &*) "%6 &q\u$ # & # T # -�@uHQ7 # G<v T;�-I7�HI:':�9;:¶ s�`x� � ) " ),HI+ � v T +5-?>W>i$ # & # - # T #E� 7N-?9 [��T#E� )c���¡ :O���� H "%6 -I@]) " &�$ #_# -iv S $'>W>�) 6 $SH # ),:EGI`

� 7:`4aAX%`:\tÐ�{Ï%`��JÐ:� ��&�&N9%>�) #E� ) #l. -b\P$ # & # -i\])X+5-?>W@P9 # ) 6 HQ7N)=<TH "%6 # k H "%6 v%S � -?: 6 &Y\P$ # &><�S k #�S�HI& #E� ),$'7& � HQ7�),&,`c��H # 917NHI:S:EG . ) � H�BI)?< ¶@* < S >�- 6Wf H "P6 # ¶A* # S >�- 6if `c~8)WHI&�&�9%>�) #E� )O7�),&�9%: # \P$ # $'&Z!H "%6 ),HI+ � @PHQ7 # Gb& � -?9%: 6µ� -?: 6 HO& � HQ7�)c! S H #_#E� )�) "P6 -I; #E� $'&q@ � HI&*)I`V~8)U+5- " &N$ 6 )C7 #l. - +CHI&�),&YHI+C+5-I7 6 $ "%[# - #E� )U-I@])C7NH # $'- " @])C7�;�-I7Z>�) 6 `

f {

Page 29: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

� T�� � �{-�+5-?>�@u9 # ) #E� ) � ac� -I;^\P$ # <�H "%6 # k ),HI+ � @PHQ7 # GF&�$'>�@u:EG¢+5-?>�@u9 # ),&�! S ¶ < S 3 # S `/��->�),&�&�H [ ),&�HQ7�) " )C) 6 ) 6 `����� � �{-T+5-?>W@P9 # ) #E� )������ -I;x\P$ # <sH "%6 # k. )O;=-?:':E- . #E� )�HQ@%@%7�-?HI+ � $ "�d f § k�e,fQg `�a�\P&*)C7�BI) #E� H #�����¥$'& #E� )U>�9%: # $E@P:'$'+CH # $'- " >W- 6 9P:E- f�k H "%6(. ) � H,BI) #E� )^;=-?:':E- . $ "1[ )JÊ 9PHI:'$ # GI`� }

S,+ w < S� 6 � }S,+ w # S� ¶ r 6

}S + w < S 6 # S�

w�� S���T �%} � < S� < T � 6 � # S� # T � >�- 6Wf� &*)C) d f § g ;=-I7 #E� )dv�9%& # $E¤u+CH # $E- " -I; #E� $S&{)JÊ�9%HI:'$ # G � `�� � )C7�)C;=-I7�) k ),HI+ � @PHQ7 # G v S +CH " +5-?>W@P9 # )�r'6$< S 6`# S\ G�$ # &�),:E; k H "%6 ),HI+ � @PHI$'7xv S H "P6 v T +CH " v�-?$ " # :EG +5-?>�@P9 # ) � < S� < T � 6 � # S� # T � HI&q$ "�#E� ) #l. -QKM@PHQ7 # G+CHI&*) k \�GT$ " BI-IoD$ "1[ >O9%: # $'@P:E) # 7NH " &*;=)C7Y@ � HI&�),&�-I; #E� )^���¡ �O���� ;R9 " + # $'- " HI:'$ # GI`

T ��Ð�Z �{Ï%`��JÐ:� � -U-I@]) " HX\P$ # # k & � HQ7�) 6 HI& # ¶ * }S,+ w #�S k�# -X@PHQ7 # G�v%S k )CBI)C7�G�v TY-I@]) " &�$ # &�& � HQ7�)Z#iT #E� 7�-?9 [��� �¡ �O���� `q� � ) " v S &N9%>W&�91@sHI:S: #E� )X& � HQ7N),& # -W-I\ # HI$ " #�`��� ^_Ñt\B� !�" +CHI&*)�H " GT@PHQ7 # G(HQ\�-I7 # &YH "%6�� -I7 6 )CB�$'H # ),&�;�7N-?> #E� )^@%7�- # -D+5-?: k HI:S:]@PHQ7 # $E),&YHQ\]-I7 #�#E� )U@%7�- # -D+5-?:�`

ΫÏ%Ð�^_Ñ�Ð�ÒÔÓ�Õ���� H-I�91IJB�I8Ç.î�za�)&�� ) K"=CB"7�H-="ÇïæöH-=C58Çï@�ñt=?K¢ä�7�>_Bt9A@AI�>_58Çð@<9Aæ � ¡yx$z @<>�9<;�= � �¡ �O���� éË;æF?H�@ADöÞõIJD="Ç5tÙ58@<>:K�9ùÞõ58Çð@ABJ@AI�7�K�ãù5�D�5�ßU9A@bå�=�5�D8å�=?H�K?58H�@1=�K?È

� � )�@%7�-�-I;U$'&iBI)C7�GF&�$'>W$S:'HQ7 # - #E� H # -I;X� � )C-I7�),> � ` f ` ��)54 #Ck ;=-I7<H " G >O9%: # $LKM@uHQ7 # G�;R9 " + # $E- " HI:S$ # G�� k�. )+5- " & # 7N9%+ # Hn@%7�- # -D+5-?: ê � #E� H # &�),+C917�),:EG07N),HI:'$EJC),&�� $ "¢#E� )b� ¡yxEz %K � G \%7Z$ 6 >�- 6 ),:�`0� � )<+5- " & # 7N9%+ # $E- " $S&HI:S>�-?& # $ 6 ) "�# $'+CHI: # - #E� ) #l. -QKM@PHQ7 # Gs+CHI&*) k )54D+5)C@ #�#E� H # &�$ " +5) . )�HI&�&�9%>�) #E� )X+C$E7N+C9P$ # +5-?>�@P9 # $ "%[ ��+5- " &�$'& # &-I;q����� H "%6n� ac� [ H # ),& k $ " & # ),H 6 -I;x������� [ H # ),& k#E� ) [ H # )5KM\�G K [ H # )W)CBhHI:S9%H # $E- "n. $':': $ " BI-IoI) #E� )�� ¡yx$z ;R9 " + # $'- " HI:'$ # G . $ #E�(6 $���)C7N) "�# ) " +5- 6 $ "%[ &_-I;{;R9 " + # $E- " &C`ΫÏ%Ð�^_Ñ�Ð�ÒÔÓ�Õ�� � =J9 � F�=Ä5«Þ®7/Ç�9A@<é�ß�5�H�9Aæª5�D5�ßU9A@<å8=?Çðæ"!.=?Ç<Çïéðä"I8H�Þª=CD)ä�7/>UBJ9A@3I8>_58Çð@<9Aæ�È ! ;�=">õê � KJ=EB"7�HE=?Çðæ®H-=C58Çï@�ñt=?K� @b>¾9<;�= � ¡yx$z é�;�æ�F"H�@3DµÞõIJD=?Çïã.@<>�9<;�=¢ß_HE=�KJ=?>_B�=�IÁäÄÞõ58Çð@ABJ@AI87/K«5D�5�ßU9A@<å8=�5D8å8="HCK"58H�@1=�K"È

#%$'& �)( �+*-,/.0*x�1*x ��32� � )UHI9 #E� -I7N& #E� H " o #E� )UH " - " G�>�-?9P&Y7�)CBD$E) . )C7N&�;=-I7�� âÃ54 1 � ;=-I7 #E� )�>WH " G<BQHI:'9%HQ\P:E)^+5-?>W>W) "�# &C`6 *879*��:*x $ *-2;=<?>A@CBEDGF9HJI?KL FNMPO/DQBSR'TVUXWZY[FNMVM�B�\^]N_`_=I�abI`]NMVced�HJf+fCF L+L?g Y g _�FhUJ]NHia-F9MPOjdkFNI=_=clamUJ]Nn�aXI=o�MPFhU g HXfpa Liq fZY[fZa-rsI=U q ] g U)UXHJf+fZaZBPtlMu8v9wyx9zS{i|?}�~�z��3���J�S�l�h�=�J�9�h�h�����X��{?���J�E��� �V�Z� nVn�B �����y�����9� � <��N����B; �y>A�0B�D^f�Fy�NfZHZB� V] g MPOVF9UXI`]NMPaG]Nd3¡�f L?g HJf�tlM�UXf+HiF L UXI`�Nf0\^]NY[n g UXI`MVoPB�tlM u)vhwhxhzS{i|?}¢~�z��3���J�S�l�h�=�J�N�£�¤��¥5¦�§C¨ª© � ��«h� nPn�B¬ ���Z� ¬ �V< � <Z���V<�B; ¬ >A�0B�D�fZFy��f+H�B�¡�f L?g HXfQ­ g _=UXI`nPF9HXUm®�R'HJ]9UJ] L ]N_�aªFNMPOj¯SfZHX]Nc�°�MP]hrs_=f�O�oNf-RªHX]�]9d�¡�®�abUXfZYAa�±�]N_`f+HiFhUJI=MPopF0 �F g _=Um®�­/I`MV]NHJI=Um®NBtlM³² �9����zSx9�´�mµA�3���J�S�l�h�`�X�N� �S¶e�N· � nVn�BS�N¸��E<��N� � <��N�P<NB; �9>A­³B�D�f+_`_`FNHXf5F9MPO¹¡EB�­/I L FN_=I�B0@8]NM�c�I=M�UJf+HiF L UJI=��f0]�ºV_=I`��I=] g a)UXHiF9MPabd�f+H¢FNMPO»FNnVnV_`I L FhUJI=]�MPa+B u)vhwhxhzS{i|?}5~�z¼�½���J�E���9�=�J�9�N¾��¥5¦E§C¨ª©¿� ÀN�9� nVn�B�¸9������¸N¸�� � �N]�_ g Y[fQ� ¬ ¸0]Nd:Á |i{?�����X|8Âp�h�l|�}¢~�zÃ�'�9Ä-�E����|+�¢ÅE{+~k|+zS{i|?� ¡�nVHJI=MVo�f+HXc�Æ:fZHX_�F9o � <��N���VB; ¸y>A�0BED�]NMVf q B�± q f0O�f L I`aXI`]NM��)IÈÇAf?clÉ-fZ_=_`YAF9M�nPHX]�ºV_=fZY"BPtlM §��J�Z{i|i|ivh~�z��9}5�bµC��ÊV|�¨EÊ�~��Xv�u)� ���9��~���Ê�Äp~k{¢ÂC��Ä[Ëi|+�j¨EÊV|i�h���Å��hÄ8�V�h}�~���Ä ¶kÌ�@Q\G¡»<+��� ¬ · � nVn�BV���y��Í ¬ � <Z���N�VB; Íh>A �B�D�] g OV]9UZB½Î:Ç L I`f+M�U0RªHX]�]9dka¢U q F9UpF»\^]NY[Y[I=UbUXf�O³@ g Ypº�fZH5Ì´I=f�a¢I`M�F9M¹tlM�UXfZHX�hF9_�B´tlM �5ÏE¥p©���¥5¦�§C¨¼Ð�Ñ�ÑNÑh� nPn�B� ¬ <������N� � �9���N�PB

f�V

Page 30: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

; ��>A �B�D�] g OV]9U^FNMPO � B�±´HiF9]�HVKfNBhÎ:Ç L I`f+M�U^R g ºV_`I L _=®pÆ:f+HJIÈTSF9ºV_`f)¡�f L HJf?U�¡ q F9HJI=MVo0¡ Liq f+Y[fZaªrsIÈU q  PF�amU^]NHª�)f+_�Fy®Nf�O��-f L ]h��cf+HJ®NB�tlM�� zZµ?�h��ÄAxh��~k�hzjxhzSvA�:�9ÄpÄ5��zP~k{ix9�e~k�9zAÅE|i{+����~��e����Å�|J{i�9zSv � zP�l|?��zExh��~k�hzSx9�E�:�9zZµ?|+�X|+zS{i|�� � � � �½ÅG� �N�9� nVn�Bh�����S<�����B; �h> � B�\�F9Y[f+MVI�a Liq FNMPO[­³B�­�I Liq fZ_`aZB�¡�f+nPFNHJFNºVI=_`I=Um®0F9MPO[f+Ç L I`f+M L ®Cd�]NHªoNf+MPf+HJI L o�HX] g n5aXI`oNMPF9U g HJf-a Liq f+Y[fZaZB9tlM u8v9wyx9zS{i|?}~�z"�½���J�E���9�=�J�9�N¾ ��¥5¦E§0¨'© � �N�N� �N]�_ g Y5f8<ZÍNÍ�ÍG]9dSÁ |i{?�����X|3Â0�9��|?}ª~�z/�'�hÄ8�S���l|?�^ÅE{+~k|+zS{i|?� n�F9oNf�a��P<Z�h��� ¬ � � ¡�nVHJI`MVoNfZHbcÆ:f+HJ_`FNo � <Z���N�PB; �h>��¢B3\�FNMVf?UXUXI�B�¡�f L?g HJIÈUm®�F9MPO�\^]�Y[n�]�abI=UXI`]NM¹]9ds­ g _=UXI`nPFNHbUm®Ã\^HX®�n�UJ]No�HJFNn q I L RªHX]NUX] L ]�_`aZB�tlM ² �9����zSx9�^�mµ¤�½���J�E���9�=�J�9�< ¬ ¶m<�· � nVn�B�<Z� ¬ ���9��� � �9�N���VB;È<��h>��¢B�\�FNMVf?UXUXI�B8MVI=��f+HiaXFN_=_`® L ]NY[n�]�aXFNºV_`f-abf L?g HJIÈUm®��� MVfZr�nPFNHJF�O�I`oNY d�]�H L HJ®�n�UX]�oNHiF9n q I L nVHJ]9UJ] L ]N_�a+BNtlM�� ÐNzEv��G©��½Å��nVn�B�< ¬ Íh�S<+��¸ � �9�N�P<NB;È<�<?>��¢B´\�F9MVf+UbUJI'F9MPO»­³B� �I�a Liq _=I`M�B�-MPI=��f+HiaXFN_=_`® L ]NY[nS]�aJF9ºV_`f L ]NY[Y[IÈUJY[f+M�UJaZB�tlM ��¥5¦E§0¨'© Ð�Ñ�ÑN« ¶kÌ�@Q\G¡���< ¬ ��· � nPn�B<Z�h����� � �N�N�P<NB;È<y�y>��¢Bª\�F9MVf+UbUJI ��� B:Ì�I`MPO�f+_`_ � �¢B��QamUJHX]h��a���®»F9M�O� CB'¡�F q F9I�B�-MVI`�NfZHJaJF9_`_=® L ]NY[n�]�aXFNºV_`fAUmr^]Nc�n�F9HXUm® L ]NY[n g UiFhUXI`]NM´B½tlMÅ:¨ª©�� � ÑhÐN� nVn�BV���9�9��¸9� ¬ � �9�N����B�  g _=_���f+HiabI`]NM I=M �3���J�S�l�h�`�X�N�j|i§���~�zP�'u)�X{iÊ�~�wh|?� �sf+nS]NHXU)�9�������<+���VB;È< ¬ >��¢B´\�F9MVf+UbUJI'F9MPO»±�B��8F9ºVI`M�B�-MVI`�NfZHJaJF9_'\^]NY[nS]�aXIÈUJI=]�M¤rsI=U q � ]�I=M�UC¡�UJF9UXf5tlM �3���J�S�l�h�`�X�N�¤|J§G��~�zP�Gu)�J{JÊ�~�wy|+� �sfZn�]�HbU�9������9�9��� ��� �!�#"%$'&!&#()" *,+.- �0/�+�1324*�/'5)*!6 &P� �9�����VB;È<Z�9>��¢B�\^HiF9Y[f+H�B'­/]�O g _�F9H��)fZaXI`oNM ]9dC¡�f L+g HJfj®�f?U£RªHJF L UXI L FN_)\^HX®�n�UJ]No�HJFNn q I L RªHX]NUX] L ]�_`aZB:R q B �0B^± q fZaXI`aZB�\87 t[F9MPO-MVI`�NfZHJaXI=Um®A]9d� -YAamUJf+HiOVF9Y � <��N����B;È<y¸y>��¢By\^HiF9Y[fZH � t�BZ�QF9Y[o:9FNHJO � F9M�O � BZ@8I=fZ_`aXf+M´BZ­ g _=UXI`nPFNHbUm®C\^]NY[n g UiFhUJI=]�M)d�HJ]NY ± q HJfZa q ]N_�O�É-]�Y5]�Y[]NHJn q I L Î'M L HX®�n�UJI=]�MtlM u)vhwhxhzS{i|?}^~�z/�3���J�S�l�h�`�X�N�h�y�����J���3���J�S�¢� ÑN«9� �N]�_ g Y5f��9�9��¸�]9dSÁ |i{?�����X|:Â0�9��|?}^~�z"�'�hÄ8�S���l|+�^Å�{?~k|+zS{i|+� nVn�Bh�9���y� ¬ �N� �¡�nVHJI=MPoNf+HXc�Æ:f+HJ_`FNo � �9�N�P<NB;È<�Íh>��¢B�\^HiF9Y[f+H � t�B9�QF9Y[o:9FNHJO � F9MPO5DQBN¡ Liq ]�f+MVYAF)�Nf+HiaZByRªHX]�]Ndka�]Nd�nPFNHbUJI`FN_ ��MV]hrs_=f�O�oNfGF9M�OpaXI=Y[nV_`I=TPfZO[O�f�abI`oNM5]9dErsIÈUJMVfZaJaq I�O�I=MPo5nPHX]NUX] L ]�_`aZB�tlM u)vhwhxhzE{i|�}¢~�z��3���J�E���9�=�X�N�h�P��¥5¦�§C¨ª©%� � ��¶eÌ´@�\G¡ � ¬ ��· � nPFNoNfZaQ<h�y�h�E<Z��� � <Z�N�N�PB;È<h��>��¢BE\^HiF9Y[fZH-F9MPOjÆ0BE¡ q ] g n�BS¡�I=o�MPFhU g HXfCa Liq fZY5f¢º�FNaXfZOj]NMjU q f0amUJHX]�MVo��8¡ F�aXa g Y[n�UJI=]�M�BVtlM up��; ¨��JxhzV}+x�{?��~k�hzP}�hz � zZµ?�9��Ä[x9�e~k�9z�x9zSv�Å��y}���|+Ä Å�|i{?����~����=<�up��; ¨ � Å�Å��Q�,> ¬ ¶ ¬ ·?�=<�ÍV<+cJ<���¸ � �9���N�VB;È<��h>£\)B�\^HPKfZn�f�F g B�Æ:f+HJIÈT�FNºV_`fpO�I�a L _=]�a g HXfp]9dªaXf L HJf?Uia)FNMPO�FNnVnV_`I L FhUXI`]NM�a+B�tlM �����X��{+���J�S�£� À��9� Ì´@�\G¡/� ¬ � � nVn�<Z�V<¢��<Z�P< �<Z���N�VB;È<��h>£\)B�\^HVKf+nSfZF g � � B��hF9M¹O�f�@QHJF�Fhd � FNMPOA CB�±½F9nVn�B�\^]NY[Y[I=UbUXf�O¤]�ºV_`I=��I`] g aQUJHJFNMPamd�fZH¢F9MPO»nVHJI=�hFhUJfpY g _=UXI=c�nPFNHbUm® L ]NY5cn g UiFhUXI`]NM´B)tlM u)vhwhxhzE{i|�}5~�z �3���J�E���9�=�X�N�N¾ ��¥5¦E§C¨ª© � �B9� �N]N_ g Y[fp��Í ¬ ]Nd�Á |i{+�e���J|QÂp�h�l|�}p~�z �:�9Ä-�E����|+�CÅ�{?~k|+zS{i|+�nPF9o�fZaQ<N<��y�E<�� ¬ B�¡�nVHXI`MVo�f+HXc�Æ:fZHX_�F9o � ���Z� ¬ <C g oPB�<��N��¸�B; �N�h>[t�B��QF9Y[o�9F9HiOCF9MPO � Bh@-I`f+_�aXf+M�ByR3f+HXd�f L U q I�O�I`MVoQFNMPO¢nSf+HXd�f L U½ºVI`MPO�I`MVo g MVI`�NfZHJaJF9_`_=® L ]�Y[n�]�aXFNºV_=f L ]NY[Y[I=UXY[f+M�U:a Liq fZY[fZarsIÈU q�L ]�MPabUJF9M�U0fED�nPF9MPaXI`]NM�dkF L UX]NH�B´tlM ��¥5¦�§C¨ª© Ð�Ñ�ÑhÐ ¶kÌ´@�\G¡Ã�h������· � nVn�B:¸9�V<?��¸N�NÍ � �9�N����BF _`F9UXfZHC�NfZHJaXI=]�M¹I=M�3���J�S�l�h�=�J�9�"|J§G��~�zP�3u)�J{JÊ�~�wy|+� �sfZn�]�HbU8�N�N�V<4�h���V<NB � �#�#"%$�&!&()" *�+.-!�0/�+�132.*0/G5)*!6H&P� �9�N�P<NB; �V<?>[t�B��QF9Y[o:9FNHJO � FNMPO � B½@-I`f+_�abfZM�B�-MVI`�NfZHJaJF9_`_=®¹\^]NY[n�]�aXFNºV_`f£Î:Ç L I`f+M�Up­ g _ÈUJI=n�F9HXUm®³\^]�Y5n g UJF9UXI`]NM¹d�HJ]NY ± q HXf�a q ]N_�OÉ-]�Y5]�Y[]NHJn q I L ΪM L HJ®�n�UXI`]NM�B�tlM u)vhwhxhzE{J|?}¢~�z��3���J�S�l�h�`�X�N�h�P��¥5¦E§0¨'© � Ñ�I9� �9��� ¬ B; ���y>�¡EB�Î'��f+M � �pBH@Q]N_�O�HJf+I Liq � F9MPOJ CB�_`f+Y[n�fZ_ � K�-FNMPO�]�Y5I`W+f�O£RªHX]NUX] L ]�_Vd�]NH-¡�I`oNMPI=MVo�\^]NM�UJHJF L UJaZB�tlM �'�hÄ5Ä5��zP~k{ixh��~k�hzP}�mµ���ÊV|�up��;"� �9�H� Í ¬ ��clÍ9���[¶m<��N��¸�·�B; � ¬ >AÎ-BP  g4L I�aXF)��I½F9MPO"±�B��C�yFNY[]9UX]�B�¡�UiFhUXI�abUXI L F9_´W+f+HJ]���MV]hrs_`fZO�o�fQnPHX]NUX] L ]�_`a�UJ]�nVHJ]h�Nf�Y[]�O g _`FNH-nS]N_`®�MP]NY[I`FN_�HJf+_�FhUXI`]NM�a+BtlM u8v9wyx9zS{i|�}¢~�zÃ�3���J�E���9�=�X�N�h�P��¥5¦�§C¨ª©%� �V�Z� nVn�B´<ZÍhc ¬ � � <Z�N����B; �9�9> � BH@�FNHJFy®[F9MPO R3B�­�F L °�f+MVWZI=f�B�\^]�M L?g HJHJf+M�UM�)ºV_`I`��I`] g a^±�HJFNMPabd�f+H�B�tlM �G©��½Å�Ð�Ñ�ÑNÑh� nVn�B ¬ <Z�9c ¬ �9� � �sf�O�]NM�O�]5D�f�F Liq �\0 � @8]h�NfZY0ºSf+H-�9���N�VB

� 1

Page 31: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

; ��¸y> � B�@�F9HiFy® � R3BS­¤F L °�fZMVW+I`f0FNMPO�°[B � F9MVo�BE¡�UJHXfZMVo9U q f+MPI=MVo/¯Sf+HJ]9cl°�MV]hrs_`fZO�o�f¢R'HJ]9UX] L ]N_�a g aXI=MVo"¡�I`oNMPF9U g HJfZaZB�tlM u8v��wyx9zS{i|?}ª~�z"�½���J�E���9�=�J�9�h�h�����X��{?���J�E��Ð�ÑNÑ�IN� 7�FNHJaJFyr � R3]�_`FNMPO � Ì´@�\G¡¢�9Í�¸NÍ � nVn�B`<y�N�ycJ<��9� � �9��� ¬ B?  g _`_���f+HiabI`]NM�Fy�hF9I`_`FNºV_=fd�HX]�Y �3���J�S�l�h�=�J�N�j|i§���~�zP�'u)�X{iÊ�~�wh|?� �-f+nS]NHXU)�9�N� ¬ �9� ¬ � ��� �!�#"0$�&!&#()"!*�+.- �0/ +41324*�/'5)*#6H& ������� & ����� � �N�N� ¬ B; �NÍh> �pB!@Q]�_`OVHXfZI Liq B�¡�f L?g HJfQ­ g _ÈUJIÈclR:F9HXUm®�\^]NY[n g UiFhUXI`]NM»¶ 7�]NH ��I`MVo[�8HiFhd�U � Æ:f+HiabI`]NM�<�B ��· � ­�FNH Liq �N�N�N�PB 8�yFNI=_�F9ºP_=f)d�HX]�Y� �#�#"%$�&!&��%/��+��� 5�� /�3(H+�� 1)-#-%/G1H2�/ +��!&�� 5� (� H&�"!"%/ � ����� B; ����> �pB�@Q]�_`OVHXfZI Liq � ¡EBE­/I L F9_`I3F9MPO 0B 7 I`o�OVf+Hiab]�M�B�É8]hr UX]£nP_`Fy®"F9M�®"Y[f+M�UJFN_�o�F9Y[fC]NHQF L ]NY[nV_`f?UXfZMVfZaJa8U q fZ]NHJf+Y d�]NHnVHJ]9UX] L ]N_�a:rsIÈU q�q ]�MVfZabUªYAF L ]�HXI=Um®NB�tlM «9�N��Ê[up��;%ÅS�hÄ-�P�y}�~���Ä �hz"��ÊV|�¨EÊV|i�9���A�bµ0�'�9Ä-�E���e~�z��9� nVn�B���<��y�V�N�9� � <Z������B; �N�h>�¡EB�@Q]�_`OVr�F�aXaXf+HCFNMPO³Ì'B½Ì�f+��I`M�B� �F9I`H L ]NY[n g UiFhUXI`]NM³]NdsoNf+MPf+HiF9_:d g M L UXI`]NM�aCI`M³nVHXf�abfZM L f£]9dsI=Y[Y[]NHiF9_ªY[F L ]NHJIÈUm® � tlMu8v9wyx9zS{i|?}�~�z��3���J�S�l�h�=�J�9�����¥5¦�§C¨ª© � ��Ñh� nVn�BS����cl� ¬ � ¡�nPHXI`MVoNfZHbc�Æ:f+HJ_�F9o � <Z���V<�B; �N�h>�¡EB:@Q]�_`O�rGFNaJaXf+H¢F9MPO � B½Ì�I`MPO�fZ_=_�B�¡�f L?g HJf£\^]NY[n g UiFhUXI`]NM 7 I=U q ] g U 8oNHJf+f+Y[fZM�U�B�tlM <ZÍhU q�� � Ž�3� ¡�nVHJI=MPoNf+HXc�Æ:f+HJ_`FNo¶kÌ�@Q\G¡"�N¸N�N��· � nPF9o�fZaQ<y�yc ¬ � � �9������B; ¬ �h>�¡EB�@Q]�_`OVr�F�aXaXf+HCFNMPO³Ì'B½Ì�f+��I`M�B� �F9I`H L ]NY[n g UiFhUXI`]NM³]NdsoNf+MPf+HiF9_:d g M L UXI`]NM�aCI`M³nVHXf�abfZM L f£]9dsI=Y[Y[]NHiF9_ªY[F L ]NHJIÈUm® � tlMu8v9wyx9zS{i|?}�~�z��3���J�S�l�h�=�J�9�h����¥5¦E§C¨ª© � ��Ñ9� nPn�BS�N�yc�� ¬ � ¡�nVHJI=MPoNf+HXc�Æ:f+HJ_`FNo � <Z�N�P<NB; ¬ <?>AR3Bs­�F L °�f+MVWZI=f�F9MPO ­³BM�sf+I=UXfZHZBs±Gr�]9clR:F9HXUm® @Qf+MPf+HiFhUXI`]NM ]9dp��¡ ¡�I`oNMPF9U g HJfZaZB�tlM u)vhwhxhzS{i|?}»~�z �3���J�S�l�h�=�J�N�h���¥5¦E§C¨ª©¿� ÑN«h� nVn¹< ¬ �Z�S<y¸h� � �9���V<�B; ¬ �y>�¡EB´­/I L FN_=I^F9MPO¹R3BF�s]�o�FyrGFy®NB�¡�f L+g HJf \^]NY[n g UiFhUXI`]NM ¶ -ºPabUXHiF L Ui·�B´tlM u8v9wyx9zS{i|�}A~�z �½���J�E���9�=�J�9�»����¥5¦E§C¨ª© � ��«9�nVn�B ¬ ���y�����N� � �9�N�P<NB; ¬�¬ >A±�B%�QaJF9Y[]9UJ]³F9MPO ¡EB0 Liq I`®�FNY[FPB� �MVfZr¿n g ºV_`I L c ��f+® L HJ®�nVUX]�aX®�amUJf+Y FNa�aXf L+g HXf�F�a[dkF L UX]�HXI`MVoPB:tlM u)vhwhxhzE{i|�}�~�z�3���J�S�l�h�=�J�9�h�h�����X��{?���J�E��� �NÀN� nVn�B ¬ ���y� ¬ <Z� � <��N���VB; ¬ �9>AR3B�R:FNI=_`_`I=fZHZB�R g ºV_`I L c �Nf+® L HJ®�n�UX]�ab®�abUXfZY[a�ºPF�abf�O¢]NM L ]NY[nS]�aXIÈUJf^O�fZoNHJf+f'HJfZaXI`O g f L _`F�aXaXfZaZBZtlM u8v9wyx9zS{i|�}G~�z/�3���J�S�l�h�=�J�N�h������X��{?���J�E�5� ���9� nVn�B �N� ¬ ��� ¬ � � <Z���N�VB; ¬ ¸y>A±�BVR3B�R3fZO�fZHJaXf+M�B�@8]NM�c�tlM�UXf+HiF L UXI`�NfQF9MPO tlM�d�]NHJYAFhUJI=]�M�c�± q f+]NHJf?UJI L ¡�f L+g HJf)Æ:fZHXI=T�F9ºP_=f¢¡�f L HXf+U8¡ q F9HJI`MVoPB�tlM ��¥5¦�§C¨ª©«9����«h� nVn�B�<y�9�h�S<+��� � <Z���V<NB; ¬ Íh>ADQB�R'TVUXWZY[FNMVM � ­³BV¡ Liq�g M�UXfZH � F9MPO£­³B7¹F9I�O�MVfZHZB�R'HJ]h�hF9ºV_`®5¡�f L?g HXf¢\^f+HXUXI=TPfZO�­¤F9I`_eB�tlMJ��� ;¿¥)|�}+|ixh�J{JÊ[¥)|��P�h���´¥��I9Ð�Ñ�� <��p�)INÐ#B)I�>jÑ9Ð �E« � ��ÑNÑh� tmDG­ �sfZaXfZFNH Liq �8I`��I`aXI`]NM � ¯"!g HXI Liq � g o g abU)�9���N�VB; ¬ ��>ADQBNR:TVUJW+YAF9MPM[FNMPO[­³B)7¹F9I�O�MVfZHZB�\^]NY[n�]�abI=UXI`]NM[F9M�OpI`M�UXfZoNHJIÈUm®pnVHXf�abfZHX�hF9UXI`]NM0]Nd�abf L?g HJfGHXf�F L UJI=��fsab®�abUXf+YAaZB9tlM up��;�'�hzZµ?|+�X|+zS{i|5�hz��'�9Ä-�E����|+�0xhzEv��'�hÄ5Ä5��zP~k{ixh��~k�hzP}QÅ�|i{?����~���� <i���½Å�Ð�ÑNÑ�Ñ >h� nPn�B��h��¸��V�N¸h� � �N�N�N�PB; ¬ �h>ADQB�R:TPUXW+YAFNMVMAF9MPOA­³B7�FNI`OVMVf+H�B ­/]�O�f+_Pd�]NH� )ab®�M Liq HJ]NMP] g a��sf�F L UXI`�Nf8¡�®�abUXf+YAa^F9MPO[I=UJa8 -nVnV_`I L FhUJI=]�M[UX]p¡�f L?g HXf­/fZaJaJF9oNf�±�HJFNMPabY[I�aXaXI`]NM�B�tlM � �'�ª� Å��hÄ8�V�h}�~���Ä¿�9z�ÅE|i{+����~��e�£x9zSv5§���~�wyx�{?�h� nVn�B´<Z�9�9���N�N� � �9���V<NB; ¬ �h>AR3BH�s]No�Fyr�Fy®�B�± q f¢HX] g MPO L ]NY[nV_`f?D�I=Um®�]9d3abf L?g HJf�nVHJ]9UX] L ]N_�a+B�R q B �pBP± q f�abI�a � ­�tm± � � g MVfA<Z���V<NB; ���h>� CB � FN]PBVRªHX]NUX] L ]�_`aªd�]NH8¡�f L?g HJfp\^]NY[n g UJFhUJI=]�M�B�tlM �G©��½Å�«9��À9Ð�� nPFNoNf�a)<�ÍN�h�S<ZÍN� � <Z������B

# # ���%$�*��u'&)("*����:*��]� $ # 2½2���� ���]���� "2~8)X7�)CB�$') . &*-?>�)U-I; #E� ) " 9%>�\])C7*K #E� )C-I7�) # $'+^HI&N&�9%>�@ # $'- " &�9%&*) 6 $ "T#E� $'&�@PHQ@])C7,`

�De

Page 32: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

ΫÏ%Ð��U\�Ñ�^UZ�� � � � `��H�tXYÒr�%\�]<^_Z�Õ � � )õ¥ # 7�- "1[ ��¥1� HI&�&�9%>W@ # $E- " $'&cH [ ) " )C7NHI:'$EJ,H # $E- " -I; #E� )�& # H "%6 HQ7 6�À¥D�¡HI&�&�9%>W@ # $E- "�.N� $S+ � � $ " ;�-I7N>iHI:':EG � & # H # ),& #E� H #U[ $EBI) " H " ��¥1� >�- 6 9%:S9%&��AH "%6 H " )54D@]- " ) " #��Ik $ # $S&+5-?>W@P9 # H # $E- " HI:S:EGX$ " ;=),HI&�$E\u:E) # -^¤ "P6�#E� ) � K #E� 7�-�- # -I;�H�7NH "%6 -?> w ` !�" ;=-I7N>WHI:S:EG kQ#E� )Y& # 7�- "1[ Kl�À¥D� HI&N&�9%>�@ # $'- "& # H # ),& #E� H # $ # $'&Y$ " ;=),HI&�$E\u:E) # -i¤ "%6 H "ú58HJF?@A9AH-58H�æ�" - " K # 7N$EBD$'HI:�7�-�- # -I; HW7NH "%6 -?> w{`

� -I7�)�;=-I7N>WHI:':EG kh. )q&�H�G #E� H # $'&{H K?5-ä"=Yß_H�@<Þª= $E;D\]- #E�� H "%6<��� e����Qf HQ7�)q@%7N$'>�)I` � � ) " :E) #.º }:|{�-�Y³W2 ��e��Q�\])^H�@%7N-I\PHQ\P$':'$S& # $'+�@]-?:EG " -?>W$'HI:EK # $'>�)�HI: [ -I7N$ #E� > #E� H #�[ ) " )C7NH # ),& # . -i7NH "%6 -?>�� �Qf KM\u$ # &�HQ;=)^@%7N$'>�),& H "P6���kH "P6 -?9 # @P9 # &�� 0 �� `� �H��X%Ò ��\L]b^UZ � Õ3Ö����U\tÑL^_Z���� � � ��� ø�I8Hõ58>Pæ­>UI8>Péi7/>4@ ä"I8H�Þ'ßUHÁIFC5F?@<Çð@lK?9A@AB½ß�I8Çðæ8>_I8Þ®@358ÇïéËK�@�ñt=�BJ@bH-B"7�@A9 � ã)9<;�=ä"I8Ç<Ç�I�!�@b>�Ù«ß_HÁI�FE5F"@bÇð@<9Aæe@bK½>U=1Ù�Çï@ÂÙ�@1F?Ç�=¨@b> ���

� 7 d � 0 º }:|{�-�Y³W2 ��e � ��¿ w 0! #"$ ¿��&% y �h� 0 � ��e � y-w y'� �ù�(%*),+ w<>�- 6 �.- �0/ f g

� � )ö¥ # 7�- "1[ �À¥D��HI&�&N9%>�@ # $E- "¢. )�$ "�# 7N- 6 9P+5) 6 \�G j HQ7N$21+(H "P63� ¤ # J,>WH "P" dLe gMk H "%6ò� HI& \])C) " 9%&*) 6 $ "&�)CBI)C7NHI: HQ@%@u:'$'+CH # $E- " & � &�)C) d fI�Dk e � g=� ` !�# $'&YHi& # 7�- "1[ )C7�HI&�&�9%>�@ # $E- "(#E� H "(#E� );�*& # H "%6 HQ7 6�� ��¥1� HI&N&�9%>�@ # $'- "kGI) #�" -i>�) #E� - 6 $'&�o " - .�" ;=-I7Y\%7�),HQoD$ "1[ $ # - #E� )C7 #E� H " ;RHI+ # -I7N$ "1[ ��`

ΫÏ%Ð 5 `4]Aa<aA]<Ð/Ñ6[�Ñ�c �%\t^��"c �J\tÐ/Ò `4Z�4Ý\�Ï%Ð � Ð/[]L�t]<^_Z � ^UÒr��^:�t]b\tÐ � Ð8�t]&4YX%^��t]b\�c `��H�tXYÒ ��\�]<^UZ�Õ � � )� HI$':S:'$E)C7O) " +57�G�@ # $E- " &�+ � ),>�) d �h�Ig $'& 6 )C¤ " ) 6 HI& ;=-?:':E- . & k�.x� )C7�)65 � � � $'& #E� ) à HQ7Z>W$'+ � HQ),:q;R9 " + # $'- " -I;7� kH "P698 $'&iH�;=9 " + # $E- "3#E� H #<# HQoI),&T$ " @P9 # ),:E),>�) " # &i;=7�-?> #E� )m&*) # �;:=<>� z(? : + e >�- 6 � ��H "%6 7�) # 917 " &8�� : �®¶A@ -w$ `8� � $'& 6 )C¤ " $ # $'- "�6 $��])C7N&�;�7N-?> #E� H # $ "/d �h�Ig - " :EG0$ "¢#E� H # . ) 6 )C¤ " ) #E� )<>�),&N&�H [ )(&*@PHI+5)T;�-I7@u91\P:'$'+�oI)CG � ¶ �B�syDCu%_HI& dE X� � �e����Qf y � � �e����Qf g_� BI)C7Z&�9%& $ $ "0d �h�Ig=�Zk H "%6T. )^7�),& # 7N$'+ ##F�# -�\�) e ��`� � )U&*),+C917N$ # G<-I; #E� $S&Y+57�G�@ # -?&*GD& # ),>�7�),:S$E),&_- "(#E� ) á®=EBJ@lK�@AI�>3G�I�ÞNß:ILK�@A9�=¨Ú¢=?K�@3D87PILK�@A9AæõÜÀK�K�7�ÞxßU9A@AI�>%k � à ���X`

Q -I7qoI)CG [ ) " )C7NH # $E- "k + � -�-?&*)�7NH "%6 -?>�� �Qf KM\P$ # @%7N$'>�),& y � k &*) # � ¶��� k H "%6 &�) #,F 0 e ��`�� � )Y@P91\u:'$'+oI)CG¢$'& �B�my F %XH "%6¢#E� )i@%7Z$EBhH # )boI)CG�$'& �B�sy F yH5 � � � %Z`8� -m) " +57�G�@ # Hn>�),&�&�H [ ) = . $ #E� @P9%\P:'$'+OoI)CG��B�my F % k&�),:E),+ # Hi7NH "%6 -?>JI É "$ H "%6 +5-?>�@P9 # )X! 0 C D I $ >W- 6 � z `�� - 6 ),+57�G�@ # Hi+C$'@ � )C7 # )54 # ! . $ #E� &*),+57�) # oI)CG�B�my F yH5 � � � % k +5-?>W@P9 # )^= ¶LK ANMPO�QSR�T ¡�UWV $YX C

K ANZ O�Q[R�T ¡�UHV $ X C>�- 6 � k H "%68#E� ) 6 ),+57�G�@ # $E- " $'&|= $'; =]\ � � /e����Qf�k H "%6

- #E� )C7 . $'&*) #E� ) 6 ),+57NG @ # $'- " $'&w= ��` � HI$':':S$E)C7 d �h�Ig & � - . & #E� H # \]- #E� !�^ A $ C >�- 6 � z H "P6 C_^ A $ C >�- 6 � zHQ7N)�),:E),>W) "�# &�-I; #E� )O;=-I7N> ��e � �a`0+ $ X e u_� k H "%6m#E� 9%& #E� ) 8 ;R9 " + # $E- " +CH " \�)�),HI&�$':EG�+5-?>W@P9 # ) 6 ;�-I76 ),+57�G @ # $E- " `

b �/�� "2P����� $ ��N ".dcs�� ��Y��� $ ���?2~8)�@%7�),&*) " #�#E� ) 6 ) # HI$S:E) 6 +5- " & # 7N9P+ # $E- " &V-I;&*-?>�) è KM@%7�- # -D+5-?:'&�$ " #E� $S&�@PHQ@])C7,`{~8)�¤%7N& #V6 ),&N+57N$E\]) #E� ) 6 ) # HI$':') 6+5- " & # 7N9%+ # $E- " -I; #E� ) è KM@P7�- # -�+5-?: ;�-I7�@%7�- B�$ "%[ o " - . :E) 61[ )X-I; 6 $'&�+57�) # )�:E- [ HQ7N$ #E� >�\ GöÅcHQ7NH�G ="9ë58ÇÂÈVd fI¦hg `�~8)@P7�),&*) " # H " )"Íi+C$') "�#Àè KM@%7�- # -D+5-?:�;=-I7 ¬ $ "öQ $ [ 917�) V `_� � )U+5-?>W>W- " 7�)C;=)C7�) " +5)�& # 7N$ "1[ +5- " &�$'& # &�-I; # . -<@PHQ7 # & ���e�� H � HI$':S:'$E)C7X@u91\P:'$'+ioI)CG � ¶ �B�sy F % .N� )C7�)e��$S&OH " �À¥D��>W- 6 9P:'9%&XH "P6fF É "$ X . $ #E� �

? k�jbn³8j��gFu�ZkH "P6t��f?� H " - #E� )C7<�À¥D� >W- 6 9P:'9%& . $ #E�Ff0[ ) " )C7NH # -I7N&(�ih�¢y F w,y F z0%Z`�� � )(@%7�-�BI)C7TH "%63#E� )�BI)C7N$E¤%)C7b& � HQ7�)sH+5-?>i>�- " $ " @u9 #j%�k�.N� $':E) #E� )�@%7�-�BI)C7OHI:'&*-To " - . &½w k &�9P+ ��#E� H # C_k ¶l% ` !�"�#E� )�¤%7N& # >�),&N&�H [ ) k #E� )�@P7�-�BI)C7&�) "%6 &XH " ) " +57NG @ # $'- " -I;)wF9%&�$ "1[�#E� ) � HI$S:':'$E)C7U) " +57�G @ # $E- " oI)CG ��`s� � ) " Hnm�KM@%7�- # -D+5-?:�$S&X9%&*) 6¢# -n@%7�-�BI)#E� H #�#E� )O@u:'HI$ " K # )54 # $ "m#E� ) � HI$':S:'$E)C7�) " +57�G @ # $E- " $'&c$ "%6 )C) 6m#E� ) 6 $'&�+57N) # ) :'- [ -I; % `�� # ),+ �P" $'+CHI: 6 $ÂÍi+C9P: # Gs$S&#E� H #Y#E� ) 6 $S&�+57�) # )X:E- [ HQ7Z$ #E� >�H "%6�#E� ) � HI$':':'$')C7�) " +57�G�@ # $E- "(. -I7�o($ "(6 $��])C7�) " # >�- 6 9%:'$M`��{-i- BI)C7N+5-?>�) #E� $'& k Ho " - .�"(# ),+ �P" $�Ê�91)^-I; H 6P6 $ "1[ H�+5-?>W>W$ # >�) "�#�# -õwm9%&N$ "1[ #l. - [ ) " )C7NH # -I7Z& �gF w,y F z � - BI)C7�H #E� $'7 6 >�- 6 9%:'9%&oh�-I;�9 " o " - .�" ;=HI+ # -I7Z$EJ,H # $E- "¢d § k { kufI�DkP�DeIk�fI¦ g . HI&�9%&*) 6 `

¥D$ " +5) #E� ) # ),+ �P" $�Ê�91),&�9%&*) 6 $ "�#E� )(@P7�- # -�+5-?:�$S& 7NH #E� )C7W& # H "%6 HQ7 6k�. )(-?>W$ # #E� )(@%7N- -I;�-I;�&*),+C9%7N$ # G¢;�-I7&N$'>�@P:'$S+C$ # GI`

��)54 #Ck . )�@%7N),&*) "�# H "«è KM@%7�- # -D+5-?:I;=-I7�@%7�- B�$ "1[ o " - . :E) 61[ )x-I; @PHQ7 # $'HI:Q)JÊ 9PHI:'$ # G�-I;�7�)C@%7�),&*) " # H # $E- "k�¹ ) º ) ¹�� w:<QyC <QyDC�w,y-w�w,yDCIzQyDCqp � `ù¥D)C) Q $ [ 917�) e 1D`

�If

Page 33: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

±8jlkL²L³8j ²L³8jS ��³8jI�y�� .0! "$

� .0! ����< .0! ��$#;.0! ����$

%�� 0 C� �>�- 6�� 0 F k I $ >W- 6 � z��� 0 F �� $ >W- 6 � z

r 0 �gF w � k �gF z ��� >�- 6 h�r � 0 �gF w � �gF z � Q >�- 6 h�

% � y � y � � yCr�yCr � �! .0! �

� !� w 0 !�w �

� z 0 I M �s>�- 6 �� p 0 ! < #

� w,y � zQy � p �� w��É ����% M %�� �+ C����x>W- 67� M � � �+ F � � � � z � $ >�- 6 � zr M r � �+ �gF w � � � F � Xz >�- 6 h�

Q $ [ 9%7�) V/�'è KM@%7�- # -D+5-?: ;�-I7 #E� ) 6 $'&�+57�) # )F:'- [ 7N),:'H # $E- " ¬�����¶ � �*�&% yDC � y-w ��� %!+ C k >�- 6 �?` à -?> K>W- " 7�)C;=)C7�) " +5)<& # 7N$ "1[ $S&XH � HI$':S:'$E)C7^@P91\u:'$'+OoI)CG0H "%6 H¾¥ # 7�- "1[ ��¥D�¡>�- 6 9%:'9P&^HI:E- "1[s. $ #E�8#l. - [ ) " )C7NH # -I7N&�*� �my Fu� y � h��y F w�y F z �*� `

�I�

Page 34: Q - Cryptology ePrint Archive · Ù ÚPÏ Íè_Î Ú ÊnÚ_ίã ØÊ Ô Î ÖµÍè ٠ͧÙ]ãPÙ ØĮ́ßlÕ Ó Ù Ø ã ØÊ Í ÊºßàÊnÓ³Ê ãDÎlØjÙ ÍÌ Ú_þÌ

±8jlkL²8³8j ²L³8jS ��³8j

� <Qy � w,y � zhy � �< y � � w � �z 0�� "$��<hy��w�y��uz 0! � �� 0! ��$

� � 0! ��� �$´�< �ï¶ C���� 6 � $< >�- 6 � z´cw �ï¶ C � � 6 � $w >�- 6 � z´Yz �ï¶ C�� X 6 � $z >�- 6 � z

´ �< �ï¶ C��� 6 � � �< � $ >�- 6 � z´ � w �ï¶ C � � 6 � � � w � $ >�- 6 � z´ �z �ï¶ C�� X 6 � � �z � $ >�- 6 � z

% < �ï¶ C ��< 6�C � �w >W- 6 % w �ï¶ C ��z 6�C � Xp >W- 6

� �ï¶ F � �< 6 F � �w 6 F � Xz 6 F p >�- 6 h�� �U�ï¶ F ��< 6 F � �w 6 F � Xz 6 F �p >�- 6 h�

´ < y-´ w y-´ z y-´ �< y-´ � w y-´ �z y % < y % w y � y � ��! 0A �

� !r < �ï¶ ! 6 I < �:<rIw �ï¶ ! 6 IVw ��wr z �ï¶ ! 6 I z � zr p �ï¶ ! 6 � � �

KN< �ï¶ � M< 6 � �< >�- 6 � zKNw �ï¶ � Mw 6 � � w >�- 6 � zK�z �ï¶ � Mz 6 � �z >�- 6 � z

r < yCr w yCr z y>K < y>K w y>K z �r@< �É ����r?w �É ����r z �É ���r p �É ��� �$C ��� 6&K $< �+ ´ M< 6J´ �< >W- 6 � zC ��� 6&K $ w �+ ´ M w 6J´ � w >W- 6 � zC � X 6&K $z �+ ´ Mz 6J´ �z >W- 6 � zC � �< 6�C � �w �+ w M< 6 % <_>�- 67C � �z 6�C � Xp �+ w M w 6 % w�>�- 67F ���< 6 F ���w 6 F � Xz 6 F � �p �+ � M 6 � � >�- 6 h�

Q $ [ 9%7�) e 1 � è KM@P7�- # -�+5-?: ;�-I7 #E� ) @PHQ7 # $SHI: )JÊ 9PHI:'$ # G -I;A7�)C@%7�),&�) "�# H # $E- " 7�),:'H # $E- " ¬���������� ¶� �*� w < yDC < yDC w y-w w yDC z yDC p � y � I < yWI w yWI z �*� ? w < + C � �< 6*C � �w >W- 6 - w w + C � �z 6�C � Xp >W- 67 �?` à -?>W>�- "7N)C;�)C7�) " +5)F& # 7N$ "1[ $'&�H � HI$':':'$E)C7n@P91\u:'$'+0oI)CG H "%6 H ¥ # 7�- "%[ �À¥D� >W- 6 9P:'9%&sHI:'- "1[/. $ #E� ;�-?9%7 [ ) " )C7NH # -I7N&�*� �myDC � y � h�3y F < y F w y F z y F p �*� `

�h�


Recommended