¼C | h g ª#$ #$ % &(' ) *+ , &- ./, ./ 01#2. 3 4 65 Ñ 4 gÔ# h F hÑ Ñï¼ 87 4 4 4 # # ÷ :¼?...

! !"#$ %&'()*

+',.-/-#0214365738,:9<;

=?>A@CBD@FEHG8I7JK>AI7J(L7M?NPOQ@RG8L7M?NM?STI7M?NUCVKWKXRV @RNYL7M?@CZYG X >A@\[CSTNRNFM?IANR]_^`G#a X M?IL7M?bdc`G8]_^2efM?@gM?ST]h^jikG X ^7M?b"G X S W

L7M?@ClYIASnmKM?@FNRS X_o#XPUCV @ X bp>AIALm V @gJ6M?EqMJ X

m V I

UCr @ X MsptfuYG8vAvfM

w tDxy>7JK>AN X w8z6z| ] V @F@gM?] X M?LmKM?@FNRS V If~

/#4g g4h\/q4Y4 ?g:h h#8R4K

2¡.¢¤£.¥¡§¦©¨ª£:«F¬:_£.¥¡¯®7°2¡.±g²¤³2 <³´`¥¬:µ¥¡¶?·`¥¸¹£.¥¡§¦©¨ª£:«F¬:_£.¥¡¯®7°2¡.±g²¤³2º³°2«g£:«g¡¤¸¹»

¼C|hgª

½¿¾`ÀYÁPÂÃ¿Ä\ÅÆPÇÈÅÉÊÅÁË

Ì #Í§4ÎÏÐ#ggªY#©#g§Í©4#Ñ #hÒ|RÓ4DÔ6ggy64Ô#ÑÍ#4/©#D##64:hA\Ó4ghÑÕ6g4#ÑDÍÒ#4Ï<Í§4#Ñ ÑÎh|h#ÎC#ghÖ :h`hÑÑKÏÐÍ©4#Ñ ÑÎ4×|h#ÎC¼C#6g:Ó?4Øfh.8ÙfÔ6\|_ ÛÚ gÎhgq4##84::Ü#ChÑÑÕ#ÝÈÔ¼Û#D Ó?×h Ôª¼&4Ó?#CÈ#Ýqg 4ÞC¯/#Ñ4\Ó4ghÑ4#64:##ggÏ<hßh¯q#Ñ8¼×#g©g4gh ÝØfh./à4hgá#gØA_h×q4©Ô8g#Ý¼×¯Üg|hÑ8\qgghâq4<ªÓ?#DÒÈg4ª?#¼gFh¯C_<#ã#Ó4g.¼hKä<ghÑÑgh q4ÒhåKg#q# ##ÏfÍ©|_#Î©¼4 |_#?4Ñ8æ:çÛ_#gY_## #4g#g#:Íg#|_nÑ¯#çQ\# gqè#Õ#g§é_gè#Ô6g:h4### Ï¤/q4Y_4#:Ó4gÑ##_êfh #©ég?Ü# Ö 4¯##44gg#|_n×ëTé Ö7ìÈí q4î||h#ghÑÕ##64:F Ì #\¼QªFÓªÈD#84.ÜÔ#Ñï¤¼yYq?g#4ðÝ¼QgFhâ2Cg\D¯/6g:DÔª¼ÛÓ?#Yg4/qgg#ggÈh |h:hAÝ¼CgFhÛPq4g4Pg4#ggÏ§Í§4#Ñ g8ggñhÑÑ¼QÑÎhòY|h#ÎÛØfh.6ÏóÓ_hôéhªõh h Øfh.ö4|hÑ ð÷ hgq4#6g:Ó?#ò ##DÝ¼È§×ø#gâªq4fYhª¼Ý#gÑ/q#Ñ#g4gªgâhÑÍ§F¼?<|FÓ?#q4§Dh ×h#:Í©g¼Yá#g:4#gâ8h q4§|RÓ?##D FÝh7g4# g#Ý#Ù ÷ ØÐùCh Ô#h#¯###44hgA # 4|hÑÑï¼4âªÏ`|h#Î#Ùf#48FhPã#4q4<#gî||h#gñhÑ##64:FÖ #:#g4hâ2ÏÍ©4#Ñ ÑÎ4C|h#Îôù\Ó4g ì hÑÔ#hïÛq4#gÑ/q#Ñ g#4#ÈhÔ64/#×gÑÑ/Ýg#:Ó4YØAhÑÑg#ghÏ|h#ÎôçÛ_#ñhòúDh#hâÕ##g4\Èûg##âAöhÑì gÎYh#Qh ¼Cg4ÑÑFh4#g_Ò# Ú ù?ÏÐq4§q#q#Ñ g#4#gÚ g g|_Ï©|h#Îy¼yg4ÑÑF_4#ggâKàh4#ðç&_Ñ4#¯Ü.ü2ghâ6¼yÔ8¼qg ýh hÑÑ#òh#gÒq4ÒFh\q#ÑÑ¼CF #Ý¼Y#gh g4g\#×¼PÙf#4ÑÖ |hÑÑï¼4â8Ô#/D4:64:hªÏÒ|h#ÎyÝ¼Q|hgªDh hÑÑÕ¼Q4? qg qhÔ6g##g§q4AhAÏÕRÍ§§6ggñhÑ?|h#Î`D ý©q4fhÑÍ§F¼?`:R¼?#D_f¼ Òh hÑÍ§F¼?##64:#×Dg6ggñhÑÑ¼CÔª¼Pg4#ÝÍyég#YhÎ8

Ì |h#ÎC¼44ÛhÑÑHþ

éÿ4:Èöh#6

"!#$#$%&(')*+,&-./,./01#2. 34 65 Ñ4gÔ#hFhÑÑï¼87444## ÷ :¼?/4:¼?gÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ95:4 4 ##ÑF_4#Òh Øf46g:gh;7444:## ÷ :¼?/4:¼?:gÈÈÈÈ95<

=>*+#$&()-?A@BC'EDFG,&-01HJIKIAILDFG,&-01M-. N 65 Ú h###Øf44hÈÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ 4G4N 4 ÷ 4#/#ÝÍQÙf#g:¼?/ é_ÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ 4ON N ÷ 4#/#ÝÍQÙf#g:¼?/ Ö ##\4#ÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ NGON P úD\¼QùÍ§h###(ÈÈòÈÈÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈPQ5

N P#65 ÙA#h#ÑgòÈÈÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈP N

R>SUT+'-01.V'-W)XY0Z.V%[\'-0]DFW)W&-)^*_-`%0bacIL)W)WC0Zd'-01#$0]ILeC.S8&&(W))GIK. RgfP#65 Ï¤ª #\4PÙfÑÑ/ ÷ #:Ó4gÓ4gö#4

Z/NsZ

ÈÈòÈÈÈÈÈÈÈ<:P# 4 Ì #ÈÙfÑÑ/ ÷ #:Ó4ØAhÑÑgù?#g ÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ< NP# N ì g#ghÑ_4Ch7#ÈÙfÑÑ/ ÷ #:Ó4ØAhÑÑgù?#gYòÈÈÈÈÈÈÈ< OP# P Ì #g#4Ñ ä<g4Qh7#ÈÙfÑÑ/ ÷ #:ÓªØAhÑÑgù?#gÈÈÈÈÈÈ<Gh

P# P#65 ü2g#h?Ü Ö Ñ¯/Ô#ÑäÐhñhª ÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ O <P#i< ?/Ññ_:¼CØfh?g4Ñ(ÈÈÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ O h

P#i<?65ΣÜ.Øfh?g4ÑßÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ O h

P#i<? 4 Øfhg4Ñq4§#ÈÙfá|hÑ¤¼Ch`ég\ü244h#ÈÈÈÈÈÈÈ OGjP#i<? N ÷ #gÎPh ÷ ##g:¯?#g ÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈh 4P#i<? P Øfh`h ÷ 4g\çQ#Ñ#ÑF_4 ÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈh NP#i<?i< Øfh`h`ØfÑñhª¯?úD#RÍÑ 4 òÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈh OP#i<? O 5\Ü¤4/:Ü¤hñÜ 4 Øfhg4Ñ ÈÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈh O

P# O ##ÑF_4CYçQ#Ñ|h:¤¼ ÷ 4#/_4 ÈÈÈÈÈÈÈòÈÈÈÈÈÈÈhGhP#ih ##ÑF_4CÙfÑg\4#ä<h#PÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈlkG:P# k ##ÑF_4C ÷ 4×gªù?¯#ggkòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈlkm<

P# k/65 çQï/ &÷ 4gªkÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈlkGkP# k/ 4 ù?8ggñhÑÈçQï/ j÷ 4×gyù?¯#gg Ú h 4

q n #¯Üóû F¼7444## ÈÈÈÈÈÈÈòÈÈÈÈÈÈÈÈòÈÈÈÈÈÈÈ j :P# k/ N Ì #ÈÙ ÷ ØÐùQhhQÙA#h#Ñq4

q n #¯Üóû F¼7444## jm4

3>*+mW).._ILeop&q01srt-01./. u%3

v w(W)C%?%IQ&(', ux

-0my Zz3

|~

$

Ì # gYh q4ò#Óhh\¼ôh 4hÑ _Qh h§4×g4#Ñ¯ô:#\#gÝÑÎhhÑ4_Ü#Y|hCÔ6gg4&:4#4g ##p#&Ññh:Yq\ÍÞ¼4Fhg Ì #C4?gC|h |h ÍÛ#4RÍÒ&h<g4Ý##F_4Q#\¤Í§4ÎÑÎhò#Ï¤g#\òh #ÈÓhh:Ñ¼Q4RÍ§Ü##ÝÔ6ghgÑg\4# \Ó?ggg n (#Q4#Q|h #g \Ó?ggYg|hÔ#ÑQ4F_Óhh\¤¼ph_hÎ4 4_ÑÐ4? ×h 4 #Ph#g|h #\¼ hCÓ?#Ñ#ghÔ#ÑY_hÎ# hD#Yh###Ññ_hô4 g:#\4ph _Qh #×#\nh§g#Ó4/q4Y_4 Ö 4:4#yh F # _Cgg#gÑ¼ð#g¯/:È\Ó4ghÑf64Ô#ÑgC4|hhgÝ#Ó_h\¼Û#h _Yg#g:¼?/4_ h6gg:gªò|h Í§hh Ì ##4Ô#Ñg Ô6gg4gÈ4×g4#Ñ¯ðÍ#g hÎ#Qq4D#84Ô#Ñ¤¼ÛQg4#/Ûëq##Ô?ÜÑgÑ¼ í Í(#Óh_ ¯ð4ô n¼q##\4#×4×hÑ44#×(#&ÍÒF¼ |_#\¼PhD:ÑÑ6¯/gg/hÔ#ÑÍ#Ñ#g§#Óhh\¼Y§g## Ì #§©Í#gò#444##g:¼?/4:¼?:gFhyÔ6ò# #gD#\¼Pg|hÔ#Ñòg4#/_4#©Íyg#g:¼?/ & _/Ï¤b5 j hköïÓªg:Ò\hÑH hPÍ©gD#î|:Ò×hÎðëq#ÑgïÑï¼ í q4§#444##g#g:¼??Ü4Û¯#gggòã/q4:#|_gÑ¼y#gÒæ¤#Óhh\¼Q#444##.êDÍ§gÝÔ#4ÎhgY\4##Ñh7¼4FhÑñ_g§Ô¼ Ú Î4\ÑÑ2h J hg4Ô#5 O HÌ #Dá#g:4C4DhªhY^5 jGj 5Í#g Ö g4gÔ|h#¿h çQgF NGj 2hÎh AæcCL/ g ) +¡¢1g ) E()

¢Q C ¤£C E(x+ y) ¥ E(xy) + /¦§ ¨\ ¢ §¡ ¨ E(x) ¥ E(y) © ê Ì #\¼Í©gDhÎ#¯/#ÑgÑï¼Ýq4ÐFhÑÑ hÑ4gÔ#hFhÑÑ¼Ý#4_Ü4##Dg#g:¼?/4Û#gggã/q4:#|_gÑ¼4â##gÈ|hÔ6ggÛÑÑD#44gY ò \g##òÍ#\#g§#Pg#g:¼?/4C#gg§¯/:©|_§h/ªggÒh gg#hâ|hÑïÜ#4#4Ðf4#hK#g#gñhÑ|48gY#4Ô#Ñg<×g:¼?/44h#ª¼4`ûpFh##hÐ\Ñ#

«§¬)®¯1°±Z²)³¯¬ 4

á#g:4Û#gÔ#/Í§È#Ó4ò|_64Ô#Ñò4Ô/hÛhÑ4gÔ#hFhÑÑï¼P#444##g:¼?/4:¼?:gò4Ó4gôY#444##Èg:¼?/4:¼?:g 4ôY6ggñhÑA#4?Ü.hÔ6gÑñhÛhh#Ì #<YF¼Ô8Ó?\Í© hÐDî|:Ð:gYh#:Í©g#ò#hÔ6RÓ4á#g:4´K7RÍ©\Ó4ggâ/##gÍ© C#h\4# g##4Ô#Ñg&h \hÑHÏ¤#:F Í©ÝYh#Ñ¼Cq?g#4h##ÑF_4#§h`#h44##g:¼?/4:¼?:gg

µ>¶¸· Áº¹ Å¼»-½PÄ¯Ë»

Ì #È#gÈ4ªh# ªQ#gYYh|h:gyÏ¤p#î|:Ý|h:Í©Yg4# gÝhÑ4¯ÜÔ#hFhÑÑï¼ô#444###gg#gY#\¼ hCh#Ó4hhÑÐ64:h#g g4##6RÍ©g:q#Ñ2g:¼?/44h##ò#hg4Ñg7RÍ©\Óªggâ6h§Í§ÈhÑF ¼C\4# Ò#hgÑFh¼4\Í#\#g# ëq/ªggªÝh gg# í #gg\Ó4g ¯/:F Ì #×î|:Ýg4ªÔ#/4h©#D#gÈÈy#\Íßh##ªhy##4Ô#ÑgQDg4#:#\#&hÑ4gÔ#hFhÑÑï¼&#4_Ü4##g:¼?/4:¼?:FDq4Þg#g:¼?/4ð¯#gg|ÈhÝ#444##4ð6ggñhÑ#4?Ü.hÔ6gÑñhY44##Èëqgù?g\4 4 65 í Ú h 4yÝg4#:#\4yh Ú g?Ü n h Q÷ Ñ\Óª5G56h DTh\<q4)#¾:g\Ó44g4\:¼Í©4Ô#g:Ó4|_§hYhÑ4gÔ#hFhÑÑï¼Ý#444.Ü##g#g:¼?/4Y¯#gFhYÔ6g4#:#\ q4ßÈ#444##g#g:¼?/4Y#g4'#ô:¼?\&44# 4 \Ó4gjgÑggªgl7g#ghâ#ôFh q4yhÑ4gÔ#hFhÑÑï¼#444##¯#g\FhpÔ6 #g y#Fhq4ò#444##Ý¯#ggÈ46ggñhÑ2#4?Ü.hÔ8gÑñhP44##gÏ¤p#Cggh |h:ÈÍ©Ph|hÑ¼?Y64Ô#ÑYh##ÑF_h#ÈhhÑ4gÔ#hFhÑÑï¼ô#444###ggg n Í#ÛQ#CÑñhÎôhÒ#¯ #ggÍ§ g4 4Ñ/4#ÈÔ|h 4 #4_Ü4##Ý¯#ggÈ4#Ñ¼h ÷ ÑFhÑ¼4â#gFh##hÈÔ8YhD6RÍ§\q#Ñ<hò4Ñ/4#Ô|h 4hÑ4gÔ#hFhÑÑï¼ô#444##×#gggA¿\Ó4g:#gÑggâ<Í©C#ggªhÛgg4 g4ïÜÔ#/4PÝ#RÓhhÔ#Ñ¼Ygg#hâ|#4?Ü¤ªgh\Ó44Ñ/4Yq4©g#g:¼?/#q##¯4#§4ÓªgCÔ¼64Ñ¼?#4ñ_ÑÕÔ#h###C#44hh q4g#g:¼?/# _|_g4#/ Ôª¼Û#q##\4#Èëqg ÷ |h/g N í Ì #ÒÒÝ#\Í h##ªh¯C×g#Ñh4#D\h7g#g:¼?/hÔ#Ñh :ÑÑ/¯/gg/hÔ#ÑÒq##\4#gAû 4#f#h?g4Ñ/Í©hhÔ#Ñ§Dg#g:¼?/Ðq##\4#fh#DgÑñh§h784Ñ¼?#4ñhÑ8Ô#h#¯###44hg Ì ?#gâ#Ò§Ý#\Í h##ªh¯C#RÓ44#hÐ#Yhðh##ÑF_4#hÐ#444##g:¼?/4:¼?:gg Ö #:#g4hâÍ©×?Ü #gòÝ#\Íj#48g:.¼|ÒhåKgÒ464Ô#Ñï\§\4#gg##×g4#/_4#ÐÍg#g:¼?/ Û _h g#g:¼?/ q##\4#gâ#g8g\ÓªgÑ¼4fûp g4#:_¯#h#ÑgÒh

«§¬)®¯1°±Z²)³¯¬ N

#444##¯#gg|FÓ?##Ò#46g:¤¼C|_Í©òFhÑÑÀm/¦/Á m 1Â Ï¤y#D# |h:ÒÍ© ggÔ6ÈhQ¯#h#ÑDhf#444###gò|_hÑÑRÍ§q4FhÑ#P#4Ñ/4#ò#gg ##\Ó?4#|h:gCûp4g#ghÑ#È#gð4Ô/h ÛÍ YÓhh\¤¼phh##ÑF_4#gçQ4Pg4#g\gÑ¼4âÐhÛî||hÑ§g4ªÔ#/4Í© \Ó4gÑ4&×#g#4Ñ Û gg:¼?/4QÓªg4ÛhA#ÈgÑÑ/Dg#:ÓªÝØAhÑÑg#g8 P N HÌ #Óªg4ôFhôÔ6×#RÓ4gôyÔ6×hògg#hÈCgghÑ #gÍY# #ÑñF¼ªg<Í#Ý6g:q4<# gg:¼?/4â/Hhâ?ÐFhCÔ6#RÓªgYÔ6gYhªFhÑÑ¼gg#hªh#:ÒÝ:_D Ó4gh:¼PY#h 4¿4h\Ñ gÑH7ûpDg4#:#\Ò#©#g#4Ñ #gÛ#ð×Í§F¼y|_g8ggñhÑÑ¼P# q4\Ó4ghÑ`h##ÑF_4#g Ú h 44#Ý#\Í¿¯#gCÍ©P#ggÓhh4#Ý#hg4Ñ|_×hY#Cî|:×gÑÑ/Yg#:Ó4PÓ4g.Ü4#hf#Î Ì #g#h?g4ÑÔ##Ñ #ÝYhy?4ÑÕq4h##Ñ¼?#Y4#¯#g åKggªgg|h4#&hÝ#Ñ|h:¤¼Cg4#/_hQh Óª_#×¯#g\F Ö |hÑÑï¼4â/Ô¼ n¼?#Q#C44|hÑ<gÑÑ/×g#:ÓªPØ`hÑÑgÈ#gYÍ§PhChÔ#Ñ&Ô|hPÛ6ggñhÑï/ g4×gªÒ#gÈ4yF

ÃÄ½CËÕÄ · ÁPÅ ÂYÅËÆ · »9ÇÆPÅ¼» · »

ûpÈÔ8g4yÍÛhQª #\4C#444##Dg:¼?/4:¼?:g ÷ |h/g 4 Í#gÍ©Y#Ó? \î|#4#Èh ¯/#Ññh#\Ññ_4##Ô8\.Í©gg #444##h hÑïÜ4gÔ#hFhÑÑï¼&#444##Ýg#g:¼?/4p#ggpù?g\4 4 654âAh Ô#/È#¼ð #g4DhA#DYhyh##ÑF_h#ÒQù?g\4 4 4 Ï¤ ÷ |h/g N Í©#ggy#\Í4Ñ/4ðQ4#×h©#Yhh##ÑF_4#Dh§#4_Ü4##g:¼?/4:¼?:gÒæ¤g4#/#Íyg#g:¼?/ Û _êQù?g\4 N 4 h gÑ4gÑ¼gÑñ_ æ¤g4#/#yÍ g#g:¼?/ q##\4#.êY ù?g\4 N N Ûngá#ÎÑ¼pª_Ü #g# #gô#gPÍ§ð#?gg Í ggÔ## #ôYh F Ô8g# Ô#_#¯###44hg`ûpò#gQ#ggª.Í©#RÓhhÔ#Ñ¼Cgg#Ýh #4?Ü¤ªgh\Ó4D#h?g4Ñ§ÍÉ g_Ü¤g4|_hÑÑRÍjg4#/_4yÍÛg#g:¼?/ ð _Yh q##\4#gâ8g6g\Ó4gÑ¼4Ö |hÑÑï¼4â8Í© \î|#×Y#\Í!#46g:¤¼Ûh<#444##È#gg|_Í©FhÑÑ7Îh\¼&:Í§h?Ü###ûp g4#:_Í¼ð#D#46g:¤¼ôD#\q#Ñ<ô#×g4ª¯?h§gh#/_4#Íg#g:¼?/ _Qh q##\4#g Ö #:#g4hâ2Í§4Ó4×¤Í©Û¯#h#ÑgÈh§#4_Ü4##¯#gg|_##64:Îh\¼y:Í§h####Ï¤ ÷ |h/g_PQÍ©C#ggª×Û#gY|_|h¯#h\Ñ¼ô#Y#h6g:g#g q44#

«§¬)®¯1°±Z²)³¯¬ P

4Ñ/4y#ggª ÷ |h/g N ì hÑÔ#hï ÒgÑÑ/g#:ÓªØ`hÑÑg§¯#gB P N H<ûpÔ6g4ÍÝ#Yh##`gYhªFhÑÑï¼ògg#hâ#4Ô|hÔ#Ñ:hâh # ÓªgÑ¼ò#444.Ü##ò#gh Ò4g#ghÑ_4PÔª¼ ì hÑÔ#hï Ì #gF_ngÍ©Èg4#:#\q#:#ggYhªFhÑÑ¼ògg#4g#ghÑ_4ù?g¯4pP# P#â4|hgÑ¼#g#4Ñ gg:¼?/4Ó4g.Ü4YhfÍ©gÑÑ8hÐòÑg#h?ÜÊÈ|¯/Ô#Ñ©ÓhhñhªF Ì #f#g#4Ñ g:¼?/4:¼?:FßfÔ##Ñ<×#òÍ§F¼×|_©ÐÐ#hÔ#ÑÒq4<64:hªÐh##ÑF_4#gfù?#Ô#gá#gªÑ¼4â?Í§ g4Y\Ó4ghÑh?/Ññh.¼Y#h?g4Ñ§Ûù?g\hP/i<×#RÓ? #òÔ|h§q4h##ÑF_4#§h74#Ò4g#g.ÜhÑ gÑÑ/Ýg#:ÓªØ`hÑÑg#gh Ì #gYh##ÑF_4#Dh×g4# g pù?g\4#P# O âLP#ih?â8h P# k/ Ì #\¼Q#gÑ ò×Ý#Ñ|h:¤¼C#hg4ÑHâ|×#h?g4Ñq4ÒgÑg\4#Óªh:Ü##â/h Ýg4ï×g©#gq4ÐÍ#CÍ§|RÓ4Ñ4ªÑ¼× n¼×## gÑ¼?#gÑÑ/g#:Ó4ØAhÑÑg©¯#ghÖ |hÑÑï¼4âh ÷ |h/gË<Í§Òg4#gÑ §#AÍ©4ÎÝh 4Ó4Òh4/Ñ4Îò4Ýq#:#gfá#\h#|_hòhAªgg:F

|~ Ì

Í Î Î Ï~Ð JÑÒ~tÑÓtÎ

é##C#Ññh:q\Í ¼ªFhD#444##g#g:¼?/4ô#ggò|FÓ4Ô6ggô: ¯Üg#Ó4gÑ¼C#g#\¼C|FÓªDÔ8gg4D4h 464:hª§CYh¼ åKggÒg:¼??Ü44h###h?ghÑ©#yhgâ?h#âÓªh##hg4Ñg`Ï¤Y#©|h/gÐÍ§ #g#_Ü44##g:¼?/4:¼?:gY#gD:g#ÈëñæóÍ|_:êâæ¤#RÍ§êâ#h æóÍª¼ªê í |_Ò/È|g\Ò#Yhðh6g\gûp:h:òÔª¼ \î|##y#444##g:¼?/4:¼?:FÈh hÑ4gÔ#hFhÑÑï¼#444##g:¼?/4:¼?:gg Ì #gyÍ© \Ó4gÑ4QÝ\# g4#:#\hÑ4gÔ#hFhÑÑï¼#444##Q#ggC4Ó4g6ggñ_Ñ#444##y#ggg Ö |hÑÑï¼4âÐÍ© ggÔ8h##ÑF_4#ÐhÕ#444###gggÐj4g#ghÑh 4 \hÑ ª #\4#444##g:¼?/4:¼?:FFhQÔ8Dq4# A h N HHJ0mÔ;CÕ Ö6zÖ)LÖ ü2\©#gh4|hg

(M, )Ô6òî|#ëqgïÜ í 44#âh Ñ\

σÔ6Ò#gg#¤¼×|hh\ggA×L¨B¨BÊL1 ¢ £ØÙ /Ú Àm/¦ g ) L/¨Bëq4¤%¨B Ú¨BÊL1 Y §//§/¨ í 4 M

§Ýá| ##Ñ(K,E,D,A)

h7#4Ô|hÔ#Ñ:hâ?¯/8g\ 64Ñ¼?#4ñ_Ñ6DhÑ44ï#gâ/_:n¼?##Û 0m,ÝÜJ01-0ZIKgÞ n ##/ 1σ

#hÑh4#K4/#/<hg#g:¼?/4%ß gg:¼?/4

Îh\¼y|h(ke, kd) = k ∈ K

Í#gK g#hg#òÎh\¼y|hghÙà

DFG,&-Þ n ð##/ 1σâke

â2h h&gÑggªm ∈ M

#Ýg#g:¼?/4ôhÑ44#E4/#/g##g:¯?

c ∈ CÍ#g

C g#hg#òg##g:¯?|hgh

áãâä)åæç ç è/éZêgëìæí ëïî ðñ ëòí ëóäñ ëóôYî ð+ñ õëìí åððî ðö\ñ î ÷ïëìøùgñ õë¼æç ö/ø/í î ñ õ÷Kæ/äqæúùåðû§ñ î ø/ð_øù

σíÊæñ õëòíñ õæð

log σé/îýü)ë/ü éêgëêgæðñQñ øæç ç øêpëãþÿëóû§ñ ëóô¤ÿø/ç èVðø/÷ïîiæçVñ î ÷ïëî ð¤ñ õë-ä)ëóû§åí î ñèÿæíÊæ÷ïëãñ ëòí

σñ øqö/ëòðëãíÊæñÊëæ/ëòè/üõëòí ëãùø/í ë/éñ ëóûÊõðîiûòæç çièFêëïðëòëóôFñ ø\ñ õî ðFøù

Kæ/äGëòî ðö\ö/î /ëòð

σî ð+åðæí ècðøñÊæñ î ø/ðZüõîiä;îiäôëòðøñ ëóôè

1σü

F¯J¯J¯® Q³C²(®(§¯V O

HJ01G,&-Þ Ì # gg:¼?/4phÑ44#D \g#:h n ##/ 1σ

âkâ7h

hYgÑggªc ∈ C

Ð4/#/§hgÑggª©×#gh4|hgM

È|_Ðq4ÐhÑÑm ∈ M

#4Ñ gÒc = E(1σ, ke,m)

#gôØf4Ô[D(1σ, k, c) 6= m]

#g4Ñ4Ô#ÑhâHhâ##4Ñ §|_ØfhÔ

[D(1σ, k, c) 6= m] ≤ 2−σ

!#$#$%&(')8aY%&q0Z,qÞAhðhÑ44ï# |_4&##/

1σ, ke

âKh gÑggªc1, c2 ∈ C

4/#/ÈhôgÑggªc3 ∈ C

P|_Dq4DhÑÑm1,m2 ∈ M

D#4Ñ gm3 = m1 m2

h c1 = E(1σ, ke,m1), c2 = E(1σ, ke,m2)

#gØf4Ô

[D(A(1σ, ke, c1, c2))] 6= m3]

Ò#g4Ñ4Ô#Ñh

Ï¤/q4YhÑÑï¼8FhÎ##â#È#h44##Òg:¼?/4:¼?:g©Èg:¼?/4:¼?:gßÍ# # ïÜ4|hÑ6#46g:¤¼Y|_§#gD¯/:hyªgghÑ44ï#g4#/DhPg#g:¼?/4yh#Ý#Þ4#Ý# #\FâKhf.Í©Cgh4gD4Ó4gÛ#Ý##Ô#ÑÈÎ4\¼&h #Ýg#\¼?/h#h7#ògh4gÔ#/#h#ògh4gÒ#ggÑÓ4ggÏóM

hQ # Ó4×ëqgïÜ í 44#Y#gy#ò#gòÒFhÑÑ m¥m¥ C)/ØÙ_L¨B¨BÊL1 h #hÑ44ï#

AFhÑÑ # n #g:Í#Ý#gÝFhÑÑ ¨ ¢ ØÙ) ØÙ )/ØÙL¨B¨BÊL1 h #ÈhÑ44ï# A

ÒFhÑÑ çQ#ÑF

01#$IQ Ö6zÖý Ö

54ú¿h|_§q4ÒÝ#444##g#g:¼?/4P¯#gÔ8D/ªggª§§g#gñhÑ6YhÎh#×|_È# É h©#g##g:¯?gYh#ò64Ñ¼?#4ñ_ÑÑ¼&Ô84# #ògg#¤¼P|hh\g

σ ##×g8F_ g4#/_4#g

4 Ì #Ýgg#.¼&h6g\gâ \î|#4#gâh gÑh<#444##Èg:¼?/4:¼?:FhD#DhÈh#|hÑÑ¼q4Òg:¼?/4:¼?:Fg

Ïó©#g#g:¼?/4 hÑ44ï#E4\h # 4|hÑA##/y##q4 h h #ÝÔ6g

rhÐY\

Zâ6#Ýg#g:¼?/4ð#gF_ÑÑ G£ £CØÙý/) h#g:ÍFhÑÑ ¥ Ú

§/¨¦ ý/) t7g#gY§Pg:¼?/h¼?:g ò#4Ô|hÔ#Ñ:Ý#g×Ô6gÑ4#y¯ÓªghÑ åKggªg##g:¯?§Ý4#gh4 g6g #4Y#h 4ß?#ÝÔ6g

r ∈ Z Ú /Ò#h|_

F¯J¯J¯® Q³C²(®(§¯V h

hÐÔ6\q4# gg:¼?/4ChÑ44#!gYh# \g#:hâªHhA#g¾:#:©4#g.Üh4Ô6\Ñ4#4#È4ÓªgCg##g:¯?FëHù?gDÔ8gÑÍkÙA#h#Ñ 4 :/ P# íÝÖ #:#g4hâ?P#4Ô|hÔ#Ñ:hâK#444##Ýg:¼?/4:¼?:g #×hÑ44#

A#4#Ñ Ô6#4Ô|hÔ#Ñ:hâ

?&ð# Y#P##/g##g:¯?g Ö 4Ý#:h#ghâf#ÝFh(Ô8yFhÑ Ô¼ h##Ñ¼?#YÔ#Ñ #YhÑ44ï#ëqgÝé\î|#4 4 :/ih í 4&ðë \g#: í \4#/_4Ûh`#g#g:¼?/4Qh7#ò# #\h hA#D#Qâ|g6g\Ó4gÑ¼4 LIKÏ Ö6zÖý=Ö Ï¤p#Yq4ÑÑRÍ#yÍ§YÍÑÑ<4ò#Cgg#¤¼p|hh\g

σh #

##Ô#ÑÈÎ4\¼ÛÛ# gg/4&h`#ÝhÑ44#×gûpÍÑÑ2ÍEke

(m)4E(m)

q4E(1σ, ke,m)

h Dk(c)

4D(c)

q4D(1σ, k, c)

Í#g×#D# g:h #`84Ô#ÑhÏó##gÐ#4Ô|_Ô#Ñ:ÒÍ©ÍÑÑ8hÑòÍ

Eke(m)

4E(m)

h<Í©gÑÑ6hEke

(m, r)4E(m, r)

q4E(1σ, ke,m, r)

Ö #:#g4hâ/Í©DÍÑÑKÍ

(E(m), E(m′)) = E(m m′)

g#hò|_Ò#hÑ44ï#¿¿ëqg#gÒ # 4çQ#Ñ í h##Ñ 4y¤Í©Yg#\¼?/h#hA#gh4g

m,m′ ∈ (M, )h 4/#/hÛg#g:¼?/4Ûh

m m′â8Hhâ8#4Ñ

|_D(

(1σ, ke, Eke(m), Eke

(m′))) = m m′

¯/gg/Íy#g4Ñ4Ô#Ñ#4Ô|hÔ#Ñï¤¼4D\yIL#$&(W0 Ö6zÖR-Ö 7gÍ§×4Ó4×hô¯#h#Ñh§ \g#:hâÝ#Ñ#ÑF_ïÓ4gÑ¼Û#_Ü44##ð#g h hk¯#h#Ñðq4y(#4Ô|hÔ#Ñhâ # Ó4gÑ¼#444###gh

54 Ì #Ð\ÑñhFhÑöù? #g\ hG<ªÕhÈ¯#h#Ñfh# \g#hâR#Ñ#ÑFïÓ4gÑ¼#444##Òg:¼?/h¼?:g4

M = (Z NZ, ·)âÍ#g

N<## #\©hK.Í©

Ññh4D#gg<g##g:¯?|hgòÍ©È|FÓªC = (Z NZ, ·)

h hÎ4\¼y|hgòÍ©|FÓª

K = (ke, kd) = ((N, e), d) | N = pq, ed ≡ 1 mod ϕ(N).Ì #Èg#g:¼?/4

hÕÈgh4m ∈ M

\î|# hEke

(m) := me mod Nh q4 gg:¼?/4Ch

×g##g:¯?Eke

(m) = c ∈ CÍ©Èg4#/

Dke,kd(c) := cd mod N = m mod N.

n ÔªÓ?4#Ñ¼4â`#Pg#g:¼?/4 h#P# #\h¤Í§ðgh4gFh(Ô8P/ªggªÑ¼

F¯J¯J¯® Q³C²(®(§¯V k

g4#/ Ôª¼PÝ#Ñ#Ñï¼?##òg4g64 #g##g:¯?gâ|Hhâ

Eke(m1 ·m2) = (m1m2)

e mod N

= (m1e mod N)(m2

e mod N)

= Eke(m1) · Eke

(m2)

Í#gm1,m2 ∈ M

7g#ghâ|#ÈhÑ44ï# çQ#ÑFhyFhÑ¼CÔ8ò#Ñg×g hçÛ#Ñï

(Eke(m1), Eke

(m2)) := Eke(m1) · Eke

(m2).

ã|hÑÑ¼YC#òöù?*#gÈh§Í©gÑÑ2hÒPhÒ#ggÔ|h 4C# 6ª\#Ñ.¼hTh\4#ð#ygg#¤¼(|hh\g

σ#yÔ#×Ñg#h(h

N Ö 4#:h#ghâ

σ = 1024g44Pgg#¤¼y|hh\gg

4 Ì # ì 4Ñ Í§hg.Ü:çQFhÑ##ghâ?#464 P O 6ÐhY¯#_#Ñh2ò#4Ô|hÔ#Ñ.Ühâ2 # Ó4gÑ¼Q#444##g:¼?/4:¼?:g 4

M = (Z 2Z,+)Í

C = Z =

(Z NZ)∗Í#g

N = pq§#ò# #\h7¤Í§×Ññh4D#ggAûpò|RÓ4

K = (ke, kd) = ((N, a), (p, q)) | N = pq, a ∈ (Z NZ)∗ :

(a

p

)=

(a

q

)= −1.

ù?#g#§#gò§#4Ô|hÔ#Ñ:hâ#Dg#g:¼?/4QhÑ44ï#ß4\_ # 4|hÑ##/h 4¿ÓhhÑ#

r ∈ Zfûp \î|#

Eke(m, r) := amr2 mod N

h

D(ke,kd)(c) :=

0,

cá|h

1,gÑ

Ïó#4Ñ §|_

Eke(m1, r1) · Eke

(m2, r2) = Eke(m1 +m2, r1r2).

Ì ##ÈhÑ44ï# # FhyÔ6ò/ªggªÑ¼P#Ñg×gª h#Ðh #

(Eke(m1, r1), Eke

(m2, r2), r3) = Eke(m1, r1) · Eke

(m2, r2) · r32 mod N︸︷︷︸Eke

(0,r3)

= Eke(m1 +m2, r1r2r3)

F¯J¯J¯® Q³C²(®(§¯V j

Í#gm1,m2 ∈ M

h r1, r2, r3 ∈ Z

;¿h§|_<hfF ¼gª4# #fhÑ4_Ü#*#4#Ñ Ô8©#4Ô|hÔ#Ñ:hâHhâh74Ô/h#Ah 4*#ÝÔ6g

r3hA # 4|hÑ

##/F##Ô#Ñ¯Ü¤Îh\¼"#444##yg#g:¼?/4¯#gÛ4''ëqgïÜ í #

(M,+, ·)Fh'Ô8

\î|# ðhðh|hÑ444#Yh##ggòù?#ô#ggDg4#:Dhf¤Í©yhÑ44ï# # h çQ#Ñq4×#Q#444##P#46g:¤¼"#:F h4#&hÑ44ï#

Aâ§Hhâ©YY # ïÜ

Ó4gÑ¼Yh Ý#Ñ#ÑF_ïÓ4gÑ¼×#444##_Ð#hh<ù?#P#gghFhÑÑ Ø Â V£ ØCØÙ_L¨B¨BÊL1 HJ0mÔ;C Ö6zÖý3Ö Q # Ó4gÑ¼#444##\#g:¼?/4y¯#gD4QyëqgïÜ í #(M,+, ·)

<FhÑÑ Ø ïL¨B¨BÊL1 6#g¯/:©È#4Ô|hÔ#Ñ:hâ4¯/6g\ 64Ñ¼ªÜ#4ñhÑ`YhÑ44# çQï/ Ü:çQ#Ñ|_Ý4 ##/

1σ, ke

âs ∈ M

h h gÑggc ∈ C

4/#/hÛgÑggªc′ ∈ C

|_q4hÑÑm ∈ M

#4Ñ |_F©m′ = s ·mh

c = E(1σ, ke,m)#gÛØf4Ô

[D(çQï/ Ü:çQ#Ñ

(1σ, ke, s, c)) 6= m′]#\4Ñ4Ô#Ñh

Ì #ð FhÑñ_Û#444##ô¯#g× ÛÛ64Ô#Ñg4#/ h g#g:¼?/4E(1σ, ke, s · m) = E(1σ, ke,m

′)hÒQ# #\h§¤Í§Ûgh4g

s,m ∈ M4Óªg#

##Ô#ÑÝÎh\¼ke

h hðg#g:¼?/4c = E(1σ, ke,m)

_©4#Ýgh4mh #h#g

gh4sh#Ññh¯?F

n ÔªÓ?4#Ñ¼(h¼ ¯#gy|_hÑ4gÔ#hFhÑÑï¼ #444##P×FhÑñh×#44×4##hâ?# LIK Ö6zÖ Ö ûpDÍÑÑ g#hòÔ¼

çQï/ Ü:çQ#Ñ(m,E(m′)) = E(mm′)

D(çQï/ Ü:çQ#Ñ

(1σ, ke,m,Eke(m′)) = m ·m′

#4Ñ Ò¯/gg/ÍQ#g4Ñ4Ô#Ñ#4Ô|hÔ#Ñï¤¼4HJ0mÔ;C Ö6zÖ xgÖ £ØÙ ¥ 1Â$ Ø Â CC1¨ ×ô#4Ô|hÔ#Ñ:hâ<64Ñ¼?#4ñ_Ñ©yhÑ4_Ü#QâKÍ#4ô##/

1σ, ke

âÕh c ∈ Eke

(m, r)Í#g

r ∈ Z_ 4Ñ¼Û¯#4g

4/#/òh#h#gDg#g:¼?/4c′ ∈ Eke

(m, r′)hmÍ#g

r′ ∈ Z#4gô##q4Ñ¼

_h 4Q

!"$#&%'$( *)®,+³²-+ ' ' .F¯J¯J¯® Q³C²(®(§¯V, 5:

Ö 4Ò#:h#ghâ|y×#4Ô|hÔ#Ñ:hâ/#444##g:¼?/4:¼?:g4(M, )

#òÔ#Ñ #hÑ44ï#FhðÔ6FhÑ Ôª¼ðh##Ñ¼?#Y#hÑ44ï#

A4&#Ýg##g:¯?

ch h

g#g:¼?/4Qh7#ò gª.¼YgÑggM

01#$IQÒ Ö6zÖýfÖ ÏóM

y44##&Z nZ

M

Cî|#hâ4PZh#g:Íhâ

#gQ#ÝhÑ44ï#çQï/ Ü:çQ#ÑF_&FhÑ¼PÔ8#Ñggª ##Y 4#Ô#ÑÈh #hÑ44ï#Q Ì #©<g4ÝÔ## ÍCÈÔ#Ñ #hÑ44ï#)K##g<#4Ô|hÔ#Ñ: 4 <Hq7g#ghâ?\Ó4g:¼Y # Ó4gÑ¼Ý#444##Òg:¼?/4:¼?:g4

Z nZ4

ZÐhÑòFhÑñh

#444##yh #ÛhÑ44ï# çÛ¹/ Ü:çQ#ÑYFh"Ô6Q/ªggªÑï¼(#Ñg×g ëqghÑ h O í 01#$IQÝ Ö6zÖýuÖ ù?#gÈ4:hA#¯/:#Y # ÓªgÑ¼C#444##Dg:¼?/4:¼?:Fëqh#âL PZ<?âk/â OGN â OGO â O k?âGhk/â Nm4 âP N í h \î|# 4

Z nZq4A4

n ∈ N#§hÑ44#

çQï/ Ü:çQ#ÑFhQy4g#ghÑ2Ô8ò#Ñggª /ªggÑ¼4 /

02143 ½Ä\ÇÈÅ65.7 ¶¸· ¾ ¶ ÄgÄ-8:9 ÂÉÊÂÉÊÂ;7=<Æ · ¾?>@78A<PËÕÂ+»B88»ËÕÅÉ »

ÝhÑF ¼ôgª4# #Y¯/:g#gChh /ªggªh gg#PhÑ4gÔ#hFhÑÑï¼ð#4_Ü4##Ýg:¼?/4:¼?:g |hDÔ6ggpPÑ4#P:h #P46gôá#g:4Ï¤ð#Dg\4ðÍ§î|:§#ggª§gÑñ_ Í©4Î×g4# g#ò#Ð#4Ô#ÑgQ Ì #gF_ngÐÍ§ ggÔ8#gÑñÜ4##×Ô6\¤Í©ggPhÑ4gÔ#hFhÑÑï¼Ý#444##Ò#gg§h #444##Ò#ggÐ46ggñhÑª#4?Ü.hÔ6gÑhò44##g`çQ4<#gggÑ¼4âÍ§<#RÓªf|_AÒ#444##`g#g:¼?/4#gP4p#Y#4?Ü.hÔ6gÑñh44#

(S7, ·)âA#C:¼?\44# 4 \Óªg(gÑggªgâ

hÑÑRÍyg4#:#\ÝhhÑ4gÔ#hFhÑÑï¼Û#444##Ýg#g:¼?/4ô¯#g×4(F2,+, ·)

PhÑ4gÔ#hFhÑÑï¼#444##g#g:¼?/4P#g4

(F2,+, ·)FhPhÑÔ64Ô/h#

q4!#444##Ðg#g:¼?/4¯#gÒ4Ý#§6ggñhÑ/Ñ#Fh744#(SL(3, 2), ·)

RÓ4gF2

Ö #:#g4hâ|##g #×#g4:¼yh&_Ñ4gÔ#hFhÑÑï¼Y#444##g#g:¼?/4Q4h&hÔ#h:¼Cî|#È#4î|gÑ g4#Ñ Ô64Ô/h# 4Ó4g&#444##òg#g:¼?/4#g4C4#h#g#4?Ü.hÔ6gÑñhhh##F Ì #gD4Ô#g:Ó__4#§\4#Ñ Ô8Dòî|:§g4ÑÓ4f#<#4Ô#ÑgjÍ#\#g7/ªggªAh gg#§hÑ4gÔ#hFhÑÑï¼#444##f#gg¯/:F×çÛhª¼ðh/#4D|RÓ4 P4ÑÓªÝ##4Ô#ÑgQÈÏ¤Õ5 jGjGO â Ú 4#gh ü2/4|FÓª#RÓ4gô|_D# gòCFh4|hÔ#Ñh#/4ð\Ó4g:¼ \g#:hâhÑ4gÔ#hFhÑÑï¼

CED+ëqôëòðøñ ëFVèNñ õëä)ëãñøùQÿøä)î ñ î /ë;î ðñ ëòö/ëòíÊäòü

!"$#&%'$( *)®,+³²-+ ' ' .F¯J¯J¯® Q³C²(®(§¯V, 5G5

#444##g:¼?/4:¼?:g FhðÔ6Ô#4Îhgô&#Ô?Ü¤¯/64#gªñhÑAB5VPH Ì #DYF¼Ô6Ygg(hÝQ#gª_Óg#Ñg4#gg##Û#Y¯/:g#gPhÑ#4#4p4:¯/:#Ûg:¼??Ü4:¼?:gQâ`h#âA#Cöù?¯#g×ôëqgCÙA#h#Ñ 4 :/ P í 4ò#CÙfÑ ì hYhÑf#gðëqgù?g\4 N P#65 í âKFhðÔ6ÝÔ#4Îhgð&#Ô?Ü¤¯/64#gªñhÑ7hâ8?# Ö #:#g4hâ6fÍ©hggÎ#Ûq4hÑ4gÔ#hFhÑÑï¼ð#444##×##Ô#Ñ¯Ü¤Îh\¼ô¯#g×gÝ4 YhÑÑ`î|gÑ 4È#4#&h

M = F2

â|4ÔªÓ?4#Ñ¼C#&#gÈ|hÒ×Ô6ò#4Ô|hÔ#Ñ:Ô8ògg#hn #g©h/#4Ð|FÓ4 òî| gh _g<q4ÐhÑ4gÔ#hFhÑÑï¼Ý#444##Ò#gggÏ¤A5 jGjGN â Ö gÑÑÍÒh ú4Ô#Ñ É #gg hQhÑ4gÔ#h##Ô#Ñ¯Ü¤Îh\¼Pg:¼?/4:¼?:gFhÑÑ Ø74ÑÑ¼ ÷ hÎhgp Pm:HDÏóhÑ4gÔ#hFhÑÑï¼y#444##Èh #RÓhhÔ#Ñ¼Qgg#hÈã/q4:?Ü|_gÑ¼4â7#Y#gY|hQ#Ô8gh 6ªg#Ñgh È#hÝ/ªggªÝg4#gg##Û#g##g:¯?òÑg#h Ö :Ñ¼4âØ74ÑÑ¼ ÷ hÎhgDDC84Ñ¼?#4ñhÑ¹Ü¤Ô|_ :¼?:\Q Ì #g\q4hâg4#/#hP\#g:¼?/4yh2#D# #\

E(m1 ·m2)hÕ¤Í©gh4g

m1

h m2

Ô¼Ý#Ñ#Ñï¼?#C#g4g84 #Ûg##g:¯?84Ñ¼?#4ñhÑ

E(m1)h

E(m2)â7ÑF ò

hð¯/84#gñhÑAÔ#ÑRÍ#ðÛ#Ý#ÝÔ6gh<4#4ñ_ÑgF7g#ghâ ##Cg6F_ ghÜ#/_4#§#gòhQ¯/6h#gñhÑ2Ô#ÑRÍ#yP#òg##g:¯?Ñg#hù?gg4 Ñ¼4âhÑÑÕ¯/:#Y#:hªñ_4#_ÐØ74ÑÑ¼ ÷ hÎhg/åKgq4 q#:#g RÍÔ|hÎëqgÛh#âF O : í Ì #\¼hÛg#g#gg#Û#gQ#\¼"#gg#ÝÔgg:h_h¯Îgâ#\¼(hP?ð#/ªggôÔ6y#h\FhÑHâf4Ý#\¼ Ñ?4C#QhÑ4gÔ#hFhÑÑï¼p#444.Ü##ò#48g:.¼4¤7g#gÈ§ThÒq4¿gÑFh#RÍ #ÛÎ h`#ggg4#Ñ Ô8ò## ªÝ/ªggh gg#ÈhÑ4gÔ#hFhÑÑï¼#4h4##g#g:¼?/4y#ggg< \hÑ h|hÑ¼?h Q gg/hQh7#gò¯#ggFhyÔ6Dq4# A Om4 HÏ¤ 4 :G: 4 â7à/Õé4#4_Ü Ö gg¦ N h \Óª\Ñ46 P#4Ô|hÔ#Ñ:hâÕhÑhgÔ#hFhÑÑï¼Û#444.Ü##Ègg\:Ü¤Î4\¼Ûg:¼?/4:¼?:gQ Ú /hÔ6\q4È##g#h/ªg\#gÈ#gDhð¯/64#gªñhÑ`Ô#ÑÍÒ#ô&#\##g:¯?ÈÑg#h ##Cg6F_ Ý#Ñ#ÑF_4#gÖ #:#g4hâ#ÒÍ§hgggªÑ¼PÔ#4Î4gQÔª¼Pûph4#gh ÛÚ h k 4 â N HÌ #<g4# g##444##©g#g:¼?/4¯#ggf444##`#:F h6#4`gg×4C#4#Ûð4ÑÓ4Y#P#ggª á#g:4Ïó×Ô##4Ý#gÑ4gð:#\#g|_C|FÓªÛÔ6gg'#gggq#ÑÑ¼# g:¼?/44h#¼h Ì #Ûq4ÑÑRÍ#ô#g4g #RÍ|_ g #PFh(q4hÑ4gÔ#hFhÑÑï¼#444##Y#gg×Fh(Ô6P #g #òFhQq4Ò#444###gg4Q6gghÑ2#4?Ü.hÔ6gÑñhP44##gT+'-0Z%01# Ö)LÖ)LÖG L ¡ ØCØýÁq 1Â / §/¨B C ¦IH ¢ Øý KJ

LM G L/_ONGý/C b Ø Â V£ ØCØÙ_L¨B¨BÊL1 g ) L/¨B+ (F2,+, ·)M

!"$#&%'$( *)®,+³²-+ ' ' .F¯J¯J¯® Q³C²(®(§¯V, 5 4

PQM G L/¼ONGý/C L¨B¨BÊL1 g ) L/¨Bì CL/¨p¨B/) qÂ ¢ (S7, ·)M

Ì #òq4ÑÑRÍ#Ý#?hAÒÔ|h 4Ûhy Fh Ì 4Yhù/h ggR ¡ M 1 ⇒ 2

Ì # g\4phÒ#C#?hÒq4ÑÑRÍ× ñ_gÑ¼h Ý#4Ñ Èq4ÝhhÔ#h:¼Qî|#Ý44#ô#g46g_4#h<î|#44##DFhhÑÍ§F¼?DÔ6#Ñggª Ôª¼ Ú ?4ÑFhygg#gÐü2\

S7

Ô6Èg#ggª h×#Ô#\h0, 1l

â#Í#g_#l = 21FhðÔ6Ý#4gTSâh Ñ\

CÔ6Ygg#Íð # 4&h Ý#Ñ#ÑF_4Ûª_g|_

hÎhgh§##/©#DÔ#|h:¼Yg#ggª_4#Òh7gÑggªm1,m2 ∈ S7

h 4/#/§#Ô#|h:¼Èg#gg_4h

m1m2

AÏó8Í©§|RÓ4Òh×hÑ4gÔ#hFhÑÑï¼È#4×44##Ðg#g:¼?/4#g

(K,E,D, #

,çQ#Ñ

)h

(F2,+, ·)#gÍ©ÒFh \î|##444##©#g

(K, E, D, ˜çQ#Ñ ) 4 S7

Ôª¼ \î|##E(m) = (E(s0), . . . , E(sl−1))

Í#g(s0, . . . , sl−1) g#hg×#yÔ#|h:¼ g#ggª_4"h

m ˜çQ#Ñ××g4#:#\ Ôª¼(#Ô#:/#ð# # 4pª_g

CÔª¼p # h #CÝ#Ñ#ÑF_4ª_gÝÔª¼ çQ#ÑF

Kh

Dh

\î|# P#ò4ÔªÓ?4#§ÍÒF¼42 ⇒ 1

Ì ##h©|hD¤Í©Q:g#g Ö :Í§×#Pg4#:#\4h Ú g?Ü n òh p÷ Ñ\Ó45G5(#RÍ|_P#ðî|gÑ ëF2,+, ·

í FhkÔ8ôg#g '#ð6ggñhÑDÑ#FhY44#(SL(3, 2), ·)

RÓ4gF2

Ì #gÍ©Yh##Ñï¼ôC#g4g q4#¾:g\Óª×4\4\:¼&y#Í|_

(SL(3, 2), ·)#Ô#44#Qh

S7

Ì ##RÓ4g#ògÑñhyûpòYh

(F2,+, ·)encoding→ SL(3, 2)

x 7→

1 0 x

0 1 0

0 0 1

=: M(x).

ÏóÒFh:¼CgD|_F54M(x)M(y) = M(x+ y)

4 ∃ T ∈ SL(3, 2) :

TM(x)T−1 =

1 0 0

0 1 x

0 0 1

h

UIVîiðûãëlog2(7!) ≈ 12.3

éVñÊõëq÷ïî ðî ÷ìæçZÿGøä ä)î ç ë(ðVå÷ëòí-øùî ñÊäîiäl = 13

ü

!"$#&%'$( *)®,+³²-+ ' ' .F¯J¯J¯® Q³C²(®(§¯V, 5 N

∃ S ∈ SL(3, 2) :

SM(y)S−1 =

1 y 0

0 1 0

0 0 1

:

M(xy) = [

1 y 0

0 1 0

0 0 1

,

1 0 0

0 1 x

0 0 1

],

Í#g[A,B]

g#hg#òg4Ý/_4§h7#òY_ggAh

B

Ì #g\q4ò#ò46g_4#W!h ·

F2

FhQÔ6ò6g:q4 4M(0)

h M(1)

###ôg4#:h

Sh

Tâh #ô#Ñ#ÑF_h''#ô44#

SL(3, 2) ì Ó4gk

#444##g#g:¼?/4P#g(K,E,D,

çQ#Ñ)4

(SL(3, 2), ·)Í©FhC#§FhÑ¼

g4#:#\_PhÑ4gÔ#hFhÑÑï¼#444###g4(F2,+, ·)

##E(S), E(T )

â/h #ÈhÑ44ï# çQ#ÑFûpC#RÍ¿#Y#CT_\Ý|_

SL(3, 2)È#C44# hg4ÑÑ#F_4#È4 #C#¾:\\Óª

#Ññh#Ó4gF2

ëqgòh#ú < N í fÒ#§#Ññh#|húhÝ64ªgâSL(3, 2)

FhyÔ6DgÝÔ6 # S7

â#Í#Qg4#gÑ gÒ#ò#?h. 01#$IQ Ö)LÖý Ö "h|hÑ¼?Ýh#P#?hh Ì #g4g 4 65465y#RÍ|_×##&# ggÔ6 g4#:#\4Y#Y_gg

SâTâ/h

M(x)h#\ggh:¼q4<#g#g ##

ûpò|FÓª〈M(1), S, T 〉 = SL(3, 2).

7g#ghâ`#gC#&#48gÝ#Ô#44# hSL(3, 2)

|_ÝFh Ô6C# Ûg#g F2

#yhÔ8Ó4CÍ§F¼4ûpP # 4|hÑÑï¼#hC|_

S7

#PYhÑÑg:Ý:¼?\Y44# Í#

SL(3, 2)FhyÔ8DgÝÔ6 # #g

SL(3, 2)g4ªh#ÒhPgÑggÒ_74 gúh? Ö 4

#:h#ghâ##Dq4ÑÑRÍ#gÑggª|h4 g\h?

0 0 1

0 1 1

1 1 0

.

ôªgg:#Pá#g:4ðÍ#\#gD#g×hæ¤YhÑÑg.êÈ4æ¤#Ñg.êÈ44##|hS7

!"$#&%'$( *)®,+³²-+ ' ' .F¯J¯J¯® Q³C²(®(§¯V, 5VP

h SL(3, 2)

|_hÑÑRÍ g#g ò#Dî|gÑ F2

Ö 4#ò#?hAh Ì #g4g 4 65h65òÍ©ò4Ô/h g\Ñ¼#Dq4ÑÑRÍ#Ýg44ÑÑñh:¼4*+%W)WIQ, Ö)LÖý=Ö Ì #òq4ÑÑÍ#:_ggªhògá#ïÓhhÑgªF

54 Ì #gÈ¯/:hQhÑ4gÔ#hFhÑÑï¼Y#444##g#g:¼?/4Q#gÈ4(F2,+, ·)

4 Ì #gQ¯/:ð#444##Cg#g:¼?/4"#gQ4 #y6ggñhÑÒÑ#FhÝ44#

(SL(3, 2), ·)

01#$IQ Ö)LÖR-Ö Ï¤¯#h\Ñ¼(#QhQÍ§F¼hY(#Û#?hh Ì #g4g 4 65465ÛïY64Ô#Ñ×g#g D#Dî|#Dî|gÑ

(Fp,+, ·)P#ò6ggñhÑ2Ñ#Fh§44#

(SL(3, p), ·)

7444##yg#g:¼?/4#ggCh44##|FÓªÛÔ8gg'¯?g#Ó4gÑ¼": Ö 4#:h#ghâÍ©Ð|FÓ4Ð#444##`¯#gg74ò44##

(Z MZ,+)âq4

MÔ8g#Ò?h

#Ô8gÒëqh#âL P O â j â OGN í â_q4M = pq

Ô8g#höù? #Ñ#Ë O k/âP N Hâªh q4744##((Z NZ)∗, ·)

Í#gNDhöù? #Ñ#gÑÑAÎ#RÍ/ªggªh gg#×¯#gg

F¼h©#444##<4hÔ6gÑñhÈ44##gâ4#RÍ©\Ó4gS7

h SL(3, 2)

h©#4?Ü.hÔ6gÑñhù/h ggâ 4##h J ## hkÕ| hÎh ¯/#ÑgÑ¼Í#\#gÒ#gò#444##g#g:¼?/4y#gD4P#4?Ü.hÔ6gÑñhY44##g Ì #D #\4yhÔ6RÓ4DhÓh_gÐ|_§#` g h64:hª`á#g:4×h |_f#4?Ü.hÔ6gÑñh44##f#4#Ñ Ô6Ò: q4#Ò6g6g\Ó4hã/q4:#|_gÑ¼4â?Í§DFhy#h ggÔ6ÈhP/ªggª#444##g#g:¼?/4y#gD4×#4?Ü.hÔ8gÑñhP44#Q#gÈg#gg<Ñ#4#4y#4?Ü.hÔ6gÑñhP44##| Ô8ggÛ#\Ó?4#Ñ¼# Pg4#:#\Èg#g:¼?/4ô#ggCëò < j â O hâk N â-<: í #g#Ñ#P#ggDÍ©g#h#444##òy#Èg#|_Í©#g q4gh#/#/ªggªÑ¼Q4Qg#g:¼?/ _Qëqg ÷ |h/g N í Ï¤A <: ì 44\ÓYh ØA4#4Yhg#Îh#464òÝ#\Í \î|#4Ch7#444##g:¼??Ü4:¼?:gò4ðÍ#ð#¯¼ðÔ|hP\# yg4#:#\È#444##g:¼?/4:¼?:FRÓªgÕhÔ#h:¼î|#`44##2#gÑ #§#4?Ü.hÔ8gÑñh44##g Ì #g2g4#:#\4ò\# ÐÔ|h 4#Th\Ð|_§\Ó4g:¼î|#44#C©hYg#4##Yh4hÕòqg# #\hÐî|#×\¼?gÑ44##g×ÏóÈ#gÈ¯/:#y#444##Ýg#g:¼?/4#ggÈ4ôî|#\¼?gÑÒ44##<hfÔ##Ñ #Ô#Ñ?ÎfD4Ô/h#444##©g#g:¼?/4#gg<4hÔ#ïÜh:¼î|#44##gÐù?#g#g##g:¯?§|hgh#4Ô/h# g#g:¼?/4Y#g

!" !X% T ' ³²-+K)³¯¬TY+¬°.Zq®¯4®V)³[,c¯\=F¯J¯J¯® Q³C²(®(§¯, 5<

qg# #\h<44##h&¯/64#gªñhÑAÔ#ÑRÍ#&hf#g##g:¯?DÑg#h# ##g6F_ g4#/_4#Ðq4ÑÑRÍg Ì #òFh4Q©|_Ò#DÑg#hPh7#D# #\h7¤Í©gÑggª

xh

yh&qgy# #\×(hg#ghÑÒ#y# h#yÑg#h(h

xh

#CÑg#h hyb7g#gC#gÈg##á#P/åKgÝq4 #Ph× FÍÔ|h¯ÎphÝÔ6\q4

h # ?g#hÑF hQ/ªggg:¼?/4:¼?:FQï¿hD|_##×#g4#:#¯Ü4Y§©64Ô#ÑÝg4#:#\#444##g#\¼?/hP¯#g4Y#:¼?\44#

S7

h 4×#6ggñhÑ|Ñ#Fhf44#SL(3, 2)

`ÏóKÍ©g4Ô##§#fÍ Ì #g4g4 65465©Í©§F_g4#:#\<hhÑ4gÔ#hFhÑÑï¼D#444##<g:¼?/4:¼?:g)4#Ðî|#Ðî|gÑ (F2,+, ·)

fã/q4:#|_gÑ¼4â?#¯/64#gªñhÑ6Ô#ÑRÍ#CRÍ#È#g4#:#\4C\# Q#Ý#4×44##òg#g:¼?/4&¯#g4

S7

h 4SL(3, 2)

âKg8g\ÓªgÑ¼4â6Í©4#Ñ ÑF DhÝ¯/84#gñhÑ?Ô#ÑÍÒ#

F2

_ #g#g§ÑFRÓ4gA#©á#g:446gÝ8h/ªg\hÑ4gÔ#hFhÑÑï¼Y#444##\¼?/4:¼?:g4

F2

¯/:gÏ¤ <15/ ì 44\Óh Ø74#4Yhg#Îh&#464Ch#h#gÝ\# &g#g:¼?/×hÔ#h:¼î|#ò44###444##FhÑÑï¼4 Ì #\# Ô|h 4y# 6ªg#Ñ¤¼Ph7#ÈgÜÔ6g#k#4Ô#Ñg q4P44##yhªg4gPY_gggâÍ#Ñ& <:DyPÔ|h h # 6ªg#Ñ¤¼ChÕTh\4##q7RÍ©\Óªggâ8hÒÔ6\q4#Ò#gò§#h/ªggªF Ö #:#g4hâb <15/ÕhyhÑ4gÔ#hFhÑÑï¼#44×4##g:¼?/4:¼?:gRÓ4g§î|#Dg4×/_ïÓª#4§#464 â|Ô#/RÍ#×Òg#D É DÒ/qFhÔ#Ñ×#Ñg×gF

021I0 ½]<^<yÄ · ¾ ¶ Ë · ÂÁ» ¶ ÁyÆ _`7`Âa<ðÅb77Ë · Å¼» ÂYÅc9 ÂÉÊÂÉÊÂ;7=<Æ · ¾>@78A<PËÕÂ+»B88»ËÕÅÉ »

k##gg FÍÔ|hÎ'hÝ#444##ð\¼?/4:¼?:gÛP|_Û_hÎQ4ªP¯Ü#Ñ4#g # 4|hÑ2:#\#h Ö 4#:h#ghâ6##Y#ÑñhÛöù? hG<7q44###â|#Ý#Ñ#ÑF_4ðh©¤Í©Û4|#gD¼?gÑ ÈPÓhhÑ 4|_#×hÐ## #\hÐ#×¤Í©g4g64 #Qgh4ggYÑ#4#4ð#gYh×Yhª¼&Í§R¼?òÛFÓ44 #¯ _hÎòq4#:h#gÔª¼#h##ÑF_h×hK|hq##\4#gâª##h # h#\¼×4<#4Ô|hÔ#Ñ:#gggâª#76hgªñhÑÍ©FhÎ#g<ÑF A#A#©á#g:4ÝÍª¼Ý#444##Ð#ggh# 4|_4#f#:F hKg4Ó4gª4|hÑ8g:¼?/4:¼?:gg Ì #YhFh4q4§#òªgg:P#444##g:¼?/h¼?:gÒ§Í òh##ÑF_hPg46h Ì #gh×#g4\FhÑÐhÈÍ©gÑÑÐh#h¯FhÑÐh##ÑF_h#ò åKggªÝhFh_Òg:¼?/44h#¼h

!" !X% T ' ³²-+K)³¯¬TY+¬°.Zq®¯4®V)³[,c¯\=F¯J¯J¯® Q³C²(®(§¯, 5 O

Ï¤&#Èq4ÑÑRÍ#×Í§Ñ:4hf#YhÛh##ÑF_4#h #46g:gh<#444.Ü##ò#ggh #Yh#ò FÔ6g# #\QfûpDÍÑÑ ggÔ6òhòh7#gh##ÑF_4#§y4 \hÑK#4#4#4/Ò#Dq4ÑÑRÍ#Ý|h/gg

•Øfhg\4yhA4Ô#ÑDh4ggn #hK#4:Ðªgg:#h Y gYh #_##ÑF_4h2#444##Òg:¼??Ü4:¼?:g`Õ#©#hg\4h8×4Ô#ÑÐh4gªgA`ÕÍÒhA#RÍ Ì #g4g 4 65465Ý#444##g#g:¼?/4y#gD4QÝ6ggñhÑ#4?Ü.hÔ8gÑñhC44#PÍ©4#Ñ ÑF h(hÑ4gÔ#hFhÑÑ¼#444##Cg:¼?/4:¼?:g 4 #Cî|#Cî|gÑ

F2

ù?#ghÑÑÕg4ªÓªgª4|hÑ7g4#/gDh#g\#gDhÔ|h 4ðÔ#|h:¼y:#hDh 4#Ñ¼gá#ÝÝ#Ñ#ÑF_4&h # 4â6#¯ô#444##Èg:¼?/h¼?:gÍ§4#Ñ håKg#Ý84Ô#Ñ¤¼PCg#g:¼?/òÍ#4Ñ#44hY|_:ÑÑÕ¯/gg/hÔ#Ñh7g#ghâ`Èg4#Ñ Ô6# Û#hg\Ý4Ô#Ñh4\hªh#:ÈYhÑg4#ò#4:ÈÔ¼g#g:¼?/#×#gQÐçQ4 \hÑÒhÔ64/Ò# FÝFhQÔ6Dq4# b hGhH

d ÷ 4#/#ÝÍyg#g:¼?/ q##\4#gÌ #©©8ggñhÑ6Fhh2##hg\4YhÕ×4Ô#Ñh4gªgfÏ¤P#Pgg|h.Ü4Ðògg\©q##\4Ð##Ô#ÑgÑ¼Ý\ÓhhÑ|_ ×#PDÍ§R¼|_Ð#q##\4gYh#Dgg\FCã#Q#444##g:¼?/4:¼?:gÈ#×g#g:¼?/ q##¯Ü4QFhðÔ8Ý¯ÓhhÑ|_ Í#&4|hhgg#Óhh\¼4 Ì #ÍÑÑÕÔ6Èq#:#g g# P#ò#¯?|h/gg

d ÷ 4#/#ÝÍyg#g:¼?/ Û _/74h4##D#gghåKgÒ#È84Ô#Ñ¤¼Yg4#/ò##Ô#ÑgÑ¼YÍQ¯Üg\ _D#|_<fgYh#fgg\F Ì #fFhÔ6 4#Ôª¼g#g:¼?/#ò# _×Û Ó_h#gÝh #gÛ¯/#Ñ4##È#444##D#46g:¤¼PYghÜ#/CÍÒ g#g:¼?/ _/ Ì #yhFQæ¤g4#/#ÛÍ g#g:¼?/ " _êFhyÔ6ò#Íy×Ô6ògá#ÓhhÑgª#òFhòhÕæ¤g4#/#ÝÍyg#g:¼?/ q##\4#.ê Ì #Ý FÔ8g# ÍÑÑ2Ô8q#:#g¯/#Ññh# Q#Èq4ÑÑRÍ#|h/gg

•çQ#Ñ|h:¤¼Cg4#/_4Ï¤ Ý#Ñ|h:¤¼ðg4#/_4#ggÝ\Óª\_Ñ§|_\ÈÍ§hª&g4#/YÛghÜ4â##Ô#ÑÛq##\4 4'#gP##/CÍ#Ñ&Î4gg## #gP Ó? |hÑ##/#Óh_h Ì #fÔ6gÑ4#4`D#hFDh8æ¤g4#/#Í×g#g:¼?/ Y _ê`ã|hÑÑ¼

!" !X% T ' ³²-+K)³¯¬TY+¬°.Zq®¯4®V)³[,c¯\=F¯J¯J¯® Q³C²(®(§¯, 5h

ðÝ#Ñ|h:¤¼Qg4#/_4&#h?g4ÑÍ©×|FÓ4C\Èhn ≥ 2

#ÑñF¼ªgÍ#gFh g4#/#ÛÍ g#g:¼?/ " _Ûgg|h4

n = 2 Ö #:#g4hâf Ý#ÑïÜ

|h:¤¼×g4#/_4×#h?g4Ñ`#q##\4×|_©#4#Ñ Ô6g4#/ <##Ô#ÑgÑ¼Î#RÍâ/Í#gFh§#DhFÈhæ¤g4#/#òÍYg#g:¼?/ y _ê©©#Óh_##/Èh©4#|h:¤¼4ß#h?g4Ñ`q4D#Ñ|h:¤¼&g4#/_4#ÍÑÑ`Ô84Ó4gù?g\4P# O

•ù?gg\|h#¯#gggÏ¤&gg\|h##g\Ó4ghÑ7|h:g|hg\¯D|_#Y|h:¤¼yFhgg4#:#\Ò#gg\ÒÔ¼YgÑ.âÔ#/Ò2Yh¼|h:g©g?48g_#\¼ChDhÔ#Ñgg4#:#\F7ghâ/#ò#444###48g:.¼C#Ñg©|_§#òg4844h7#ògg\g8|hghò|hgh7#ògg\g6g4844

•Ì #g#4Ñ #gggù?gg\|h#Y#ggDh Ý#Ñ|h:¤¼Pg4#/_4y¯#ggh¯#h#Ñgh#g#4Ñ #gg|_hòÔ|h 4y#ò#444###46g:¤¼4<Q¯#h#Ñh`Ý#g#4Ñ Q gg:¼?/4Q#gÈÒ4Ó4gy ÷ |h/g¼P#

• e g_Ü¤Î#RÍÑ 4D#?hqgÌ #©©òq# hgªhÑ6#ïïÓªh2g:¼?/44h###hg4Ñ©h #§hC¯#hÜ#Ñ`h/§#g4\FhÑ4h##ÑF_hh/#44×4##Ag:¼?/4:¼?:gg e g_Ü¤Î#RÍÑ 4#?hq©h# Ý#RÓ4Î#ÍÑ 4h24#Óh_/q4Y_4 Ö 4Ð#:h#ghâg4# g<#F_Í#gDò#g©|h<#RÓª#Ð gª¤¼ÝÝÈ#4:©Ôª¼Ñ444#òÍC#©hgg4#Òh #Óh_|h:Í©4 n ÔªÓ?4#Ñ¼4â?Y#¯y#hg4Ñ8##\Í§hªf#f#Óh_§/q4Y_4yëqHhâª#f|h:Í©4 í ò:F¼Ý#Óh_h #hfDÔ6ÑFhÎh ###§#h?g4ÑH e g_Ü¤Î#RÍÑ 4©#?hqA4|hhg©|_A#§#h?g4Ñg4Ý##F_g¯#h\Ñ¼Û#Î#RÍÑ 4Ý|_DÍ§hDg âÕh #ë É g í ¯?Î#RÍÑ 4hô¯#h#Ñh©hô#4#g::ÜóÓ4gî|g É g_Ü¤Î#RÍÑ 4×#?h<|_ò#g#Ý#444##È#46g:¤¼&4Ó4gðôù?g\4^P#i<? N Ö 4q#:#g¯#h#ÑghÉ g_Ü¤Î#RÍÑ 4ò#?hqÒ##×#ò#h44###46\¤¼PgB 4N H

•ÙfÑg\4P#gggÏ¤ôgÑg\4ð#ggD#×#444###46g:¤¼ð#RÓ? gòC4ÑAP4Ô/h&#hÑÑ¼ô4Ó4g #Cg#g:¼?/ Ó4hgÝÍ#4/ gg:¼?/#&#P Ó? |hÑ<Ó4hggôÏ¤ù?g\4P#ihÍ§DÍÑÑK¯/#ÑñhP#§q#:#gg

!" !X% T ' ³²-+K)³¯¬TY+¬°.Zq®¯4®V)³[,c¯\=F¯J¯J¯® Q³C²(®(§¯, 5k

•û _gYhÎ#×h î|#4\#ª##gggé4hÑfÍ§_gYhÎ#ðh î|#4g##Û#gggÔ8 # 4|hÑ</q4YÜ4' 4hÑ _/ Ì #ð#444##Û#48g:.¼C# ( # YhÎY#\Ó?4#Ñ¼yg#g:¼?/ ð _/Ï¤Û4g#ghÑÕÍ§_gYhÎhÈ# Y gn¼C#RÍ#gß_gÑÑgAh 4hÑ44? Õg##©#©g4ª¼?4ªR7Ï¤î|#4g#ª##gg#Ò6g4×Í#òÔ#/¼?f# _#4#Ñ Ô6Ò gªî8hÔ#Ñ§Ôª¼Ý#Òg|hªfòg##|_ _òf#hÐÑÑgªhÑÑ¼ :Ô#/ Ö #:#g<#46g:g<h#Y¯#ggÐFhÔ6Dq4# A h1524Y 4 H

• n Ô#ÑÓ?h#§h#:qggn Ô#ÑÓ?h#Òh#:qgh#h#gg:¼?/44h##È#ïïÓª_§ã|hÑÑ¼PQ×¤Í©_Ü¤|h:¤¼5\Ü¤4/:Ü¤hñÜ 4 4Ô#ÑÓ?4#§h#:qg#h?g4Ñ#Dî||h:¤¼yg Ô#×#ògg4 |h:¤¼ (#&Í§F¼ |_#Pgg4 |h:.¼ gggÓ4g×ÍÒ(#4Ô|hÔ#Ñï.¼

1/2â

Í#4/Ò#òî|:|h:¤¼PÎ#Í#Í#\#g4Ò#h#ògg4 |h:.¼PgggÓ4 #Ô#F&¯#h#ÑÈh<#ô#h?g4Ñ2|_#g##444##ò#48g:.¼QFhÔ6Dq4# A O 5H

•÷ 4×gªÒ#ggg÷ 4×gªÕ¯#g\fhÐh#h#gÕq# hgªhÑª#ïÓ4<h#g:¼?/44h##<#_Ü?g4Ñ7#gh¼hÈÏ¤ôCg4gª#g×Y#ÑñR¼4gYhÎhgDCg4×gªFF7hÔ#ÑÝy#?4CÓhhÑ#Ýq44\h g4y#D#4g×#|_ò#Fh'# Ñ4#4gY|h#4&#C 7 ?gC#hP|FÓ4& \ÓªFhÑ#C#4gðhÑïÜ#4#4Q#ÈYF¼ C_4òÑñ_gÒ64ªF ÷ 4×gÒ¯#gg|_#ò##444###46g:¤¼CÍÑÑÔ8ò #g Ûù?g\4P# k/

•ü2hg:¼P#hg4Ñgã|hÑÑ¼×g:¼?/44h##§Ñhg:¼Ý?#ÝÔ6g`64ª##§Í###Îh\<|hYÔ8¼¾:4ªÑ¼yh h 4Ñ¼C¯#4g&Ôª¼ÛhÑÑ2|h:g|hªgÒã#C#444##g#g:¼?/4Û#gò#FhQÔ6ÈFhÑ É hÒq4ÑÑRÍgÐÙ<hÛ#ÑñF¼ªg#?4g×h?Ü 4 #Ô8gDÍ#p#×g#g:¼?/g Ì #gp##y#×#444##Ý#46g:¤¼ð#g#g:¼?/4Q_7#ò# h7#Dh 4ÓhhÑ#gFhyÔ6D/ªggÑ¼Cg4#/ Ì #g4ÝÔ#|_4Ph7#h #g#4Ñ Q gg:¼?/4Q#gCëqÑÎ4h#Ð#484 ù?g\4P# P í ÑF Ò# g q##\4|hÑï.¼h<ù?gB P 4 2q4§q#:#g \hÑg

!" !X% T ' ³²-+K)³¯¬TY+¬°.Zq®¯4®V)³[,c¯\=F¯J¯J¯® Q³C²(®(§¯, 5 j

•çQïÜ¤#\gçQïÜ¤#\hY#h?g4Ñò|#RÓ? Ch#4¼?¤¼ðq4g g×Ôª¼pg4ÑÑg\#Qg?Üg:¼?/ gh4gq4\Ó4ghÑ7#gg Ö 4Ò#:¯_#ghâ64#òFhÛg4# gïÜ¤#\|_`g4ÑÑg\`g##g:¯?<h 4/#/`#§g4g64 ##Ññh¯?AÝh 4Ñ¼6gÝ/ 4 ggÏ¤&#¯ôgg|hC#Óhh\¼Q4Ô/h# Ôª¼Qgá##|_#6gÝ/_4â/Y_####/§×4/#/gâ|Ògg\hª¼44#ò¯/gg/#òïÜ#\F©Ï¤Û|h:g#Ññhgâ \g##×g4g\##/ß_4/#/|hgâ|Hhâ6×g##g:¯?ÍÛg4g64 #C#Ññhª¯?Fâ8#4#Ñ #hÔ64È\åKg\Ó4#g&4#g#C4#_h 4Q ghÔ#ÑÈ#46g:¤¼QCÔ##Ñ #ôïÜ¤#\¯Ü¤g#g:¼?/4ÍÑÑ<Ô6Cg4# Ô8gÑÍòâ`##Ó? Ô¼#C#Ph#444##g:¼??Ü4:¼?:ghÒÔ##Ñ #Ô#Ñ¯Î8 Ö 4Ò#:h#ghâ|gt P j Õh Nm4 q4 \hÑg

ÙA#h#ÑghA#\q#Ñ2#46g:ghA#444###gghD#Dq4ÑÑRÍ##

•ö¯Ü¤h 4 É hÔ#Ñg#g:¼?/4%ß_¯Ü¤g#g:¼?/4ö¯Ü¤h 4 É hÔ#Ñ§g:¼?/4:¼?:gëqg_#ì < 4 í hÒ#4Ô|hÔ#Ñ:©g:¼?/4:¼?:FÍ&#× # 4|hÑ7#46g:¤¼&|_D4Ó4gð###Ô#ÑÝÎh\¼

ke

h hôg#g:¼?/4Eke

(m, r)hÕÈgh4

m ∈ M# \©###Ô#ÑÎh\¼

ke

h Èh 4#Ô8gr ∈ Z

ÝÝ84Ô#ÑYð/ªggªÑ¼pg4Ó4g:Eke

(m, r)ªðh#h#gg#g:¼?/4

Eke(m, r′)

|_ÝÝ8g:qg\Ñ¼p :#4#|hÔ#Ñ×q4 yæóqg?êCg#g:¼?/4 hm# gÒ#ò##Ô#ÑDÎh\¼

ke

Ì ##48g:.¼PhÑ×FhÑÑ ¯Ü¤g#g:¼?/4n ÔªÓ?4#Ñ¼y\Ó4g:¼Q#4Ô|hÔ#Ñ:D#444##Dg:¼?/4:¼?:gÞ¯Ü¤h 4 É _Ô#Ñhû #4/Ñ4hf4g#ghÑ¤¼CÍ§h#È|_#g:¼?/4:¼?:gÞ # Ó4gÑ¼P#_Ü44##h ì Óªg

Eke(m, r)

h ###Ô#ÑÎh\¼ke

Í©FhCg4#/Eke

(0, r′′)q4h 4 #ÝÔ6gr′′h #g#gÈg4#/

#(Eke

(m, r), Eke(0, r′′)) = Eke

(m+ 0, r′) = Eke(m, r′),

Ír′Ô6g#Yh&h##4#ñ_òh 4#ÝÔ6ggì¿hò|_#¯#h\Ñ¼PÍ|_

Ô#Ñ #×_Ñ44# ëqgÈé\î|#4 4 :/ih í ?gg

•öh 4¿gÑñÜ¤ #gÔ#Ñï¤¼6Ñ4# Ík#84Ô#Ñ¤¼hÝ¯Ü¤g#g:¼?/4jg4gy##46g:¤¼ hÝh 4gÑñÜ¤ #gÔ#Ñï¤¼Qg4#gg##P#×#4Ô#Ñg_©g4#/#C##Ññhª¯?q4Þ#

!" !X% T ' ³²-+K)³¯¬TY+¬°.Zq®¯4®V)³[,c¯\=F¯J¯J¯® Q³C²(®(§¯, 4 :

g##g:¯?Fô¿g:¼?/4:¼?g FhÑÑ h 4gÑñÜ¤ #gÔ#ÑYhª¼ hÑ4hï#|_òFhÔ#FhÎðP#4?ÜóÓ?ñhÑÕqh\4ôhÐg##g:¯?ÈFhphÑPÔ#FhÎðPh 4#:h#gÒÍ4#î|Fhª<#4Ô|hÔ#Ñï.¼4 Ì #<#46g:¤¼×fq#:#g g# ×h# NGN 24¸ hkH

•ä<gî8hÔ#Ñg#g:¼?/4#ßThÒg#g:¼?/4#gÏóAhPg#g:¼?/4yÐÓªgî8hÔ#Ñ§#Ó? gÒÝg|h#Ý¯#gÎY#g4g\#ghg#g:¼?/ _&Í#4/\4#4#&ggg\¼ª Ö 4×#:h#ghâ<#×#¯Üq#Ñ7ÛÓ4h#C#ggCg4ªÓ?#ghª¼Û4Ô#g:Ó4g|_#Ýg#g:¼?/ |hhÐFh _hâ/Hhâ?#Dg#g:¼?/ ÓªhD§ g C#DÑ:§h7Fh _ggf*g:¼??Ü4:¼?:gÍp#Ý#48g:.¼|_ÝÔ|h 4 #444##g#g:¼?/4#ÝFhÔ6×q4# h 4 ëqgYhÑ&ù?g\4AP#ih í 8¿h|_Èô#Ñg_#ÝÓ4gî8hÔ#Ñg#g:¼?/4#hÈhÑFhÑÑ Th§g#g:¼?/4#g

|~ f

Î ~ÐÓhg i j JÑÒ~ k | |

|Ï j JÑÒ~ l $

Ì #FhCq4©hC/ªggª_Ñ4gÔ#hFhÑÑï¼#444##\¼?/4:¼?:g¿©Ñ4#È:h Ü#P46gô#4Ô#ÑgQÈY¾:4hÓh_4&##_\\4ôhÐ4Ô#Ñhhgòh #gg#&g4#/_4(Í"g#g:¼?/ q##\4#4g#g:¼?/ ' _phYhÑF ¼(g?Ü4# Ûù?g\4 4 4 <ãªÑ#RÍj#×/ªggªhÑ4gÔ#hFhÑÑï¼#444###gò|hÔ6ggq4# ÑÑ`Fh _gòq4#4#gg/åKgDq4#h FÍÔ|hÎ"ëqgù?g\4 4 65 í A#\¼Y|FÓ4 FhÑ6ÍChC¯/64#gªñhÑKÔ#ÑRÍ#CY#g##g:¯?ÒÑg#h ##Èg6F_ Ý#Ñ#ÑF_4 Ì #Ð<#Yh×Fh4YÍª¼×#gÐh##ÑF_hg48g:\ q##\4#Ò|FhÛÔ6ò#Ñggª Ôª¼

NC1gg#gâ|Hhâ|gg#h

Ñ4ªhï# g/ Ì #h#g\# hò#g ×g|hÔ#ÑDgg#\4#/_4#gûpYg4Y#pÍ Q#\Í¿h##ªh¯pq4ò#È#hÔ#ÑgQQÏ¤#:F h \Óª\Ñ4##ðQ#\Í#gÛ4g4# g#pgg#Í§yhÎ4ÛgÑ4gÑ?4Î _YÔ#h###p#44hhCg4#/_4|hÑ# gÑ|q4fq##\4#gfã#¯/:#È/ªggªÒ # Ó4gÑ¼#444##g:¼?/4:¼?:gÑÎhØ`hÑÑgg ©g:¼?/4:¼?:F O k2#RÓ? g?4Ñ6g#g:¼?/Ô#_#¯###44hÛ4 gYg|hÔ#Ñgg#Ýg4#/_4#g Ì ?#Q#|h/gÍ©Ý#4hg:Ý#Ó_hÔ#Ñ¼Y\g#h #4?Ü¤ªgh\Óª\# g4#/ÍPg#g:¼?/ Q _h g#g:¼?/ q##\4#|_DhÈ4Ó4g&Ôª¼Û84Ñ¼?#4ñhÑ2Ô#h###Y#44hg Ì ##î|:È4Ñ/4&##yÔ#h###y#44hÍ©h×FÍÒhh. Ö #:#g4hâ2DòC#\Íh##ªhÛYg#Ññh4ò#gÑñhh`q##\4#|_h/ªg\Ñ¼Û_ #4?Ü¤ªgh\ÓªgÑ¼g#g:¼?/hÔ#ÑÈh :ÑÑK¯/gg/hÔ#Ñòq4

NC184Ñ¼?#4ñhÑ6Ô#_#¯##×#44hF PH

mQ"$#;n ®,+¬²O Q³6¬ ( Zq®/¯ ( ®,+* 4G4

çQ4gRÓªggâ©Í© \î|#&ô#\Í#48g:.¼(h#444##Pg:¼?/4:¼?:g|_Í©QFhÑÑÎh\¼&:Í§h#### Ì ##48g:.¼QhåKg#\Í 84Ô#Ñgyæ¤g4#/#×ÍÛg#g:¼?/ _êòh æ¤g4#/#ÝÍQg#g:¼?/ q##\4#.êDgg|h4g Ö |hÑÑï¼4â?Í§ò#ggª¤Í©¯#h#ÑghA#444###gg##64:#×Îh\¼y:Í§h####

oA143 p 7 ¶ Áy¾qÆ · ÁPÇq_r7AÂÈÇ;7 ¶ É »

Ï¤Û#\\4ÛÍ©ª #gò#Ô|hÈg4#gg/hfÔ#h###Y#44hÒ|Í§ÈÍÑÑ#g #q4ÑÑRÍ## Ì # \î|#4×Í§#gg§Í§h©gggÑ¼ª #g Ôª¼ ÷ hg\ÈhÑHKÕ 4 hHÏóò\q\PÔ#h###y#44hÓ4gòhÔ#h:¼Û#4gÈÏóD#gÑ g##h4Q#|hÑÑ¼P4Ó4gÛQÑg_#hâ|h#â6$ kPAh6ggñhÑÕFhh Ì #g\q4È#È#\Í \î|#4pÝh6RÍ§g:q#ÑÐÍ# #gg|_ÝÎ#Í Th\Ý:ÑÑ<#4Ñ q4È#Y#\Í \î|#4Qh |_Ô64# 4ªÒÔ8ò#Ó4 HJ0mÔ;C=Ö)LÖ)LÖ Ä£ g 1 1Â Â ¨ h7 É λ 4Q##/ x1, . . . , xn ∈ Rn

â/Í#gRChhÔ#h:¼(##âÒCpá| ##Ñ

BP = (G,w, s, t)Í#g

G = (V,E)Y

g\ h\¼?gÑ4h#ÍVÔ8g#È#\§hKÓªg:gg§h

EÔ6g#È#\§h2 4ggâ

wÑñhÔ6gÑÑ#q##\4Qh

sh

t_¤Í§×6ggñhÑÓ4g:gghA#D4h#

ûp×YF¼ðh#|_V = s = 1, . . . , t = λ

h |_Dq4DFh 4(i, j) ∈ E

#4Ñ ò|_

i < jyü2\

wh4ÛFh 4Y g4g¯Ü5Y64Ñ¼?#4ñ_Ñ`Ó4g

Rô#

##/ÒÓhhñhÔ#Ñggâ/Hhâ

w(i, j) =n∑

k=1

alkxk + bl, alk, bl ∈ Rh

1 ≤ l ≤ λ2.

Ö 47Fh g\ |_φ = (i1, i2, . . . , ik)

q4i1ikGâ_#\Á¤/ Â 12h φ \î|# ×Ô6D#ò# #¯

w(i1, i2) · w(i2, i3) · · ·w(ik−1, ik)W (i, j)

g#hg#p§ ØÁ¤/ Â 1hAhÑÑ g\ |_#©q4

ijq4

i < jënÓ?\Í© hÒq##\4Ch

x = (x1, . . . , xn)í

Ì |_W (i, j) =

∑

φ(i,j)

k−1∏

m=1

w(im, im+1)

Í#gi = i1

h j = ik

h φ(i, j)

g#hg|_yq4ij

mQ"$#;n ®,+¬²O Q³6¬ ( Zq®/¯ ( ®,+* 4N

Ì #òq##\4f : Rn → R

g4#/ Ôª¼y×Ô#h###×#44hBP

\î|# Ôª¼

f(x) = W (1, λ)(x) = W (s, t)(x).

D\yIL#$&(W0 =Ö)LÖý Ö ü2\RÔ8òÝg4Ý/_Ó4#Í]54fÒhy¯#h#ÑhAÝÔ#h#?Ü

#ô#44h h É <ð4(##/ x1, x2, x3, x4

g4# g#Pq4ÑÑRÍ#ð4h#â<Í#gx1, x2, x3, x4, a, b ∈ R

1

2 x 1

x 4

5

3

4

x 3

x 2 + bx 3

a+ x 4

ûpò|FÓª

w(1, 2) = x1, w(2, 3) = x3, w(3, 5) = a+ x4, w(1, 4) = x4, w(4, 5) = x2 + bx3.

7g#ghâ|#DÍ©g4ªh7#ò|_φ = (1, 4, 5)

\î|# ×Ô6

w(1, 4)w(4, 5) = x4 · (x2 + bx3).

Ì #ÝhhÑÕÍ©g4ªW (i, j)

gá|hÑx1x3(a + x4) + x4(x2 + bx3)

_ ?##q##\4g4#/ Ôª¼

BP

f(x) = x1x3(a+ x4) + x4(x2 + bx3).

ÏóRYg4Ý/_Ó4È#×Í254â8#g4#/_4Ûh

f(x)FhðÔ8ÝYh#8 Y#

g4#/_4Ph7# \g|hªh`Y_ïCP#Dq4ÑÑRÍ#ÝÍ§F¼pëqgòh#ú <P í ü2\

x = (x1, . . . , xn) ∈ Rn Ì #g &h* ¾hgg#\¼jY_ï

H(x)Ô6gÑ4#4#(

Fh'Ô#h#¯##p#44hBP = (G,w, s, t)

Í#gG = (V,E)

ÐÏóDÍ©ðh#Û|_V = s = 1, . . . , t = λ

h |_Dq4DFh 4(i, j) ∈ E

ò#hÑ D|_i < j

â2#gH(x)

Dhλ × λ

Y_ïQÍ#g\:¼(i, j)

gá|hÑF:C(i, j) /∈ E

h w(i, j)

h#g.ÜÍh©ù?#g

G g\ h\¼?gÑD4h#

H(x)hQ##8gñh#4#Ñh§Y_ïYÍ#

ò#Ñ6hgªFYÏórD##Ô8gÈh§ 4gh©#Ñ4#4g:È|_q4

s = 1t = λ#g

H(x)r+1 = 0û #gð#P# g:h #Y64Ô#ÑÈÍ§ÍÑÑ2Í

H#:F

hH(x)

ü2\

k Ú C g#hYy|_|_g4#:Èh§¯#h\Ñ¼ k 4gg Ì #g:¼ (i, j)h©#

mQ"$#;n ®,+¬²O Q³6¬ ( Zq®/¯ ( ®,+* 4 P

6RÍ©gHk

gá|hÑÒ#DhhÑKÍ©g4ªh`hÑÑ g\ kÜ¤|_#Òq4

ij7g#ghâ

H∗ :=∞∑

k=0

Hk =r∑

k=0

Hk

§#òY_ïYÍ#4(i, j)

gª:¼y\á|hÑW (i, j)

©ù?#gHr+1 = 0

Í©È4Ô/hC|_

(I −H)H∗ = H∗(I −H) =r∑

k=0

Hk −

r∑

k=0

Hk+1 = H0 = I

⇒ H∗ = (I −H)−1.

Ö #:#g4hâA#Y##6gÝ4gª:¼(1, λ) = (s, t)

hH∗â g#h Ô¼

H∗1,λ

â`gá|hÑW (1, λ)(x)

Ì # \g#gÐ#4/#/f(x)

h2#q##\4f : Rn → R

g4#/ Ôª¼

BP Ì g4#/#Dg:¼

(1, λ)hH∗

Í©òFhÛh##Ñ¼ ÷ hgg #Ñh7g#g

H∗1,λ = W (1, λ) = ((I −H)−1)1,λ = (−1)λ+1 detM

det(I −H),

Í#gM g#hg©##Ô#Y_ïYh

I −H4Ô/h# Ôª¼ gÑ\#È#î|:Òg4Ñ#Ch

#ÈÑñh:ëλÜó í RÍòù?#g

HhÛ##8gñh#4#ÑñhÒY_ïCÍ©|FÓª

det(I −H) s 547g#gH∗

1,λ = (−1)λ+1 detM = f(x).

01#$IQE=Ö)LÖý=Ö ÏófÒ Ú 4ÑFhCq##\4C#gP# \g|hªÒgá|hÑÒg#gì:4

54â|Hhâ#g4×#/#Dq##\4PÓhhÑ#òÒKªggª gg ò7#òY_ïMh

q#ÑÑh#Î8D\yIL#$&(W0 =Ö)LÖR-Ö Ì # ¾hg\#\¼ Y_ïhÈ#ð4h# hÝÙA#h#Ñ N 654 4 Ñ?4ÎPhq4ÑÑRÍg

H =

0 x1 0 x4 0

0 0 x3 0 0

0 0 0 0 a+ x4

0 0 0 0 x2 + bx3

0 0 0 0 0

mQ"$#;n ®,+¬²O Q³6¬ ( Zq®/¯ ( ®,+* 4 <

⇒ H2 =

0 0 x1x3 0 x4(x2 + bx3)

0 0 0 0 x3(a+ x4)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

,

H3 =

0 0 0 0 x1x3(a+ x4)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

, H4 = 0.

ù?#gH∗ :=

∑Hkâ|§q4ÑÑRÍ©|_

H∗1,5 = W (1, 5)(x) = x1x3(a+ x4) + x4(x2 + bx3) = f(x).

Ì #§Fh4ÈÑ4ÎDgÑ4gA_AÔ#h#¯###44hÕÕ|_`\Ó4g:¼Èq##\4fFhÝÔ8©g4Ü

#/ Ôª¼CÝÔ#h###Ý#44hQâ/h Yhª¼ut¯|_#_Ñtq##\hCThÑgëqC|h:g#Ññhg4#ÑñhÝÑñh#4|h4g í Fh"Ô6Pg4#/ Ô¼ Ñ#Fh.Ü¤ É CÔ#h###ô#44hg Ö #:#g.Ü4hâ6#ÝÔ#h###C#44h É Ý#PÑñh4g|h&#g4g64 #Cq4Ý#Ññ× É hf 4 hHú7RÍ§\Ó4ggâK64Ñ¼?#4ñ_ÑïÜó É òÔ#h###Y#44hhÈ#4Ô|hÔ#Ñ¼y#h6RÍ§g:q#Ñ

g#4#4"/ªggªÑ¼(g4#/QhÑÑÒ84Ñ¼?#4ñhÑ¹Üó×Pg4#/hÔ#ÑPq##\4#g Ì #RÍ#yP#×Th\ò|_D#6RÍ©gÈh©64Ñ¼?#4ñhÑ¹Üó É Ô#h###Q#4h_òg4#g gÍP|_h åKggªÒÓhhñhªÒhAÑ4_Ü¤|hgDg4#/_4b <G<H 01#$IQÏ=Ö)LÖý3Ö Ì #P##6gÔ64# q4#C É PhÔ#h###ð#44h×Ýg4#/#Ú ?4ÑFhòq##\4#

f : 0, 1n → 0, 17¯/64#gªñhÑH Ö 4Õ6ggñhÑgÑñhgAh#q##\4#

×YhÑÑg§Ô64# FhÛÔ8È#RÓªg©Q¯#h#Ñòh7q##\4#|_FhÛÔ8Èg4#/ Ô¼Ñ#Fh.Ü¤ É §Ô#h#¯##D#44hfh Ú ?4ÑFhÝq##\4# f(x) = 1+

∑J⊆1,...,n

∏i∈J xi

7g#ghâ

f(x) =∏n

i=1(1 + xi)h #gQ¯/:YôÔ#h####44h h É n + 1g4#/#

f

D\yIL#$&(W0^=Ö%Ö$Ö ü2\

f(x1, x2, x3) = 1 +∑

J⊆1,2,3

∏

i∈J

xi = 1 + x1 + x2 + x3 + x1x2 + x1x3 + x2x3 + x1x2x3,

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ 4O

Í#gxi ∈ 0, 1

q41 ≤ i ≤ 3

Ì #g

f(x1, x2, x3) = 1 + x2 + x3 + x2x3 + x1(1 + x2 + x3 + x2x3)

= (1 + x2 + x3 + x2x3)(1 + x1)

= (1 + x3 + x2(1 + x3))(1 + x1) = ((1 + x3)(1 + x2))(1 + x1).

Ì #òq4ÑÑRÍ#ÝÔ#h#¯##×#44h¿hA É ¸P×g4#/g f

1 2 3 4 1+ x 1 1+ x 2 1+ x 3

01#$IQ=Ö)LÖ xgÖ gg:#(h##ÑF_4hÔ#h###(#44hÜ|hgÑ¼"#g4#:#\4&hf_Ü¤FhÑÑ h 4 É #64Ñ¼?#4hÑÒÜ©Í§hª #g 2 <G<?â< O `h q#:#gPhg#ghÑ 4 hH Ì #gôh 4 É #p6hÑ¼?#4ñhÑChð g#ggª_4h©q##¯4#òÔ¼ðÑRÍ§Ü g4g×64Ñ¼?#4ñ_ÑÍ#òg8ggñhÑÑ¼&#\q#Ñfq4D# g4hgg#ÈÝ#Ñ|h:¤¼Yg4#/_4P#h?g4Ñg

oA1I0 > ÂÉ<½CË · ÁPÇÃ · ËÆ|kÁy¾78A<PËÕÅÆ:~ ¶ Ë ¶

Ï¤#Yg\4Í©&g4# gCh64:hªYh##ÑF_hhD#444##yg:¼?/4:¼?.Ügg ¨\ ¢ ) 1Â ÁqCC g § ¥B¥m `ûpî|:Ò4Ó4DhC #\4Y#§#4Ô#ÑgÔª¼ ggÔ##yÈh 4Ó?#P¯#h#Ñgg Ì #gÍ© \ÓªgÑ4py#hg4ÑA|_È4ÑÓ4gD## gÑ¼?#ô#4Ô#ÑgQ n ##h?g4Ñ§Ý#4?Ü¤gh\Ó4hâ</ªggª×q4Ý64Ñ¼?#4ñ_ÑïÜó É Ô#h###Q#44hgâ7h #RÓhhÔ#Ñï¼ôgg#h Ì #gâ`ïÈÈy#\Íßh##ªhyî| Ü#RÓª 4Ñ/4#§q4©#D# gÑ¼?#×#4Ô#ÑgQ÷ 4# gÝ#Cq4ÑÑRÍ#ôù?gg|h 54ÏóÍ§h #g hB /¢ C /¢ C+- Ø ¢Q ) 5Òëqg×hÑ^ k O âPZh í h FhðÔ6q4# Û#ÝÑg_#òÍ&\Ó4ghÑ åKggª#h4#gâ8h#â6$ hk/â-5hH¼¿hò|_&ù?g\4 N N Í©g4# ggg|hYÎ#RÍðh ¨\ ¢ ) 1Â ÁqCC] g § ¥ú¡¢1g ) \â/|_Ògá#Ó_hÑgÒYù?gg|h54dG01-IQLÞÑgô|hÛ q##\4

f : Rn → R|_Û# ?gQ#hQÍ§hªQ\Ó4FhÑh 4

gg\##/x1, . . . , xk ∈ Rk

Ú 4Ôô|hò4gg\##/y1, . . . , ym ∈ Rm

Í#gk + m = n

ÑgÍÑÑ#Cyg4#/f(x1, . . . , xk, y1, . . . , ym) =: f(x, y)

4ô#g

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ 4 h

##/xi, 1 ≤ i ≤ k

h ðÚ 4Ô gg\D##/yj, 1 ≤ j ≤ m

|_4#Ñ¼ Ú 4ÔÛÑFh##Ò4/#/<h8#©q##\4 Ú 4Ôh Ñg§#4#Ñ ÑFh#h##DhÔ64/`#Òh#gf4#g##/ÒÓhhÑ#gÔ8g g#ò/q4Y_hC|_Ò#ò4/#/\Ó4FhÑgXYIQIL.Þ54 Ö 4`4h##ÑF_h#AÍ§Òg4#Ñ ##ÎÝhÔ64/

fÔ8g#ÈD##Ô#Ñ©q##\4Ûëqgh#

k O âQPZh í Ì #È4Ñ/4P4Ó4gyÔ6gÑRÍ FhQhÑ×Ô6ò# 4ÑÓ4D#Ògg|h#4 Ï¤Q4DFhgÑgD#4#Ñ ÑFhP#ò4/#/Fâ/#N n #Ñ¼CÑgD#4#Ñ ÑFhP#ò4/#/ f(x)

01#$IQ=Öý Ö)LÖ n ÔªÓ?4#Ñ¼ Ñgh ##/FhhÑðÔ6yggh&î// |h:×h#gq##\4

f7g#g

k = 0hÑ×64Ô#Ñh

D\yIL#$&(W0^=Ö6 Ö6 Ö ¿ÍjÍ©ò4Ó4ò¯#h#Ñgh7#ÈhÔ6RÓ4ògg|hYh §Óhhñhªg54Ýh ¯#h×#ÑYhù?gg|h$5Í©Cg4#Ñ ##ÎôhÔ64/Ñg|FÓ?#&h /ªg\hÑ44ï#Þq4DT_\4#P|_È##gÈQFh4#\¼&Íph #òÍÒhªòÎhgg&#Óh_h Ú 4ÔÛ|hhQ##/|#òÍ§h|RÓ4òTh\4 É ©ÑgòhåKgÚ 4ÔP×Th\4Ò#Ò##/Òq4Ò4#\¼4

4 D.¼?#FhÑ7¯#h#Ñhf#î|:ÓhhñhªÍ©Ýg4#Ñ ##ÎÛhÔ64/ÑgòÍÒhª#×gÑÑ2#gg4#/#6RÍ©gÒ Ú 4ÔPq4##Ô#ÑòhÑ44ï#Q

N Ïóf×##Ô#ÑYÜh×(ä<hñhJ5PÜh Ô6h"Ñgyh Ú 4Ô"#4#Ñ ÑFh #

4/#/ÜhÛäÐhñhª 4 Ü§Í§Ý|FÓª#Ýgg|hC#|hÑÑ¼Qg4# \ 4 Ü¤|h:¤¼g4#/_4#h?g4ÑgQÝh ¯#h#Ñ×Í§Yg4# g

fhÔ6g#Û#q##\4

|_©g4#/gÐ#Y/Ý#)h.Í©Ý##/g`Ñgh CÚ 4ÔC#RÍ #fY4 g

gg Í#Phf#gÞFh#D4È4#\¼4ÈçQ#Ñ|h:¤¼Qgh#/_4ÛhÑ/4#q4D#¯ Pgg|h åKgDq44#ò4Ñ/4ð#gð#q4gD4##4Ñgh<ÑgÝh ôÚ 4ÔôhÝ#|hÑÑ¼Û:¼?\hâKHhâ6#g4#/_4##\¼Û|RÓ4Ý6g:q4 hf#FhÑ¼#<hhAù?g©ù?g\4¸P# O q42#ÒÝ#Ñ|_:.¼g4#/_4#h?g4Ñ2h q#:#g \hÑg

P#ÑgÒÍ§hª#È ÝÈg4##g.Ü¤4##4×64ÑÑ|ÐhY¯#h#ÑhäÐhñhª N q7ghâÑg×g4ÑÑg\È##/ÈÓhhÑ#gh Ú 4Ô'ëTh h#gÈgg í &\Ó_hÑ|_×#\#ïÜÓh_gÑï¼4

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ 4 k

÷ ÑFhÑ¼Èù?gg|hY5ÐFhÈÔ6Ð4ÑÓ4 ÍÈ4g#ghÑª#h?g4Ñ2q4Õg\#<q##\4È\ÓhhÑ|_4ëqgÒh#â% k O âPZh í ;7RÍ§\Ó4ggâªÍ©§Í©4#Ñ ÑÎhÐ|RÓ44Ñ/4|_``#4?Ü¤ªgh\Ó4hâHhâ`Ñ\Yh Ú 4Ô #4#Ñ |FÓ4C4#Ñ¼ô4#Y4# h _Û¯/|h#4Ph #4#Ñ #h|FÓª×Û\4##F_Ýq#:#g ##y#\ÓhhÑ|_4ôh©#q##\h Ö #:#g4hâÍ©4#Ñ Ô6Ý4/YhÑÕP#RÓ? ×C4Ñ/4&ÛÍ#¯&#ÝÍ§hÎÑª |_òÑgÝ|h6g:q4 g8g Ð4#Ñ¼4#Ò/ªgg#\¼h8#Ò# gÑ¼?#Dg#g:¼?/4×¯#ghâÔ#/<#h4y#æ¤ É ¯êDh f â|Hhâ/#ò É òh`gg#g4#/# f ù?4f#4?Ü¤ªgh\Ó4`4Ñ/4#|RÓ4<Ô6gg ggÔ8 #fÑg_#hâRg<h#¤ hk/âG5hH7RÍ§\Ó4ggâ##4#hÕ#ghåKg§ g6g g#¯¼Ph2#æ¤ É ¯êh f Ì #DÔ8g:Î#RÍy#4?Üªgh\Óª4Ñ/4&y4#D\g|hQ ggÔ6 Ô¼ôé Ú FFÓªgòE O H×Ïóò#Ó? gÈ4Ñ/4×q4Ð#4 \g#:Ñ4_Ü¤|hgq##\4#ë)¿ü n ì ù/Ø6 ÷ Ù í h ©ÐÔ|h 4Ûg4#:#\4 Ôª¼ ù/h g\hÑH$ hkH Ì #CÑñ_g ggÔ8 Û#h?g4Ñ<##&#RÓªgÑg4#:#\4#©Í#Pgá#g§4#Ñ¼4#4# _Õgh\4Ý\Ó_hÑ|_

NC1gg#

h Í#ð4Ó4g Ú 4Ô&g4#/_4|hÑÕ#Ó_h\¼Qq4###/yh ÑgÈ/q4Y_4?Ü

#g4\f#Ó_h\¼q4#g2q##\4f7Í©\Ó4gÕÑgh Í©4ÎÑª D g6g Õ¯/64#gªñhÑÑ¼

4ô# g/dhÐ#×gg#

C Ì #g\q4hâÕ#D4Ñ/4ðDÔ64# ygg#Dh

Ñ4ªhï# g/&ÜÎ#RÍphNC1

gg#F Ú FFÓ4gt O f#RÓªg#4Ñ/4&4Óª\Ôª¼Ûù/h g\hÑH¤7Èª #gg#RÓ4gÑ7|hhÑÑgÑ É _4Pg##á#gh g4Ô##\#gÍ×#\# ggÔ6 hkH Ì #ÐÑg fò#hg4Ñ#ÍYDÑñ_g#\¼h;5|_#gÈ#g#?g\#yQh#\Ó4P4Ñ/4&q4¸¿ü n ì ù/Ø6 ÷ ÙÒûp×#RÍ)g4#ôÍP#\Íh##ªhôy#RÓ4Ý#g#ÑDq4U hkH n #òh##ªh¯ôD#4?Ü¤ªgh\Ó4Í#4/#g#?gg#Yh Ò/ªggq4Ò64Ñ¼?#4×ñhÑKÔ#h###×#44hgn ÔªÓ?4#Ñ¼4â7y|_#hÑfFh _Q4ÑÓ4Ô8hpgá#\gYëq#Qgh\4ôh ?Ü g6g g#\¼ h§#Y É Yh f í Í©4#Ñ Ô6CQ#4Ô|hÔ#Ñ:hâ7hÑ4gÔ#hFhÑÑï¼ð#44h##g#g:¼?/4#g©#gÐÕ#RÓ? gAÒ?4Ñ4q4Õg4#/_4#24g#g:¼?/ Ý _/(7Í©Ü\Ó4ggâ6hQ/ªggªh gg#hÑ4gÔ#hFhÑÑï¼Y#444###gò|h#hÔ6ggyq4# ¼4\F×ÈhÑF ¼&gª4# ù?g\4 4 65hÑÑ`Î#RÍ#ggÈ|FÓª FhÑAÍh¯/64#gªñhÑÐÔ#ÑRÍ# ##Q\6F_ g4#/_4#g Ì |_ÝòÍª¼#gh##ÑF_4g46©7g:\ q##\4#7|_AFhÝÔ8©#Ñggª Ô¼

NC1gg#F7Ï¤J </ÕÍÒh

#RÓªg|_§hª¼ Ú ?4ÑFhgg#<h2Ñ44h#× g/YFhYÔ6/ªggªÑ¼×Ý#Ññ_ Ô¼Dgg#<RÓªgÐhhÔ#h:¼î|#Ò#4#4ÑÓhhÔ#ÑÒ44#;7g#g##464 g#\¼?/h#hA#h?Ü.hÔ6gÑñhC44##Òh ì 44\Ó\hÑH/b <:/â%<15/Ðëqgù?g\4 4 65 í FhyÔ6Èh##Ñ Ý4Ô/hg#g:¼?/4#§h

NC1gg#gq7RÍ§\Ó4g§Í#ÈÈ#¯/84#gñhÑ6Ô#ÑRÍ#Ch

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ 4j

#gÐ#g#\¼YFh##h Ýh¼×Ô6\gÐ|hNC1

gg#g Ì #gâ|h#h#g©h##ªh ghÔ#Ñ×g#Ññh4#ògÑñhhAg#g:¼?/hÔ#ÑDq##\4#gn #4Ñ/4P#gg Ô6gÑRÍkq4ù?gg|h5òÒÔ|h 4QhQ FÝh`ö4|hÑ Û÷ hggÏóC|h.Í© #4 gªgkûðh#Q|_Y#Qq##\4

fY4Ó4gÔª¼Ô#h#¯##

#44h¿P#Dq4¿h7#òg4g64 # ¾hgg#\¼yY_ïK Ì #gyÍ§òg#g:¼?/|_Y_ïY##Ý#4Ô|hÔ#Ñ:hâ/ # Ó4gÑ¼h FhÑñ_§#444##g#g:¼?/4P#gëqg ÷ |h/g¼Pq4hQ¯#h#Ñ í Ì È#RÓ4Ò#gg#¤¼hK4#<4Ñ/4q4YhÑÑï¼ÈÍ©|RÓ4Òò4|hhªgÒ|_

R<î|#

î|gÑ âªhÑ#4#4Ý#h\g©Í©Ò#|hÑÑ¼YF¼Ýh#©|_`#Ò#Z NZ

Í#gNA#

# #\hA¤Í©YÑñh4D#gÔ6g|FÓ4gÑÎhò×î|#Dî|gÑ ëqgÝögYhÎ N 4 N í Ì ?#Í§g:\f#g4# g g\|hDD#FhÒÍ#gÒ#Òq##\4

f : Rn → R \î|#

4Yî|#î|gÑ R Ö #:#g4hâ#g×hÑÑ7Î#RÍ/ªggªÈ # Óª\Ñ¼Û#444##

g#g:¼?/4ô#ggÈh \î|# 4Z kZ

q44k ∈ N

DÍPg:\R

Ô6g#CY#òî|gÑ Ì #Í§ÝR¼y#Ý#4Ô|hÔ#Ñ:hâ6 # Ó4gÑ¼y#444##òg:¼??Ü4:¼?:gÈh# ì 4Ñ Í§hg.Ü:çÛFhÑA#g P O f4

M = (Z 2Z,+)â24¯#ggÈh

# n Î_hh_Ü.ã#¼ªhYò#g_ OGO K4 M = (Z pZ,+)Í

pÔ6g##h`ÏóÕ # ïÜ

Ó4gÑ¼Ûh FhÑñh#444##òg:¼?/4:¼?:gÍÑÑÕÔ8 \Ó4gÑ46 4ðhÔ#h:¼Pî|#î|gÑ ©ÜÐ_#g§|hy#î|gÑ §Üf#gQh#Ò4Ñ/4PFhy× ñ_gÑï¼YÔ8ò4g#ghÑ 01#$IQJ=Öý Öý=Ö ÏóÕ2hÑÒ64Ô#ÑAÒg4# gq##\4#

f : Rn → RÍ#g

R = Z NZh N = pq

D## #\h©¤Í§QÑñh4×#gg ì Ó4gx ∈ Z NZ

##4Ô|hÔ#Ñ¤¼|_

gcd(x,N) = 1 φ(N)

N=∏

p|N(1 − 1p) = (1 − 1

p)(1 − 1

q)Èù?#gÈ##4Ô|hÔ#Ñ¤¼

§#g4Ñ4Ô#Ñï¼gÑ4J54â|hÑ4:§hÑÑ6gÑggÒhÓ4g:Ô#Ñh Ì ?#gâ|C#h\gÍ§DYF¼h#Q|_

Z NZÔ6g|FÓªgYÑÎ4Qðî|#yî|gÑ 7RÍ©\ÓªggâÒ(#FhQ#Qgg#.¼

hÒ4#È#h?g4Ñ<Fh #hÔ6q4YhÑÑï¼ð#RÓ4g&ÙA#h#ÑgÈhÒ#4Ô|hÔ#Ñ:hâ7 # ÓªgÑ¼#444##×¯#g×gÍ

M = Z NZhYØ`_ÑÑgg ò¯#g×] O k§h #YgÑÑ/

g#:Ó4ØAhÑÑg§#gt P N 2Í#yÍ©DÍÑÑ ggÔ8ÈQù?g\4P# 4 LIK]=Öý ÖR-Ö ü2\

MÔ6§Y_¹ÈRÓ4g7#©gh4§|h\

Mh

EB : M → CÔ6

h×g#g:¼?/4q##¯4×# g<##Ô#Ñ§Îh\¼ke = B

Ì ##h_4EB(M)

Fh#A|_EB

h##Ñ 4ÛFh¯&gª:¼Û_Mh |_#g#ÑÍg&ÛYY_ïQhªh

û #gÛh##Ñ¼?##ÈhÑ44ï#© # 4çQï/ Ü:çQ#ÑÒ4y#¯Q\##g:¯?Y_gggâ?Í§h##Ñ¼Y#g4yFhQg:¼Pg|h_gÑ¼4Ö 4QÔ8\g# g:h #ÛÍ©î|:Ý4Ó4YQÔ|h×Óª\h hÒ4#È4Ñ/4â7h #g

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ N :

#RÓªÒ/ªgg#\¼Q_ng:Í§h gv IL.V)^dW)Þ54 Ú 4Ô g#g:¼?/Ý#È##/Ó_hÑ#g

yj, 1 ≤ j ≤ mÍp#È##Ô#ÑÎh\¼

B##&

#4Ô|hÔ#Ñ:hâ # Ó4gÑ¼Û#44h##g#g:¼?/4ô¯#gE : M → C

Í#gM = (R,+)

h RÈP#Ýî|gÑ Ýûp g#h#pg#g:¼?/4#òÔ¼

EB(yj)Í#g1 ≤ j ≤ m

Ú 4Ôyg #gÈg#g:¼?/4#Ò×Ñgh4 Ì g4#/ÐhÈ4/#/

f(x, y)|_7FhÈ4#Ñ¼DÔ8ÐF Ôª¼ Ú 4ÔÈ4Ó4gÈ#Õg#g:¼?/

##/EB(yj), 1 ≤ j ≤ m

_ Ñgh Ý##/xi, 1 ≤ i ≤ k

âfÑg g#q4ÑÑRÍ##ëT í ü2\

BP = (G,w, 1 = s, λ = t)Ô8CÛÔ#h###Û#44h hÒ É λ g4Ü#/#CÑgh q##\4

fÈü2\

HÔ6#Ýg4g64 #

λ × λ ¾hgg#\¼

Y_¹K Ì #gªgòhHh g4g¯Ü5×64Ñ¼?#4ñhÑRÓ4g

Rð#×##/

Ó_hñhÔ#Ñgxi

h yj

â/Thâ#\¼gá|hÑ ∑ni=1 alizi + bl, ali, bl ∈ R, 1 ≤ l ≤ λ2

Í#gzh ∈ xi, yj|1 ≤ i ≤ k, 1 ≤ j ≤ m, 1 ≤ h ≤ n

ëqgù?g\4 N 65 í Ñ\ gÑ\gÐ#î|:§g4Ñ#h #Ññh:ÐRÍ'h

Hh 4Ô/h#ÐÈY_ï

H−ù?#yg#Ññhgg×Fh¯"gª:¼ ∑n

i=1 alizi + blhH−

Í(g#g:¼?/ gª:¼

EB(∑n

i=1 alizi + bl)Û4Ô/h h g#g:¼?/ Y_ï

E(H−) ¿h

|_Ð#ÐFhCÔ6 4#Í#ÈÈ##444###48g:.¼h#g#g:¼??Ü4Û¯#ghÏó

zh ∈ xi|1 ≤ i ≤ kâHhâ`#gª:¼&hf#ÝY_ï

H−

×ô64Ñ¼?#4hÑ©"#g×##/Fâ<#gÑgPhÔ#ÑPg4#/EB(alhzh) g\Ñ¼ ## Ú 4Ô ##Ô#ÑPÎ4\¼

B¿hy|_ÑgPÎ#RÍ

alh

h|h:hÒ#gÈq##\4

fQÏó

zh ∈ yj|1 ≤ j ≤ mâ`Hhâ7#Ygª:¼ g8g 4

Ú 4Ô È##/Fâ7#g 4Ó4g g#g:¼?/4#EB(yj)

h Ú 4Ô ##/hpg#g:¼??Ü4

EB(alhzh)Fh&Ô6g4#/ ###ÝhÑ44#çQï/ Ü:çQ#Ñh`#

g#g:¼?/4Q#gD×4Ô/hçQï/ Ü:çQ#Ñ

(alh, EB(yj)) = EB(alhyj) = EB(alhzh),

q41 ≤ h ≤ n, 1 ≤ l ≤ λ2

Ì #g'4Óªg"#&g#g:¼?/4#EB(alhzh)#yg#g:¼?/×Fh¯

blh #g#QhÑ44ï# # ô4\h(g#g:¼?/4

EB(∑n

i=1 alizi + bl) Ì #ò#DhÔ#ÑÈCg4#/×hðg#g:¼?/4

E(H−)hA#gëq #g í ¾hgg#\¼yY_ïôëqgÝù?g\4 N 65 í

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ N 5

ëqÔ í Ñ\ gÑ\g`#©î|:<g4Ñ#Ýh #ÒÑñh:`RÍ"h8#λ×λ

gª.¼ÈY_ïI4Ô/hP#òY_ï

I− Ì #g&#È\4#/ghQg#g:¼?/4

EB(I−)h

I−Ôª¼Pg#g:¼?/#Fh¯Qgª:¼:h 5È## Ú 4Ô ##Ô#ÑÎ4\¼4

ëq í é#È##444###48g:.¼h#g#g:¼?/4C#gÑg©hÔ#Ñôg4#/

EB(I− − H−) =: EB(M)gª:¼(Ôª¼(gª:¼jëqg&ù?g\4 N 65 í

#g #

(EB(I−),çQï/ Ü:çQ#Ñ

(−1, EB(H−))) = EB(I− −H−) = EB(M)

Í#gD#ÈhÑ44#×§ # h çQï/ Ü:çQ#ÑhÈh##Ñ 4yFh¯Qgª:¼4ë íPÌ ÈÔ#Ñ âªHhâ4È# §#gg

(EB(M))i,j

h6#fY_ïKâ4ÑgÒ#?4g¤Í©Ph 4

(λ − 1) × (λ − 1)Y_gg

S, TRÓ4g

RâY|_

det(S)h

det(T )hÈ##

R§Ïó

f#h Ú ?4ÑFhyq##\4y#g&Ñgò#?h\

S, TÍÒ

det(S) = (det(T ))−1 Ì #gP#g4#/g

EB(S ·M ·T )é#

#© # Ó4gÑ¼òh FhÑñhÕ#444##<#48g:.¼òh##Ðg#g:¼?/4#g#Òg4#/_4PFhyÔ6 4#h<Ïó

(EB(SMT ))i,j = EB(∑

l

∑

k

si,k ·mk,l · tl,j) = EB(∑

l

∑

k

si,k · tl,j ·mk,l).

Ñ\D#g#ÈhÑ44ï# çQï/ Ü:çQ#ÑÒgh#/D#DÓhhÑ#gçQï/ Ü:çQ#Ñ

(si,k · tl,j, EB(mk,l)) = EB(si,k · tl,j ·mk,l).

¿hQ|_CÑgQÎ#ÍY#ÛÓ_hÑ#g(si,k · tl,j) ∈ R

Ì #g #&#gC#hÑ4hï#¿ # Y4Ô/h

EB(SMT ))i,j

ù?#gR×î|gÑ

EB(S ·M · T )§hyg#g:¼?/4PhAÝh 4¿Y_ïhÕ#Dhh#ÎChMëqgÈü2gY

N 4 ih í ëq í Ñ\Dg #òg#g:¼?/4

EB(S ·M · T ) Ú 4Ô

N Ì 4Ô/hf(x, y)

Ú 4Ôy#g#Ògg\Î4\¼P gg:¼?/#ògghEB(SMT )h ##ò4Ô/h#

SMT

ÏófÐ Ú ?4ÑFhq##\4Ð<Kªggª©Èg4#/Ò#h#ÎhK# gg:¼?/

Y_ïð##Qh/ªgghÑ44#yÏ¤§ògá|hÑλ − 1

âÕ#×4/#/hf(x, y)gá|hÑ¸54â#h#g:ÍY:QëqgögYhÎ N 654 N í

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ Nm4

ÏófÍ§h \î|# RÓªgA#fî|gÑ

Z pZÍ

p 6= 2#g Ú 4ÔÝh##Ñg7h/ªgg

hÑ44ï#ß×g4#/

detSMT = detM = (−1)λ+1(I −H)−11,λ = f(x)

4Ô/hY#4/#/ÒhÕ#q##\4ëqgÈù?g\4 N 65 í q¿h|_Ò§#4Ñ ©|_det(S) = (det(T ))−1

01#$IQ=Öý Öý3Ö Ïó

EB(M)Í§h×#4##4#ÑñhÈ#g"4#Ph 4#4##4#ÑñhÝY_ï

Í©4#Ñ |FÓ4&Ô8gg'KªggªPùg 4 pÔ#Ñ #Ûgªg(EB(M))i,j

âÒÔ#/C#gÑg gD#hÎ#RÍ #Ýgg\òÎh\¼4â##hDhÔ#ÑÈ gg:¼?/òh Cg4#/#h#Î8

Ì #òq4ÑÑRÍ#ÝYh4#Yhg§#ò FhA4#Ò4Ñ/4

f , x = ( x 1 ,..., x k )

D B ( E B ( f (x, y ))) = f (x, y ) f (x, E B ( y )) = E B ( f (x, y ))

E B ( y )

Alice Bob

E B ( f (x, y ))

y = ( y 1 ,..., y m )

Ö 4# N 654 ÷ 4#/#Íyg#g:¼?/ Û _

Ï¤ph#gÈÍ©4 ò#Ô|h FyhÒh#È4Ñ/4D#q4ÑÑRÍ##òÑg×|hQ#Óh_q##\4

fh #Óh_D##/

x Ú 4Ôy|h#Ó__D##/

y|_#Dg#g:¼?/##

#ò##Ô#ÑÎh\¼Bh g È#g#g:¼?/4ôQÑghYù?#\ÓhhÑ|_gò#gòq##\4

f4(#g#Ó__P##/×h "Ú 4Ô g#g:¼?/ ##/F ù?#y î|g#yg#ÑÝô# #gÒq##\4Qh 4Ô/h#ÒhQg#g:¼?/ æóq##\4PÓ_hÑ#¯ê

EB(f(x, y))©ù?#òg #

ÓhhÑ#Ô|_¯Î Ú 4Ôq7<#Í hÔ#ÑÒ gg:¼?/§Ð###Ðgg\§Îh\¼Ch Èg4#/#ò4/#/

f(x, y)

01#$IQ]=Öý Ö Ö Ñh74Ñ/4FhÔ6§4Óªgq4`g4#ª#Dh g4## #Ñ_ÜpÔ#h####44hg Ì #gDÔ#h#¯##Ý#44h©g4#/q##\4#

f : 0, 1n → Nh f : 0, 1n → Z pZ

â#g6g\Ó4gÑ¼pëqgt <P#â<G<7h < O 2q4 \î|#ï4# í

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ NGN

Ì #RÓ4D/ªgg#\¼yhA4#Ò4Ñ/4CÍ©ÈÑ4ªÑ¼Y n¼YP#Dq4ÑÑRÍ#ÍÒF¼6ûpÎhgg&#gÑFh:¯?ÓhhÑ#ghÑ4#Ch84Ô#ÑòY #g#g4#/_4|hÑ2ghÜ#Ñ¯/¤¼P#gò#|hÑÑ¼P # 4#Òh Ý#Ñ#ÑF_4#ÒhAgÑFh:¯?Ó_hÑ#ghò4D/ªÜggª|hP#ò#h4òh7#ÈhÑ44#×§ # h çQï/ Ü:çQ#Ñ4Qg#\¼?/ ÓhhÑ#gg #$&1`01.W)Þ54ÝÔ6\q4 Ú 4Ô g#g:¼?/×#Ý##/

yj, 1 ≤ j ≤ mÍ #Ý##Ô#ÑÎ4\¼p##

Q#4Ô|hÔ#Ñ:hâ7 # Ó4gÑ¼ð#444##×g#g:¼?/4p#ghâfh g È#gg#g:¼?/4#

EB(yj)×Ñgh

4 Ì g4#/<hò4/#/f(x, y)

|_ÕFhò4#Ñ¼Ô8<F Ôª¼ Ú 4Ôò4Ó4gD#fg#g:¼?/ ##/

EB(yj), 1 ≤ j ≤ mh Ñgh ##/

xi, 1 ≤ i ≤ kâ2Ñg#Í g

#Dq4ÑÑRÍ##ëT íyÖ :FâfÑg gÑ\g#Pî|:×g4Ñ#(h #PÑñh:Í¿h#y ¾hgg#\¼

Y_¹Hh 4Ô/h#×ôY_ï

H− Ì #g#Ûg#Ññh\g\Óªg:¼ # g

alhzh

Í#gzh ∈ yj|1 ≤ j ≤ m

h7#Ègª:¼ ∑ni=1 alizi + bl

H−

Íg#g:¼?/ gª:¼

EB(alhzh)## Ú 4Ô ##Ô#ÑÈÎh\¼ðh #_Ñ44#

çQï/ Ü:çQ#ÑF(ÑgPÎhgg##Û # g blh #Û # g

alhzh

Í#gzh ∈ xi|1 ≤ i ≤ k

h#\¼(hh(ég#hQÔª¼EB(H−)

#yY_¹ #4Ô/h#gù?#gÈ#Y_ïy#hgªgÑ¼Qg#g:¼?/ Í© C#h g#hÔª¼

EB(H−)

ëqÔ í Ñ\ gÑ\g`#©î|:<g4Ñ#Ýh #ÒÑñh:`RÍ"h8#λ×λ

gª.¼ÈY_ïI×4Ô/hP#òY¹

I−

ëq íPÌ #g(#Pgh#/gEB(I− − (H−)) = EB(M)

gª:¼ Ôª¼ gª:¼4 Ì #Ý 4#×Ôª¼ô##P#hÑ44# # h çQï/ Ü:çQ#ÑDyg4#/#×g?Üg:¼?/ ÓhhÑ#g©Í#gFhÒ#gYh##gÑFh:¯?§ÓhhÑ#g§C#§Y_¹YFhÔ8g4#/ ð g\Ñ¼4 Ì #D4Ýh<#_¹K gªgDhÝg#g:¼?/ â4h|h:Ñ¼g#g:¼?/ h #gYh##DggÐhgÑFh:¯?<ÓhhÑ#gg

ë íPÌ DÔ#Ñ #g`Y_¹HÑgÐ¯#4gA.Í©h 4

(λ−1)×(λ−1)Ygg

S, TÓ4g

RâÐ|_

det(S)h

det(T )_y##×

RÍ

det(S) =

(det(T ))−1Ðù?#ÈF_Q#Í \4#/

EB(SMT )fÔ6\q4DÍ©ò|RÓ4

(EB(SMT ))i,j = EB(∑

l

∑

k

si,k · tl,j ·mk,l).

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ N P

¿RÍ 4<h?#<Ó_hÑ#gmk,l ∈ EB(M)

h<g#g:¼?/ h h#gAh<|h:Ñ¼g#g:¼?/ 4gÑFh:¯?ÓhhÑ#gg Ì Pg4#/

(EB(SMT ))i,j

#g4#/g#C# #\×h #ÝhÒ#P\ÑFh:¯?ÝÓhhÑ#gh #gÝ#PhÑ44ï#çQï/ Ü:çQ#Ñ×h # ðg4#/C#Ph#gÝ# #\h #gpng|_#g#g:¼?/#ÈgYh##gÑFh:¯?ÓhhÑ#ghA#Y_ïyh #g#DhÑ44ï#ß # Ý4Ô/hY#_Y_ï

EB(SMT )hÒY#DhÔ6RÓª

Ô|hD4Ñ/4ëq í ù?#òg

EB(SMT ) Ú 4Ô

N Ú 4Ô ?gÒ#òhÈhhÔ8Ó4D×4Ô/hf(x, y)

D m01,qÞ÷ 4# g#Y#g4Ý##F_4Qg4#Ñ¯/.¼y4##hg4Ñ74/YhÑ2#g#4?Üªgh\ÓªhâHhâ_Õ#g %¾:#:74#Ð4# h#g4Ý##F_4 ÷ 4#\g###<g4#/Ü4|hÑg4#Ñ¯/¤¼È g8g <h#Ñ¼ò4#© É §h|#©# gÑ¼?#Ô#h####44hhAÍ©gÑÑ#hA4Ý#§/ªgg#\¼Ýh|#§#444##Ðg:¼?/4:¼?:FQ Ö #:#g`gh#/_4#ÑÎh#gÑg\4h§h 4 Y_ggò ùg 4 4D#g4#/_4ôh©# \g.Ü|hªÈ4òh#Îð ùg N FhpÔ6 4#/ªggÑ¼4 n ÔªÓ?4#Ñ¼4âÕ#/ªgg#\¼h§##h?g4Ñ#`g:\ Dq##\4#`|_ÐhÒg4#/hÔ#Ñ§Ôª¼64Ñ¼?#4ñ_ÑïÜó É ©Ô#_#¯###44hg Ì #ÈgÑñhh`#Ûq##\4#Ò|_hÈg4#/hÔ#ÑDÔ¼yÔ#h####44hhK64Ñ¼?#4ñ_Ñ/ É Ò#gÑ gf#\Ññh<h NC1

q##\4#g Ì #gâ? # 4D# Fh Ú FFÓ4gF O 6|_§#RÓ? gÒ4Ñ/4×q4ï¿ü n ì ù/Ø6 ÷ Ùjëq#4 \g#:Ñ4_Ü¤|hgq##\4# í 4#g#Ñ×gg4 g#Ññh4ggÒh7#DgÑñhh

NC1q##\h#F

*+%01-01.V.ÞÌ #Yg4g\#ghÒ4#È#h?g4Ñfq4ÑÑRÍÈ× ñ_gÑï¼&q4 #Yg#Ñ#ggª ù?g\4 N 654â2q4Þ##48g:gòh©#hÑ44ï# # h çQï/ Ü:çQ#ÑFâq4#Ý#Ñ#ÑF_ïÓ?¤¼Yh7# \g|hFâ/Hhâ

detSMT = detS · detM · detThÒÍ©gÑÑ2h

q4 #ÈTh\|_Ý#Ñ#ÑF_4QÔª¼Ûh 4#4##4#ÑñhY_gg ?g#h|h#4#òh#Î8d01m-C,qÞn #64Ô#Ñæ._hÎ4ê|_ÒÑg§hÔ#ÑÝ8g:q4 ¥ Ø ¡ / \â/Ôª¼Y#hÒg Ü#×hª¼YÓhhÑ#Ô|h¯Î Ú 4Ô7RÍ©\Óªggâ|Pgg|h4§Í#gòÑgÍ§hªÒ4\|h q4g # Ú 4ÔP#òg#g:¼?/ 4/#/Fâ/#_hÎCÒ##ÑÎhgÑ¼4

mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ N <

#h#g_h¯ÎðÍ§4#Ñ Ô6Y|_Ñ\Yg QÍ4#ðg#ÑÝ Ú 4Ô Ì #ÝÝ&4g?ÜghÑ#4Ô#Ñg¿h7#Q#h?g4ÑÒhÑ#4#4PPYhª¼Cgg|h4 Ú 4ÔyÒhÔ#ÑÓ4gn¼Y#g4g\#gDh`#4/#/F Ì PFÓª4 #¯ðhð_hÎ Ú 4ÔÛg4#Ñ hÎQÑgDq4 É \ÜÎ#RÍÑ 4ò#h7Óªgn¼Y#òg4g\#ghA#gg4#/_4ëqgÈögYhÎ N N P í û |_fÍ©g4# gf#gfgg#¤¼g4#g\##ò#Ò#Óhh\¼Ýh Ú 4Ô <##/<h Ñgh q##\4ëTh #gò##/òhò|h:òh©#gq##\4 í ÝÏóòq4ÑÑRÍ× ñ_gÑï¼Q|_D##Óhh\¼h Ú 4Ô Ý##/gá#ÓhhÑgªÛ#Cgg#¤¼ h§#C# gÑ¼?#&g#g:¼?/4#g#gÈÑgD4#Ñ¼P4Ô/h# Ú 4Ô g#g:¼?/ ##/g©Ïó`#È#444##Dg:¼??Ü4:¼?:gfgYhªFhÑÑï¼gg##Yhgâh#fØAhÑÑgg `#ghâª#gÑg§f#hÐhÔ#ÑÛªhphª¼ô/q4Y_4hÔ64/ Ú 4Ô È##/g Ö #:#g4hâ7#gYÑgò#hÝhÔ#Ñ gg:¼?/#×gªgòhÐ#×Y_gggâÕ#×ò#hÈhÔ#ÑÝQg4#/Ý#h#Î&4# \g|hªh

SMTh #h

Mâ`Hhâ`#CÈ#h_Ô#ÑÛg4#/#Y4/#/

f(x, y)

Ú 4ÔPgggÓªghyg#g:¼?/4y_Ah 4Y_ïSMT

hAî// h#ÎCh hAî// \g|hªFt7g#g#D#hhÔ#ÑÝ gÓ4h¼ð/q4Y_hôhÔ64/ò# ¾hgg#\¼Y_ï

Hh #hÔ64/ÝÑgh Èq##\4

fÜD/gg/× É h(ë)¿hY|_Í©CFh

hÑ#Ó? Ý×#h?g4ÑÕyÍ#&#È4#Ñ¼y4\Î#Í*#&##6gÔ64# hA# É hâg×ù?g\4 N N í Ö #:#g4hâ6#Ý#hhÔ#ÑÈ gg Í#&ggÔ6gÑ4#CÑïÜgh ×##/g Ì #h4#gªÍ¼(Ñ\h q##\4(gYh##Ó__PÍÑÑ©Ô6Pq#:#gq4YhÑ É P#Dq4ÑÑRÍ##ûpòg:hÔ#Ñy#ò#Óhh\¼PhAÑgh Òq##\4P#4#4y#Dq4ÑÑRÍ#ÝÑgY/fÏó#RÍ|_Ý#CY_ï

EB(S ·M · T )|_ÝÑgChÔ/¯_#_ngÝÔ#Ñ #Û#gY_ïh

g#g:¼?/4ôh©Ch 4ÞY_ïQ|_D|h#hÝh#Î&h #h \g|hªhM Ì #gªgDhf#Y_ï

EB(SMT )h#4#Ñ¼yÓhhÑ#g|_ Ú 4Ôð\ggÓªg

q4ÞÑghF7g#ghâ2f#Y_ïQDh&g#g:¼?/4ôhÐCh 4ÞY_ïKâ Ú 4Ôð#hhÔ#Ñ×ªhQ_¼P¯?×/q4Y4yhÔ64/Ñgh §q##\4CÜÐ¯/gg/Ò É h 01#2#$I8=Öý Ö xgÖ ü2\

MÔ8Ò

n×nY_ïòÓ4g

Fq

h Ñ\Sh

TÔ8§##ïq4Ü¤h 4

#4##4#Ññhn× n

Y_gg©Í \(T )·

\(S) = 1

RÓ4gFq

Ì #gyq4§î// Mgh¯

64Ô#ÑDY_ïSMT

gg#ÍP#òhD#4Ô|hÔ#Ñ¤¼4R ¡ M ÷ 4# gA#§44# G = (S, T ) |

\(T )·

\(S) = 1 ⊆ GL(n, q)×GL(n, q)

Ì #44#ð48g_gD4&#Ý\

M := M | M

n × nY_ïCRÓ4g

Fq_M 7→

S ·M · Tt¿h|_ò#D46g_4 gÈ#hÈ|h#4# \g|hÈh h#Îðh

mQ"m ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^-±¬²)³¯¬T NGO

M ∈ M ÷ 4# gò#×##44#

GM

hGÍ

GM = (S, T ) ∈ G | SMT = Mh q4M ∈ M

#ò#Ô#\G.M := SMT | (S, T ) ∈ G

hM Ì #gy#òYh###

G/GM → G.M, (S, T ) ·GM 7→ SMT

7ÒÔ# ¾:g\47é#<#<gá#6hg#\¼òh?#<4Ô#G.M

h #<##q4 :Ô#/4hSh

T#ògÑñhßq4ÑÑRÍg

Ï¤ô4#D#h?g4Ñ`DggñhÑA|_D##444###g×D#4Ô|hÔ#Ñ:ÈÍ#4|hhªggD#ïÓhh\¼Ûq4 Ú 4Ô D##/#gÑgÝ#hòhÔ#ÑPg4|h åKggªDg?Üg:¼?/4#g Ì #Ðg4 4×ÐhÑF ¼#gÑ ×4#<gg#.¼h|hÑ¼?<hÔ6RÓª#gÒÍ§gá#YP\YhFhÑÑï¼&gg#Yg#g:¼?/4#ghJ7g#ghâA4#ò#h?g4ÑfòhÈgg#hÒ#ò#4Ô|hÔ#Ñ:hâ?#444##g:¼?/4:¼?:g# 01#$IQb=Öý ÖýfÖ Ïó

ff64Ñ¼?#4ñhÑh g4g

d|_<Ñg©Í§hªfD# hâª#Ò#4#Ñ

#h4/#/4|hdÓhhÑ#gÒ Ú 4Ô ëq4Ò4g4#ÈgÑ í n #g:Í Ú 4ÔQg4#Ñ ##Dq##\4PÓhhÑ#g§×4Ô/hPÑgh Ò64Ñ¼?#4ñ_ÑKÔª¼yüÕh4h#4h ªg64Ññ_4

oA1Eo > ÂÉ<½CË · ÁPÇÃ · ËÆ|kÁy¾78A<PËÕÅÆ:8½yÁy¾ÐË · ÂÁ»

ûpÛ#Í|FÓ4ðgÑ4gYÑ?4Î(_#ÛgÑ4gÑ¼"gÑñ_ #4Ô#Ñg FhÑÑ ¨\ ¢ ) 1Â ÁqCC g § ¥\¡¢1g ) ª Ì #ògg|h§#Dq4ÑÑRÍ##dG01-IQ ÞÑgò|hq##\4

f : Rn → Rh ÛÚ 4ÔÛ|h×gg\##/

x = (x1, . . . , xn) Ú 4Ô

`ÍÑÑ#Èg4×#/f(x)

Ñ?FhÑÑï¼4#f##/x×#CÍ§F¼|_<4#Ñ¼ÝÑg§ÑFh#

#Ò4/#/<h8#©q##\4 Ú 4Ôh Ñg§#4#Ñ ÑFh#h##DhÔ64/`#Òh#gf4#g##/ÈÔ6g gò|_ÈÍ|È#4/#/\Ó4FhÑgCÏ¤|h:g#ÑñhgâÕÑgh Dq##\4#4#Ñ gYhPgg\F¿hÒ|_ g6g #È4#Òq##\4

f#4/#/

f(x)g4#Ñ \ÓªF_Ñ|#gª##/

xÏóhÑ64Ô#Ñ×g4# gÒ#Dq4ÑÑRÍ#Ý¤Í§ÓhhñhªÒh7#Ògg|h#XYIQIL.Þ

mQ"m ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^-±¬²)³¯¬T N h

54 n #Ñ¼ Ú 4ÔQ#4#Ñ ÑFhP#ò4/#/ f(x)

4 ÑgDh ÛÚ 4ÔQ#4#Ñ ÑFhP#ò4/#/f(x)

D\yIL#$&(W0^=Ö6=Ö%Ö

54Ph'¯#h#Ñ&hù?gg|h 4 Í©ðg4#Ñ ##ÎhÔ64/yp4Ô#ÑÛh4gªf|_

g4ÑÑg\ _xDÏ¤&#FhÍ§Í©4#Ñ ##ÎQhÔ64/ Ú 4Ô D##/

xh##Ô#ÑgÑ¼

Î#RÍ Ö 47#:h#ghâ4Ñg<Í§hªAÔ6?4ÎÈ¼È|4Ah Fh¯#gAq47#©#Fh6g:È|4ª_È#g g h Ö 4D#ò##84Y#g y4Ô#Ñ×h4gªÈy#g:Ó4gh åKggªÈhÑ#gÍ#Qg4ÑÑg\#h##4#ñ_ _Cyî| 4/D##Fh6g:;È|4ªF7ng7|_`7\##AÑgh Ì #hg\`#×h_4g`hªh#:YhÑgh#Ò#4:ÒÒg#g:¼?/ Òù?gt hGhq4§q#:#g \hÑgÏó`#4Ô#ÑÈh4gªDhÑ×gÑÑ#ÝhÑ#òg:Ó4g#¯#Fh8g:FÈ|4ªFâ|äÐhñhª 4Ò#Ñg×g

4 h ¯#h#ÑYhÒäÐhñhªB5Phù?\g|h 4 Í©Cg4#Ñ ##ÎhÔ64/h /ªg\hÑ44ï#ßq4©Th\4#|_ÑgÍ§hªÒ×gÑÑ6 Ú 4ÔPÍ#4/Ò#RÓ? ##ÍÛ#hÑ44#yÏ¤ðg4ªh:yÙA#h×#Ñ N 4 4 #È#4#Ñ Ô8 Ú 4ÔÍ#6g:q4Ò4:h7#òg4#/_4#g

Ñhgg|h Í§hCg4# Ôª¼\Ó4ghÑòh/#4gâÔ#/CÍ'Ñ4ªÑ¼ åKggª#h4#g Ö 4ò#:h#ghâAÔ| ©h Ö g4gÔ|h# 5Ðg4# g Qgg#ò|_#Ñ¯Ügª©#q##\4

f#:F hÕg4# g#

fgÑ.<ù?#ygg|hÐFhÑÑ /¢

C /¢ C+- Ø ¢Q ) M Ö #:#g4hâfÔ| Òh Ö g4gÔ|h# 84ª &#Cq4ÑÑRÍ#gÑñ_4pÔ6\¤Í§g\"g4#/#ÛÍ g#g:¼?/ " _(ëqgQù?gg|h 5 í h g4#/#Íyg#g:¼?/ q##\4#ÈëqgÝù?gg|h 4 í æúý Ø Øý ¡ ¨ C1ý ¥ ) C úC% ¥ ý/) g ) £/)Á¤ )¥m T+¥ C Ú/¢ CC ý ¢1L M ¡6[ ØÙ x\ YCL £CØÙC)t§p1 ¥ C /¢ C¤CL ^ Lx A Ø 1 ¥ p¨B¸ § ¥m \/C¨\ØÙ]£ ¥ ÁqC 1ÂT Cc §^CL C /¢ C M //ØÙc § Ø p Á(1 [ ØÙ xY pCL £CØÙC)ã81 ¥ ¥m Lx Õ Ø B1 ¥ ] C /¢ CC1 ¢GÂ _¥ /§ ¢ 4J [ ØÙ x ¢1 CL;g§ Ø - ÀmúCL C /¢ C ¡ f §_£ Y¢1 / Ø C /¢ C ¥t¢ g ¥ 1Â ¡ C% C /¢ C LxcÁ G ¸§¦1 ¥ Y ¢ M ê

mQ"m ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^-±¬²)³¯¬T N k

ù?#g×Í©g4# gÈÔ#h###Q#44hÈhyg4#/_4|hÑA gÑ`q4Dq##\4#È?Ü:F h7gg#gâ/#§g#ÑÒFhy#h g\Ñ¼YÔ8òh##Ñ 4#§Fhh Ö #:#g4hâ|æ¤##Ó4ghÑ2Ô#h###×#44hê|h#hÔ6gg \î|# ¼4\FûpÝ4ÑÓ4×ù?gg|h 4 q4q##\4#

f : Rn → Râ6Í#g

RC#òî|gÑ Û#q4ÑïÜ

ÑRÍ#ÍÒF¼4Ë¿h|_hÒÔ6\q4DP#h\gD4#§4Ñ/4CFhyÔ6D4g#ghÑ #4R = Z NZ

Í#gN = pq

Ô#/Í#gò#gòÒ#×q4YhÑK#?h7h`gg#¤¼PÎ#RÍ

dW)Þ

54Ñg×g#g:¼?/#Y ¾hgg#\¼pY_¹HÔ8gÑ4#4#yy#YÔ#h###Û#44h

|_Èg4#/gò#×q##\4f Ì Q#âA##gÝy#4Ô|hÔ#ÑhâÕ # ÓªgÑ¼

h FhÑñh#444:##Dg#g:¼?/4&#g4Rh g#g:¼?/#Ègªgh

H##×#g##Ô#ÑDÎh\¼

AÐù?#ò4Ô/h#Ýæ¤Y_¹ê

EA(H) :=

0 E(a1,21 ), . . . , E(a1,2

n ), E(b1,2) . . . E(a1,λ1 ), . . . , E(a1,λ

n ), E(b1,λ)

0 0

E(aλ−1,λ

1 ), . . . , E(aλ−1,λn ), E(bλ−1,λ)

0 0 . . . 0

Í#gw(i, j) =

∑nk=1 a

i,jk xk + bi,j

_ λÒ#È É ÈhA#ÈÔ#h####44hQ

Ïó(i, j)

Ò#hhQ 4òG#ò\

ai,jk = bi,j = 0

q41 ≤ k ≤ n

h g#g:¼?/#gòÓ_hÑ#ggÑgg

EA(H) Ú 4Ô7g#ghâ Ú 4ÔY4\ÐÎ#RÍ'# É h#Ô#h#¯##

#44hQYÏó§Í©Y4#Ñ¼&Í§h#ÛÑFhphp##8gÔ64# h©# É ×#gÍ§Fhô##Y_gg

Uh

N \î|# Ô6gÑRÍ!ögYhÎ N N 4 #:F hÐ#

Y_ggHh

M

4 Ú 4Ôy#g#ÈhÑ44#×§ # h çQï/ Ü:çQ#ÑÒ×g4#/D#òY_ï

EA(H(x)) = EA(H)(x) =

0 E(∑n

i=1 a1,2i xi + b1,2) . . . E(

∑ni=1 a

1,λi xi + b1,λ)

0 0

E(∑n

i=1 aλ−1,λi xi + bλ−1,λ)

0 0 . . . 0

mQ"m ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^-±¬²)³¯¬T NGj

Í#gx = (x1, . . . , xn)

##ÓhÈ##/Fc7È#gðg4#/gh&g#g:¼?/4EA(I)

hA#(λ− 1) × (λ− 1)

gª¤¼CY_ïI##×Ñgh Ò##Ô#ÑDÎh\¼4©ü2\

EA(H(x)−) g#h#ÝY_ïÛ4Ô/h# Ô¼ gÑ\#Y#Ýî|:òg4Ñ#ðh #

Ññh:RÍ hEA(H(x))

Ú 4ÔyÒ#RÍ*hÔ#Ñ×g4#/

EA(M) = EA(I −H(x)−) = #

(EA(I),çQï/ Ü:çQ#Ñ

(−1, EA(H(x)−))).

Ì YÔ#Ñ #ÈgªghEA(M)

_ ## ò#gg\##/x#È#?4g

¤Í§Ýh 4ß#4##4#Ññh(λ− 1)× (λ− 1)

Y_g\S, T

RÓªgRÍ

det(S) =

(det(T ))−1h g4#/g

EA(S ·M · T )DhÑF ¼ ggÔ6 ðù?g\4 N 4

#Èg4#/_4ôFhpÔ6 4###Q#YhÑ44ï#D # h çQï/ Ü:çQ#Ñh#ò# gÑ¼?#×g#g:¼?/4Q#gh Ú 4ÔQg

EA(S ·M · T )×Ñgh

N Ì È4Ô/hf(x)

ÑgÒ|h<Èg4#/(−1)λ+1

#Ñï#Ñ Í×# \g|hª4©

f Ú 4ÑFhY#h#ÎhÕ# gg:¼?/4Ph2#Y_ï

EA(S ·M · T )<ù?#

#g#gÝgg\Î4\¼ gg:¼?/#Cgªgh h##ÑgÝh /ªgg×hÑ4hï#|_g4#/gÒ# \g|hªh h#Î8â|g8g\ÓªgÑ¼4

Ì #ÈYhhÓ?|hÑ É g§#ò FhA4#Ò#h?g4ÑH

Alice Bob

f x

E A ( f )(x)= E A ( f (x)) D A ( E A ( f (x)))= f (x)

E A ( f )

E A ( f (x))

Ö 4# N 4 ÷ 4#/#ÝÍQg#g:¼?/ q##\4#

7g#ghâ#ðÔ|h& F hÈ4#C4Ñ/4PhCq4ÑÑRÍgÑg&|hypq##\4f|_

#g#g:¼?/òÍ#gò##Ô#ÑÎh\¼Ah g ò#×g#g:¼?/4

EA(f) Ú 4Ô Ú 4Ô

\ÓhhÑ|_g`#fg#g:¼?/ q##\44#fgg\Ð##/x×#YÍÒF¼Ý|_<#4Ô/h#

hg#g:¼?/4EA(f(x))

Ö #:#g4hâª#Ô#Ñ f#<##/©h g <#g#ÑÐÔ|hÎQÑgh×Ñ\×D#RÍßhÔ#ÑÝ gg:¼?/ò##Q#gògg\ÝÎh\¼h Qg4#/#4/#/hA#gÒq##\4

f(x)

mQ"m ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^-±¬²)³¯¬T Pm:

01#$IQ=Öý=Öý Ö é\î|#Y_ï

U =

(O O

O H

),

Í#gHÒ#È ¾hgg#\¼QY_ïPÔ8gÑ4#4#Y×Ô#h#¯###44h¿|_g4#/g

#×q##\4fâ7h

Oò

r × r É g_Ü¤Y¹K×Ïó H Èλ × λ

Y_ïÛ#gUò

(r + λ) × (r + λ)Y_ïK<ù?#g

HÒ#Ñ6hgª

UÒ#Ñ8hgFâ??#fÏó#4Ñ §|

U i =

(O O

O H i

)

h Í© \î|#U∗ :=

∑∞k=0 U

k<Ô8\q4Cëqgù?g\4 N 65 í Í§ò|RÓ4

U∗ = (I − U)−1

h det(I−U) = 1

`ù?#gf#<gª:¼(1, λ)

hH∗ =

∑Hk

gá|_Ñ2#<g:¼(r+1, r+λ)h

U∗â#Í©ò|RÓ4

U∗r+1,r+λ = W (1, λ)(x) =

detN

det(I − U)= detN = f(x),

Í#gN g#hg©##Ô#Y_ïh

I−U4Ô/h# Ôª¼ gÑ\#ò#

(r+1)ÜóYg4Ñ#

h #(r + λ)

ÜóôÍò Ì #g\q4hâ7#q##\4ôÓ_hÑ#f(x)

FhpÔ8 \g# Ôª¼g4#/#ð# \g|h×hDÑñh4g É Y_ï N #:F hg4#/#ð# \g|hªÐh

Mëq#Ñ#Ñ Ôª¼

(−1)r+λ+1 í q7g#ghâ/#:F h## H Ñgg4#Ñ hÑC#Ý#Y_ï

U&4##h?g4ÑHû #gôg#g:¼?/#

U#ÝÍ§4#Ñ g#g:¼?/ò#

Í#4Ñð##6gPñh#4#ÑñhYY_ïKâHhâ#gÑ # # É g4y#ô##6gPñh#4#Ññ_Y_ïQh

U Ì #g Ú 4Ôð|hCg4#/h&g#g:¼?/4ðh

N#:F h

Mh |h

×¯|h#4D#ò É gh7#òY_gg S h T gá|_gÑ¼hÌ #C Óhhªh4h§##Uh

N#:F h

Hh

Mò|_ Ú 4Ô¾:#:ÝÑFh#Èh

##6gAÔ64# q4Õ#©Ô#h#¯###44h* É §h #h7#©¯#h\f É h n |g4#hâ4#7_©#g4:§h7 # 4|hÑ|g4#/#È6RÍ§g©#g#Y_gg©hÑñh4gf|hCÔ6\q4h 01#$IQ$=Öý=Öý=Ö Ï¤ 4 h ÷ hgf\©hÑHA#464 DÍ§R¼ÝÈªhÔ#ÑÒÔ#h###ò#44hÍ#h \î|# RÓªgDhÔ#h:¼Q#4g Ì #gg##á#FhhÑCÔ8×h##Ñ P4##h?g4ÑHù?#gÝ4hÔ#Ñ Ô#h###Y#44h|h×q#ÑÑ2Y_ïy#:F h<hÛ##6gñh#4#ÑñhY_ï h× ¾hgg#\¼(Y_ï#ÝÍ©4#Ñ ÑF ðÑg/ªggª#h?g4ÑÍ yÑ4ªÑï¼ð|h#4 \Yhh Ì #g\q4#Dg##á#Y#4#Ñ #hÔ6#

mQ"X O v +T³ý¬ ( PQ5

4##hg4ÑH

d01m-C,q*+%01-01.V.(D m01(, Ì #Dg4g\#ggâ|/ªgg#\¼Ph gg#¤¼Ph|hÑ¼?©h2#§#hg4ÑK©¯#h\Ñ¼#Dhhh`#È4#Èðù?\\4 N 4 ¼7g#ghâKhÔ6\q4È4##h?g4ÑÕ#RÓhhÔ#Ñ¼ygg#hâK#4?Üªgh\Óªòh /ªggªq4Ò84Ñ¼?#4ñhÑ¹Üó É Ô#h####44hg 01#$IQ=Öý=ÖR-Ö

54ÏóY×KªggªC#Qg#g:¼?/4hÑ44#EYô:¼?\QhÑ44ï# h

gYh#<gg\F Ö 4<#g4#/_4# Ú 4ÔY4#Ñ¼×#g ©#hÑ44#×< # h çQï/ Ü:çQ#ÑF;¿h§|_<

Ef:¼?\©#g×Ñg§|h`òg h×g#g:¼?/4

E(I)h7#ò gª.¼YY_ï

Iâ##

4 ÑgÝ\4#Ñ hÎ Ú 4ÔðqhD#RÓ?##âh# É g_Ü¤Î#RÍÑ 4ÝTh#4"ëqgYù?g\hP#i<?65 í â4|_<#Î#RÍÐ#f##/

xâ4|_

x<ÓhhÑ ##/Fâh |_f#Òq##\4

E(f)\Ó_hÑ|_ 4

xgá|hÑ

E(f(x))Ðù?gt 4N 2q4Ò#Û#hg4Ñg

oA1 Åb8×Ã ¶ <^< · ÁPÇ

Ï¤Û#g\4ÛÍ©ª #g#\Í#48g:.¼y|_Í©FhÑÑ(Àm/p/Á m 1Â ©ûp#4:Ñ¼ g4#:_DÍª¼P##48g:.¼Ph`g:¼?/4:¼?:gÒ#\q#ÑHâ|g6ggñhÑÑ¼CP#òg4ª¯?hÐ#D|h/gg Ö #:#g4hâÍ©×#RÍ!|_D#ghÑF ¼&¯/:òÝ#Ñ#ÑF_ïÓªgÑï¼&hÍ©gÑÑÕh # Ó4gÑ¼Y#444##hâ/#4Ô|hÔ#Ñ:g:¼?/4:¼?:g|FÓ?##Ò#46g:¤¼4HJ0mÔ;CÒ=ÖR-Ö)LÖ ì Ó4gk g:¼?/4:¼?:F

(K,E,D,A)4#ôgh4ð|hg

MÍ#gyq4#Pg##g:¯?|hgQ#4Ñ Ý|_C ⊆ M

(ü2\Eke

(m) g#hQh(g?Ü

g:¼?/4Ûhfgh4m ∈ M

g#g:¼?/ ÍÛ##Ô#ÑòÎh\¼ke

ü2\aÔ6ò#È##Ô#Ñ

Îh\¼ hÛ|h:¤¼Ah

bÔ6#Y##Ô#ÑÎh\¼phh#h#g|h:¤¼

B Ì #Cg:¼?/4:¼?:g

ÒFhÑÑ Àm/_/Á m 1Â 7#gD¯/:hy/ªgghÑ44#¿úD\¼ªÜ:ùÍ§hy|_4Ó4gQhg#g:¼?/4

Ea(Eb(m))g4#/ghyg#g:¼?/4

Eb(Ea(m))

k#4Ô|hÔ#Ñ:hâ # Ó4gÑ¼h FhÑñh<#444##Òg#g:¼?/4Y#g|©_:î|gÎh\¼y:ÍÒ_###g4#Ñ Ô6ò# ×4ÑÓ4D#Dq4ÑÑRÍ#äÐhñhF5òhfù?gg|h 4

mQ"X O v +T³ý¬ ( P 4

ÑgÈ|hq##\4f : Rn → R

h ðÚ 4Ô&|hh&##/x ∈ R

q4f|_#Í§hª

È\ÓhhÑ|_h Ú 4ÔC_ Ñg#4#Ñ ÑFh#h##hÔ84/Ð#h#gÐ4#gÐ##/gâHhâfh

x#4#Ñ gYhògg\fh 4#Ñ¼ Ú 4ÔÈ#4#Ñ ÑFhò#<4/#/

f(x) Ö #:#g4hâ

Ú 4ÔQ#4#Ñ y ×4:|h:h7#òg4#/_4P4y#ÒÍì Ó4gôC#4Ô|hÔ#Ñ:hâK # ÓªgÑ¼Qh FhÑñh#444##Èg:¼?/4:¼?:g|D_.Üî|g<Î4\¼:ÍÒh####âD4Ñ/4Ñh7D#4#g ggÔ6 Yù?g\4 N 4 h ù?g\hN N FhQFhÑ¼CÔ8ÈhÓªg Ì #È FÝ #ÑñF¼4 P#Dq4ÑÑRÍ#Ýî|4#h

Alice Bob

f x

E a ( f )(x) = E a ( f (x)) E b ( E a ( f (x))) = E a ( E b ( f (x)))

D a ( E a ( E b ( f (x)))) = E b ( f (x))

D b ( E b ( f (x))) = f (x)

E a ( f )

E a ( E b ( f (x)))

E b ( f (x))

Ö 4# N N <úD\¼P:Í§h###

Ñgg#\¼?/Ý#gÈq##\4fëqHhâ7#Yg4g6h #&Ô#h###Û#h4h í Ô¼#g

##Ô#ÑÎh\¼aÐù?#Dg Ò#Dg#g:¼?/4

Ea(f) Ú 4Ôâ/Í#\ÓhhÑ|_gÒ§4y#Ò##/

xC4Ô/h

Ea(f(x))F7#gD###Ô#ÑÎh\¼

bCg#g:¼?/#g#Ñ

Ea(f(x))h

4Ô/h#Eb(Ea(f(x))) = Ea(Eb(f(x)))

Í#y#g Ô|hÎÝÑgh`Ñg#RÍk#g#gÒgg\Î4\¼Y gg:¼?/Òh 4Ô/h#

Eb(f(x))Í#y#Dg Ô|hÎ Ú 4ÔPÍ#

hÔ#Ñ gg:¼?/h hÔ/¯_P# g q##\4PÓh_Ñ#f(x)

n ÔªÓ?4#Ñ¼YäÐhñh N hfù?gg|h5ÈFhyÔ6ò4ÑÓ4 h|hÑ444#Ñ¼4 01#$IQÕ=ÖR-Öý Ö #h#g§64Ô#Ñï¤¼×Ý4ÑÓ4#gDgg|h4§Ð#q4ÑÑRÍ## ì Ó4gÝq##\4

f : Rn → RÍ#g

RÝî|#D## Ì # #Dq##\4PÓhhÑ#

f(x)q4

x = (x1, . . . , xn) ∈ RnÍ©ÈYF¼Pg4#/

f(x) + rÍ#g

r ∈ R#4g&_h 4Q

Ì #gf(x) + r

h 4¿Ó_hÑ#òRâ#?#

Ö 4#:h#ghâÐ äÐhñhªJ5QhDù?gg|h 4 ÑgPg4#Ñ g#g:¼?/Y#gq##\hfÍ

ô#444##Pg:¼?/4:¼?:g ###g×##Ô#ÑPÎ4\¼ah g

Ea(f) Ú 4Ô"Ïó

Ea(f)(x) = Ea(f(x))#4Ñ gâ Ú 4ÔôFhô\ÓhhÑ|_Ý#Ýq##\4ð4ô#D#Ó__##/

x

Ö #:#g4hâ#Fh¯#4òh 4!ÓhhÑ#rh g4#/ #

(Ea(f(x)), Ea(r)) =

mQ"X O v +T³ý¬ ( P N

Ea(f(x) + r)ÏóÐ#Ýg #CÑg#ÝhÔ#ÑÈ gg:¼?/òC4Ô/h

f(x) + rÍ# ÝÛh 4 Ó_hÑ#h&Ïó#Pg Ô|hÎô Ú 4Ô #CÝhÔ#Ñ&#Ô/h\r

4Ô/hC#q##\4PÓhhÑ#f(x)

Ï¤yhQ_|hÑ444#©Yh##g©äÐhñhª N hAù?gg|hJ5DFhPÔ6D4ÑÓ4 Ì #g Ú 4Ôyg#g:¼?/#Ò##/

xÍQ#44h##g:¼?/4:¼?g##×#Ò##Ô#ÑÎh\¼

bh g #

g#ÑEb(x)

ÛÑghPÏóf(Eb(x)) = Eb(f(x))

#4Ñ gâAÑgFh \Ó_hÑ|_f(x) + r

âÍ#gÝ#Ý#?4g

r_h 4Qâ84

Eb(x)C4Ô/h

Eb(f(x) + r)Ïó<#Ýg #

ÓhhÑ#Ô|hÎ× Ú 4ÔC#§hÔ#Ñ gg:¼?/©Y4 g©4Ô/hf(x) + r

Í#P©hªhðh 4 Ó_hÑ#h Ïó#Pg

f(x) + rÔ|hÎpôÑgP#yFh"#Ô/h\#g×gg\Fâ

h 4¿Ó_hÑ#r×4Ô/hC#ò4/#/

f(x)

Y[E ¡ Y¢£¥¤§¦¨ª©

Ï¤ô#òg\4ôÍ©×#ggÈ¤Í©Q¯#h#Ñgòh©#444##Ý#ggò|_È_:n¼ðÎh\¼:Í§h####

T+'-0^DFWÊÜJIL#$ILWïd'-01#$0

Ì #×ÙfÑ ì hYhÑÕ#g N kAÍ§hî|:ò##Ô#Ñ# 25 j kP#DÏóDÔ|h 4&# 6ª\#Ñ.¼hf#g4#/_4Qh g\Ñ4ªhï#gÏóY64##Ññh¯#h#ÑÈh<Y#4Ô|hÔ#Ñ.Ühâ#Ý#Ñ#ÑF_ïÓ4gÑ¼Y#444####Ô#Ñ¯Ü¤Îh\¼C#gh Ö #:#g4hâ#Ò##64:Îh\¼:Í§h###YhÒÍ©DÍÑÑK#RÍ Ô6gÑRÍòÌ #ÙfÑ ì hY_ÑKg:¼?/4:¼?:gÍ©4ÎhÒq4ÑÑRÍgÛ 0m,ÝÜJ01-0ZIKgÞ

•ì g#g_Ññh4©h 4!#

ph 4g#g_4

g_6#ÒÝ#Ñ#ÑF_ïÓ4©44#

(Z pZ)∗

•ù?gÑg\h 4 ªg4g

x, 1 ≤ x ≤ p− 2h g4#/

y = gx mod p.

Ì #È##Ô#ÑÎh\¼yhAa = (p, g, y)

h #ògg\Î4\¼yx

mQ"X O v +T³ý¬ ( PGP

DFG,&-ÞÌ g#g:¼?/Ýgh4

m ∈ M = (Z pZ)∗gÑg\h 4ªg4g

1 ≤ r ≤ p− 2h

g4#/c = (γ, δ) = (gr mod p, m · yr mod p) =: Ea(m, r).

HJ01G,&-ÞÌ gg:¼?/

Ea(m, r) = (γ, δ) ∈ CÍ#g

C = (Z pZ)∗ × (Z pZ)∗g4#/

γ−x = γp−1−x = g−xr mod p

#gm = (γ−x) · δ mod p.

!#$#$%&(')8aY%&q0Z,qÞÌ ##gÈÒÝ#Ñ#ÑF_ïÓ4gÑ¼Y#444##_`Ïó

Ea(m1, r1) = (γ1, δ1) = (gr1 mod p,m1 · yr1 mod p)

h Ea(m2, r2) = (γ2, δ2) = (gr2 mod p,m2 · y

r2 mod p)

#g

Ea(m1, r1)·Ea(m2, r2) := (gr1+r2 mod p,m1·m2·y(r1+r2) mod p) = Ea(m1·m2, r1+r2).

7g#gÛ#&hÑ44ï# çQ#ÑYFh#Ñ¼ Ô6Û#Ñ\gª Ôª¼"#Ñ#Ñ¼?#ô#Ûg4¯Ü64 #×#Ññhª¯?h ##×Ô#Ñ #×hÑ44ï# ëqgé\î|#4 4 :/ih í Û 0m,Ýdg@_IQ&&()-?-Þì Ó4g(#y##Ô#ÑPÎh\¼?

a = (p, g, y)_Dð|h:.¼

Ah

b = (p, g′, y′)hô|h:¤¼

BÍ#gy = gx mod p

h y′ = g′x

′

mod pÍôg4g84 #Qgg\ÈÎ4¯¼?

xh

x′â

g6g\Ó4gÑ¼48¿h|_ò#×#p#4#Ñ Ô6×#×h×ôÔ6hÎh\¼?gâAHhâ2Í§h

Í©4Î# #&hÛ44#âÔ#/C#&4g#g_4gYF¼"Ô6&#4g åKggªÑ¼4jûp

î|:y|RÓ4& \î|#ð#ðg#g:¼?/4 # gP#ð##Ô#Ñ&Îh\¼bh 4Ó4g g##g:¯?

c = (gr mod p,m · yr mod p) = Ea(m, r)â|Hhâ/Í©È|FÓ4ò×¯?g #ògh4ò|hg

mQ"X O v +T³ý¬ ( PZ<

q4(Z pZ)∗

(Z pZ)∗ × (Z pZ)∗

Eb(c, r′) = Eb(Ea(m, r), r

′) = Eb((gr mod p,m · yr mod p), r′)

:= (g′r′

mod p, gr mod p,m · yry′r′

mod p)

= (g′r′

mod p, gr mod p,m · gxrg′x′r′

mod p).

¿RÍjÍ©ÈFh \î|#D#ÈhÑ44#¿úD\¼ªÜ:ùÍ§húD\¼ªÜ:ùÍ§h

(Ea(Eb(m, r′)), r) =

úD\¼ªÜ:ùÍ§h(gr mod p, g′

r′mod p,m · gxrg′

x′r′mod p)

= (g′r′

mod p, gr mod p,m · gxrg′x′r′

mod p)

= Eb(Ea(m, r), r′).

ù?#â4Ó4gEa(Eb(m, r

′), r)#hÑ44#úD\¼ªÜ:ùÍ§h8¾:#:D:Í§h#D#î|:.Í©Pg?h ¹Ü

|_gh7#Ò#Ñ×4Ô/hEb(Ea(m, r), r

′) Ì #FhQhÔÓ?4#Ñ¼CÔ8 h#ò¯?gÑ¼

/ªggªÑ¼4ì Ó4gÝhÝg#g:¼?/4

Eb(Ea(m, r), r′)#©g4g64 #gg\fÎh\¼È#©##Ô#ÑÐÎh\¼

bFhCÔ6# Ý4Ô/hEa(m, r)

Ì #©©:h4ªq4:Í§h `Ph|hÑ¼?Ðh2#gg#¤¼h7#ÈÙfÑ ì hYhÑ6#gÈh q#:#g \hÑ§FhQÔ6Dq4# A N kH

T+'-0]HJIL#$?«QB¬I-[d'-01#$0¿RÍ*Í§4Ó4Ýh#h#g¯#h#ÑÈh<h&g#g:¼?/4&#g|_Îh\¼Û:Í§h#### Ì #f77#4Ô|hÔ#Ñ:hâ # ÓªgÑ¼Dh FhÑñhÕ#444##`g:¼?/4:¼?:F h #g#gFh Ô6P# &hÑÓªY#Cgg|h4#ggª p#PÑñh:Ýg\4#gôÏóÍ§hÝ\ggÑ¼##Ô#Ñ# Ôª¼PÏ8éhªõh h ç6àh#Î^ Nm4 H Ì #È#gòÍ©4Îh§q4ÑÑRÍgÛ 0m,ÝÜJ01-0ZIKgÞ

•÷ #?hÈhÛöù?* #Ñ#

N = pq = (2p′ + 1)(2q′ + 1)Íy#g

p, p′, q, q′

•ù?gÑg\ÛhjgÑggª

g ∈ QN

Í#gQN

g#hgQ#44#khhÑÑá|hgÛ(Z NZ)∗

÷ #4α ∈ Z τZ

â#Í#gτ = p′q′ = |QN |

•÷ 4#/

h = gα mod N

mQ"X O v +T³ý¬ ( P O

Ì #È##Ô#ÑÎh\¼ya := (N, g, h)

h #ògg\Îh\¼yα

DFG,&-ÞÌ g#g:¼?/gh4

mâ|#?4ÈhQªg4g

s > 0×|_

m ∈ Z NsZâ8h #?4

h 4r ∈ Z nZ

Í#gn = 4log2 N

Ì #òg##g:¯?

Ea(m, r) = (gr mod N, (hr mod N)Ns

(N + 1)m mod N s+1) =: (G,H).

ü2\Ls

g#hÝq##\4PÍ

Ls((N + 1)m mod N s+1) = m mod N s.

hÑ44# |_7g4#/gÕ#2q##\4âHhâ|_7FhÑg#Ññ_g# g\ÐÑ4ª_ï#Íyg6g\#ògÑggª

(N + 1) ggÔ6 A N 5H

HJ01G,&-Þì Ó4gPÈg##g:¯?

c = (G,H) = Ea(m, r)âsFhCÔ6 #g q4)#Ñg#hYh

c4

_h# #òg#g:¼?/4 Ì #gQ#ògh4mFhQÔ6Dq4# h

m = Ls(H(Gα mod N)−Ns

)

= Ls((gαr mod N)Ns

(N + 1)m(grα mod N)−Ns

)

= Ls((N + 1)m mod N s+1).

!#$#$%&(')8aY%&q0Z,qÞÌ #Ý#gPÝ # Ó4gÑ¼#444###gC4Ó4g

Ea(m1, r1)h

Ea(m2, r2)Í§

FhQg4#/

Ea(m1 +m2, r1 + r2) = (gr1+r2 mod N, (hr1+r2 mod N)Ns

(N + 1)m1+m2 mod N s+1)

= Ea(m1, r1) · Ea(m2, r2).

7g#g×#×hÑ44# # Fh/ªggªÑ¼&Ô6×#Ñggª Ô¼&#Ñ#Ñï¼?#Y#×##/g##g:¯?h h##Ñ¼?#Ô#Ñ #hÑ44#yÛ 0m,Ýdg@_IQ&&()-?-Þì Ó4g×###Ô#ÑÎh\¼?

a := (N, g, h)Í#g

h = gαh

b := (N, g′, h′)Í#g

h′ = g′βÍ"g4g64 ##Óh_yÎh\¼?αh

βâ©g6g\Ó4gÑ¼4Ý¿hy|_

N = pqÝ#:

mQ"X O v +T³ý¬ ( PZh

Ô6C#yhCq4Ô8h"##Ô#ÑCÎh\¼?g Ì #g(#Qéhªõh Ü:àh#Î#gy_:î|g×Îh\¼:Í§h####ûp×î|:|FÓ4 \î|#Yhpg#g:¼?/4phÒyg##g:¯?FâAHhÍ©|FÓª×Qg#Ññh4#gh4ò|hgh

Ea(Eb(m, r′), r) = Ea(g

′r′

mod N, (h′r′

mod N)Ns

(N + 1)m mod N s+1)

:= (gr mod N, g′r′

mod N, (hrh′r′

mod N)Ns

(N + 1)m mod N s+1).

Ì #g

Eb(Ea(m, r), r′) = (g′r

′

mod N, gr mod N, (hrh′r′

mod N)Ns

(N + 1)m mod N s+1).

7g#ghâ©4Ó4gEa(Eb(m, r

′), r)#QhÑ44ï#úD\¼ªÜ:ùÍÒh(Fh":Í§h(#Cî|:×¤Í©ôg_Ü

4 |_gÝhÒ##Ñ×ðg4#/Eb(Ea(m, r), r

′) Ì ð4Ô/h

Ea(m, r)4Ó4g(h

g#g:¼?/4

Eb(Ea(m, r), r′) = (g′r

′

mod N, gr mod N, (hrh′r′

mod N)Ns

(N + 1)m mod N s+1)

=: (F,G, I)

Í©Èg4#/

I(Gα mod N)−Ns

= (hrh′r′

mod N)Ns

(N + 1)m mod N s+1(gαr mod N)−Ns

= (gαrh′r′

mod N)Ns

(N + 1)m mod N s+1(gαr mod N)−Ns

= (h′r′

mod N)Ns

(N + 1)m mod N s+1

= (g′βr′

mod N)Ns

(N + 1)m mod N s+1

=: H.

7g#g(F,H) = Ea(m, r)

§# g Q gg:¼?/4Ö #:#g#48g:gh§#È#gPh Q#?h§q4ÈÈgYhªgg#¤¼# g¤Í©Fh4|hÔ#Ñòh#/4#hD4Óª\QA Nm4 H

|~ ®

¯ ° J ²± ³> J ¥´ jµ±ª±~

.¶l · |Ïª±ª± ¸] Î |Ï

¯ ~~¹±Õ|

Ì #§ØAhÑÑg2#gc O k?7hÈ¯#h#Ñ<h|ÒÓªg:¼ò/ªggFâ_#hÔ|hÔ#Ñ:hâ_ # Ó4gÑ¼Dh FhÑñhÒ#444##g#g:¼?/4y¯#gòÔ|h 4Qh#\g§C#ò#hAªg4g #Ñ

N2Í#g

N## #\òh<¤Í§PÑñh4Ý#ggÈÏóDÍ§hò##Ô#Ñ# 5 jGjGj

h h|hÑ¼? h ¯?g Ô¼ô\ÓªghÑ©h/#4Cëq# hB Nm4 â 4 :/â N 5hâ5 j â 4 5/ í n #h`#g¯?g#4#Ò#gÑÑ/Dg#:Ó4ØAhÑÑ\#ghâÙ ÷ ØÐùyq4#4:Fâ8Í#QÍÒhgggªÑ¼&##Ô#Ñ# Ô¼ôù6 ì hÑÔ#h P N H Ì #Ù ÷ ØÐù&DC4g#ghÑ_4Ûh©ØAhÑÑ\F g#g:¼?/4ô#gq4#ªg4gD #ÑPCá|hPgÑÑ/g#:Ó4gòÓ4gD#4gØAhÑÑg#gÑf P4g#ghÑ#¯#g×Y#gÑÑ/Èg#:Óª\#PÔª¼&##h#4YhÑ4#ÒgÑÑ/Dg#:Ó4gÓ4g#4¸ OGj Hâ8Ô#/ ì hÑÔ#hCq4# gg#.¼JÈ8FÍy#4g#ghÑ_4] P N 2Í#gFh#ÈÙ ÷ ØÐùPFhQÔ6ò#RÓªgQgYhªFhÑÑ¼Ygg#ÈgÑñ_Ó4Y#\Í \î|# #4Ô#ÑgQÏ¤Û#ÝhÈÍ§R¼ÛhÏKéhªõh h çàh#ÎyYh|h4 4g#ghÑ#È44|hÑØ`hÑÑg§¯#gD×#4#gÒ #Ñg§g|hÔ#ÑÈÍ gh##ÑFÜ4Cg46 ì hÑÔ#hï \Ó4gÑ48 Ý4g#ghÑ_4YhÕ#DgÑÑ/g#:Ó4ÈØAhÑÑg©#gëqgB P N í Ì #×ØAhÑÑg¯#ghâK#Ý#\Í!gÑÑ/Èg#:ÓªÓªg4ðÔª¼ ì hÑÔ#hï&hÍ©gÑÑ`h#q#.Ü#gP4g#ghÑ_4'h&¯#h#ÑgCq4C#4Ô|hÔ#Ñ:hâ # Ó4gÑ¼"#444##Ûg:¼??Ü4:¼?:ggâ7Í#phhÑyFhÑñhò#44×4##hâ2Hhâ2#\¼ô|FÓª##48g:gò|_

%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®6L²O ,.&+¬°8³Êx % T ' ³²-+K)³C¯¬T P j

hò#gggh:¼Pq4§#ò4Ñ/4#§#ògg|h4#ggª C#òÑñh:|h/ggÌ #×8g:q4Yh#gh<#×Ù ÷ ØÐùôh hÐ4g#ghÑ_4ðhÝÔ¼ÛThÑRÍ§g|hð#44|hÑ/ØAhÑÑgA¯#gÒ44\#g`Í#4g#ghÑ_4Ýhéhªõh h àh#Î#g#\¼Y46g_4PgÑÑ/g#:Ó4gÒ #ÑÈÑñh4?#ÝÔ6gg7g#ghâ/#gÑÑ/g#:Ó4Ó4g.Ü4C©Yh#Ñ¼×h2#g4\FhÑ6ªgg:F n #ªgg:#Ý64ª§Ð|_©#gÑÑ/g#:Ó4Ó4g4 ÈÔ|h 4 QÑ4ªÑ¼ åKggªh#/4|h Ø`hÑÑgg Èh4|hÑ`Ó4g4Ì #©h#×/4YYF¼hÑÈ#4Ñ \Ó4gCK#44|hÑ|ØAhÑÑg<h#/4Í©gÔ#4ÎhgÏ¤#Y|h/gYÍ©&Ô8g4Í4Ó?# #4:C #\4"pgÑÑ/yg#:ÓªgCÓ4g#4 gFhÑÑ#&Ô|hÛTh\|_yhÛ#g q4#Ûq4ÑÑRÍ##ggg Ì #gâ_ng#Yh#Û#QÙ ÷ ØÐù(h ì hÑÔ#hï Ý4g#ghÑ_4pÍ© \Ó4gÑ4"&q#:#g4g#ghÑ_4Ph7#Ù ÷ ØÐù6|ûpòÍÑÑKg4#:#\#g#4Ñ Q gg:¼?/4QÓ4g4QQhh|hÑ444#ÒYh##ghéhªõh h àh#Î Ô|h 4ÛØAhÑÑgg Ò¯#\B N 5H Ì #4##\Í¿#g#4Ñ Ù ÷ ØÐù &#g#4Ñ Ó4g4(h#yÙ ÷ ØÐù 4(4#P yh hgÑÑ/g#:ÓªÓ4g4PhÕ##g#4Ñ ØAhÑÑg©#gDÔª¼Péhªõh h àh#Î4C#h#g h

ØAhÑÑg→

Ù ÷ ØÐù↓ ↓Ì #g#4Ñ ØAhÑÑg

→T+'-01.V'-W)DY*+a¦d

n #<#g#4Ñ Ó4g4ChÑÑRÍ<h¼×#Ô#\§h k 4/©h ` #ÑñF¼ªgf gg:¼?/ÒÈg##g:¯?g4g\Ñ¼h /ªggªÑ¼Í#ÑÒ hÑÑRÍÐhª¼×#Ô#\§hÑgf|hk#ÑñF¼ªgfÈ4Ô/h

hª¼Q/q4Y_4QhÔ64/#Ègh4h Ú h 4Û4#Ò#g#4Ñ Ó4g4QÍ§hhÔ#ÑD ggÔ6Óhh4#©h##ÑF_4#ÐCÑhfÍ§R¼Ch©ÐÍÒh h#Ô|h 4C##g#4Ñ Ó4g4Qh`Ø`hÑÑgg §¯#gYëqgB Nm4 â 4 <â NGN â N P í Ö :Í©¯/#ÑñhQ#ÍÑg#h?ÜÊÈ|¯/Ô#ÑÓhhñhªh`#È4g#ghÑ Ù ÷ ØÐùQFhÛÔ6g4?Ü:#\ âAHhâAPÓ_hñhªò|_hÑÑRÍ \g##× É h©#×g##g:¯? ##g#g:¼?/4 Ì #f<#\q#Ñ/?4Ñ?q4 åKggª<h##ÑF_4#A#Yh`ïÜ¤#\g Ö #:#g.Ü4hâÍ©#RÍ)#Í ##g#4Ñ Ù ÷ ØÐùôFhôÔ6×# \Ó?×4g#ghÑfÝ#Ñ|h:¤¼g4#/_4'#h?ghÑgâHhâ## #g#4Ñ k gg:¼?/4'Í§ô#ggª& Ý#Ñ|h:¤¼g4#/_4Y#h?g4Ñ6Í#yhÑÑRÍÐÝg4#/DhPg#g:¼?/4PhÕ## #\ÒhÕ¤Í©gh4g4Ó4gy#Èg4g64 #g##g:¯?Ô#/#h#ògh4g#ggÑÓªgg©

Q"$# «§¬g)®¯Z°±Z²)³¯¬A§¯Ây ' ' ³()³C²ï±®À,F¯ÃÀ®Ä¼³ý¬ ( Z/NsZ

<:

q#:#g<h##ÑF_hÍ§ ggÔ6#RÍD#hg4Ñ/q4`gÑg\4#©Óªh#DFh×Ô6Ò4Ô/h# âÛÍ#&Fh¯ÛÓªhg#Ñ¼y64:#Ô|hÑÑh4ðYÔ##ÑÑ\QÔ6ªh Û#¯ôÍ§R¼Q|_#î||hÑ6hÑÑ¼×FhPÔ6Dgh#/ h§#¿h7hÑÑ8Óªhg§Í#Ñ©g4g\#g©Ó4gî8hÔ#ÑÛhª¼ð4Ô#g:Ó4gÝh ÍÒ#ÑÝ##Ó_h\¼ðh©#× Ó? |hÑ7Ó4hgÈD4|hhg ×Ññh:òh##ÑF_4ÛÍ©#ggªYÍÒF¼ÛPÔ##Ñ C6\gñhÑ`g4×gª¯#gÔ|h 4#gÑÑ/Ýg#:ÓªYØAhÑÑgD#g×|_È|hÈ4×ªgg:#Q#46g:ggâ7h#YòFhÔ6P#:hªñ_ g#gÝ6g:qg\Ñ¼ # #&h8g:qg\Ñ¼ Ô# ## Ö 44:h#gh##ÑF_4#4Ýq#:#gD#Ô?Ü¤#h?g4Ñòh#g Í#h ggÔ6 pù?g\hP#i<? Ì #g×#hg4Ñh#î|:ògÑÑ/g#:Ó4ÝÓh_hh<#Ý#hg4Ñ \ÓªgÑ48 ##Ø`hÑÑgg § F/

.143 Å ÁÝËª7AÂYÆ½y¾ÐË · ÂÁ ËÕÂ»| Ä\Ä · <PË · ¾Æ>Ï½¡7ÇÝÅ¼» ÂÇÝÅb7Ý¹ · ÁPÇ+»Z/NsZ

ûpC#4:Ñ¼ô #gYgÑÑ/×g#:ÓªgÝÓ4gÝ#4ò&#RÓ? #YÔ|hTh\|_Ýh#g Ý# g:h ì hÑÔ#h ÐgÑÑ/g#:ÓªòØAhÑÑgÐ#gòh©Í§gÑÑ_Ò#©4g#g.ÜhÑ_4Qh 4#q#:#g4g#ghÑ_4 Ì #ª #\4yYh#Ñï¼PÔ|h 42 P N HÖ 4§q#:#g \hÑÒh 4g#ghÑ2ª #\4C×gÑÑ/g#:Ó4gÍ§ò\qgÒ kK5Hü2\

RÔ8òÝg4Ý/_ïÓª#Í^5òh

R∗#D\Òh7ªÓªg:Ô#ÑgÑgg§

Rf

/ØCØÙ ) _/¢ y2z = x3 + axz2 + bz3

RÓªgR \î|# Ôª¼Q|h

a, b ∈ R#Q|_

6(4a3 + 27b2) ∈ R∗§#|hÑ2#È\

hRÜóÓhhÑ# 84ªD g#h Ôª¼

E(R)h \î|# yÔ6#×\Èh©gá#Ó_hÑg#g

gÑñhgòhÐ64ª(x : y : z)

Íx, y, z ∈ R, y2z = x3 + axz2 + bz3

h #¯ô|_#È FhÑÕ4\#g_ Ôª¼

x, y, zRâ6h q4Í#Q#òq4ÑÑRÍ#×gá#ÓhhÑg#gÈgÑñ_4

#4Ñ g

(x : y : z) ∼ (x′ : y′ : z′) ⇔ ∃ λ ∈ R∗ : λx = x′, λy = y′, λz = z′.

Ö 4ò# gg/4ph§#YgÑÑ/×g#:Ó4PØAhÑÑgD¯#g×Ph È4g#ghÑ_h#DÍ§#g yg4# gD#4

R = Z NsZÍ

NÔ6g#P## #\òhÐ¤Í§y#g

ph

qâ§h

s ∈ N Ö h×##4#Q#|hÑÒ¯#4 h h#4g46\4"#RÓ? g

44#ôÑñRÍ 4E(Z NsZ)

ÍO = (0 : 1 : 0)

hD gª.¼ÛgÑggªR Ú ¼&# ÷ ##g


<15

ögYh g Ì #g4g §q4ÑÑRÍ©|_E(Z NsZ) ' E(Z psZ) ×E(Z qsZ)

Ì # ¤Í©64ª

(x1 : y1 : z1)h

(x2 : y2 : z2)4QhPgÑÑ/g#:ÓªòRÓ4g

Z NsZÍ©òFhy#D#

#|hÑq4Ý#Ññh`q4z1, z2 ∈ (Z NsZ)∗

Ì FÓ44 #4Ô#ÑgÕz1

4z2

hÐ#hAªÓ4g:Ô#Ñ`ÜhAAÍÑÑhngÝgg#`Ýh#`\#ÜÍ©§YhÎ4Ð#©g#:Óªgfª#§###

(x, z)−#Ññh#

Ôª¼Pgá##Ý|_y = 1

¿h|Òq4z#hÔ6g#×##

R§q4ÑÑRÍ©q4#

g#:Ó4×gá|_4&|_x#hòC##Fâ6# Ì ?#D&#Fh

y|hPÔ6P##Dh

Í©Fh\y = 1

t7g#ghâÕÍ©g4# gòg#:Ó4gÈh©#q4z = x3 + axz2 + bz3

×64ª

(x : y : z)#RÍ*Ô8gg4g

(x/y, z/y)h #È gª¤¼PgÑggªg#ggª Ô¼

(0, 0) Ì #òªÓ4gòhAÝ64ª

(x, z)§4Ô/h# h

(−x,−z)fÑh#×h§# Ó?4

46g_4#Ðh \î|# ×###âª##(x3, z3)

hK¤Í§È64ª(x1, z1)

h (x2, z2)q4

x1 6= x2

F_QÔ6òg4#/ Ôª¼C#Dq4ÑÑRÍ#44#Pq4#Ññ_h

x3 = x1 + x2 +(z1 − λx1)(2aλ+ 3bλ2)

1 + aλ2 + bλ3,

z3 = λ(x3 + x1) − z1,Í#g

λ =z1 − z2

x1 − x2

.

Ïóx1 = x2

#Q|_ÒÍ§ 4#Ô#Ñò64ª(x1, z1)

Í©ò4Ô/hP#ò84(x3, z3)

h

x3 = 2x1 +(z1 − λz1)(2aλ+ 3bλ2)

1 + aλ2 + bλ3,

z3 = λ(x3 + x1) − z1,Í#g

λ =3x1

2 + az12

1 − 3bz12 − 2ax1z1

.

Ïóò# Ó?4#Ch&#h \î|# Í©ð#g #ðh#gYq4#Ññ_h Ì (4Ô/h#q4Ý#ÑñhÍ©ò|RÓ4D×gFhÑÑ4Th\hÔ84/#

pÜ. #g4:¼PhAgÑÑ/g#:Ó4ggâ8g

kK5q4§q#:#g \hÑgfü2\

E1(Z NsZ) := P ∈ E(Z NsZ) : P #gg

(0 : 1 : 0)E(Z NZ).

Ì #gâ|q4hQgÑggª(x : y : z) ∈ E1(Z NsZ)

#4Ñ §|_N | x,N | z

h N - y

Ùà÷ 4# g©#gÑÑ/g#:Ó4

z = x3 + axz2 + bz3RÓªg

Z NsZ ì ÓªgPhª¼

xÍ

N | xáÈ ñëE/ëòðcõø/çÙôäñ õæñ

N3 | zü


< 4

Í©òî| ##á#ò4Ñ/4CÍ#gN | z

fÏó|hÒ#Dq4Q

w(x) = x3 + ax7 + bx9 + 2a2x11 + 5abx13 + (5a3 + 3b2)x15 + 21a2bx17 + · · ·

ënq4#h#ÑÈ4ÑÓ?#×#Ègá|_4ÛÔª¼Qg_4 í Òù?#gN | x

#4Ó4gÛ#w(x)

î|#D

Z NsZh

N3 | w(x)

ü2\ψ : N(Z NsZ) → E1(Z NsZ)

Íx 7→ (x : 1 : w(x))

Ì #ÐYh4fh/#Ð#Ô#44#N j(Z NsZ)

# gψ§#ò#Ô#44#

Ej(Z NsZ) = (x : 1 : z) ∈ E1(Z NsZ) : N j|xh

N 3j|z

hE1(Z NsZ)

n ÔªÓ?4#Ñ¼4â Es(Z NsZ) = (0 : 1 : 0).Ì #ÝYh

ψÔ# ¾:g\Ó4Ýh |h

gg:hô#444###46g:gg×Ñ#4#4ðòD#hÈy44#ô#444##gÑq4ÈhÑÑ

sâ7È #ggy44##444##Þq4

N j(Z NsZ)/N j+1(Z NsZ)y#

44#Ej(Z NsZ)/Ej+1(Z NsZ)

<Ï¤Q|h:g#ÑñhgâÍ§ò4\

|Ej(Z NsZ)| = N · |Ej+1(Z NsZ)|

q4ÒhÑÑ1 ≤ j ≤ s− 1

7g#g|E1(Z NsZ)| = N s−1

¿h|_§q4s = 2

Í§D4Ô/hPh44##Ô6\¤Í©gg

Z NZh

E1(Z N2Z)

¿RÍÍ©Fhð4Ó4h&¯/#ÑgqhÝ#Ññqh##Þh`#Ý84(x1 : 1 : w(x1))

h (x2 : 1 : w(x2))

Ì #ò#§#ò64ª(x3 : 1 : w(x3))

Í#g

x3 = (x1 + x2) + a(−2x1x24 − 4x1

2x23 − 4x1

3x22 − 2x1

4x2)

+b(−3x1x26 − 9x1

2x25 − 15x1

3x24 − 15x1

4x23 − 9x1

5x22 − 3x1

6x2)ëP#65 í

+a2(−2x1x28 + 8x1

3x26 + 16x1

4x25 + 16x1

5x24 + 8x1

6x23 − 2x1

8x2) + · · · .

ÝhÑF ¼ðgª4# âE(Z NsZ)

Èyî|#×44#Íp gª¤¼(0 : 1 : 0)

Qù?#gE1(Z NsZ)

Ò#DÎ4g#gÑÕh7#ò #\4yYhPq4E(Z NsZ)

E(Z NZ)

â#Í©ò4\

|E(Z NsZ)| = |E1(Z NsZ)| · |E(Z NZ)|

= N s−1 · |E(Z NZ)|.

Ö 42# gg/4h##Ðg:¼?/4:¼?:FAÈ#<q4ÑÑRÍ#g\4#7Í©©#g g4# g

Q" !wº &y ' ' ³()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. < N

#§64ªPi = (Ni : 1 : 0)

4E(Z N2Z)

Ì #\¼h§hÝ64:hª7ThÑï¼ò_864ªA4E(Z N2Z)

Ì g4#/m ·Pi

Í©|FÓª##hÔ6RÓªÙfá|_4BP#65#g Ó?4#hC#h \î|# ûpP4\Ý|_

mPi = Pmi

ù?ÑhÑï¼4âAÍ#g Í©4Î#ðE(Z NsZ)Í©Y#×#64ª

P1 = (N : 1 : w(N))âAh gh#/

mP1

h##Ñ¼?#QÙfá|_4$P#654Ö #:#g4hâ#ghQÔ6ò#RÍQ|_

N s−1P1 = (0 : 1 : 0).

.1I0 ÇÆPÅÉ| Ä\Ä · <PË · ¾c>Ï½¡7ÇÝÅÊ_ ¶¸· Ä\Ä · Åb7¹C¾;ÆPÅÉÊÅ

Ï¤#<g\4×#gÑÑ/Òg#:Ó4ØAhÑÑgf#gÔª¼Yù6 ì hÑÔ#hïÍÑÑ#Ô8#Yh ÏóA7|_#hÑª4g#ghÑ_hòh|ØAhÑÑgg 2#4Ô|hÔ#Ñ:hâR#444##f##Ô#Ñ<Îh\¼Èg:¼??Ü4:¼?:g9 O k/gÑÑ/<g#:ÓªgfRÓªg`#4g Ì #`#g©ÜÕhAÍ©gÑÑ/hÀ4g#ghÑ_4#|_Í©ÈÍÑÑ#ggªÛù?g\4P# N h P# PÜ©|FÓ4È¯#h\Ñ¼P#È#46g:g#g q44##hg4ÑÒg4#:#\ P#òÑñh:|h/ggù?g^ P N <q4 \hÑòh gPù?g\4AP#65×q4È4×Y_#gY_FhÑATh\hÔ64/ÈgÑÑ/g#:Ó4gRÓ4g#4h hÔ84/#

pÜ. #g4:¼PhAgÑÑ/g#:Ó4gg

Û 0m,E?%010ZIKÞÌ 4g#g_ÈÎh\¼

•g4#/ò× #Ñ#

N = pqh# #\h7¤Í©#g

p, q > 3

•÷ #?h×Ch 4gÑÑ/Èg#:Óª

E : y2z = x3 + axz2 + bz3Ó4g

Z NZâHhâ

gcd(N, 6(4a3 + 27b2)) = 1

ü2\M = |E(Fp)| · |E(Fq)|

Ô6D#D4 ghE(Z NZ)

Ì #gÛÎ#RÍÑ 4òhM

64Ñ¼?#4hÑ¹Üó×gá#ÓhhÑgªÈQÎ#ÍÑ 4h§#×Th\4_hhN = pq

ëqg P N í Ö #:#g4hâ|

p, qhòÎ#ÍÒÛ#g

MFhÛÔ8Èg4#/ Q64Ñ¼?#hñhÑ

###ù?¯#hñÜ¤ÒÎ?Ü.ÙfÑÎghÑ44ï# ëqgÈh#ú5 4 í

•÷ #?hP84

Q = (x : y : z)ÍÒ

ord(Q)|ME(Z N2Z)

×ù?#gÍ©×|FÓª|E(Z N2Z)| = MN

ëqgôù?g\4P#65 í â§#Y64ªCFhÔ6Qq4# Ôª¼"hÎ# h 4 84ª

Q′ = (x′ : y′ : z′) ∈ E(Z N2Z)h \#

Q = NQ′

ü2\P1 := (N : 1 : 0) ∈ E(Z N2Z)

Q" !wº &y ' ' ³()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. <P

Ì ###Ô#ÑÎh\¼Cg4#:§h2# #Ñ#NëTh #g#g#84

P1í â#g?/ªggª

(a, b)h7#ògÑÑ/g#:Óªhâ8h #ò64ª

Q

Ì #Ègg\Î4\¼yÒ#ò4 gMhA#ò44#

E(Z NZ)

ÝhÑF ¼ôgª4# ù?g\4$P#65Y#4Ñ ò|_mP1 = Pm = (mN : 1 : 0)

q40 ≤ m < N

Ðù?#g

(m+N)P1 = ((m+N)N : 1 : 0) = (mN : 1 : 0) ∈ E(Z N2Z)

Í©§FhhÑ \î|#mP1 = Pm

q4m ∈ Z NZ

Ì #ÀhÑÓhhÑ q47#©4g#ghÑ_4#4Ó4gQP#DqhÑÑRÍ#Ýg\4#gDFG,&-ÞÌ g#g:¼?/×gh4

m ∈ Z NZ#?4×h 4 g4g

1 ≤ r < Nh g4#/

#ò64ªC = rQ+mP1 = rQ+ Pm.

Ì #Èg##g:¯?§#ò64ªC ∈ E(Z N2Z)

HJ01G,&-ÞÌ gg:¼?/#òg##\¯?

C#ò#ògg\Îh\¼

M×g4#/

MC = r (MQ)︸︷︷︸O

+MPm = PmM = (mMN : 1 : 0).

ì Ó4g #xÜ¤g?4 |_

mMNg#\

ZÍ© Fh Ó? Ôª¼

N'4Ô/h

mM ∈ Z NZh #gÝ#Ñ#Ñï¼ Ôª¼(#QÓ4gÛh

M mod NpggRÓªg#Ûg.Ü

h4m ∈ Z NZ

yë)¿h|_Í§×|FÓ4Z NZ ' E1(Z N2Z)

hDg4# ù?g\4P#654 í!#$#$%&(')8aY%&q0Z,qÞÌ #Ð#gÐ # Ó4gÑ¼Ý#444##Ò#g4Óªgg#g:¼?/4#

C1 = r1Q+m1P1

hm1

h C2 = r2Q+m2P1

hm2

hg#g:¼?/4(C1+C2) = (r1+r2)Q+(m1+m2)P1

h(m1+m2)

FhÝÔ6©g4#/ ¾:#:`Ôª¼Ý # ##©g##g:¯?C1

h C2

;7g#ghâ4Í©§Fh \î|##DhÑ44# # h© #

(E(m1), E(m2)) := E(m1) +E(m2) = E(m1 +m2)4h #

(E(m1), E(m2)) := E(m1) + E(m2) + r′Q = E(m1 +m2)

Q" !wº &y ' ' ³()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. <G<

Í1 ≤ r′ < N

×4 g<ÈÔ#Ñ #g#ÑF<ù?#gÒ#gh4\M

gá|hÑZ NZ#g:¼?/4:¼?:g §hÑFhÑñh©#4×44##h #DhÑ44ï#çQï/ Ü:çQ#Ñ§FhCÔ6

#Ñg×g ##Ûg8F_ Ñ¼ô#YhÑ44# # h 4Ô#Ñ #QhÑ44# hhÑF ¼Cgª4# ÷ |h/g 4 01#$IQJR-Öý Ö)LÖ Ö 42g#g:¼?/4ÈÍ©ÐÝ#:74|hhªgf|_Õ#Ðh 4jÓhhÑ#

rÕ#4g

q4!\f|_fAÑñh4©g#4#4âªHhâhÍ©Ò#?4r ∈ 1, . . . , N−1

7û #g # #¤Í©g##g:¯?

C1 = r1Q+m1P1

h C2 = r2Q+m2P1

Í©4ªÐ4Ô/hCÈÓ_hÑ#r1 + r2Í#ðÑñh4g|h

N − 1ûpFh&4#4ò#hÑ4#Chq4#ÝC4Ô/h# #\Í

g##g:¯?C<#4Ñ f|

C 6= (mN : 1 : 0)âÍ#YFhYFhÑ¼ÝÔ6Ó4gî| n #g:ÍÍ©|FÓªÝÔ#Ñ #gh4g§Ôª¼C # #

r′QÍ#g

r′ ∈ 1, . . . , N − 1Ðh 4Ñ¼

#4gQû #g ##Q#Y#444###46g:¤¼#q4ÑÑRÍ# ëqHhâ`hÑQ#4g#ghÑ_4# í #Ò#gÎyhÑÍ§F¼?#ÑgÑï¼Y#gÑ

Ì #Ègg#¤¼yh|hÑ¼?Ò§Ó4g:¼yÑhÐ×|_h`ØAhÑÑgg §44|hÑ6#gh¤¿hD|_#ÛgÑÑ/Pg#:Óª

E(#¯#gÛ#4g'h 4Ñ¼4Ý7g#ghâ gY#hCÑFhÎ

hª¼Û¯?P/q4Y4QPhô Ó4gh:¼ÛÍ#¯&Í©4#Ñ #gÑ&YTh\4N¸7ghâK#

gYhªðgg#¤¼ëTh \î|# PZ< í #ðFhôhÈ|hÓ4 Ó4ghg g8g 4p#C|h #g_Ò#Yq4ÑÑRÍ#Q#4Ô#Ñg Í# Í©CFhÑÑ<#b/ØCØÙ ) /¢ R CØCØÙ / ¢ ¨\) ì Ó4g&×64ª Q ∈ E(Z N2Z)

hf4 g Ó? #|E(Z NZ)|

Í#gN

#§# #\`h|.Í©Ññh4©#ggâ4h 4Ó4g×h 464ªC ∈ E(Z N2Z)

\g#Í#\#g

CÑgÒP#ò#Ô#44#Q4g#g_ Ôª¼

Q

¿h|#|h #g×h#/4 Ñ4ªÑ¼ åKgÈq4 R CØCØÙ /TË ¢ ¨\) âAÍ#'#ðh#gY#&|h #gPhD#Ûq4ÑÑRÍ#p#4Ô#ÑgQ ì Ó4g

N = pqh

#Ô8gc|_Dg#gh 4

(Z N2Z)∗4Ch 4

N ð6RÍ§gD

(Z N2Z)∗ gg DÍ#\#gcÑgÒP#ò#Ô#44#Qh

N Q6RÍ©gg

Ïó64Ô#ÑòCh##Ñ¼Q:h h \# Y4Ô/hÛ4È4Ô##:gg#¤¼Û#48g:gq4 pgYhªFhÑÑï¼ \g#ðg:¼?/4:¼?:gQ Ö 4#:h#ghâÒ× h:g4#:#\4 ggÔ6 ÍÒ#¯ðg4ªÓ4g:òØ`hÑÑgg #gÝC|_DghFhÑÑï¼ygg#×hªh#:hÛ h/Ó4D#4gÛg##g:¯?_h¯ÎYP#òh 4 4hgÑD× gÑHù?gB P N 2q4§q#:#gÒ\g#.¼ g#4#4§q4Ò¯#h#Ñgg

Q"m%ÍÌ ¬®,+ ' ³ ,+K)³C¯¬ ¯\¤x §y ' ' ³()³²ï±®À&ZÁ+³ ' ' ³®YL²O I. < O

.1Eo ½ ÎÞÅÁPÅb7 ¶ Ä · » ¶ Ë · ÂÝÁ ÂYÅ)ËÆPÅÏ| Ä\Ä · <PË · ¾:>Ï½¡7ÇÝÅÐ_ ¶¸· Ä\Ä · Åb7Y¾;ÆPÅÉÊÅ

Ï¤ÝÑhÍ§F¼hAéhªõh h àh#Îp N 5/?|RÓ4©4Ó4gÝ4g#ghÑ_4òh/#©44|hÑØAhÑÑgò¯#gY&YhÎhY4Yªgg:#Qq4Ýh##ÑF_h# ì hÑÔ#hp4g#ghÑ #Ù ÷ ØÐù P N H7Ð4g#ghÑ_4q4f#gÑÑ/Òg#:Ó4FhÍÑÑ#Ô8#gg ×#g\4h Ññ_g74Í©©ÍÑÑ4g#ghÑ©7q#:#g7D#g#4Ñ Óªg4 Ö #:#g4hâÍ©ÍÑÑ#4Ó4òÑg#h?ÜÊÈ|¯/Ô#ÑÒÓ_hñhªÐhK#4g#ghÑ gÑÑ/Òg#:Ó4ØAhÑÑgf#ghÌ #gC4g#ghÑ_h#ò#Y#4#gÈ8Í©gh

Nh |RÓ4Ygg:hp Óhhªh4gÝhÈFh

Ô6ggÕ N 5/Hù?#â#:F hÐg4# g#C#g##g:¯?ò44#E(Z N2Z)

Í©×#RÍg4# ggÑÑ/òg#:Ó4gÓ4g

E(Z Ns+1Z)q4

s > 0Ï¤Q##?ggÍ©|FÓ4ÈhÎ4

FhòhA#Ô/Ñ\ggÑñ_##Dq4YhÑK44#â|gÝù?g\hP#654â kK5Õh P N HÛ 0m,E?%010ZIKÞÌ 4g#g_ÈÎh\¼

•#?4& #Ñ#

N = pqhC# #\Ch.Í© #×g4F_g|h N h

#?4s > 0

ë Ì ##ògh4D¯ÍÑÑKÔ6D#ò44#Z NsZ

í

•÷ #?hPÛh 4gÑÑ/g#:Ó4

E : y2z = x3 + axz2 + bz3RÓªg

Z NZâ`Hhâ

gcd(N, 6(4a3 + 27b2)) = 1Ðü2\

M = |E(Fp)| · |E(Fq)|

•÷ #?h64ª

Q = (x : y : z)Í

ord(Q)|ME(Z Ns+1Z)

Ì #64QFhðÔ8q4# Ôª¼QhÎ#PYh 464ª

Q′ = (x′ : y′ : z′) ∈ E(Z Ns+1Z)h

\#Q = N sQ′

ënûpò|FÓ4|E(Z Ns+1Z)| = MN s

ëqgÝù?g\4P#65 í í¿RÍÑ\

P1 := (N : 1 : w(N)) = (N : 1 : N 3 + aN 7 + · · · ) ∈ E(Z Ns+1Z)

ûpÈ¯_Î4ÈgÛ#zÜ¤g?4 |_È#ªÑ2# g4gÝ4F\|h

s + 1ëqg

ù?g\4P#65 í n #òFhQ#RÍ |_ P1

|h4 gN sâ#Hhâ

N sP1 = O

Ì ###Ô#ÑÝÎh\¼&g4#:DhN, s

âK#g/ªgg(a, b)

h<#gÑÑ/Èg#:ÓªhâÕh #64ª

Q

Ì #Èg4g84 #gg\Îh\¼y§#ò4 gMh7#ò44#

E(Z NZ)

DFG,&-ÞÌ òg#g:¼?/©Dgh4

m ∈ Z NsZ#?4h 4!ª\4g

1 ≤ r < N s_ g4#/

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. <Gh

#84C = rQ +mP1

##CÙfá|_4P#65hÐù?g\4 P#65Cëq#g Ó?4#h#h \î|# #g í Ì #Èg##g:¯?§#ò64ª

C ∈ E(Z Ns+1Z)

HJ01G,&-ÞÌ ggRÓ4g#ògh4

m ∈ Z NsZg4#/

MC = r(MQ) +mMP1 = mMP1 := m′P1 = (m′N + · · · : 1 : (m′N)3 + · · · ).

Ì #gm′ ∈ Z NsZ

FhðÔ6Ýg4#/ g_Ó4gÑ¼Ûh _ngÝ#Ñ#Ñï¼?#×#Ýg#ÑÔ¼M−1 mod N s

Í©×4Ô/hÛ#gh4hm = m′M−1 mod N s

+¿hÝ|_DRÍ#C#CTh\|_q4

s > 2#yYh

ψÝ#hðhh#(44##Ô#/× #gg×4#Ñ¼

#D44#C44##q4N j(Z NsZ)/N j+1(Z NsZ)

Ej(Z NsZ)/Ej+1(Z NsZ)

â Ó?4#Òhò#h \î|# #ghâ8gù?g\4P#654Ì #×g_4ðòhDq4ÑÑRÍgûpÍ

m′ =∑

im′iN

iðgDhÐDÔ|h¯Ü

Ng#¯Ü

gª_4<ü2\#ò64ªm′P1 = (x : y : z)

Ô8ò4Óªg Ì #xÜ¤g4 |_Dh7#Ò84

gá|hÑ ∑im

′iN

i ·N + · · · = m′0N +m′

1N2 + · · · + · · ·

fûÈFh \g##DÓhhÑ#hm′

0

hm′

0 = xN

mod N`ûpòFhy#gQ#Ô/h\

m′0P1

q4m′P1

ëq##YÙfá|_4P#65 í &4Ô/h Q#\Í¿84ª

(x : y : z) Ö 4 #84ÈÍ©CFh ggRÓ4g

m′1 = x

N2

mod Nh #ò#ggÒg_

¿hD|_Òq4s = 1

Í§ò4Ô/hC#òÔ|hògÑÑ/g#:ÓªÈØ`hÑÑg§#gh

n ÔªÓ?4#Ñ¼(#4g#ghÑ_4"|h#ÛhQ#444##y#46g:gCh#ÛÔ|h#gh Ì #È×gYhªgg#.¼Ô|h 4p#Ph# |h #gh§#q4ÑÑRÍ#yh#/4âÍ#ôÍ§×FhÑÑ Â / ØÙý ¥ /ØCØÙ ) 8/¢ R CØCØÙ / ¢ ¨\) ì Ó4gCÈ64ª

Q ∈ E(Z Ns+1Z)h2h \ Ó? #

|E(Z NZ)|Í#g

N<## #\

h`¤Í©YÑñh4ò#gh 4Ó4g&h 46hC ∈ E(Z Ns+1Z)

\g#òÍ#\#gCÑgÒP#ò#Ô#44#Q4g#g_ Ô¼

Q

.1 ½ ÇÆ¡7AÅ¼»-ÆPÂÈÄgÆÐÑ*Åb7;» · ÂÁdÂYÅ ËÆPÅÒ| Ä\Ä · <PË · ¾Ê>Ï½¡7ÇÝÅh_ ¶¸· Ä,ÓÄ · Åb7¹Y¾qÆPÅÉÊÅ

Ï¤ N 5Aéhªõh h àh#ÎP#464 ×#g#4Ñ & gg:¼?/4ÛÓ4g4&hA#\4g#g.ÜhÑ ØAhÑÑg¯#g×Ô|h 4h FPÔª¼ù?#4# kG:H Ì #D#g#4Ñ Ó4g4|h

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. <k

Yhª¼ôªgg:#Q#46g:gh È#\q#Ñfq4È#g4#h##ÑF_4#È_òÍÒhÈ#RÍh/ N 54â 4 <â N P#â NGN HkÏ¤#g\4Í©ð 4/Y#gYh##ªh h##Ñ¼"#×4g#ghÑ gÑÑ/Ýg#:Ó4YØ`hÑÑg#gh Ì #òÑF DQ

k4/òh

`#g#4Ñ

gg:¼?/4ÛÓ4g4Ûh`#È4g#ghÑ Ù ÷ ØÐù6â8Í#&FhÛÔ6h##Ñ &×Ñh§Í§R¼Ó_h4#Yh##ÑF_4#gâÐgðù?g\4#BP# O h P#ih? n #Yh##ÑF_4#hQÔ|h 4\Ó4ghÑ`#h?g4ÑÍ#&Í© \Ó4gÑ4ðôù?g¯4]P#i<? Ì #g#hg4ÑhÈ#î|:gÑïÜÑ/Dg#:Ó4òÓªg4#hA#ÒÎ Ö #:#g4hâ|Ûù?#Ô#g\4 P# P#65DÍ©òÍÑÑ ggÔ8ôÑg#h?ÜÊÈ|¯/Ô#ÑCÓ4g4"h#P#g#4Ñ Ù ÷ ØÐù(|_×YR¼ Ô6y#\q#ÑÒÔ##Ñ h#ïÜ¤#\gÌ #ò gÝÔ6g# #g#4Ñ g:¼?/44h#¼C© :Ô#/#D8Í©gÒh74#òh/#4¤¼×44#PhÕ#ÑñF¼4gg Ö 4§#:h#gPÈ#g#4Ñ P gg:¼?/4P#ghâ/#gg\Îh\¼|h y#¯&Í§F¼P|_4#Ñ¼y×44#QhA#ÑñF¼ªghÔ#Ñ gg:¼?/D×g##g:¯?h #h-¾:#:©4#6g47ghâÍ©#44g:Ò

k4/©h

`#g#4Ñ Y gg:¼?/4Y#ghâ

Í#Q#h?g4ÑK|_ :Ô#/g§#ògg\Îh\¼yh7#ò4\#ghÑ Ù ÷ ØÐùP×\h`#ÑñF¼4g&|_hÑÑRÍÝh¼p#Ô#\h_ÝÑFh:

k4/h

`#ÑñF¼ªg gg:¼?/

ðg##g:¯?g4g\Ñ¼(h /ªggªÑ¼4â<Í#ÑC hÑÑRÍ×hª¼ #Ô#\YhÑg|hk#ÑñF¼ªgÐÝg4#/ gg:¼?/4P4ÐÝ4Ô/hChª¼Y#\q#ÑK/q4Y_h Ì #§#48g:.¼

#4#Ñ hÑ#4Ñ ÕhC Ó4gh:¼Yg4#/©4#Ô#\ÒhÑgÐ|hk#ÑñF¼ªg<Í§4Î#

44\#gg n ÔªÓ?4#Ñ¼Q##4#Ñ Ô8 4#ÝÍ#ÑÎhgg##P##444###48g:.¼h Í#4/ g4 #Ý#Dgg#.¼Ph7#ò:¼?:gQÌ #C FQhÒ#g#4Ñ #ggÝ4?gÔ|hÎôðù?|hòÍ# ggÔ6 Ý h j ©#RÍß|hDgg\<Ó?ñòüÕh4h#4h fªg64Ññ_47Ïó<fÔ|h 4#§Th\f|_ÐD64Ñ¼?#4hÑh g4g

k − 1§##á#gÑ¼ \g# Ô¼

k84g Ì #4Ó4g

k4§4D84§h

84Ñ¼?#4ñhÑ6h g4gk − 1

Ò§64Ô#Ñgg4#:#\§Í#ÑÑgÒ|hk84ª

#hÑFhÎChª¼P/q4Y_4yhÔ84/#ò84Ñ¼?#4ñhÑTÏ¤ kG:©ù?#4#ô#h64 h/ªg\ò#g#4Ñ ÓhhñhªDh§öù?4|_#ggp7×g?Ü4# ¤Í§Yhy#46g:gh`#ð×#gÈÍ#&hÑ#4Ñ q44##ghÔY LÚ¡ Â £CØÙC)&Fh#D|_ÈÑgò|h k #ÑñF¼4gh×#hhÔ#Ñyq44y4|_#Í#ÑG£ ¢ / Fh#A|_`g4#/ #ÑñF¼4gfh§#h<hÔ#ÑÐD#\Ó4gªÐ##g4#/ #ÑñF¼4gq4¿4g#g_#4|_#gg n #Ò#gòFhyÔ6D#RÓªgygg#ÈC#Dh 4¿4hgÑ gÑ<h##P#|h #gÈh©#4g#ghÑ gÑÑ/g#:Ó4CØAhÑÑ\òh#/4Ö #:#g4hâÕ4#D#g#4Ñ Ù ÷ ØÐù#RÓ? gò#×#46g:¤¼ôhù?#4# ò#g#4Ñ _Ü

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. < j

|_#P¯#gP #Pg#y|_|h gg:¼?/4"h Óªgî|F_4(hPgh#Ñ\gÑ¼#4?Ü¤ªgh\Óªhâ<h |_Ý#y É Phh( Ó? |hÑÐ|hy g8g g×q4##Ô8gh#ÑñF¼ªgg Ì #gy#46g:gÝ#4Ñ A/ ) ¢ ) ¨B ¥ /Ø¤âAÍ#gyh Ó4gh:¼y#?4gÍ#Q#ÑñF¼4gÒ×g4#/_Ò#DÓ4g:¼yÔ6g4###h7#ò_h¯ÎCh Ò#hhÔ#Ñ|h#4ò#Ò P ###ò¯/gg/4QhA#ò#hg4ÑHÌ #ÒYh|h:fhKù?#4# `#g#4Ñ öù?4|_#§#gÒ`#h?g4Ñ|_fg|hÔ#ÑgC\DhÐ#ÑñF¼4gP/ªggªÑ¼&h g4ÑÑg\Ó4gÑ¼QhÝhð##/yYgg\ò¯/64#gª

d #ÑPhpöù? #Ñ#Nâ2Hhâ24ô##/

xFh#ÑñF¼ªgDg4#/gÈP|h×hÐ#

g#ÑF"Ñ4#ôÍ #×4?gô#?hhg4g\#gFâ§Hhâ©ô#?h|_×#Q#ÑñF¼ªg#4#g:Ñ¼C# #§|hDyÝg4g\ÒÍ§F¼4 Ì #Ò##g§|_ #4#g:h g4#/ #ÑñF¼ªgFh Ô6Y gî| Ôª¼ hª¼p#ÑñF¼4gÝh 4Ô#g:Ó4g×h #g á|hÑî| ðù?KªÜggªÑ¼ Yhª¼ g4g\|hgh#yg#ÑFhÔ8Q/ªggªÑ¼ g4ÝÔ## ôg4#/xd mod N

Ì #Ð\# Ôª¼Cù?#4#Í§h§hÑÈ# N 5/Kh FhYÔ6h##Ñ |hg4ÝÔ###ð(4#×Fhh7gy#y#ÑñR¼4g×Ý#Ñ#Ñï¼ h(##/g##g:¯?×Í(#ggg\Ò|hY#yÈÍ§F¼×|_©#gg#Ñ©ghCÔ6g4ÝÔ## Ý4Ô/h×## #\MC

h7#òg##g:¯?Ch #ògg\ÓhhÑ#

MÐù?#Ô#gá#FªÑ¼4â|#ògYh##Ý|h:

h6# gg:¼?/4#?g #FhÔ6 4#ÒFhÑ¼Í#4/<Î#RÍÑ 4hM`ÐhÑF ¼

gª4# #<\# #g §h©ò#Ô#4/#ò#?hhKg4g\#ggË7g#<#?hÒ#?h7h2#gá|hÑ.¼h g\Ñ4ªh#×Ð|_§#Ó4g§|_§##ÑñR¼4g§Ô8g|RÓ4 #4#g:Ñ¼yh g4#/ #òg4g\ÓhhÑ#hfÏóÍÑÑKÔ6ò4Ó4gQQù?g\4P#i<?

n ##g#4Ñ #g#g D×#: & FhÑgy#Îh\¼Û4g#g_hÛ#|hÈY\##§Î4\¼?g Ì #`f4#gÒh q4fhÑÑ?46g_4âhHhâh#©#: × FhÑgA`#h`#gggh:¼q4#g#g:¼?/4âK|h gg:¼?/4ð4|hÝg4ÝÔ####|hh Ö #:#g4hâ_ng :Ô#/##Î4\¼?##: |h:.¼QFh gÑ\hÑÑÕgg\ò/q4Y_4\7RÍ§\Ó4ggâÍ©FhC4\ÒhÑ4###Ý#Ñ|h:¤¼g4#/_4g##á#g©Í#¯PhÑÑRÍ#© #Îh\¼y4g#g_4PÍ#4/Ý#: |h:¤¼4|hÑ444#Ñ¼ðôù?#4#pÍ©CFh #Yh4# gÑ©h gg#¤¼pgá#ggÝhq4ÑÑRÍgc ) Cð4#D#gÍ§×|FÓ4P\Èh ` #ÑñF¼4ggâ7C#: FhÑggâÕh h Ó4gh:¼4 Ì ##Ô8ghÐgh4Ý|hgD#g gg:¼?/Yg##g:¯?ò

kh

#&?#ÝÔ6gChòg4#/ #ÑñF¼4gPt*ûp&gá#Û|_

t ≤ k − 1h |_C#

Ó4gh:¼ðgÑg\òP#Ô#\Èht#ÑñF¼ªgPg4#/Dq4#Ô6g4#### n ##g

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. O :

FhÔ6 Ó? ªP#q4ÑÑRÍ#BL ªDÏ¤ô#JÀm/ Â / ) L ## FhÑg4g#g_gPp##Ô#ÑyÎ4\¼

ke

hÑ4#ôÍgg\PÎ4\¼|hgs1, . . . , s`

âh #Ó4gî|F_4(Îh\¼?

Vh

V1, . . . , V`

ngÝ|_Í©yFhh#C|_#Q Ó4gh:¼Î#RÍÐ#gg\§|hgÐhK#g4#/ #ÑñR¼4ggâ###Ô#ÑÎh\¼4â#h #Ó4gî|F_4Îh\¼?g(é##ð#$ g ) L yðgh4PÝg#g:¼?/ # g#P##Ô#ÑCÎh\¼4Ì #gP#¸ ¥ ) L VFhC#ÑñR¼4gÐg4#/g§Ègh4|h4Ó4g#g##g:¯?F Ì #©gh4|hÐ##Ô#Ñ# 44\#gÐÍYÈ#?h2h#g4g\#gh2#g4#/_4AÏ¤C#Y ¨t£ 1Â L k gh4|hg§q4ÐÍ#C#g¯/:ÈP#h©hÐg4g\#ghg4ÝÔ## ##C#×##Ô#ÑÝÎh\¼ðôh \DPgggÓ4# g Q gg:¼?/4pën#ògh4 í h7#Dg##g:¯?FÐÏ¤y4#Ò gÑK#È Ó4gh:¼hÑÑRÍ© ×#Ô# gg:¼?/4Ûgá#g:×#ò##g4#/ #ÑñF¼ªgÒq4Òg##g:¯?h2#Ð#4ghfã64C#PÈgá#g:Fâ/##ÑñR¼4gÐ4/#/ÒÈ×g_4|hq4<#4Óª\g##g:¯?FÛûCF¼|_#C Ó4gh:¼ ¡ Â ¨B Â YÒ#YÝhÔ#Ñ×&4/#/ÓhhÑ gh4g4g64 #ÈÝòg##g:¯?©|_<ÍÒhÐ#h©#Ô# h© gg:¼?/4gá#g:Û_òÑFh:

k − t##g4#/ #ÑñF¼4gg Ì ##g LÚÙ¡ Â £Øý×©ïÈ

g4#/_4|hÑÑï¼ /qFhÔ#ÑCq4#Û Óªgh:¼(ôq44Ûôgh4hâ§Hhâ©#yÓ?\Íhò#ô Ó4gh:¼|_Pg4#/y#

t#ÑñF¼ªg gP#hyg|hÔ#Ñ&# g4#/

gg:¼?/4#×4 #RÍ n #Ý#g#4Ñ ¯#gPÝh ôÔ8bG£ ¢ /Ò#y#h#g:#ÑñF¼ªghPhÔ#Ñðg4#/PÛÓ_hÑ gg:¼?/4pÜò#ðY_gÈÍ|_Ý#Cg4#/ t#ÑñF¼ªg &ëqgÈ_#¼ <Gh í

ûp#Í g4È# gg/4YhÕ4#Ð#g#4Ñ Ù ÷ ØÐù6â Ì Ù ÷ ØÐùYq4Ð#4:F Ì #Îh\¼4g#g_4yh7#ò4g#ghÑ Ù ÷ ØÐùy|hÒ×Ô6ò¯?g hÒq4ÑÑRÍgÛ 0m,E?%010ZIKÞ

•÷ #?h

N = pqâ#Í#g

p, q > 3hò#ggâ|h 4×

s > 0

•÷ #?hPÛh 4gÑÑ/g#:Ó4

E : y2z = x3 + axz2 + bz3RÓªg

Z NZâ`Hhâ

gcd(N, 6(4a3 + 27b2) = 1Ðü2\

M = |E(Fp)| · |E(Fq)|

•÷ #?hDÈ64ª

QÍY4 ë

QíÕME(Z Ns+1Z)

fü2\P1 := (N : 1 : w(N))

E(Z Ns+1Z)

ë Ö 4©# \î|#4yhw(N)

gÝù?g\4P#65Èh kK5/H í

•ØfÎ

M ′ô_:n¼

M ′ = 0 mod Mh

M ′ = 1 mod N skënûpyYF¼ hÑÍ§F¼?

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. O 5

h#D|_gcd(M,N s) = 1.

í ÷ 4#:#\#ò84Ñ¼?#4ñhÑ

f(x) =k−1∑

i=0

aixi ∈ Z[x]

Ôª¼P#¯Î#h 4ai ∈ Z

q40 < i < k

|_hDÑñh4Dg#4#4T/h a0 = M ′

Ì #Pgg\Î4\¼ |hCh#ÑñF¼4gisi = f(i)

q41 ≤ i ≤ `

h #C##Ô#ÑÎ4¯¼ (N, s, (a, b), Q)

Ö 4ÝÓªgî|F_h"h#Qh\4#×h#Q#ÑñF¼4ggâÐÍ§Pq#:#g#g #Pq4ÑÑRÍ#&î// ##Ô#Ñ×Ó_hÑ#gg

V ∈ E(Z Ns+1Z)âAÛ|_

〈V 〉 = Gh #Y\¼?gÑ44#

G#4#Ñ

|FÓªÑñh4Ò4 ggâ?h#|G| = N s

Ö #:#g4hâªq4<FhC#ÑñF¼4g<Í©#g DÓ4gî|F_4Îh\¼

Vi := (∆si)VE(Z Ns+1Z)

â7Í#g∆ = `!

7g#ghâf_ngò#YÎh\¼p4g#g_4#|hhâ/###Ô#ÑÎh\¼Pg4#:§h

(N, s, (a, b), Q)h #Dg\¯Îh\¼P|hh7#ÑñF¼4g

isi

Ö #:#g4hâVh hÑÑ

Vi

Fh&Ô6Èggôh|h:h`#È##Ô#ÑÈÎh\¼ÛhÍ©gÑÑH Ú /#hÒ|_f###Ô#Ñ§Ó_hÑ#g

Vh hÑÑ

Vi

hÒ4#Ñ¼Ý#g Y ##D#|h gg:¼?/4#|h×ëqgÔ6gÑRÍ í h #h©g4#/hYg#g:¼?/47g#g#\¼FhCÔ64 4:h7#òFhgh 4:h7#ò#hg4ÑÒ#ggª Ûù?\\4P#i<? 01#$IQ R-ÖR-Ö)LÖ ¿hÝ|_Dq4Dhª¼&#Ô#\h

k64ªD

1, . . . , `#ÝÓh_Ñ#gòh

f_Ð#gk64ªÐ##á#gÑ¼ \g##g?/ªggª

ai

q40 < i < k

hfh #g#g

#Ó_hÑ#hf_Ò_¼h#g§84©

1, . . . , ` Ö #:#g4hâq4§hª¼Y#Ô#\Òh

k− 164ª1, . . . , `

# :Ô#/4#h`#ÈÓhhÑ#ghf_#gÝ84h##qh

h Ý/|hÑÑï¼C \6g gªF

Ì #§g#g:¼?/4#|h§h|4#Õ#g#4Ñ #g§gYh#2#©h©hAÈ#©4g#ghÑ Ù ÷ ØÐù6DFG,&-ÞÌ g#g:¼?/<gh4

m ∈ Z NsZâh#?4§h 4*ªg4g

1 ≤ r < N sh g4#/

#òg##g:¯?hC = rQ+mP1 ∈ E(Z Ns+1Z)

Ì # gg:¼?/4(#|hP Ó? ªÛ.Í©ô:g#g Ö :t ¥ ) âf Í#Fhô#ÑñF¼4gg4#/gD#|hhf# gg:¼?/4ð##P#|hhf#gg\òÎh\¼4

CEÈ ùs, ai ∈ Z, f(0) = M ′

æðô|M ′| := K

ñ õëòðËî ñ%ûòæðÖGëä)õøêðñ õæñai

êî ñ õ|ai| = log `!+K+k+σ

éêõëòí ëσôëòðøñ ëóägñ õëä)ëóûãåí î ñè¼ÿæíÊæ÷ïëãñ ëòíóéæí ëqä)åî ñÊæç ë/ü

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. Om4

Ì #g ##p ¨t£ 1Â # åKggª©|hg©h# gg:¼?/4Phg4ÝÔ## 4Ô/hC#Dgh4hd'-IQ0]-01G,&-Þì Ó4g(#yg##g:¯?

CâÐ#ÑñF¼ªg

ig4#/gYh ##Ô#Ñ#g

Ci := (∆si)C44\#g

ÍP#h?g4Ñ6#YÍ#¯C#Ó4g©|_logC(Ci) = logV (Vi)

h |_ ?g§#h§\Ó4FhÑhª¼Y # 4|hÑ#/q4Y4 Ì #Ðg4ªÓ?#ggÐ#h#g©#ÑñF¼4g©h<Í§gÑÑ6h©hª¼×¯?g|hÑ4Ô#g:Ó4gÈ|_È#ÑñR¼4g

i|hò g Ý#Ñ#Ñ Ôª¼ô#ògg\Ý|h

si

YÏ¤ ù?g\4AP#i< #4?Ü¤ªgh\Ó4ð#hg4Ñq4C#ChÎ"ÍÑÑÔ6ð#gg Í# PÔ|h 4'#g4g64 #×4#òA kG:HÖ 4hª¼P#Ô#\

Shk64ªÒ

1, . . . , `h q4

i ∈ SÍ§ \î|#

λS0,i = ∆

∏

i′∈ S\i

−i′

i− i′∈ Z.

Ì #gÝÓ_hÑ#gh g\Ñ¼ gÓ4 q4üÕh4h#4h ªg64Ññ_4yq4Ý#Ññ/ù?#gÈ# g#4|_4 Ó? g

i!(` − i)!Í# ô# Ó? g

`! = ∆#gYÓ_hÑ#gÝh¯Ü

4gg Ö #:#g4hâ4#\¼FhYÔ8gh#/ /ªggªÑ¼4 Ö 4ßüÕh4h#4h <ªg64Ññ_4q4Ý#ÑñÝ§q4ÑÑRÍ©|_

∆f(0) =∑

i∈S

λS0,if(i) =

∑

i∈S

λS0,isi.

d'-IQ0]G#]w())-?-Þ#y|_×Í©Q|FÓ4y#Qgá#

k4×4P|hg

Ci

44\#gÍô#hhg4g\#gg©ûpÈFhy#gÛ4Ô/hP#òg4g64 # gg:¼?/4QÔ¼Pg4Ô###Ý#gQâHhâ#Í©DhÎhÈ×#Ô#\

Shk|hgh g4ÝÔ###g¿

C ′ :=∑

i∈S

λS0,iCi = ∆

∑

i

λS0,isiC = ∆2f(0)C = ∆2M ′C.

ù?#gord(Q)|M ⇒ ord(Q)|M ′

h N sP1 = (0 : 1 : 0)

h M ′ = 1 mod N s

Í§ò|FÓª

C ′ = ∆2M ′rQ︸︷︷︸O

+∆2M ′mP1 = ∆2M ′mP1 = ∆2mP1,

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. OGN

Í#gm`# g #Ññh¯?F(7g#gÍ§Fhg4×#/

mg_Ó4gÑ¼Ýh<Ô8\q4ëqg

ù?g\4P# P#65 í h #Ñ#Ñï¼Y#Èg#ÑÔª¼(∆2)−1 mod N s.

¿hò|_Í©ÈFh&hÑÍ§F¼?h#D|_

p, q `

ù?#g×#g#g:¼?/4 #g #YgYh#ò##¯|h#4 ÈF_ FhÑ¼ðÔ6ggp|_È##g#4Ñ Ó4g4|h`#hÒ#4×44##©#46g:g<hf#Ò44|_Ñ/gÑÑ/§g#:Ó4ØAhÑÑg§#gh§hÑF ¼gª4# âª##h2hÕg4g\#gÒ_

logC(Ci) = logV (Vi)g|hÔ#Ñg©\Óªg:¼

#ÑñF¼ªgÝh 4Ô#g:Ó4gh§#C#gYÛÓ4gn¼#Yg4g\Ô8g|RÓ?4#hÒ#ÑñR¼4gih

#g#g á|hÑn¼Ý#!6##h ?\Ð#hÐ#ggg Ì ##?h6ÍÑÑ#Ô6 ggÔ6 ù?g\4_P#i<?AÏó`#g<g:¼?/44h##©|hÝq##\4ÈYhÎ4©A#4?Ü¤ªgh\Óªh Ì #g¯Üq4Ò4#<#hKhKgg#.¼ÝÍÑÑ/Ô64Ó4g×#h 4)4hgÑ§ gÑH Ì #<Fh#f|_g:¼?/44h##ò|hyq##\4#hòg#Ññhg Ôª¼y×h 4 4hgÑh Ì #h 4 4hgÑ gÑfÍ§hÝ/q4YhÑÑ¼ðª #g Ô¼ Ö ñ_h ù?|ht PQ5§h Ññ_gòq4YhÑ É Ú gÑÑñhh ö4ªFÍ§R¼ hH Ì #gF_ngòÍ§hÈ h/ ôY_¼&|h8ggÏóÐÍ§ÝÍ§4#Ñ Ý##?hq§hÕg4g\#ggh\Ó4gÑ¼#:F â?Í©Dg4#Ñ 4Ð#h 44hgÑhûp#RÍ!#RÓ4Ý|_òh##C#Ý|h #gòh<#Ý4g#ghÑ gÑÑ/g#:Ó4×Ø`hÑÑgh#/4(#Ý#g#4Ñ gg:¼?/4(Óªg4"h#QÙ ÷ ØÐù"h×gg#&hôgg?ÜhÑ #gÍÒY4##: #ÑñF¼4g<Í#Ý6g:q4<# gg:¼?/4â/Hhâ?h©gg#hf#4g#ghÑ Ù ÷ ØÐù6 Ì #<#?h6ÍÑÑ#Ô6 4#YD:_g4#/4× gÑ/Í#ghô Óª\h:¼&g4#/D#&

k − 1#ÑñF¼4gq4#ÝÔ6g4####ûp#×C:h h

g##á#ò #g#gg#.¼hK4#f#g#4Ñ Óªg4×D#gg#¤¼×hKf#4Ñ¯Ü gg:¼?/4?Ü¤g:Ó4gÝg4#ªg|h:FâAHhâÕ#4g#ghÑ Ù ÷ ØÐù6ÕûpY¯/#Ô#QÝ#Ññ_hÍ#P|h§#hggg§hª¼Ygg\Ò/q4Y_4Ô#/§Í#P|h§hP4h\Ñhggg§##4Ñ¯Ü gg:¼?/4?Ü¤g:Ó4gòFhÑ_hðh©#×# gÑ¼?#yg:¼?/4:¼?:Fëqgh# <Gh í Ïó7#È Ó4gh:¼y§#hhÔ#Ñ :#4#Y#DÓ?\Íjh`hy¯/gg/4Qh7##g#4Ñ #h?g4Ñ7h #Ññ_ ¯/gg/4Û|_#gh4hgÑD#È#4Ñ¯Ü gg:¼?/4?Ü¤g:Ó4gFhÑ_hâ/4#§#g#4Ñ #gÈhgg#hÒ#D4g#ghÑ Ù ÷ ØÐù6T+'-0Z%01# R-ÖR-Öý Ö ¢ ¨BcCLF ¥ ¨ Z Øýc¨B ¥ /Ø ¥t / ) ¸Z¥ / _C Ú ¢ C ¢ $§ k − 1

Ø m/ ¡ ¨CL¦£ Â L 1Â MY× ¢ L/¨B, ¢ ¨BYCLF ¥ ¦ ¡CL Â / ØÙý ¥ /ØCØÙ ) _/¢ R CØCØÙ / ¢ ¨\) MG L CL £*ú L/¨B\ý q M M Â ^ LL/§ONGqCL ¥ ) § Ø ¢ Ù ¢ C+CL Ø §ONGìON §ÁqCC Â ØÙ Â )£Øý¤G£ £CØÙC) M

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. O P

Ø Z Øý_C \ BÂ LL/§ONGï/ ¢ ¦CL §L ¥ 1Â Ø §ONGìCLm¥ / ªË /ÁÝ ¡ CL ¥ ) § Ø £YxÙ )ØÙ¸/C¨ ¢ Ø § ¥ ÁqCC / )ý Ú) ØCØÙt ¥ ý/) 1Â¢ ý £Øý ¥ ý/))£ ¢ ) MYÚ g YCL £/*¸C1LØ ¥ L/¨BFý F /¢ YCL Â / ØÙýV ¥XÛ RÜ G£ ¢ / ¥t LÚÙ¡ Â £Øý MÌ #È#h7q4ÑÑRÍ©Óªg:¼PgÑ4gÑ¼C#ò\4\84 ##?hqÒA <k7h kG:HR ¡ M ÏófÍ§Ýh#|_hð Ó4gh:¼&Fh&4#Ñ¼yg4ªÔ#/#g4g\Ó_hÑ#gq4#Ci

AÍ#g4Ñ4Ô#ÑÐ#4Ô|hÔ#Ñï¤¼4âhg4g\#g<h|#Ò#g§q4ÑÑRÍA _gÑï¼4 Ì #gâQ#âKg## Ôª¼Q#6ggñhÑA4# #gDhf##h?g4ÑÕq4#Ýgá|hÑ¤¼Qh .Üg\ÐÑ4ªh#×24Ó4gòq4ÕFh

Ci

âÍ#¯òÍÑÑ4Ô6Ðª #g h #Ó4gÝù?#Ô#g\4P#i<? 4 #Qh"4hgÑC4Ó4g(|_×4"ð4Óª\"g##g:¯?Yh##/×\##×#yg4¯Ü64 ##Ññhª¯?F Ì ×Ý#Ññ_#ò Ó4gh:¼6 Ó?\ÍkhÕ# gg:¼?/4y#h?g4Ñ6Í§:h:q4¿###Ô#ÑòÎh\¼

(N, s, (a, b), Q)Òü2\

i1, . . . , ik−1 ∈ 1, . . . , `Ô6ò#\h

#g4#/ #ÑñF¼4ggûpFhô#RÍ)Ý#Ññ_È#× Ó4gh:¼6 òÓ?\Í!Ôª¼ð#?4#C#|hg

si1 , . . . , sik−1

h©#×g4#/ #ÑñR¼4gÈhòh 4g4ggûp×|FÓ4YhÑF ¼h4# ëqgyögYhÎbP# P#65 í |_#gCÓhhÑ#gÍÑÑ<Ô6C:_:FhÑÑï¼ô :#4#|hÔ#Ñq4#FhÑfÓhhÑ#ggPögghÑÑ`|_

M ′ = f(0)h

si = f(i)h

M ′Dq#ÑÑ¼ \g.Ü

# Ô¼ð#Y#4gh§#Y##Ô#ÑÓhhÑ#gN, s,

h (a, b)

n #g M ′ \g#

h #ÓhhÑ#gsi1 , . . . , sik−1

h#4gP#|hgsi

q4i ∈ 1, . . . , `\i1, . . . , ik−1q4Û# ##g4#/ #ÑñR¼4gðh # 64Ñ¼?#4ñ_Ñ

fhpî// hÛÍ©gÑÑH û |FÓª

f(i1) = si1 , . . . , f(ik−1) = sik−1

â§Ô#/M ′

h fFh##hYÔ8Qg4#/ Ô¼ #QÜ

#Ññ_4ge7g#ghâÒ#Û|hgsi1 , . . . , sik−1

h#Ûg4#/ #ÑñF¼4g #hYÑFhÎ"hª¼/q4Y_4YhÔ64/§#|hg§h2###g4#/ #ÑñF¼ªggâ?#gg\ÒÓhhÑ#

M ′4Ð#

64Ñ¼?#4ñ_Ñf Ì #Fh4PÐ|_©#D Ó4gh:¼Y©#hÒhÔ#Ñ :#4##ÓhhÑ#g

g4Ý##F_ y ##Ý#ò#h?g4ÑKq4 h 4ÓhhÑ#gg¿RÍ*Í©Ýh4#ò|_#òÓhhÑ#g

Vj

h`###g4#/ #ÑñF¼4gh #g|hgCj

h×g##g:¯?

C|_FhQÔ6ògg ###È|h gg:¼?/4Q#|h #h4Ó4Èhª¼

¯?Û/q4Y_hô&h Ó4gh:¼4Q ggÔ6 Ô8gÑRÍ#gÓhhÑ#gFh ghÑ¼ôÔ6Ý#Ññ_ Y|_#\¼Qî#Y#ÝhÔ6RÓª|hgDh`#Ý##g4#/ #ÑñF¼ªgg Ö 4#|h:Èh©##?h©Í©Yh#×|_È#Ý#Ññ_hD|hhgggÈÛ gg:¼?/4p4hgÑhâHhâ#ggªhÑ Q g\¼?/hyÓªg4CÜ<#D4g#ghÑ Ù ÷ ØÐù6ûp<#RÍpg4# g2#`Óªgî|F_4DÎh\¼

VhÕÒg##g:¯?

V = rQ+m0P1

E(Z Ns+1Z)

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. O <

q4AÍ#Ý#ÒÝ#Ññ_h`_Î`#§4hgÑ©q47#Òg4g84 #D#Ññh¯?m0

Ú ¼ 4#×Í©òFhQg4#/

f(0)V = M ′V = M ′rQ+m0M′P1 = m0M

′P1 = m0P1.

ü2\SÔ6D#ò\

0, i1, . . . , ik−1â8h Ñ\

λSj,i = ∆

∏

i′∈ S\i

j − i′

i− i′

Ô6È#ÝüÕh4h#4Èg?/ªggªq4ªg64Ññ_#×#ÈÓ_hÑ#gh<Y6hÑ¼?#4ñhÑ2Û84jëng

∆í q4ïÓ_hÑ#gÛ84

Sâ6Hhâ

∆f(j) =∑

i∈S λSj,if(i)

Ì #gÛÍ§FhFhÑ¼&g4#/ÝÓhhÑ#g

Vj

ÍVj = ∆sjV = ∆f(j)V

q4#×##gh#/ #ÑñF¼ªggâHhâ

j ∈ 1, . . . , `\i1, . . . , ik−1â|h

Vj =∑

i∈S

λSj,i · (f(i)V ).

û #gÈgggÓ?#Òg##g:¯?764ªCh2##/FâRÍ§ÐhÎ#f4hgÑ`q4#fg4g64 #

#Ññhª¯?m Ì #Ò4Óª\#Ò#ò64Ô#Ñï.¼×g4#/#ò64ª

f(0)C = M ′C = mP1

Í#PFh#©|_§Í©DFhPªg64Ññ_h g4#/#g4Ô#/4#Ci = ∆siC

q4#×##g4#/ #ÑñR¼4gòð#hÝÍÒF¼hÈhÔ8Ó4h Ì #Èh4#gªÈ#RÍòÍª¼ðÍ§ \î|# #ò|hg

Ci

×Ô6∆siC

#:F h.â|F¼4âsiC

n ÔªÓ?4#Ñ¼Y#ò4Ô/h# ÓhhÑ#ghDg4g\Ñ¼ :Ô#/

[E ÝµÞX¨ªßàÁáâ6ã*äA¦¨¾ Yåæ§¦¨`çè¢FéTå¢Ößêá

Ï¤b N 52éh×ªõh h àh#Î#484 Ñg#h?ÜÊÈ|¯/Ô#ÑÓ_hñh©hÕ#g©4g#ghÑ_4hØAhÑÑgg `g#g:¼?/4#ghfÏ¤YDÑ_AÍ§F¼ÝÍ§#RÍ#f#Ô#g\4#Í<64Ô#Ñf ¾:#:A#ÐÔ#Ñ¯ÎÑg#hÈh|4#2#g#4Ñ #ghâ4HhâÍ©©#464©ÒÓhhñhªh<h# Ì Ù ÷ ØÐùÛ|_F_ð/ªggªÑ¼Q|h Ñ\hhg

mhÐhÔ#h:¼yÑg#h

m < N s′

Í#gs′ ≤ s

âªÍ#gFh©###Ô#ÑÎh\¼Ch #gg\§Î4\¼|hg§gYhhî// É gg

Q"%»º Q®,O ¯ ' °½¼¾®Oó³¯¬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. OGO

çQ4Y#gggÑ¼4âAÈÈ64Ô#Ñ× gg Y4 yÓhhÑ#×q4s′ ≤ s

âAQ|_È#Ygh4|hg

MÍÑÑ`Ô8

Z Ns′

ZÍg4g84 #Û\##g:¯?|hg

C = E(Z Ns′+1Z)

_hª¼64ª©_ng<#Îh\¼?§|FÓ4Ô6ggC4g#g_ <Ù`Óª\C#g g©Fh \g 4#úÈ#¼4CòÓhhÑ#qh

s′Í#gPg#g:¼?/#ÝÈgh4hÐù?#PÈÑg#h?ÜÊÈ|¯/Ô#Ñ:¼?:gßÍ©4Î§h

q4ÑÑRÍgÌ #Îh\¼&4\#g_4&FhðÔ6 4#ÝQ#ÝhÍ§F¼&hDhÔ6RÓ4Í#g

s#4#Ñ Ô8Ý¯#_Ü

g'Ññh4Qg#4#4 ü2\s′ ≤ s

×¿hQ|_#&4 gM

Y g6\ gPhsh

|_q4Q64ªQÍp4

(Q)|ME(Z Ns+1Z)

ïÝhÑÛ#4Ñ ò|_Ý4 (Q)|M

E(Z Ns

′+1Z)

Ö #:#g4hâ2q4P1 = (N : 1 : w(N)) ∈ E(Z Ns

′+1Z)

ïÈ#4Ñ ò|_N s′P1 = (0 : 1 : 0)

ëqg ù?g\4 P#65 í âDh q4M ′ = 1 mod N s

QhÑ(#4Ñ P|_M ′ = 1 mod N s′

DFG,&-ÞÌ Èg#g:¼?/§ògh4

mg#ggª hÐò#4?Ü¤#gª_Ó4Òg4ggâª#?4

s′ ≤ sÈ|_

m < N s′â`#?4CQh 4 ªg4g

1 ≤ r < N s′h g4#/

C = rQ + mP1

E(Z Ns

′+1Z)

fÑ4#ÍP#òg##g:¯?#DÓhhÑ#Dhs′§h#

ù?|h gg:¼?/4h |h×g4ÝÔ###YÍ©4Î&¯#h\Ñ¼&#hÝÍ§R¼ðhòÔ8\q4×##s′#:F h

s

Ïóò× ñ_gÑï¼yq4ÑÑRÍ|_#×hÔ6RÓ4ÝÓhhñhªDgYhªFhÑÑ¼Qgg#×<#44|hÑ#gÈÒgYhªFhÑÑ¼Ygg#hÖ 47#:h#g_âh#7#48g:.¼Èh#Ñg#h?ÜÊÈ|¯/Ô#Ñï¤¼44\#gÕÍÈ#©#444##<#4?Üg:¤¼ph§#Y#gCÈ#\q#Ñ<q4ÈgÑg\4p#gghÈÍ§\ÑÑ©hÈÛÔ##Ñ ïÜ¤#\Cëqgù?g\4 4 4 í ÛçQïÜ¤#\ÝhY#hg4ÑòÍ# #Ó? Yh#4ª¼?.¼&q4g gÝÔª¼pg4ÑïÜÑg\#D\#g:¼?/ gh4gfq4 \Ó4ghÑ|#\g Ì #gYDg4ÑÑg\4Ýh6g:Óªg<#?ggg#ghâY|_D_#g ##Ññhª¯?gh4gDh4/#/Ûh 4Ñ¼y6gÝ/ 4 gh gª#Dg gggÓ4gFË7g#ghâ8ÒÒ#h64Ô#ÑhgòÔ|hÎY#gh4gfq4 #ÒgggÓ4gfD#Òg4g64 #òg gg Ì #Ñg#h?ÜÊÈ|¯/Ô#Ñï¤¼Èh6## gÑ¼?#Pg#g:¼?/4ð#gg##gD|_#ïÜ¤#\hÔ#ÑP|h Ñgh4gh2hÔ#h:¼ É hâ?HhâhÑ#4#4hÑÑ#gh4gÐ#Ô# ò#4ÑÒ#h6#ïÜ¤#\Ý#:|FÓªò#òhòÑg#hyQ4 gÒ×#Ó? Èh#4ª¼?¤¼4â/#g44yÑg#hyFhÔ6C¯#4g qggÑ¼q4Fh¯ #(hÒ#PïÜ¤#\FâAÍ#4/Ý|RÓ?#Ûð|h#4yhª¼p##Ô?ÜÑÝ/q4Y_4 Ì #òDg6ggñ_ÑÑ¼&#\q#Ñfq4D#RÓ? #yh#4ª¼?.¼4â2h#Ýq4DgYhÑg

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' O h

ù?g8 Nm4 Õq4 \hÑhfïÜ¤#\h q#:#gh##ÑF_4#h`Ñg#h?ÜÊÈ|¯/Ô#ÑD#g#4Ñ g:¼?/4:¼?:g×gÑg\4#Ó4h##

.1Ií ½Ó½Aî · Ä ·¶ 78:_r7AÂòËÕÂY¾©ÂÈÄ»

7gª4# hÔ8Ó4<Í©Ð#g hA#Ô#4/#<ò#<|h gg:¼?/4#|h©#h?g4Ñq4#Ýgá|hÑ.¼Qh g\Ñ4ªhï#×gâKHhâ2C#?hf|_D4Ó4gðÓ_hÑ#g

C, C, V, V

E(Z Ns+1Z)Í#g

〈V 〉 = Gh

|G|Ññh4hâ?©#4Ñ <|_

logC(C) = logV (V )<ù?#¯y

#h?g4ÑKÍÑÑKÔ6ò#ggª Ûù?#Ô#g\h P#i<? 4 Ï¤8 4 < ÷ hgÕ\`hÑHÕª #g #\Í(h##ªh¯ò#Ñ|h:¤¼g4#/_4òÔ|h 4#444##f#g#4Ñ g:¼?/4:¼?:Fg Ì #© gh/#7h##ÑF_hòh|#444##g:¼?/4:¼?:gÍÑÑÐÔ6 ggÔ8 (ù?g\4 P# O Ú g gÝ#P#|hÑ§#46g:gÝh#g#4Ñ #444##Ýg:¼?/4:¼?:g 4×h?/Ññh.¼Û#h?g4ÑDh#g q4#h##ªh©çQ4ò#gggÑ¼4â#Í©ò#g

•#h?g4ÑKq4Ò#g¯Î#AÓ_hÑ#òÝÓ_hÑ g##g:¯?Fâ

•&#h?g4ÑÐq4#Ó?#Û#Pg4g\#g×h#PÝ#Ñ#ÑF_4phh g#g:¼?/ ÓhhÑ#òÔª¼y×g4#:hªFâ

•#h?g4ÑKq4Ò#RÓ?##òÎ#ÍÑ 4Èh`#Ññhª¯?F

Ö 4`#h##ÑF_4hK4#f#g#4Ñ #444##§g:¼?/4:¼?:gògÑg\4#§Ó4h#Í#DÍÑÑhÔ6 ggÔ6 ù?g¯4¸P#ihÒh#h#g2h?/Ññh:¼#h?g4Ñh2#g 7Ò#h?g4Ñq42#RÓ?#§|_7hòg#g:¼?/4Èg4ªh#24#f4/Õh?¤Í©4ÓªgÓhhÑ#ggâÍ#4/2\Ó4FhÑ#Í#'4#ÛCgjÏ¤#g\4Í©ðhÛ44# \Ó4gÑ4#gð#hg4Ñq4Y4##444###g#4Ñ g:¼?/4:¼?:g (h(h|hÑ444#ÝYh##ghÍÒh 4#Cq4#È4g#ghÑ ØAhÑÑgÒ#g$ N 5hâ 4 <?âg<k?â NGN H Ì #8ggñhÑ2.¼?8hf#hg4ÑÒÍ§#g hÐFhÑÑ

ΣÜ.Øfh?g4Ñg Ì #\¼ÈÍÑÑÔ8©#4:Ñ¼ \î|# ò#©n4ÑÑRÍ##Ô#g\4

ðï6EΣ

ã*ñèéTòÁáò6ó¾òÖ¦©

ΣÜ¤#h?g4Ñhò.Í©_Ü¤|h:¤¼ É g_Ü¤Î#ÍÑ 4#hg4ÑhfY|_\#ÑñhÒq4QÒü2\ R Ô6gÑñ_4yg4#:#h`|h

(x,w)â8Í#gÈÍ©È##ÎCh

xÔ6g#C×##Ô#Ñò#:h#gÈh

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' O k

4<g4#/_4|hÑh#4Ô#ÑgQâh wÔ6g#FÁqC ªâHhâ_4Ñ/4Ò|_7#:h#gh

ûpòÍÑÑ2Ô6Èg4#gg# ÍQ#h?g4Ñh`#òq4ÑÑRÍ#q4Q Ì ##h?g4Ñ24\xh

Ûg44p##/Èq4È#Y#Ó4gPh #YÓ4gî|g

Vâfh #Y#Ó4g # 4|hÑÑï¼

Î#RÍÍÒ#gwh#Ó__ò##/#Q|_

(x,w) ∈ R

54Pg gh4

a

V

4 Vg h 4

tÜ¤Ô#Ò|hÑÑg#4

e

P

N Pg g#Ñ¼

z

P#V gg gÒhggg/4Òã¾:g\Ô|h 4P# _#ò|hggâ8Hhâ

x, a, e, z

ô=õ öÃ÷TøOù&úTû$üÃýΣþÿ ø

ùYõ O÷ùP

VøOùTøTõ õIõ õ!"Iõ ù#$%Tõ Tù4û

&ù4ùP ' ( )+*,%öÃù-.*ù4ø

Võ Tù-/012 ùTöÃù-34Tù5YõTù

wû

67,8:94;=<1;?>9A@CB DEBGFBIHJTø0Fõ -õ K%)ù$Σ L MNOPQO.R%OS 3?Ãø&øOù!Iõ

Rõ3TêùUV*Ãù

Tù53? .YõTöTý

• W Tù-TøBõ I34%Tù-.*ù53?Ãøwû

• XTOYZMS [\PQ[\]^[%_%_ ý:à3 P V3? .bTù-TøcTù5*ù-øIõed ù4ø(IV 24ù+4û

• W Tù4øOùùgfõ $h õ! þ Iõ2ùOõ Â÷!Ãø"2Tõ %i õ T÷+xj ø

% ùTöÃùeÃ÷+T÷+k4ù+Iõ Töh,*ù4øIõ l3(Tù#3?Ãø

(a, e, z)YõmTù

ùnTøTõ õeQ oTõ IøOõ T÷+Iõ kU,*ù4øIõ Uù\Qbù4ùpTùP

V

õ T÷+xq% ùTöÃù

eû W Tõ -Tøù4øQ õ -V ù _QM[%R\rGsS4tO]^[%_\PGLQu[\Ngr vw[\Nyx1[\NOLz ]ÔIS [|.0[ û

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' OGj

•Ü L Øì ¢1¥ ªp ¯#F_# #RÓªgPFhkh#:Í©gy4#Ñ¼h#ðhÈ#ð64Ô#Ñ|hÑÑg#4gg`çQ4Ð#gg\Ñ¼4â_q4*_¼Dg44ò##/

xh hª¼ò|h2h#hggg/#

g4ªÓ4g4#(a, e, z)

h (a, e′, z′)

Í#ge 6= e′

âÍ©FhY/ªggªÑ¼×g4#/w×|_

(x,w) ∈ R

¿hÈ|_##h4Û#4#g::ÜóÓ4gî|g É g_Ü¤Î#RÍÑ 4#h<ghôhÑYÔ8q4# Q#Ñg_##:F h

ΣÜ¤#h?g4Ñg

ÏóÝFh:¼pÛÓ4gn¼ô|_#C#46g:ghΣÜ¤#h?g4ÑhYªÓhhñhª# gÝ|hhÑÑgÑ

g464h Ö 4Ò4 \hÑÒh #48g:ghΣÜ¤#h?g4Ñ§gB N :Õh 4 <H

ðï6c~ ñèéTòÁáò6óòÖ¦,òFé áâ¨r)¢Ö¦åKá]ò"# å©QóéT¨¾á¨rÞXòÖàÖ¢FéTåKáâ§£²©

Ö 4C#ð|h gg:¼?/4k#|hðÍ§ô#g #Ô#4/#ô#RÓ?# |_y4Ó4g'ÓhhÑ#gC, C, V, V

E(Z Ns+1Z)

Í#g〈V 〉 = G

h |G|

ÒÑñh4hâ#Ò#4Ñ §|_

logC(C) = logV (V ).

û D#¯§#h?g4Ñ4#òg4#©44\#gÍD#f##Ô#Ñ# g#ÑCën#<g##g:¯?

|hCií hÝ(#ÑñF¼4g

Pi

â#h#gy#ÑñF¼4g ëTh hª¼'4Ô#g:Óªg í h4#P#g#4Ñ gg:¼?/4 ¯#gÍÑÑ<Ô6Yg4ªÓ?#g h§#Yg4g\#gh#Èg4#/_4 Ì ##gD#ò#ÑñF¼ªg

Pi

§#ò#RÓªgg Ì #DÓhhÑ#gC, C, V, V

hÒÍ©gÑÑ2hÒ#ò##Ô#ÑÎ4\¼yh##Ô#ÑÝÓhhÑ#gÈô4#È#hg4ÑHYØfÑñF¼4g

Pi

Î#RÍÝ # 4|hÑÑï¼∆si =: y

Í#pD#gg\##/Fûpò#g

ΣÜ¤#h?g4ÑKq4§#ògÑñ_4

R = ((C, C, V, V ), y) | C, C, V, V ∈ E(Z Ns+1Z), y = logC(C) = logV (V ).

ÏóÍ©4Îh§q4ÑÑRÍgaYLGW[Ê%B'-0 0-ILW)C,Õ%[\).VG0m0 W%?%IQC'#2. -&(mÞ

ke = (N, s, (a, b), Q), C, C, V, V ∈ (E(Z Ns+1Z),+)âAÍ#g

〈V 〉 = Gh

|G|ÒÑñh4

aYC`QIK0 )-&(t[Ê%t'-0 &(1`0ZZÞy#¯y|_

y = logC(C) = logV (V )

¿h|_Dð4#h##ÑF_4Q#ÝÑg#hðhyÍÑÑ7Ô8Ô64# Ôª¼

s|N ||M |Ô#D/g

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' h:

V ∈ E(Z Ns+1Z).7ghâ

|N |h

|M | g#h§#ÒÔ#fÑ\#hh

Nh

Mâªg6g\Ó4gÑ¼4

û ÑH # #`#gòÔ#Ñg#h#hDÎ#RÍ&hÒÍ©gÑÑH

54 Ì ##RÓ4gP#?4gòYh 4Þ#Ô8g

rh<Ô#Ñg#h

s|N ||M | + 2tâKÍÒ#g

tÝÛgg4 h:¼pgg#¤¼p|hh\gg^7Yg4#/g

A = rCh

B = rV

E(Z Ns+1Z)h g

(A,B)×#DÓ4gî|g

V

4 Ì #DÓªgî|g#4g×h 4tÜ¤Ô#|hÑÑg#4

eÍ#Q#òg ×#ò#RÓ4gg

N Pg4#/gÒ#ò#ÝÔ6g

z = r + eyh g §

V

P# Ì #Ó4gî|ghggg/##?hf<h 4#Ñ¼yzC = A + eC

h zV = B + eV

E(Z Ns+1Z)

01#2#$IER-Öý3Öý Ö Ì #YhÔ8Ó4×#h?g4ÑAq4#×gá|hÑ.¼&h g\×Ñ4ªhï#×òΣÜ

#h?g4ÑH

R ¡ M Ì #g4#Ñ\g#gÐhK#<#hg4Ñ|# g§h#4#g:©#RÓªgÐf4ÔÓ?4#g<ù?8ggñhÑ#4#g::ÜóÓ4gî|g É g_Ü¤Î#ÍÑ hy#4Ñ Ô6gFh#hâ©q4ðh 4 t

Ü¤Ô##Ô8geh

h 4s|N ||M | + 2t

Ü¤Ô#ò#ÝÔ6gzÍ©|RÓ4×|_

((zC − eC, zV − eV ), e, z)Èh

hggg/#g4ªÓªg_4ÈÍÈ#ÐhÐ#4Ô|hÔ#Ñï.¼ :Ô#/4 Ö 4Õ6ggñhÑ4# #gÍ©h#ò|_

P g#hÎ#RÍ

yÐÏó7#È#RÓ4gFhQg g4g\g84#g

zh

z′¤Í§ åKggª|hÑÑg#4g

eh

e′|__:n¼C#DÓ4gî|gg ¯#gÎCÍ#gòÍò ÑH # #

e > e′â|Hhâ

zC = A+ eC_

zV = B + eV ,

z′C = A+ e′C_

z′V = B + e′V

#g(z − z′)C = (A+ eC) − (A+ e′C) = (e− e′)C

h (z − z′)V = (B + eV ) − (B + e′V ) = (e− e′)V ,

h #g#glogC(C) = logV (V ) = (z − z′)(e− e′)−1

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' h15

# g\DÑ4ªh#×©§/ªggªÑ¼Cg4#/hÔ#Ñh Ì #Òg4ª \4C#ÍÒ|_§##RÓªgÐÐhÔ#ÑÒÝh#:Í©g§_Ð4:Ð4#¯|hÑÑg#4

eg4g\Ñ¼4 Ì #g\q4##4Ô|hÔ#Ñ¤¼

h`hggg/h#gÈh` #4#\#RÓ4g2−t

Í#¯yÒ#g4Ñ4Ô#Ñt

Ö 4#FP/ 4 ÐØfhg4Ñq4§#ÈÙfá|hÑ¤¼Ch`ég\Èü24ªh#

ã## Ö ñ_:Ü:ù?|h×7#g#:c PQ5/Í#gá#g<|hÝq##\4#`#hg4Ñ/FhÔ6Y #4?Ü¤ªgh\Ó4hâÍ#gDgg#¤¼YFhPÔ6#RÓ4gPY#h 4ß4hgÑ gÑHü2\

HÔ6È|hyq##\4â#Í#4ò4/#/

tÜ¤Ô#ªg4gÈë

t = 128â|F¼ í

54 Ì #Q#Ó4gP¯#4gCôh 4 #ÝÔ6g

rhÔ#Ñ\#h

s|N ||M | + 2t7

g4#/gA = rC

h B = rV

E(Z Ns+1Z)

â7\e = H(A,B,C, C)

h g4#/g

z = r+eyq7 \î|#gÐ##hhKg4g\#g©òÔ6

(e, z)h g

#§#DÓªgî|gV

4 Ì ×Ó4gn¼Y##hV#g¯Î|_

e = H(zC − eC, zV − eV , C, C)

Ï¤#f#4?Ü¤gh\Ó4§Óªg4hK##h?g4Ñ#Í©FhY¯/gÑ V, V

###/fH#gÛ4#h##ÑF_hâ|#\¼&hÈî// h #4gðÔª¼&Y#4#\ FhÑghÔ6g#Y#

Ó4gî|F_4Îh\¼?g<ÏóÍ©h###RÓª\F ÐgÑñh ÈÔ6#hâ?Hhâª#gá|hÑ¤¼hK# g\ÈÑ4ªh#×Ò#4Ñ #gQ#Èg4g\#gh`##h?g4Ñq4ÑÑRÍ× ñ_\Ñ¼4##È#h 4ß4hgÑ gÑHâHhâ?g#Ññhg#

HÔª¼YÈh 4q##\4âªÍ§Fh

#RÍ*4# #g_ 6ggñhÑ2#4#g::ÜóÓ4gî|g É g_Ü¤Î#RÍÑ 4ÈhhÔ6RÓ4hÌ # Ö ñÜ.ù?|h<#g#:ghyhÑÝÔ6# ÝYhÎ4hÑÑ6h2#q4ÑÑRÍ#È#h?g4Ñ©h#Òg\4Q#4?Ü¤ªgh\Ó4h

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' h 4

ðï6x â¨ªó+ ò") å¤§â¨BéQá¨¾ báß¨ª©Q©¿RÍ*Í©ÈÍÑÑ ggÔ6ÝY#hg4Ñ2|_4Óªg&×ÓhhÑ#

Ch #È##Ô#ÑÈÎh\¼

ke

#gÎÍ#\#g

CòCÓ_hÑ g##g:¯?F Ö 4#×|hh\gDh©4##g#4Ñ #g×|_

Fh#|_4Óªg&84Ch ###Ô#ÑòÎ4\¼

ke = (N, s, (a, b), Q)#gÎyÍ#\#g

#gC¯/ÝQgh4m ∈ Z NsZ

h Ûh 4 gÑgg0 ≤ r < N s

#¯ |_C = Eke,r(m) = rQ + mP1 ∈ E(Z Ns+1Z)

B¿h|_D#×Ó_hÑ#gVh hÑÑ

Vi

h4#Ñ¼Û#g q4#Ý|h gg:¼?/4ô#|hð4##g#4Ñ #gh Ì #\¼ðFhðÔ64 h##Ô#Ñ##/QhÑÑK4#Òq4ÑÑRÍ#Ý#h?g4ÑgÌ #y#48g:.¼

C = rQ + mP1

4ÔªÓ?4#Ñ¼pgá#Ó_hÑgÝC − mP1 = rQ

â<HhâfC −mP1

Ô8g#yhôg#g:¼?/4Eke,r(0)

h0# \D###Ô#ÑÝÎh\¼

ke

h Ch 4#Ô8g

r

7g#ghâ|Í©òÍ©4#Ñ ÑÎh×g4#:#\ΣÜ¤#h?g4ÑKq4§#ògÑñ_4

R = ((ke, C), r) | C = Eke,r(0)Í#g

ke

§#ò##Ô#ÑDÎh\¼.

#hg4Ñq4§#ÒFhQFhÑ¼YÔ6ò4Ô/h# q4#òg4g64 #4#òA <kHaYLGW[Ê%J'-01)-?b@B'-0m'-0ZBIL01W01#$01t).¦IL01G,&-E%[

0 -&(mÞke = (N,S, (a, b), Q), CaYC`QIK0 )-&(t[Ê%t'-0 &(1`0ZZÞ

rÍ

0 ≤ r < N s×|_

C = rQ = Eke,r(0)

54 Ì #ò#Ó4gP#?4g

r′Í

0 ≤ r′ < N s_h 4Q7òg4#/g

A = r′Q = Eke,r′(0)

h g A×#Óªgî|g

V

4 V|hÑÑg#4g

PÍQh 4

tÜ¤Ô#Ò:#

e

N Pg4#/g

z = r′ + erh g

z

V

P#V¯#gÎ©Í#\#g

A,C ∈ E(Z Ns+1Z)h |_

Eke,z(0) = A+ eC7hggg/

Ah 4#Ñ¼CÕ#§#òFhh 01#2#$I¦R-Öý3Öý=Ö Ì #©hÔ6RÓ4Ð#h?\4Ñ4q4Õ#gÎ#Í#\#g`hÈgÑg×gÕ7hÈg#g:¼?/4h

0

ΣÜ¤#h?g4ÑH

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' h N

R ¡ M Ö 4Cg4#Ñ\g#gCÍ©ð|FÓª& #RÍÞ|_C#&Ó4gî|gPhÑÍÒF¼?Phggg/yD##RÓªgh #DÓªgî|gÒqhÑÑRÍ #ò#h?g4ÑH Ì ##4Ñ hÒq4ÑÑRÍg

A+ eC = r′Q+ eC = r′Q+ erQ = (r′ + er)Q = zQ = Eke,z(0).

Ö 426g\ñhÑª#4#g:7Ó4gî|g É g_Ü¤Î#RÍÑ 4fÍ©Ð|FÓ4fÝ#Ññ_`g4ªÓ4g_4 (A, e, z)Í(#yhy#4Ô|hÔ#Ñï¤¼ :Ô#/h Ì #g\q4y#QÝ#Ññ_hÝ¯#4gzh

eh 4Ñ¼"#gCg6g\Ó4 4Yh#gÏ7ð\A = Eke,z(0) − eC

h 4/#/(A, e, z)

â/Í#¯y4ÔªÓ?4#Ñ¼Pg4ªÓªg_4PÍy#òhD#4Ô|hÔ#Ñï¤¼ :Ô#/4Ö 4f8ggñhÑ|4# #g<Í©h#×Ò|_f#gh§.Í©Èhggg/#ÈghÓ4g_4#

(A, e, z)h (A, e′, z′)

Í#ge 6= e′

h Íò ÑH # #e > e′

7g#ghâ|Í©È4Ô/h

Eke,z(0) = A+ eCh

Eke,z′(0) = A+ e′C.

Ì #g\q4(e− e′)C = Eke,(z−z′)(0) = (z − z′)Q.

ù?#ge 6= e′

Í©ÈFhQ\v := (z − z′)(e− e′)−1

Ðù?#â

C = (z − z′)(e− e′)−1Q = vQ = Eke,v(0),

Hhâ/#D#gggh:¼PÓ_hÑ#g§q4ÒhPg#g:¼?/4yh0Fhy g Ô6D/ªggªÑ¼Cg4#/

Ï¤&4#h##ÑF_4Qh`#h #qhÑÑRÍ##h?\4ÑÍ©h#È|_#Ý #Ñ#N = pq

|hÝ¤Í©ð#Th\4Ýh4#4#Ñ¼#ChC É hA7g#ghâ< |N |È#PÔ#

Ñg#h×hNâªÍ©Fh×\

t = |N |/2òg##|_f##4Ô|hÔ#Ñ¤¼hD#F_#ò#RÓ4g

×YhÎ4#DÓ4gî|ghggg/≤ 2−t

ðï6[ ñèéTò6ò"Aò"n òFééT¨ªóák¦Káå¤§¦åóB¢ÁáåòÖß

ûp#Í ggÔ8òÝ#hg4Ñ6q4©#Ó?#|_ÒÝ4Ó4gPg##g:¯?Ò©## #\Òh7hg#g:¼?/ Ó_hÑ#DÔ¼y#Ññhª¯?F7g#ghâ|Í© \î|#ò#ògÑñ_4

R = ((Cm1, Cm2

, D), (m2, r2, r3)) | Cm1= E(m1), Cm2

= Er2(m2), D = m2Cm1

+r3Q,

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' hP

Í#gôhÑÑg#g:¼?/4#yh 4#ð# gY#ðhÛ##Ô#ÑÛÎ4\¼ke = (N, s, (a, b), Q)

ã##ÈhÑ44# çQï/ Ü:çQ#ÑÒ×¯/#g#ògÑñ_4C#ÒFh#Ò|_

D =çQï/ Ü:çQ#Ñ

(m2, Cm1),

HhâDh&g#g:¼?/4&h

m2m1

!#h?g4Ñ2q4#gÑñ_4QÑh§#4##464 A 4 <q4§#D#g#4Ñ ØAhÑÑg§¯#\haYLGW[Ê%8&1`-)-?2G%01t#bWCC&(W)IK -&(mÞ

ke = (N, s, (a, b), Q)h

Cm1, Cm2

, Dâ||_

Cm1= r1Q+m1P1, Cm2

= r2Q+m2P1, D = m2Cm1+ r3Q

aYC`QIK0 )-&(t[Ê%t'-0 &(1`0ZZÞm2

h r2, r3

54 Ì ##RÓ4gP¯#4g

m ∈ Z NsZh

0 ≤ v, u < N s_<h 4Q;7g4#/g

A = mCm1+ vQ

h B = mP1 + uQ.

Pg

(A,B)#DÓ4gî|g

V

4 Vg h 4

tÜ¤Ô#Ò|hÑÑg#4

e

P

N Ì #ò#Ó4gPg4#/g

w = m+ em2, y = v + er3, z = u+ er2

h g (w, y, z)

V

P# Ì #DÓªgî|g#gÎ|_

wP1 + zQ = B + eCm2, wCm1

+ yQ = A+ eD.

7Èhggg/Ah 4#Ñ¼CÕ#Ò§#DFhh 01#2#$IpR-Öý3ÖR-Ö Ì #©hÔ6RÓ4Ð#h?g4Ñ4q4Õ#RÓ?#g4g\AÝ#Ñ#ÑF_4ò7

ΣÜ¤#h?g4Ñ

#RÓ?#òÎ#RÍÑ 4hm2, r2, r3

#¯×|_Cm2

= r2Q+m2P1

h D = m2Cm1

+r3Qâ

Hhâ|#RÓ?#|_Dg#g:¼?/

m2m1

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' hG<

R ¡ M ù?#Í#×g4#Ñ\g#gÒ:h4ªqhÍ§h #g

wP1 + zQ = (m+ em2)P1 + (u+ er2)Q = mP1 + uQ+ e(r2Q+m2P1) = B + eCm2

h

wCm1+yQ = (m+em2)Cm1

+(v+er3)Q = mCm1+vQ+e(m2Cm1

+r3Q) = A+eD.

Ö 46ggñhÑ#4#g:CÓ4gî|g É g_Ü¤Î#RÍÑ 4hâ4Ó4g'hª¼"¯|hÑÑ\#4 e g4g\Ñ¼ .ÜÔ#/ g4ªÓªg_4yFhQÔ6òÝ#Ññ_ h§q4ÑÑRÍgA#òÝ#Ññ_4©#?4g#ÓhhÑ#gw, y, z

_òh 4ð#gDg8g\Óª 4Yh#g¦7Ý#\g4#/gòY_##CÓhhÑ#gA,B

##×#Dgá|_4#wP1 + zQ = B + eCm2

h wCm1

+ yQ = A+ eD

Ö 44# #gÍ©h#È|_q44òÓhhÑ#Èh(A,B)

##RÓ4gPFh&g4g\Ñ¼

h#:Í©gC.Í© åKggªC¯|hÑÑg#4ge, e′

Í#g&Íò ÑH # #e > e′

7g#ghâ#Ûq4ÑÑRÍ#gá|_4#hD_:î|

wP1 + zQ = B + eCm2, wCm1

+ yQ = A+ eD

h w′P1 + z′Q = B + e′Cm2

, w′Cm1+ y′Q = A+ e′D.

Ì ##Ñg§|_(w − w′)P1 + (z − z′)Q = (e− e′)Cm2

h (w − w′)Cm1

+ (y − y′)Q = (e− e′)D.

7g#gCm2

= (w − w′)(e− e′)−1P1 + (z − z′)(e− e′)−1Q

h D = (w − w′)(e− e′)−1Cm1

+ (y − y′)(e− e′)−1Q.

ûpPFh"g4#gÑ P|_m2 = (w − w′)(e − e′)−1, r2 = (z − z′)(e − e′)−1

h |_r3 = (y − y′)(e− e′)−1

Ì #DÒ g hQg#g:¼?/4Qh

m2m1

Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' h O

ðï6ðï ñèéTò6ò"Aò"¡ñu¦¢Öåßá¨¾ bái`ßòC ¦¨àÖ¨

ûpòÍ§h4Ô/hÛΣÜ¤#h?g4Ñ|_4Ó4g&hQg#g:¼?/4

C = Eke(m)

hm#Ó4g

|_PÎ#RÍÐ#g4g64 #gh4

m<ù?#PÈ#h?g4Ñ8<g4h# #ÑgÑï¼

(#ÛhÔ6RÓ4Q#h?g4Ñ§q4×#Ó?#ghg¯CÝ#Ñ#ÑF_47ghâ§#QgÑñ_hR

4ÔªÓ?4#Ñ¼

R = (ke, Cm2), (m2, r2)) | m2 ∈ M, Cm2

= Eke,r2(m2).

aYLGW[Ê%8&1`-)-?$&(WIL)0my2-Z@BW01-?0 -&(mÞ

ke = (N,S, (a, b), Q)h

Cm2= r2Q+m2P1aYC`QIK0 )-&(t[Ê%t'-0 &(1`0ZZÞ

m2

h r2

54P#?4g

m ∈ Z NsZh

0 ≤ u < N s_Õh 4Q7fg4#/g

B = mP1 + uQh g B

V

4 Vg h 4

tÜ¤Ô#Ò|hÑÑg#4

e

P

N Pg4#/g

w = m+ em2, z = u+ er2h g

(w, z)

V

P#V#gÎ©|_

wP1 + zQ = B+ eCm2

h hggg/©h 4#Ñ¼K#<f#Fhh 01#2#$I R-Öý3Öý3Ö Ì #hÔ6RÓ4#hg4Ñ8q4Ð#RÓ?##Ññhª¯?©Î#RÍÑ 4§

ΣÜ¤#h?g4Ñ

#RÓ?#×Î#ÍÑ hÈhm2

h r2#y|

Cm2= r2Q+m2P1

Ì #È#h7q4ÑÑRÍ§ ñgÑ¼q4 ü2gY_P#i<? P#

ðï6G ã*ò"Yáã*ò"Eã~@ñèéTòÁáò6ó¾òÖ¦

ûp#RÍ ggÔ6ΣÜ¤#h?g4Ñ#|_§4Ó4g.Í©Ý\##g:¯?

C1, C2

â#h ###Ô#ÑÎh\¼ke

#RÓ4gA|_A#©#Ó4g`Î#RÍ`4#©h|#©g4g64 #gh4gAÍ#4/A\Ó4FhÑ#Í#ð4#g Ì ##46g:¤¼ÛFhÑÑ ÁqC _ ¥ ý/) 1Â¢ ý £CØÙC)_DÔ8\q4Ý#È\á#ÓhhÑgªÈÛ#RÓ?#y|_È##RÓªgÈÎ#RÍÝ4#g4g84 #Û_ 4 ÓhhÑ#

r&.Í©ðg#g:¼?/4#×h\:/âfHhâ<C1 = Eke,r1

(0) = r1Qh

C2 = Eke,r2(0) = r2Qhg#g:¼?/4## gÈ#Yh##Ô#Ñ×Î4\¼

ke

h #Yh 4 ?#ÝÔ6gr1h

r2â

Q"e;% T ' ³²-+K)³¯¬]§¯$± ' )³¿B+®V$ ¯X(±QK+K)³¯¬ hGh

g6g\Ó4gÑ¼4âAQ#Í|_PÎ#RÍg#g

r14r2Í#4/È\ÓªFhÑ#Ûgï#\Èh§#

ÓhhÑ#ggfé#Ýü2gY¸P/i<? N h ##È#g¯##á#g©q4Ò 4O 6Í©FhC× ñ_gÑ¼Ô##Ñ #"&#h?g4ÑHÛûpCÍÑÑÐh#CÍ#4/ÝÑ4Ýh4g#ghÑ¤¼ô|_Ý#C#Ó4gÎ#RÍ

r1#|_

C1 = Eke,r1(0)

h Í#gS g#hg<##4#g::ÜóÓ4gî|g<#Ññ4

q4<##hg4Ñ8qhÐ¯#gÎ#Í#\#gÒhYgÑggª©©hYg#g:¼?/4Ch0h7ù?#Ô#g\4

P#i<? N ¬4¬%[x¬ $&LGW -&(mÞ

ke = (N,S, (a, b), Q), C1, C2aYC`QIK0B)-&(Y[Ê%Y'-0B&1`0ZZÞr1Í

0 ≤ r1 < N sÈ|_

C1 = r1Q = Eke,r1(0)

54 Ì #×#RÓ4gP#4gÈh 4Ñï¼

rÍ

0 ≤ r < N s+7×ªÓ44ÎhgD##4#g::Ü

Ó4gî|gÈÝ#Ññ_4S4p##/

ke, C2

Û4Ô/hyg4ªÓªg_4(A2, e2, z2)

Pg4#/g

A1 = rQ = Eke,r(0)h \

A1

h A2

#DÓ4gî|gV

4 V#?4g_ 4Ñ¼P

tÜ¤Ô#Ò¯|hÑÑg#4

eh \ §

P

N Pg4#/g

e1 = e − e2

âKÍ#\ÝÍò ÑH # #e > e2

âh z1 = r + e1r1

¸7g e1, z1, e2, z2

V

P#V¯#gÎ©Í#\#g

A1, A2, C1, C2 ∈ E(Z Ns+1Z)h |_

Eke,z1(0) = A1 + e1C1h

Eke,z2(0) = A2 + e2C2

7Èhggg/Ah 4#Ñ¼CÕ#§#òFhh

¿RÍ*ü2gY_P#i<? N h #ò#?h7g##á#gq4 4O 2#Ñ¼#Dq4ÑÑRÍ#×ü2gY/ 01#2#$IbR-Öý3Ö Ö Ì #hÔ8Ó4p5\Ü¤4/:Ü¤hñÜ 4 #h?g4Ñ

ΣÜ¤#hg4ÑH

.1 ½]<^<yÄ · ¾ ¶ Ë · ÂÁ ËÕÂ µ ½PÄ¯Ë · < ¶ 7AË8 > ÂÉ<½CË ¶ Ë · ÂÁ

ì Ó4g ÛgYhªFhÑÑ¼ðgg#C#g#4Ñ #444##×g:¼?/4:¼?:F h 4P_?ÜÑh:¼ #h?ghÑ×××64Ô#ÑP4Ô/h(4g#ghÑÝ#Ñ|h:¤¼ g4#/_4 #h?g4ÑÍ#&hÈgg#Ýhªh#:hÛh\ÓªÈh :_È Ó4gh:¼y|_g4#/hª¼y#4¤¼h©##ÑñF¼4ggâAg^ 4 <H Ö 4ò4#ò#g#4Ñ Ó4g4ph©#Y4g#\_Ñ gÑÑ/g#:Ó4ØAhÑÑg§¯#gòÍ©Èg4#:#\ #Èh?/Ññh:¼Y#h?g4Ñ§|_hÈ#g ëqgù?g\hP#i< í ;7g#ghâª#Ò#464 #gFhÔ6Ò# DÔ##Ñ #Ý#Ñ|h:¤¼Èg4#/_4

Q"e;% T ' ³²-+K)³¯¬]§¯$± ' )³¿B+®V$ ¯X(±QK+K)³¯¬ hk

#h?g4ÑgòÏ¤ð#g\4&Í§ÝÍÑÑ7#YhÈ#g#Ñòh Fhh<#×h##ªhÔª¼ ÷ h\\hÑT 4 <HÌ #ò Fh7#Ñ|h:¤¼Yg4#/_4 _gÔ|hÎY#ò|h8gÔ¼ ì 4Ñ g¯â|çQF_ÑHâû g4 PZh©h Ôª¼ h$ km<H Ì #\¼ôÍ©gChÔ#ÑQ#RÓ4×|

n#ÑñR¼4gÈF_p/ªÜ

ggªÑ¼Cgh#/òhQh4g nÜ¤##/§q##\4PhÕ#g§##/ÒyÝgg#òÍ§F¼4<Ï¤y#

×#Ñï|h:¤¼Cg4#/_4y#h?g4Ñ2gg#¤¼yFh#|_\Ó4g:¼44#ÝÑFh#Ò#Èg4g\4/#/Í#Ñ#Y#Óhh\¼ôhÒFh¯ #ÑñF¼4gg ##/È4|hhªg ¿h|_È#Yg¯Ü|h4§g4# g ÷ |h/g N Ô6gÑ4#Ý#òhòhF/ Ì #gÈhD\ÓªghÑ åKggª¤¼?6gÈh§ Ó4ghgò64Ô#Ñq4D#¯Î h©#4Ô#ÑgQâ2h Í©×Fh :#4#ðÔ8¯Ü¤Í§gg(h\Ó4Ch |hÓªYg4#/4QÏóh Ó4gh:¼pFh g4#/Û\Ýh#ÑñF¼4gYhÎhD#gÔ8g|RÓ4ò#òÍ§F¼y#òÍ§hªgâ|Í§òR¼C|_# Ó4gh:¼y Z )4â|gÑò# Ó4gh:¼×hÔ/¯_#Aæ¤4#Ñ¼ªê©#Òg4#Ñ\Ò/q4Y_h#gÑ Ôª¼Ý#g4#/ #ÑñF¼4gÍ#Ñ##ÑñF¼4g©:ÑÑ6hhÔ#Ñ¯/gg/##h?g4Ñ8g4g\Ñ¼4â?#©FhÑÑ /ªÏó©#\h§g4#/ #ÑñF¼4gÈDî// q4#Ô6g4####â2#Y Ó4gh:¼òFhÑÑ / ) Ïó#y Ó4gh:¼ Fh(__¼p ##&#C#_g4Ñ©¯#4Pðg4#/#\Í#ÑñF¼4gÔ|h 4&hÑÑ/q4Y_4P#|h_#Òhâ|#ÈFhÑÑ m¥m )ª Ì ##h?g4ÑfÔª¼ ì hÑ g¯\©hÑH`h ¦ hÈFhÔ6#RÓ4gYgg#hªh#:ÐhYh\Ó4hâ:_hâ64Ñ¼?#4ñ_Ñ<×YÔ64# Ó4gh:¼pÍ#ôFh g4#/×h¼p\×h_4:

n/2 − 1#ÑñF¼ªggù?\Ó4ghÑK#hg4Ñ<q4<#Ñï|h:¤¼g4#/_4#<|FÓ4Ô6ggC#464 RÓ4gÐ#¼ªFhÐ#RÓªÒ/ªgg#\¼4â/gh#¼ 4 PHÐçQ4:©hK##464 #hg4Ñ<|RÓ4Ô6ggCÔ|h 4Ó4gî8hÔ#Ñgg\|h##gggÐ²/ £Øý\ /; 1Â %/¨BDhÑÑRÍÒ FhÑgðgg#gÑ¼ :Ô#/P&gg\

sh4#ð&\h#ÑñF¼4ggâfÍ#gP# FhÑgh ß_4

4Y#ÑñF¼4gYR¼ôÔ6Y#F_## 7gCÈ4|hhªg |_Ý©# FhÑgÈÈ#4#g:Fâ#gQ#ò#F_g4Ô/hy#×/q4Y_4yhÔ64/

sâ8h hÑÑ#4#g:#ÑñF¼4ghÈhÔ#ÑD

gg4#:#\#ògg\s\Ó4g&hªh#:Ò#Èh\4#Òh7#ò#F_#×#ÑñF¼ªgg

Ì #©#h64hÑ?Ôª¼ ÷ hg7\fhÑHÕÔ|h##Ñ|h:¤¼Dg4#/_4È#h?g4ÑÕ4#g?Ü4Ñ #444##òg:¼?/4:¼?:gÞ#:F hfgg\|h#Y#ggÑF Y4È\ñÜî|ggª#hg4Ñ#gY #ggÝ#YhhÑ<#Ô8gÝhÒÔ#È|_Ý|FÓ4Y&Ô6YgÝO(nσ|C|)

âÍ#gnf##Ô8gÐh#ÑñR¼4ggâ

σ<#gg#¤¼×|hh\g©h

|C|#

É _<#gg# C g4#/#Y# g q##\4òÑÑ7#\Ó?4#D#hg4ÑÍ#Í©gÈgg#hªh#:_\Óªò Óªghggá#

Ω(n2σ|C|)Ô#§Ô#ª Fh:F

Ì # FÈh ÷ hgÐ\§hÑH`<Ô|hFhÑÑ¼×h<q4ÑÑRÍgA4Ó4gPÈghFhÑÑï¼gg###Ô#Ñ¯Ü

Q"e;% T ' ³²-+K)³¯¬]§¯$± ' )³¿B+®V$ ¯X(±QK+K)³¯¬ h j

Îh\¼×g:¼?/4:¼?:g)Í#g#4Ñ × gg:¼?/4×Í#gÒ#gh4Ò|hgM

<h# Ô8ðp##!ëqÏ¤4#YFhÛÍ©&|FÓª

M = Z NsZí Ì #Qq4ÑÑRÍ##444##

#46g:gyh&gá# 4Ó4g g#g:¼?/4#E(a)

h E(b)

hah

b# gC#

hÛ##Ô#ÑÛÎh\¼Í©ðFh'ªggÑ¼"g4#/ðhg#g:¼?/4'hD#&#E(a + b)

Ö #:#g4hâ`qh h g#g:¼?/4

E(b)h ð#Ññh¯?

a ∈ MÝ#4#Ñ Ô6PFh:¼

Yg4#/×h 4g#g:¼?/4&_E(ab)

n ÔªÓ?4#Ñ¼P#g#48g:ghì¾:#:# # Ó4gÑ¼#444###48g:.¼Yh #FhÑñh.Ü¤Ý#Ñ#ÑF_ïÓ?¤¼ÝÍ#C#4Ñ q4Ð4##g#4Ñ Q gg:¼?/4ÛÙ ÷ ØÐù6Ö |hÑÑï¼4â74Ygg#P#Ô?Ü¤#hg4ÑÝ#:ÝÔ6CFÓhhÑñhÔ#ÑhÛ#hg4Ñ<q4È¯#gÎ#ðÓhhÑ#<7ÒÓ_hÑ g##g:¯?Fâ4#h?g4Ñ4q4Õ#RÓ?##<g4g\#g`h?#ÐÝ#Ñ#ÑF_4h`hQg#g:¼?/ ÓhhÑ#DÔ¼yg4#:hFâ8h #h?g4ÑKq4Ò#RÓ?##òÎ#RÍÑ 4òhf#Ññhª¯?F Ú h 4Q4#Ò#gò#gÈ#hg4Ñ§Í©g ggÔ8 Ûù?g\4P#i<?Ì &gg#gÑ¼ g4#/Ch

nÜ¤##/q##\4Í©Cg4# gÛgg#ÈÍ # 4ph

Ý#Ñ#ÑF_4Pª_gg4#/#Ý#Dq##\4Qh \ÓhhÑ|_#ògg#ª_DÔª¼yª_hÌ :h:Õ#fÝ#Ñ|h:¤¼g4#/_4#h?g4Ñ4FhÈ#ÑñF¼4g

Pi

âRq41 ≤ i ≤ n

â##Ô#Ñ#ghjg#g:¼?/4

E(xi)hÝ#y#Óh_ô##/

xi

44\#gyÍjΣÜ¤#h?g4ÑD#RÓ?#

#QÎ#RÍÑ 4hxi

âDg ù?#Ô#g\4sP#i<? N ª¼k46g_4 Ó44ÑÓ?#" # 4 4Ý#Ñ#ÑF_4 Ôª¼k(g4#:hÛFh #gjÔ66g:q4 Í#4/yq#:#gQªgh\4##D#hÑ44ï#×` # h çQï/ Ü:çQ#Ñ<h6#g#g:¼?/4×#gh7g#ghâÍ©;¾:#:#g ò#hg4Ñ|q4<gg#gÑ¼×g4#/#ÈhYg#g:¼?/4YhK## #\

E(c) = E(a · b)4Ó4gQg#g:¼?/4#hE(a)

h E(b)

Ô#/#h#òÓ_hÑ#gah

b#ggÑÓ4gg Ì #

#QFhQÔ8 4#òÔ¼C#Dq4ÑÑRÍ##hg4ÑH

WC&IQ,EG#$&(IKE&LGWË[Ê%8#bWC&(WGIKCE%[úZ&('-0Z0my. -&(mÞ

E(a), E(b)ot&(mÞE(c) = E(a · b)

54Ù<hQ#ÑñF¼4gPi

#?4gh 4ßÓhhÑ#ri ∈ M

h Ô#ª Fh:hyg#g:¼?/4E(ri)

hÕF Ö #:#g4hâ/hÑÑ6#ÑñF¼4g§#Ó4|_§#\¼CÎ#RÍ #g©Ó_hÑ#ri

###òg4g64 #×#Ô?Ü¤#h?g4ÑH

4 ü2\r = r1 + · · ·+ rn

fÑÑ8#ÑñR¼4g©FhP#RÍkg4#/hPg#g:¼?/4E(a+ r)

Ôª¼h##Ñ¼?#Ýg_ÓªgÑ¼×#òhÑ44# # 4

E(a)h

E(r1), . . . , E(rn) Ì #ò

Q"?Y% T ' ³²-+K)³¯¬]§¯Ây ' ²)®¯¬Q³²¼¯1)³ý¬ ( kG:

4Ô/h# g##g:¯?E(a+r)

7#g gg:¼?/ ###§#h?g4Ñq4A#g#4Ñ gg:¼?/4Ë¿RÍ*hÑÑK#ÑñR¼4gÎ#RÍ

a+ r

N ØfÑñF¼4gP1

\a1 = (a + r) − r1

Í#ÑhÑÑAh#gD#ÑñF¼4gPi

q41 < i ≤ n

\ai = −ri

¼¿hò|_\Óªg:¼Q#ÑñF¼ªghÔ#ÑDYg4#/hQg#g:¼?/4&h`Fhaih |_ÒÍ©È|FÓ4

a = a1 + · · · + an

P#Ù<h(#ÑñF¼4g

Pi

g4#/gh g#g:¼?/4E(aib) =

çQï/ Ü:çQ#Ñ(ai, E(b))

h Ô#ª Fh:f`44\#g`Í×#?h8q4A#Òg4g\#gÐh8#Ò#Ñ#ÑFhÝ#.Ü##Ðg4g64 ##Ô?Ü¤#h?g4Ñªq4 ù?#Ô#g\4_P/i<? P4##/

E(b), E(ai)h E(aib)

<?ü2\

SÔ8§#Ò\<h6#ÑñF¼ªgAq4AÍ##Ò#\Ó?4#`:\#ggg â?h Ñ\

ScÔ6

#Dg4#ÑggÒhS`ûpò#RÍjî|:g4#/

E(∑

i∈Sc ai)###È_Ñ44#

# Ì #gY##ò#g#4Ñ Y gg:¼?/4Í© gg:¼?/©#<g##g:¯?E(∑

i∈Sc ai)Í# 4Ó4gÝ##YÓ_hÑ#aSc :=

∑i∈Sc ai

Ì #hÑÑRÍÈ\Óªg:¼44#C&g4#/hYg#g:¼?/4

E(aScb)Ô¼×##È#hÑ44ï#ßçQï/ Ü:çQ#ÑF Ö 4)#Ðh #

g#g:¼?/4#E(aib)

q4i ∈ S

â`hÑÑ<#ÑñF¼ªghChÔ#Ñ×Ûg4#/Ch g#g:¼?/4E(∑

i∈S aib+ aScb)â#ÍÒ#¯Q g hQg#g:¼?/4Qh

ab

§Y#Qî||hÑ:_Q#ÛÎ#Íg#g:¼?/4#Ch#Û4/#/YÓhhÑ#gYFhÔ8 gg:¼?/ ###g#4Ñ Q gg:¼?/4y×4Ô/hC#Dq##\4PÓhhÑ#hfÏ¤ª#ïïÓªgÑ¼#ÒÒgg#È## gÑ¼?#Ýg:¼?/4:¼?:F¿§g\#_â|#g×ëTh|h:©q4ß#4/#/ í _ÑÑ8ÓhhÑ#g©|_h gg:¼?/ hg#gòh 4Ó_hÑ#gÈ4ÓhhÑ#gÈhÑF ¼&Î#RÍy# Ó4gh:¼4Ì #×/ªgg#\¼ôh<#D#h?g4Ñ g6g ÈgªgÑ¼Û4ô#/ªgg#\¼ôh<#×# gÑ¼?##g#4Ñ Q gg:¼?/4Q#gÈh #òg4g64 #×#Ô?Ü¤#h?g4ÑgÖ 4Ýq4YhÑ gg/4â|×#hAhAgg#.¼yh y \hÑÒgB 4 <H

.1 ½]<^<yÄ · ¾ ¶ Ë · ÂÁ ËÕÂ | Ä\Å¾ÐËª7AÂÁ · ¾ÒÑ*ÂòË · ÁPÇ

Ï¤ð#g\4ðÍ©ÝÍÑÑ ggÔ8×#RÍ 4##444##ò#g#4Ñ g:¼?/4:¼?:F FhÔ6P# ô4Ô/h &#Ñ¹Ü._/#4ï.¼gg\:Ü¤Ô|hÑÑhÝgÑg\4 ¯#gC|_4|hhgg4Ô##:#ggâ<##ÓªghÑÐÓ4gî8hÔ#Ñï¤¼h g4#/_4|hÑf#Óhh\¼4 n #g4#:#\4

Q"?Y% T ' ³²-+K)³¯¬]§¯Ây ' ²)®¯¬Q³²¼¯1)³ý¬ ( kK5

Ô|h 4(h FÛh ÷ hggâ ì g#|hðh ù?#?g#YhÎhg 4 kÐÍ#ð#464 #h gÑg\4 #gCÔ|h 4 Q#g#4Ñ Ó4g4 hÒ#PÙfÑ ì hYhÑf#g] N kDëqgù?g\4 N P#65 í ûpÝÔ##Ñ 4#Ó4h#C#h?g4Ñ7ðhðh|hÑ444#Í§R¼&h#éhªõh \hÑH/#h?g4Ñ#gg A NGN HÌ ##g ggÔ8 Ôª¼ ÷ hgP\&hÑH§yÔ|h 4j( \ÑDÔ¼ Ú g|hÑ4k\ÛhÑH 4G4 â5:?âLkKÍ#gò#òh\Ó4D|h:gÒh Ó? ª×\h7Ó4hg

V1, . . . ,Vm

h \òhfhÑÑ¼?#Yh/#4g

A1, . . . ,An

Ö #:#g4hâ2Y_Ü¤FhÑÑ £ ¢ Ø Øý/) £ ¥ B #g Í#yhgggÔ#ÑòÔª¼yhÑÑK|h:g§\ÓªgQ|hÓ4ò4Ô#g:Óªg×h¯#\Ó4ò##Ó4ghÑÓ4gî8hÔ#Ñ¤¼4©ù?#&×Ô##ÑÑ\PÔ6ªh ÑÎhò×##Ô#ÑDÔ#ª Fh:|h##gÑÕÍyg4:¼4ÏóÐ|h`##46g:¤¼Ý|_©hª¼×h\Ó4Ò4f|hÓª|h:¤¼<hÔ#Ñ§òF #g4ª\<hKFÖ #:#g4hâ##gò g4|_ g¯4Qq4ÒFh&h\Ó4ò|h:g|hªÒÍÒ#g#òFh64:#ògh4ggJ7g#g×Í§Fh gn¼ÛÍ#p#ÑñF¼ªgòFhpgh4g4gòq4Q # 4|hÑÑï¼ï#h64Ô#ÑY gÑ\Phª¼ /q4Y_4q4 #PÔ##ÑÑ\ Ô64h Ö 4ò#:h#ghâA# QÔ##ÑÑ\ôÔ6ªh FhpÔ8Y#Ñ\gª ygg#YÍ§F¼Ôª¼##h&hÑF ¼ g#Ñ¼4 ##Ô#ÑòÎ4\¼Q/qh:#\#h ×g:Óªgg#ÑF_hPg¯##á#È#\Ó4g g#hÑ2hAg:Ó?gÈ_h¯ÎgÏ¤4#ÕÓ4h##hg4Ñ?ÒÓ4hgA#Ñï¼D84:`|h:g#Ññh2g#g:¼?/4# g7# Ì Ù ÷ ØÐùh#Ó4hÈ#Ô##ÑÑ\×Ô6ªh hgg4|_# Ôª¼YÈ#h2hKÓhhÑ ï.¼|_©#RÍ©|_#Ô|hÑÑhÐ g g4ªh#§òÓhhÑ Ó4hh7g#Ô|hÑÑhÐÐg#g:¼?/ ÍY###Ô#ÑÎh\¼h|#§h/#4ggAù?#gÐ#©g#g:¼?/4\# # A # Ó4gÑ¼ò#44×4##hâ#î||hÑ<hÑÑ¼ðFh Ô6Y4Ô/h# hòæ¤#ê×hhÑÑfÓ4hgg Ö #:#g4hâAÈÓ4gî8hÔ#ÑÔª¼ôhª¼&4Ô#g:Ó4gòhÐ#gÑg\4ðhªh#:#æ¤#êÝh©hÑÑA#Ô# Ô|hÑÑhgÈé##Y#46g:gÈhÒ4#ò#g#4Ñ gg:¼?/4 #g#YÔ6g#4p4ÈYhÑ4ðThÑ#×h4òhÑÑï¼?#h/#4gÒFhÛÔ6ò4Ñg_ Í#ÑD#È#Óhh\¼yhA#È Ó? |hÑKÓ4hgÍÑÑ<Ô8C4|__g â`Hhâ7#Cg4g\#gÝhÒ# gg:¼?/4pÍÑÑ<Ô8Ph# \Ó4g #y#gg#gÛhYhÑgh#×h/#4gÍ#"g##gY##Ó4ghÑ§Ó4gî8hÔ#Ñ¤¼4 Ï¤(##464 #hg4ÑA#×#Ó__Î4\¼ðD#\Ó4gògg4#:#\ â`h 4#Ñ¼Û# #ÑgÑï¼Í#gQ#Èh/#4gÒg?48g_D gg:¼?/#Dî||hÑhÑÑ¼4ûph#§|_f###64h6#ÒgÑg\4×`ògÑg\ÐÍ##gÐh4#

LFh _ggâ

h |_FhQÓ4hghÑÑRÍ© Ó4hDq4t < L

Fh _gg

Q"?Y% T ' ³²-+K)³¯¬]§¯Ây ' ²)®¯¬Q³²¼¯1)³ý¬ ( k 4

ñèéTòF¤¨BéQáå¨ª© ò"¡¨ª¦¨ªóáåòÖß©Ì #gYhYhª¼&#46g:gDh©gÑg\4ô#ggò|_ò#4#Ñ Ô6g4# g ûp×Ñ:#Ý#46g:g|_Í§gÝg4# g Û#h##ªhðÔª¼2 4 kH Ì #g#46g:ghÑ#4Ñ q4Ò4#Ò#h?g4ÑHDFW)?Cw(W) ,qÞ n #Ñ¼PgÑ4Ô#ÑÓªhghhÔ#ÑD×Ó4hh FhÛgÑ4Ô#ÑÓ4hgFhÛFh:#4ÑÓ4hhù?#g§Í©h#§|_<DÔ##ÑÑ\ÝÔ6ªh <FÓ_hÑñhÔ#ÑÐ#`#46g:¤¼q4ÑÑRÍ`× ñ_\Ñ¼4 C`0Z.ILWX¸0ZCÔ(IQw()W) ,qÞ Ì #DTh#g§hÕ#ògÑ\\4PFhyÔ6D#g¯Îh Ôª¼Ph¼C|h:¤¼#gÑ #y|hÓ4Y4Ô#g:Óª\g Ì |_Fh#ò|Ýhª¼|h:¤¼ôhÔ#ÑÛ#g¯Îð|_È###Ô#Ñ# î||hÑ6hÑÑï¼©g4#:gªÒÍïY#g4g\Ñ¼YFh:ÒÔ|hÑÑhg Ì #§#46g:¤¼ChÑ#gÑ gÒ|_hª¼P|h:.¼PFhQ¯#gÎPÍ#¯#\Ô|hÑÑhÒhòg4g\Ñ¼PFh:Fâ6h |_4#Ñ¼ªÓhhÑ Ô|hÑÑhÒh Fh aYC`QIL,qÞ ØfÓh_\¼'hh Ó? |hÑÓ4hðyh# hªh#:yhª¼Fh4|hÔ#Ñ¼ É gªhÑ4 h|h:g|hªyëq¯/gÑ #ð#CÓª_\×#gÑ í âfHhâ<##Ñg#P?#ÝÔ6ghg4ÑÑ #|h:gf¯/gg Ð4#f#g#4Ñ

kâ åKggªÐÔ|_ÑÑhfh :#4#|hÔ#Ñ§¯Ü

6g\Ó4Èh7#Dg4h# Ó4hgg¤ ¡ ¨ ) LÚ CL) #Ó_h\¼Ph#\Ó4 Í#gQ#Ô|hÑÑhòh× :#4#|hÔ#Ñ g6g gªÝhÒhª¼ôg:¼?/44h##Yh#/4Áh#g.ÜÍ ¨\ ¢ ) Ø7#Óhh¯¼Ph¯#¯Óª %w(./-01.V.Þ Ì #CTh#Ñ.¼Ô6g|FÓ?4#Ýg#gÝÔ8g#4 4ÝYhÑgh#hhª¼pFh4|hÔ#Ñ¼ É gªhÑ4YhÕ|h:g§FhPÔ64Ñg_ Ì #§#gÑ g©##46g:¤¼Y|_§#gªhÑïÜ4ÛhfÓ4hgh<h¼Q É Fh #/YÑñh4¯Ü¤FhÑÈgÑ\\4â6Hhâ8Í©ÝFh \g\òh Fh hª¼P#F_#Óª_\F E`L02-&(W))GIKÞ ÏóD#4#Ñ Ô8×84Ô#ÑÈPg4ª¼ôh#h#gÓªhgg Óªh\Ó4gÍ#4/Î#RÍ#×Í|_#òg4# Ó4hòg ])0ZILAw0m@_0Z012`L0Z.Þ é##È#¯/gg/4Ch#Ó4h##hg4Ñ|#Ó4hg ×#h|FÓ4ò×ªgh\ÒÍyFhÛh#ggògª4# ù?g\4bP#i<YÍ©×## Ö ñ_:Ü:ù?|h#g#:Í#g|hÔ#ÑgD#DYhÎhò4##?_q#4?Ü¤gh\Ó4gÑ¼4 Ì #È|hyq##\4y#g ÍÑÑÔ8 g#h Ô¼

h

ù?#gYÍ§CÔ|hC#CÓ4h#&#h?g4Ñ©4 # Ì Ù ÷ ØÐù6â`Í©Cq#:#gh#C|_×h ?Ü:h#gòh7##g#4Ñ Ù ÷ ØÐùCÍy##Ô#ÑÎh\¼

(N, s, (a, b), Q)|hÒÔ6ggQ\# Ì #

Q"?Y% T ' ³²-+K)³¯¬]§¯Ây ' ²)®¯¬Q³²¼¯1)³ý¬ ( k N

hÑÑ¼?#Ph/#4gAi

q41 ≤ i ≤ n

ÍÑÑ`h\h gg:¼?/4g:Óªgg Ì Q4|hhªg#òg4g\#gh7#òî||hÑhÑÑ¼Í©hÑ×|FÓ4òh#D|_

N s > mLâ#Í#g

m

##ÝÔ6gÐhÓ4hg§h L##ÝÔ6gÐh2Fh _ggq¿h|_<#ÐFhPhÑÍ§F¼?ÐÔ6

q#Ñî|ÑÑ Ôª¼P#?4#s4NÑñh4g#4#4

Ï¤D#`q4ÑÑRÍ#ÐÓ4h#Ò#h?g4Ñhq4K¤Í©Fh _gÍ©fÍÑÑ_#f#f#h_4ProofVi

(S)h<ª #g NGN Hq7gS g#hg©òÑ44ghÑ#:_ggª<Í# F_gfÍ##_Ü

?g4ÑKq4ù?g\4P#i<ÍÑÑÔ8ò# ProofVi

(S) = (e, z)ÍÑÑKÔ6ÈÔ#:#×gF_

Ôª¼Óª_\Vi

Vi

#g<#h##4#ñ_Ò#h?g4Ñ#|_ÐghYÔ6# ògh\Ó4gÑ¼Ý#RÓªSc7g4#/g#Èî|:gh4

aQ##h?ghÑHâ6g4#/g

e = h(a, S, ID(Vi))Í#gID(Vi)

##á###Ô#ÑÈ:#Y gªn¼?#Vi

\7ÈhÎhg#g#Ñeh`#

h#|hÑÑg#4Èq4#Ó4gî|gòh g4#/g#g4g84 #yh#:Í§gzY¿h

|_Í©|RÓ4È#gÑ ID(Vi)

y###/hhY#\Ó4gÓ4h ##ÑF_4 Ö #.Ü

#g4hâ#hÒ|_ÐYhÑÑ|h?/Ñh:¼Ý#hg4Ñ<<FhYhÑÍÒF¼?<Ô6FhÑ¼g4#/ Í|_a#4#Ñ |FÓ4Ô8ggp4Ó4g

S, z, eâÕ| ##?h©Ô8ggpg4g\F Ì #ò 4#Ô¼B¾:#:

#Ñ#44#zh

eªy#Ó4gî|g#gÎ8 Ì #gâ`#(y#?hÒFhpÔ8Y#g¯Îh /ªÜ

ggªÑ¼CÔ¼P#gÎ#Í#\#ge = h(a, S, ID(Vi))

¿RÍ*Í©ÝhÝhÔ#ÑDY¯/#Ññh&#hg4Ñ2q4#Fh

L = 2â6Hhâ6#h?g4ÑÕQÍ#

#ÝÓ4hgDÓ4hgq4D4#4/Dh<.Í©yFh \Fù?#gÝ#gá#ÓhhÑgQY¼4gß_#Ó4hhâ|Fh¯yÓ4hòFhQÔ6òg4# g hò ://q4Ò#Yh 654#q4§¼4ggXYL)-?A&LGWË[Ê%t@_2GIL)-IK01.

54Ù<h Ó4hgVi

gg gÝ4 #ÈÓ4hvi

]7Yg4#/gCi = E(vi, ri)

âAÍ#griÒ¯#4g&_h 4Q7ÈgF_g

ProofVi(Ci

4Ci − P1

hQg#g:¼?/4Qh;:)Ô|h 4ð#J5\Ü¤4/:Ü¤hñÜ 4 #hg4Ñ`_Òù?#Ô#g\4bP#i<? O Y764:D#g#g:¼?/

Ó4hD44\#gÒÍP#ò#?h7#òÔ##ÑÑ\PÔ8ªh B

4 Ù<hyhÑÑ¼Ch/#4.¼Aj

?gÒ#Dq4ÑÑRÍ##

•ù?¯

C = O

•Ö 4hÑÑ

iQ#gÎ #Q#hÍg"Ôª¼

Vi

4B"Ïó×Ó_hÑ â<#g\

C := C + Ci ∈ E(Z Ns+1Z)

• Aj

#gCh§#D##/g##g:¯?h ¯/gg/g#Ò|h:Òh7##g#4Ñ

gg:¼?/4Q#hg4ÑHq7ò64:#g#ÑÒ#òÔ##ÑÑ\PÔ6ª_ B

Q"?Y% T ' ³²-+K)³¯¬]§¯Ây ' ²)®¯¬Q³²¼¯1)³ý¬ ( kP

N ng Fh #&Ó_hÑ gh4ggâ<#Cg4ÝÔ###&hÒ#g gg:¼?/4 |hgÍgYÔª¼×#

Aj

©FhY#Í'Ô6 4#Ôª¼Yhª¼ª4#hâ?Hhâª#gg4#:#\4Yh##Ññhª¯?©ghg84 #

E Ö 4Ð#Ñgï¤¼ÝÍ©Dh##Í|_§hÑÑ|Ó4hg§h

ÓhhÑ Ì #gyÍ§ò4Ô/h

C =∑

i

E(vi, ri) = E(∑

i

vi mod N s,∑

ri mod N s).

7g#ghâ/#g#Ñ§h2# g\¼?/hâ#Hhâ#î||hÑ8hÑÑ¼ ∑i vi mod N s

Í# ∑

i vi

#gDÍ©ò|RÓ4N s > m

|hÑ444#Ñ¼ 4 k2Í©òFhQ#RÍ #Ó4D#Dq4ÑÑRÍ## 01#2#$I$R-Ö xgÖ)LÖ ##×#|h #ghA#È4g#ghÑ gÑÑ/òg#:Ó4ÝØAhÑÑgh.Ü#/4âR4#2gÑg\4D#g<#RÓ? gÕ##Ó4ghÑhÓªgî8hÔ#Ñï¤¼4âFg4#/_4|hÑh#Ó_h\¼4â4Ô##:#g§h #\Ó4g<Óªh ##ÑF_4 Ö #:#g4hâª#h 4)4hgÑÒ gÑ#ªgh\4PÔ6\¤Í©ggQÓªhgÒ#g

R ¡ M ù?#g§#§#?hq`h|ÓhhÑ ï.¼Dq4A#§Ô|hÑÑh Ci

h§Y §#4?Ü¤ªgh\ÓªgÑ¼hâh#\¼hÈÓªgî8hÔ#ÑÔª¼&hª¼Û4Ô#g:Ó4gg Ö #:#g4hâKhª¼Û4Ô#g:ÓªgFhð#g¯ÎQ#Èî||hÑ7hÑÑ¼Íyg6g\hÑÑKÓ_hÑ Ô|hÑÑhg Ì #4Ó4g###Ó4ghÑÓ4gî8hÔ#Ñ¤¼4##&#C|h #g×h#C4g#ghÑ gÑÑ/g#:ÓªyØAhÑÑgh#/4 #

k4/×h`Ì Ù ÷ ØÐù(Ý\YhFhÑÑï¼pgg#Ûh #g#gy#ô/q4Y_4 hÔ64/#PÓ4h

òÑFhÎh 4ÓªgE(vi, ri)

CÏó©Í§Yh#×|__4:k − 1

hÑÑ¼?#Qh/#4gòhg4#/ â#Í§ò4Ô/hPg4#/_4|hÑK#Óhh\¼Ph7#ò Ó? |_Ñ6Óªhgg¿h|_©##?hÕh2ÓhhÑ ï¤¼×Fh:ÒÍÒPFhCÓª_ÐÍ#g§ :#4#|hÔ#ÑhâHhâò4Ó4gÈ#Q#ò4ô#g4g84 #QÍÒ#gÈ# ð#×#?h. Ì ?#È ?gÈ#h#gÑPÔ#FhÎC#Ó_h\¼4ù?6ggñhÑ#4# #gÐh8#Ò#?hqfh8ÓhhÑ ï¤¼4|hhªgg`|ÐÓ4hgfFh##h<Fh:<Ô644#Ô|hÑÑhg Ì #g##g4Ô##:#gÍQg6g\YYhÑg4#§ÓªhggÒö4Ô##:#gÍg6g\&_È4:

n − khÑg4#òh/#4gòò##g q4 #4Ô##:#gÝh

#òÎh\¼y4g#g_4Qh Q gg:¼?/4Q#hg4ÑgÌ #|hq##\4

h#g q4f##hq<hKÓhhÑ ¤¼Ý4\<##gÐ gª¤¼×hÐ##/F

Ì ##\Ó4gªÓ4h ##ÑF_4

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., km<

D\y01.VE WC ¬/*+IL)-IK0DFW01C.Ï¤#:F hKYhÎ#òD#4gÔ6\¤Í©ggY¤Í©ÈFh _gf<fhnggá# ò#?44#4ò4×4/Èh§\ÓªghÑÐF_ _gg Ì #gCh×\ÓªghÑfÍÒF¼?ÈyhÎÑ#È#4Ô#Ñ\QâHhây4g#ghÑÝ#×hÔ6RÓª×h##ªh¯ð

L > 2Fh _gg)Óªg:¼ð#ÑÍ§F¼ôh

ggÔ6 E NGN <DyYhÎhL|hhÑÑgÑ7¼ªgß_#yÓªhghÈhÔ8Ó4hÓ4hgDÓª_\t5×q4

#Fh _g§#Í§hªÒh :Èq4Ð#h#gg7g#ghâ#Fh¯CÓ4hgVi

g LÓªhg§

|hhÑÑgÑ6Í#ÛhDhA#DqhÑÑRÍ#q4Q

Cij = E(vij, rij)

Í#g0 ≤ j ≤ L−1.

7ghârij

h§#4g_`h 4 h vij

Vi

AÓ4h©q4AFh #Ô8g

j¸7hÑCgF_g

ProofVi(Cij

hCij − P1

hQ\#g:¼?/4Qh;:)Ô|h 4

#F5\Ü¤4/:Ü¤hñÜ 4 #h?g4Ñ/h2ù?#Ô#g\4pP#i<? O ;7§#g×g4#/g ∑L−1j=0 rij mod N s

h 64:òC#×Ô##ÑÑ\&Ô8ªh

B Ì #D#Ó4gD|_ò#ÝÓªh q4D¯#h\Ñ¼

t4/òh

LFh _g#gPh¼p4Ô#g:Ó4g×Fh(#gÎp|_ ∑L−1j=0 E(vij, rij)

h g#g:¼?/4(ht×é#P#×#?hÐhÐÓhhÑ ï¤¼QÍ©Î#RÍ!|_ÈhÑÑA Ó? |hÑ7Ó4hgÈh×g#g¸:P4¦5Í#P#RÍ©|_©#©#g¯Î©KªggªFË¿RÍ Ô¼ gg:¼?/4Ph2#

LÓ_hÑ#g

Cij

##Ô8gh7Ó4hgFhQFh _ggggÓ4 Ò4Ô/h# ¿h|_#g4#Ñ¯/¤¼Qh<# ggÔ6 #h?g4ÑADhÍ§gÑÑAÑ#Fh&#Ý#Ô8gh7Ó4hghC#ò?#ÝÔ6gÒhAFh _gg

.1 ½]<^<yÄ · ¾ ¶ Ë · ÂÁ ËÕÂÏ> ÂÉ É · Ë7ÉÊÅÁË Y¾qÆPÅÉÊÅ¼»

Ö 4ÈQg4×gªÈ#gCQ#\Íh Ó4g:¼:4#Ûgg#.¼ô#h4FhÑÑ ¢1 / Ø ¨\L £CØÙC)òÍ§h<ª #g Ô¼ ÷ h#\8h Ö #ÑJ5kH Ì #<#h4ÝÍ§hÐ 4/ Ôª¼Péhªõh h ¿gÑg] N PKÍ##ggª #\Íjg4#:#\4Ch7##Ó4ghÑÑ¼ghÜ64hÔ#Ñ§gh×gA#gg<Ô|h 4ÝDFhÑÑ §L Ø1¨¦ N1 ¥_ ¨¦¨¦C)¨B g L/¨B\Ö #:#g4hâ?#\¼P|FÓ4ò#RÍP|_§§64Ô#Ñ×Ô|_D#ÛÝ6ggñhÑï/ ghÜgªP¯#gô4 #pØAhÑÑgC¯#gh¿ã#(#gPg#ÑPÍ§ô#Í|_Qy64Ô#ÑÔ|hÈ6ggñhÑ2×ï/ g4×gªÒ#gò4QÑ4Ñï¼Y î| Ó4g4h7#òÙ ÷ ØÐù6â6h #4P#ògÑÑ/g#:Ó4ØAhÑÑgÒh#/4P#:F h`ØAhÑÑ\F

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., k O

44|hÑ7h#/4 Ì #g4#:#\4ðhÐéD_ªõh h ¿gÑg&Ô|h 4q4#¯Ü

Í§R¼ #444##×Í# ÍÑÑÐÔ6yª #g "ù?#Ô#g\4ÕP# k/ 4 Ö #:#g4hâ#\¼ô#RÓª|_òq4D#×Û4Ô/hô8ggñhÑ<ï/ g4×gªò#gòò64Ô#ÑCÔ##Ñ #h?g4Ñq4#RÓ?# Ú ?4ÑFhÛgÑñ_4#Ô6\¤Í©ggôg4 ÓhhÑ#g \î|# Ôª¼Yh¼ Ú ?4ÑFh×q##\4ChÐÍ©gÑÑ6h©#hg4Ñ<q4Ð#RÓ?#Ý # Ó4_ Ý#Ñ#ÑF_ïÓªgÑñ_4#gngÝ¯/#Ññh##Q#C FÛÔ6g# g4×gª¯#gg×h ¯/#Ññh##Q#gYh#46g:gòÍ§Y#RÓª|_ÈÈ64Ô#Ñ×Û n¼&#CÙ ÷ ØÐùp#¯ yÍÒF¼ô|_q#Ñî|ÑÑò#Cgá#ggÝh

q4#¯ÜóÍ§R¼p#444##y]7g#ghâÐ##&#Cg#Ñ

h¸ N P<Í§Y4Ô/hpQ6ggñhÑÐï/ g4ïgªÈ#gCÔ|_ 4#YgÑÑ/×g#:Ó4ØAhÑÑgÒh#/4CÍQ¯#h\Ñ¼C#Dhò#46g:gg÷ 4×gªò¯#ggÝhyq# hgªhÑf#ïÓ4Ô6h#gh¼h #h\ghg:¼?/44h#ª¼4âfh#yq4 É g_Ü¤Î#ÍÒÑ 4C#hqPëqh#âì Pmk/â5<?â 4j í â`4g#ghÑ<q##\4\ÓhhÑ|_4ô#h?g4ÑCëqh#A PZhâPGP í âAg4ªh\:Ü¤4###â7gÑg\4#×g4gg4Èg4È|#### Ï¤/q4YhÑÑï¼p"ðg4×gª¯#gQð#ÑñR¼4g

P×hÔ#ÑCôg4p

gÑñÜ¤#4gÛÓ_hÑ#Èhf×î|#È\Q#¯ðÍÒF¼P|_#ÈFh##h¯|h#4È# Ññ_g4 Ö #:#g4hâ7# g#hÈ|RÓ4×Û\Ó4FhÑ<#È#4g×Qh#gÈ#ÑñF¼ªg

V_È#

hâ#Hhâ#×\Ó4FhÑ2×Í#yÓhhÑ#D#ÈÒg4#Ý#Ë7òYF¼C×#?4ò _4òÑñ_g©#4#4hy/q4YhÑ6¯#h#ÑDg4# g§#Dq4ÑÑRÍ##hg4ÑÔ6\¤Í§g\Û¤Í©#ÑR¼4g

Ph

V

54PÍ§hªCg4PYÔ#

b Ì P#âK#ÈÍg

b4ôY#g\Ýh<|h6gh

#/Ò#ò|h6gQÔ6FKË7òÑ¯Î§#òÔ6Fy##×| Ñ¯Î84

P4Ó4gÒ#òÔ8RP

V

N ÏóP gg g×46gy#òg4ïgªÒ#ò4Ó4g

V#òÎh\¼P#ò| Ñ¯Î8

Ì #gCh.Í©QÔ|h#48g:gÈh§# y¯#g×Í#pFhpÔ8_:î| g#gÈ#?Üg4 4|hÑÑ¼Y4ÒgÑñ_Óª×g4#Ñ¯/.¼Ph#/4 Ì #8£ ¥ 1Â L/)C\##g|_Ýy¯#F_#Q#ÑñF¼ªg

PFh##h|h#4×#Ó_hÑ##|hg4ï Û_ÝyÑñ_g

hâHhâ?#Fh##h©¯|h#4Í|_©Ð# #Ô6FK7g#ghâ?Í#gC#Ô6FYÐ48g# âÍ©Î#RÍ'Í|_§<\ÓªFhÑ FhÑÑ¼ÝÍ§hÐ#¯#4g|_

Pg4 44|hÑÑï¼4 Ì #

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., kmh

1 ¥ 1Â L/)4|hhªggÒ|_Ý#F_# V Fh##hÒ4Ô/hC#ÓhhÑ# P g4 ò#Ñ

P gg gfÈ48gY#fg4×gªf#g

VFh##hÐÑ?4ÎÈ#Ô6F×#Ñ##

gggÓ4 #Îh\¼Y#| Ñ¯Îq4Pq¿h|_§§©#h§64Ô#Ñq4§#g

Ô6#Ñ_#g4#Ñ¼×6g:qg\Ñ¼Ô# #Ýh 6\q\\Ñ¼Y# ##;7ghâ/Ô¼6g:qg\§Í©Fh|_ÒhY#?Ô84# gggÓª\

V4\ É gÝ/q4Y_4YhÔ84/©#Ô# b â/g8g\ÓªgÑ¼Yh#Ô64# g4\

PFhQ|h#4ò# hÔ64/

bÍy#4Ô|hÔ#Ñï.¼ É g#

çÛhª¼ åKggªDg4ïgª¯#g×gÈhÎ#Íô&#ÝÑg_#Cëqh#âq5 N â O P#â NGO âO <?â N < í Í# hÔ|h 4 åKggªg4#Ñ¯/¤¼h#/4#h Óhh4#È#h4#hÒgg#¤¼4Ûù?4h©#\ FhÑ×#òÔ|h×q##\4|hÑ¤¼ðÔª¼Ô|h#QÈ4p#¼?FhÑ#?gggÑÎhh#Û#4:¼|h##gÑ4á|hª#g|h#ggâAÍ#Ñh#gÔ|hCÈ4 :Ô#/#ò/q4Y_4×Ô6¯.Í©ggPYhª¼×#ÑñF¼4g©g4##g\ Ô¼È#\¤Í§hÎ8fÏ¤#g4?Ü:#\4Ô¼éhªõh h ¿gÑg#g4ïgªò#gYÈy#h?g4ÑfÔ6\¤Í§gg¤Í§y#ÑñF¼4gÈhÑ#4#4&g4×g#ggÈYF¼&Ô8×#Ñggª hòP#h?g4Ñ`Ô8¯Ü¤Í§ggQ4D#ÑñF¼ªggÌ #g#gÝ##ÓªghÑÑ¼yg4×64hÔ#ÑÈg4×gª#gÍðg4#:h¯Ü|h#4Th\4Õ|_`FhÝÔ6©#:hñ_ Ýg#g76g:qg\Ñ¼# #4A8g:qg\Ñ¼Ô# #Ó4g4#gi / Ø ¨\L £CØÙC)Ûhòª #g E5k<4|__ggD|_Ègg#¤¼ôYhªh# \Ó4gÍ#g hô#?Ô84# ?#ÝÔ6gDh©g4#gòhÐ#×¯#gh×####g4#g#gªÑ¼ððh Ó4ghñhÑÑ¼Qg4ª4ÑÑ ÍÒF¼4 Ì #òDDYÓ4g:¼ð:4#P#h4ðhgg#¤¼4 Ö #:#g4hâ/§#Ñ\©#4?Ü¤YhÑÑFhÔ#Ñï¤¼Yh gg#¤¼Phªh#:Ò h/Ó4D ÜÓ4ghggPéhªõh h ¿gÑgÔ|h #×g4#:#\4p4py#\Í#ïïÓªÝÍ##\¼QFhÑÑq¨¦ N1 ¥ ¨¦¨¦C)¨B ; L/¨Bh Í#QÍÑÑ2Ô8 \î|# Ûù?#Ô#\\4^P# k/654Ú h##©#Ñggª_4È4È#§Ù ÷ ØÐùÈÍ©©FhÝ4Ô/hï/ g4×gª7#gÍy¯#h\Ñ¼C#òhD#46g:ggÌ Dª#Ó4gÑ¼ò¯/#Ññh#ÒYh Fh`h N P/Í©§##Îòh8#§#Ñg:A¤¼?6Òh8g4:Ügª#ggÍ#\ÈÔ8hQg4#×h 48g##Yhò#4?Ü¤ªgh\Ó4h Ì g4ïD×g_4

m#§g4ïg7##`hhÑ44#*g4

K

Í#gK g#hg<##Ô#Ñ

Îh\¼Ch Í#Y|h©h©##/<#gh4m44\#g<ÍCò##q4Ñ¼Ýh 4:#

r 7g#ghâ<#Yg4#/g

c =g4×

K(m, r)h g

cÛ#YgggÓ4gg Ì &46\

#Qg4ï\Fâf#Qg4ïgg mh

rð#QgggÓ4g×Í#ôÓªgî|g×|_

c =g4

K(m, r)q7g# #ÈFh#Ð|_§hÓªg¸¾:#

c#gggÓ4g g§#h§ÑFh

mh Ô# #&Fh#Ý|_Ý#Cg4gF_##h×|h#4P#Ý Ô¼ g4#/#

m′, r′Í#g

c =g4ï

K(m′, r′)h

m′ 6= m

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., kGk

Ï¤Ct) ¥ ¨¦¨¦C)¨8 - L/¨B#gÐÈ#gghh ?4Ð/q4Y4 tK Í#h?gñ_ ×Fh¯Q##Ô#ÑÎ4\¼

K Ì #gâ|7Î#RÍâ6hÑÑRÍ©#òg4gÐ|h#4

#© n ÔªÓ?4#Ñ¼×q4)#¯/:g#ghÕ#Ch 4©/q4Y_4Ðq4ÑÑRÍ<|_y#&#gò#òÔ# ##46g:¤¼PFh##hÔ6ò_:î| ##g4 4|hÑÑï¼4`Ï¤Q4:h ?h#gg\Ó4gÛ64Ô#Ñg4#/q4

tKg4×g©|_FhQÔ6

46g# Ûh¼PÍÒF¼ g Ì #6ggñhÑ2.¼?8ÝhAh ?4¯#ggFhÑÑ IH ¢ Ú £Øý\Ö #:#g4hâ?ÐÐ64Ô#Ñg4#:#\Òg4×gªÐ¯#gg©Í#g åKggª©¤¼?6hfh ?4/qh_4

t′K¯/:gâC|_#ÝÎ#RÍÑ 4Ýh

t′K4|hhªgg|_Í§

Fhp/ªggªÑ¼ôg4#/mq4 g4

(m, r) Ì #È#46g:¤¼ô _gÑï¼&#Ñg

|_y#k¯#ggyFh##hyÔ6ð##g4 4|hÑÑï¼# ## Ì #C.¼?8ôhg4ïgª#gÈÒFhÑÑ ONG) Z £ØýªÏ¤C##gh ÷ h#\h Ö #ÑY^5k6## ##46g:¤¼Ch Ô# ##4?Üg:¤¼ð_Ô6hÛq#Ñî|ÑÑ 4#Ñ¼Qg4#/_4|hÑÑï¼4 Ì #Í#C#gg4#:#\4&Í#`gg`|_f#gÒA#g |_fA#Ñ_#g4#Ñ¼È¯?h\hÔ#Ñ§h gá#ÓªÜ?FhÔ#ÑÔ6D##Ó4ghÑÑ¼Yg464hÔ#Ñh`hÑF ¼Cgª4# b N PÕÝ#\Íjg##á#q4#Cg4#:#\4 h##Ó4ghÑÑ¼pg484hÔ#ÑCg4ïgªÔ|h 4(&_Ü¤FhÑÑ ï/ g4×g#g×#464 Ú hFhÑÑ¼ÛPï/ g4×gªD#g×&g4×gªÝ#gPÍ#(h(4Ch#CÎ4\¼?×Ý8g:qg\Ñ¼ # #ðh gá#ÓªÜFhÔ#Ñhâ6#g×Îh\¼?òhÝFhÑÑ ÙAÜ¤Îh\¼?gâÕh 4ð4Ýhf#ÝÎh\¼?ò6g:qg\Ñ¼ÛÔ# #Ch ¯?h\hÔ#Ñhâª#gÎh\¼?§hFhÑÑ Ü¤Î4\¼?g¿hÒ|_<4ÔÓ?4#Ñ¼hâ?DÎ4\¼×Fh##hÐÔ8Ô8h Üòh ÙAÜ¤Î4\¼4 Ì #PÔ|hYg4#:#\4 hÒ#Èg##á#PÔª¼ éhªõh h ¿gÑg#g#g8g:qg\Ñ¼&Ô# #Y#46g:qg\Ñ¼Û# ##¸7RÍ©\Ó4ggâÕ64Ô#ÑÈP n¼#gAÔ|h©¯#g©4Ô/hg4ïgªA#g§|_`FhÔ8§#:hñ_ Ýg#g6g:qg\Ñ¼PÔ# #4Ò6g:qg\Ñ¼P# ##

GYE ?åK Y¨ ò£Æ£²åKáT£²¨ªßá©Ì #òq4YhÑ \î|#4PhAï/ g4×gªÒ¯#gghÓªgyA N P2hÒq4ÑÑRÍgHJ0mÔ;CAR-ÖýfÖ)LÖ Ú ¼¦¨¦ N1 ¥p ¨¦¨¦C)¨B L/¨BÍ©×FhYòg4ï×gf#gg4

K

ÍY44ÑhÔ|hÑ|:¼?:gÎh\¼NâÍ# \g#gÐ#gh4|hg

MNh #×Î4\¼ô|hgKN

h©#g4gªg Ì #Îh\¼|hgg4h#D¤Í©Û\gâ7#ÙAÜ¤Îh\¼?h # Ü¤Îh\¼?gâ|q4§Í#¯P#Dq4ÑÑRÍ#Ý#4Ñ g

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., k j

Û 0m,E?%010ZIKÞn #CFh(/ªggªÑ¼p4g#g_y&:¼?:g Îh\¼

NhÑ4#QÍ #C_Ü¤FhÑÑ Üóh 4

tN n #Fhâ?4Ó4g×#:¼?:gßÎh\¼

Nâ?/ªggÑ¼4g#g_h 4)g4×gfÎ4\¼?

h h 4 Ü¤Îh\¼?g ì Ó4gQ#È:¼?:gÎ4\¼hâ6Í©ÈFhÛ/ªggÑ¼P4g#g_ÝhÛÙAÜ¤Îh\¼KhÑ4#ÝÍP#ò_Ü¤FhÑÑ ÙAÜóh ?4

tK

Û 0m,)()./-?).V'-IQw()W ,qÞöh 4 ÙAÜ¤Î4\¼?Ah h 4 Ü¤Î4\¼?7h<Ô6_òg4#/_4|hÑÑï¼ :#4#|hÔ#ÑAq4h 4 Îh\¼?hÑ4#×hÒ# Üóh 4ÒÒ#hÎ#ÍD5C`GIQw()W) ,qÞì Ó4gQhyÙAÜ¤Îh\¼

Kh hyÙAÜóh 4

tKÍ©DFhy4g#g_ThÎ4Dgh×g

câ .Ü

Ô#/ ¯#h\Ñ¼phFhÑÐ\4×ggâÕÍ# Fh Ññ_gòÔ6Y46g# hÔ#hÑï¼4âÕHhâ4Ó4gÈ§gh4

mÍ§fFhòg4#/f##q4Ñï¼h 4

rq4KÍ#¯

c =g4ï

K(m, r)

D\yILÞì Ó4g×Dg4ïgª

c =g4

K(m, r)â4Í#g

Kfh Ü¤Îh\¼4â4Í§ÒFhâª4Ó4g# Ü

h ?htNâ#/ªggªÑ¼Pg4#/

mâ#Í#g

m§##á#gÑ¼ \g# Ôª¼C#ò8g:qg\

Ô# ##

¿hò|_q4Îh\¼Q :#4#|hÔ#Ñï¤¼Pq4ÑÑRÍÒ|_hÑ4#Yh# Üó_ ?4#h`Î#RÍÝ#©¯#g§Ag4#/_4|hÑÑï¼ò# #q4AhÑÑÎh\¼?g Ö #:#g4hâhA#Ñg|_òhÑ4#Ph#g#g# Üóh 4#4#ÙAÜóh ?hÎ#RÍð##gg4#/_4|hÑÑï¼YÔ# #q4hÑÑKÎh\¼?gÖ 4ò#Cg4#:#\4 Ô|h 4p#C FQhéhªõh h ¿gÑg Qq\Í8ggñhÑ©¯Üá#ggª4y#ò¹/ g4×gÒ#gÈhò#g §L Ø(¨¦ N1 ¥^ ¨¦¨¦C)¨B Ë L/¨B×DPï/ g4gª¯#gÝÍð#Ýq4ÑïÜÑRÍ#q#:#g#46g:gg Ì #Ègh4ò|hg

MN

h #òÎ4\¼y|hgKN

hDî|#44##ÈðÍ#pòò64Ô#ÑÝQg4#//ªggÑ¼4 Ö #:#g4hâ2#_yhÒÙAÜÎh\¼?ÐRÓ4gf#§hhÑ/?#ÝÔ6g`hKÎh\¼?Ð`#g4Ñ4Ô#Ñ§h #Ò_h Ü¤Î4\¼?<RÓ4gf#§hhÑ#Ô8gh<Îh\¼?ò#g4ÑhÔ#ÑÈgÑ4^54 Ì ##Ñg|_#gÝ4#Ñ¼ÛY#g4Ñ4Ô#Ñqh\4PhAÎ4\¼?Í#QÒ#g#gÙAÜ¤Îh\¼y#4 Ü¤Îh\¼4Ñh:gá#g\##gÝ#4#Ñ Ô8Ýh<Y|h:g#ÑñhÒq4Q Ì #g## Ô¼h#:q4Y_4 Ì #\hhÈ|hghA#È_Ü¤4Ô/h# h#:q4 ¯#gÈÒ#hhâfÔ#/#CÎh\¼?hC#Íh#Yq4

(K1, K2) Ì #Pg4g64 #ôÙAÜ¤Î4\¼?×h

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., j :

|hÈhÙAÜ¤Îh\¼?×h #Yg4g64 # Ü¤Îh\¼?hY|hh Ü¤Îh\¼?g&ÝÔ8\q4Y#g4#:#\4&ÑFFÓ4g4#Ñ¼Q#g4Ñ4Ô#ÑDqh\4ÛhfÎh\¼?Í#&#g#gÙAÜ¤Îh\¼Û#4 ÜÎh\¼4 Ì g4ÒY4Ó4gÛgh4

mÍ©Èg4#/Cëqg4

K1(m1)

â g4K2

(m2))Í#gm1

h m2

hÒ##q4Ñï¼Èh 4 ÓhhÑ#g`q4AÍ#¯m = m1 +m2

`ÏóKÔ8h×Î4\¼?h Ü¤Îh\¼?gâ/#g

m1

h m2

h #mFhPÔ6g4#/ Ôª¼Y¯?h\4¿h|_

hÑÑ7h#g#46g:gDhÐC6ggñhÑAï/ g4ïgª#g×hhÑYq#Ñî|ÑÑ # g#§h#:q4Y_4

GYc~ b¤a¨ªó¾å¢Ö¦$?åK Y¨ ò£Æ£²åKáT£²¨ªßáb6óTâ¨ £²¨ª©i¡ ¢Ö©Q¨ òÖßq ¢ ß¨ã

£ ¢C¥¤`ò£²ò£²òFé¤§âå©£²©Ï¤Ï N P#ô#h4 hONG) Z £Øý q Ú Á $L¨B¨BÊL1ý/ŸðÍ§hy #g ßÏó¯?g Y# \î|#4"h

q Ú Á ^L¨B¨BÊL1ý/¨ Â / §Qq4 ÷ hgh

éhªõh 4N HPü2\Gh

HÔ8î|#hÔ6gÑñhô44##h Ñ\

H/f(G)Ô8Cy\¼?gÑ

44#ðÍô4#Ñ¼&Ññh4Ý#Th\4DôD4 gg Ì #× FPyg4# gÈhôFhÑ¼g4#/hÔ#ÑÈ#h44##

f : G → H Ö #:#g4hâ|Í#4/×h 4h 4

gÑggªÐhf(G)

#4#Ñ Ô6g4#/_4|hÑÑ¼Ý :#4#|hÔ#Ñ§q4\ÑggÐ#4gh 4Ñ¼pq4 hÑÑ§RÓªg

HE¿hP|_×##ÑgÝ|_

f×|h ôªÓªg:#

h ?hòDÎ#RÍ ì Óªgô# Ph ?4òh?gñ_ ÍfD#4#Ñ Ô6×Fh:¼ð

gg hÔ64/#Ý:_#hÐhðgÑggHòçQ4q4YhÑÑï¼4â8¯?h\hÔ#Ñ

q4#¯ÜóÍÒF¼

#444##Òh \î|# h§q4ÑÑRÍ¸ N PH ThÑï¼Yh7¯?h\hÔ#Ñ

q4#¯ÜóÍÒF¼C#444##×§Ò4Ó4gyÔª¼yÝ#hÔ|hÔ#Ñ:84Ñ¼ªÜ

#4ñhÑ/Ò4g#g_4GÍ#¯Y4##/

1σÍ#g

σ g#hgÐ#gg#¤¼|hh\ggâ

4/#/§ gg/4Yh2h8kÜó##Ñ(G,H, f, g, q, b, b′, t)

âÍ#gGh

Hh44##gâ

f : G → Hòh/ªggªÑ¼ðg4#/hÔ#Ñ#444##Qâ

g ∈ H\f(G), q, b, b′ ∈ N,h t`:#FhÑÑ #Ðh 4g`ü2\

F := f(G)AÏóÀgá# |_

gF4g#g_g

#DTh\4§44#H/F

Ö #:#g4hâ/4 (g) = |H/F |

#4#Ñ Ô6D#6g6hÑ¼?#4ñhÑK#gg#¤¼C|__\g

σâ/h#

2σâ/4

(g)|qh

bÒ##Ô#ÑÑÍ©g©Ô64# 4C4 ë

gí â

Hhâ2 ≤ b ≤

4 (g) ≤ q

Ð*4g#g_4Òh ×|FÓªc ¢ £ØÙ ¥ /Ý b =4

(g) = q

Ñb′#6\84Ñ¼?#4ñhÑ<

σëqh#

2σ/2 í h 4 gÝÛ##Ô#ÑYÑRÍ©gÔ64# 4#Y#×Th\4Èp4

(g)âAHhâ`hÑÑf#×Th\4Èp4 ë

gí hC_ÑFh:

b′yÏ¤

g4ªh:ò$ N PfÍ©×Í46g_4#DGh

H # Ó4gÑ¼4 Ö |hÑÑ¼hâ2òDgá#

|_ÒCÔ8hP44##Gh

HÍ§Fhy # â#Ý#Ñ#Ñï¼×Ô¼ChPªg4ggâ?hÎhÓ4ggh

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., j 5

h#Ñh 4ÞgÑggª&#4Ô|hÔ#Ñ:È64Ñ¼?#4hÑ24Ó4g(G,H, f, g, q, b, b′)

Ì #òî||hÑgghÑ2gá#ggªhòhÒq4ÑÑRÍg )./)-?).V'-IQw()W ,qÞ öh 4gÑggª7q4

Fh ¨\ ¢ ) ØCØÙF ¥ ý/) 1Â¢ ý Ú

£ØýDq4 h 4¿gÑggªÒq4H4Óªg

(G,H, f, g, q, b, b′)

D\yILIQw()W) ,qÞ *4g#g_4§§FhÑÑ ¡¢ ØCØÙ8ONG) Z £ØýDÕ4Óªg (G,H, f, g, q, b, b′, t)h y = ig + f(r)

Í§×Fhg4#/i mod

4 (g)

ô#4Ô|hÔ#Ñ:Ý64Ñ¼?#4ñ_Ñ7h¿hD|_4Ó4gy#YkÜó##Ñ

(G,H, f, g, q, b, b′, t)ÒFh:¼Cgh#/ò4 ë

gí

q¬)`0Zw(W) ,qÞ ì Ó4g

(G,H, f, g, q, b, b′)h

y ∈ HâfÝFh:¼pðg4#/

x

|_qy = f(x)

Ë¿h|_Ò# ?g#hg4ª \ :#4#|hÔ#Ñï¤¼6`#gq

Ý#Ñ#ÑhA4 (g)â|ÒhÑÍ§R¼?§#òFhò|_

qy ∈ F

ã#×#Ûhy¯?h\hÔ#Ñq4#¯ÜóÍÒF¼C#444##éhªõh h ¿gÑgA N P ¯Ü

gÔ6 Í§R¼òh#:q4*Aª8ggñhÑ?ï/ g4gª7#gh Ì #©gh4|hgh6#g4gªf¯#gÒÍÑÑ#Ô6

Z bZ h #Îh\¼×|hg K ÍÒÑÑ#Ô6Ò#hh#HB7g#gÔ6hphî|#×44##òôÍ#ôÍ§F_pg4#/×/ªggªÑ¼4 Ì Ûg4ï

Qygh4m ∈ Z bZ 4Ó4gpPÎh\¼

K ∈ K#?h

r ∈ Hh 4Ñ¼&h g4#/

#g4×ghc = commitK(m, r) := mK + f(r)

Ì #ÙAÜ¤Îh\¼?ÍÑÑ2Ô6È#Ý\F = f(G)

Ípg4g64 #&ÙAÜóh 4tK = f−1(K)

yé#Q#Ygá#ggª|_f4

(g)`#6g84Ñ¼?#4ñhÑ?Ý#§gg#.¼Ý|hh×\g

σÍ§§q#Ñî|ÑÑª#Ògá#ggª

|_##Ô8ghÐÙAÜ¤Îh\¼?#g4Ñ4Ô#ÑòÓ4g#hhÑÕ#Ô8gh<Îh\¼?g Ì # Ü¤Îh\¼?ÍÑÑÔ6ÈgÑggªhA#òq4

K = ig + f(rK)Í#g

iªÓ4g:Ô#ÑD

Z ø/íÊô (g)ZÍ

Üóh 4t Ú ¼p#ygá#ggª|Y_ÑÑ§#YTh\4h4

(g)hPÑñh4hâf#

_ðh Ü¤Îh\¼?RÓªg#PhhÑÒ#ÝÔ6g×hÎh\¼?Y#g4Ñ4Ô#ÑCgÑ4PÕ5Ûh×gá# q4Y8ggñhÑ7ï/ g4×g#gh Ö #:#g4hâ8# Ü¤Îh\¼?Fh&Ô6h#Ñ /ªggªÑ¼P4Ó4g

t#gò4

(g)ÒÎ#RÍò

úD\¼Ý :#4#|hÔ#Ñï¤¼òq4ÑÑRÍ g\Ñ¼Èq4#§gá#ggªf|_fh 4 gÑggªf(G)

h h 4 gÑggªÝ#4g q4hÑÑ<hHhYg4#/_4|hÑÑï¼ð :?Ü

4#|hÔ#Ñh Ì ð#Í¿gá#Ó4?FhÔ#Ñ¤¼ôÍ©yh#C|_Ý#yÙAÜ¤Îh\¼K = f(rK)

h ÙAÜh ?h

tK = rK

©4Ó4gfkThÎhg4ïgªÐ©4g#g_ hc = f(rc)

q4rc ∈ H##q4Ñï¼Ýh 4QA#|_<Í©h4Ó4g

m ∈ Z bZ fü2\ r := −mrK + rc

Ì #gr<##q4Ñï¼h 4ßh

c = f(rc) = f(mrK + r) = mK+ f(r) =g4

K(m, r)

Ì #P#RÓªgC|tK = rK

4ÓªgCgá#ÓªFhÔ#Ñ.¼hâH_âT_Î4ðg4×gªY|_yh :Ô#/ ¯#h\Ñ¼ChÒFhÑKg4×gªÒh FhyÑñ_g©Ô646g# hÔ#hÑï¼FhPÔ6

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., jm4

4g#g_ Ì #RÍ ¯?h\hÔ#Ñ¤¼Í§Ðh#f|_KÕh Ü¤Îh\¼4âhThâ

K = ig+f(rK)ÍiªÓ4g:Ô#ÑÐ

Z ø/íÊô (g)Zâ4h g4×gª

c = mK+f(r) = img+f(mrK +r)q4m ∈ Z bZ 4Ó4g&ã#Qq#ÑÑ¼ô¯?h\hÔ#Ñ¤¼4âAThâA##Û# Üóh ?4

tÍ§

Fhkg4#/im mod

h (g)

q4ch q#:#g4hâ

i mod4

(g)FhkÔ6ôg4Ü

#/ q4Kù?#g

iªÓªg:Ô#ÑDÍ©FhÛ#g&g4#/

m mod4

(g) = m Ì #

h#:q4 #gC|hÝÎh\¼?(K1, K2)

Í#(hY|hÈhÎ4\¼?×h Í©Cg4hg4

K1,K2(m, (r1, r2,m1)) = (m1K1 + f(r1),m2K2 + f(r2))

ÍÒ#gr1, r2

hÝh?Ü 4gÑggª

Hh

m2 = m −m1 mod q\7g#\hÑÑÕgá#ggªq46ggñhÑ

ï/ g4×gªÒ#ghq#Ñî|ÑÑ Ö #:#g4hâ2éhªõh h ¿gÑg \Ó4gÑ4ô/ªggªÈ#hg4Ñq4#Ó?#P É \ÜÎ#RÍÑ 4ðgÑñ_4#Ch4# g4ï ÓhhÑ#gCq4Y#&6ggñhÑï/ g4ïgª#gh ù?g] N P©q4 \hÑhÒ#gP#h?g4ÑÝh q4Èq#:#gÝ#46g:gh ##?hAhAgg#¤¼Ph7#Dg4×gÒ#gh

GYx ¦uâ¨`§ ñÍ¢Ö© ¢Öß]¡ Y¢£¥¤§¦¨¨,òFé ¢q ¢ ß¨ã £ ¢4J¤`ò£²ò£²òFé ã

¤§âå©£

ûpÈFh&#RÍ*#Í |_Ñ4ªÑ¼C î| Ó4g4Ûh`#Ù ÷ ØÐù ëqgù?g¯4 P# 4 í h ¯#h#ÑYh

q4#¯ÜóÍ§F¼ #44h##Q^7g#gPÝFh Ô6C# ðg4#:#\

6ggñhÑ2ï/ g4×gÒ#gÈh ggÔ6 hÔ6RÓªhÌ \Ó4gÑ4ô#×g4#:#\4ôhh|hÑ444#Yh##gòhòE N PfÍ©×|RÓ4y n¼#Ûh 4 g4Ô#/4"h#ÛÙ ÷ ØÐù"pÔ##Ñ

q4#¯ÜóÍÒF¼"#444## h

#g#gÈg4#:#\§D6ggñhÑ|ï/ g4×gf#ghfü2\NÔ8hYöù? #Ñ#

ëTh Ñ\s = 1

2Í§Dg4# g§#D4g#ghÑ Ù ÷ ØÐù í Ö #:#g4hâ/Ñ\P1

h MÔ6

#Yh##4#ñ_ÓhhÑ#gÈg4#/ ô#YÎh\¼4g#g_4#|hYh§#YÙ ÷ ØÐùôq4ÈhgÑÑ/òg#:Ó4Í

(a, b)h|hh\ggûp|FÓ4|_

E(Z N2Z) ' G1 × G2

Í#gG1 ' Z NZ,

?#|G1| = N

h G2

§Ý44#PhÕ4 gMÍ#P§|h Ýg4#/

##Ñg§#Th\4_4YhN©Î#Íâ#gp P N H Ú \q4 \î|##Ý##444##

fÍ©Òg4# gA#§Yh

Φ : (m,R) 7→ mP1+NRÍ#g

m ∈ Z NZh

Rfh 4

64ª4E(Z N2Z)

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., jGN

01#$IQÕR-ÖýfÖý Ö

54ú¿hÈ|_Í©Fh##h g\Ñ¼y#?4ÝYh 4gÑggªE(Z N2Z)

Í#4/#CÎ#RÍÑ 4ChÒ#Th\4h

N Ì ð4ÑÓ4#ÈÍ©Yî/ Û?#ÝÔ6gh84

Q1, ..., Qt

_<Îh\¼×4g#g_4hâªHhâ4Í#g##©Th\4<hNhÎ#RÍâ

h #gÛÑ\R = r1Q1 + · · · + rtQt

Ú ¼P###th #È É òhA##ÝÔ6g

ri

q41 ≤ i ≤ t

#ÍÑÑ4g#g_h&h##4#ñ_Èh##F/Y_4PY##q4#4gDq4

E(Z N2Z)ÖS

4 ú¿hP|_#yh 4 g4ªÔ#/4 #×Yh( \î|# åKgg|h"#y44|hÑ§Ù ÷ ØÐùjëqgÛù?g\4EP# 4 í CÏ¤#:F h \î|##ôh×ðî// 64ªQ#Ñ#Ñ Ô¼Qh 4#ÝÔ6ggâ8Í©ÈhÎhÝh 464ª

R4Û#ÈgÑÑ/

g#:Ó4E(Z N2Z)

Ý#Ñ#Ñ Ôª¼NÍ#4Óªg7#`h 4*gÑggªAÈ#Ð44#

G2

n ÔªÓ?4#Ñ¼Yq4§#òYh Φ#4Ñ §|_

kΦ(m1, R1) + Φ(m2, R2) = (km1 +m2)P1 +N(kR1 +R2)

= Φ(km1 +m2, kR1 +R2).

¿hÒ|_Ð4Ó4gYò64ªmP1 +NR ∈ E(Z N2Z)

Ð<64Ô#ÑÒòg4#/m ∈ Z NZ/ªggªÑ¼

Mëq4Ygá#ÓhhÑgªÑ¼(#QTh\4_4"h

Nâg P N í Î#RÍâ#g

Í©ðFh'Ý#Ñ#Ñï¼(Ôª¼M

4Ô/hMmP1

h #g'#?gg hC# gg:¼?/4#|h ggÔ6 ù?g\4pP# 4 Ö #:#g4hâ48#\ÑÑ/§g#:Ó4ØAhÑÑg`h#/4#4Ñ gÑggªhA#Èq4

Φ(0, R)hÈ :#4#|hÔ#ÑDq4 gÑggh`#òq4

Φ(m,R)âAÍ#g

RÛh 4 6h

E(Z N2Z)h

mÝhª¼ôî// gÑggªÝ

(Z NZ)∗

Ú h 4Q#4Ô#g:Ó__4QÍ§ÈFh&#RÍ#RÍ*|_gÑñ_Ó4D#ÈgÑÑ/òg#:Ó4ÝØ`hÑïÜÑgh#/4y# î| Ó4g4&hA#ÝÙ ÷ ØÐùÛÑF Y×q#ÑÑ¼P¯?h\hÔ#ÑÈ4g?Üg_4CÍï'##Ô#Ñ&4 gg ü2\

NÔ6ôhköù? × #Ñ#YÍ #×ÛTh\4PhÈÔ#

Ñg#hσ/2

ü2\×Ô6h(44##Gh

HÔ6

E(Z N2Z)h Ñ\

f(R) = NRÍ#¯

h(gÑg×gF = G2

h 4ÔªÓ?4#Ñ¼p/ªggªÑ¼ g4#/hÔ#Ñhü2\g = P1

âÐh UEÈCðúÿæí)ñ îiûãåçiæíóé4VèïñÊõëT©õî ðëóä)ë:ªëò÷ìæî ðôëòíªõëòø/í ëò÷ î ñùø/ç ç øê-ägî ÷ï÷ïëóôîiæñ ëòç èìñ õæñ

t = 5îiägëòðø/åö/õñ øìö/ëãðëòíÊæñ ë;íÊæðôø/÷ ÿGø/î ð/ñÊäòü¬«-øêëE/ëòíóéî ñûòæðcæçiä)øêGëqä)õøêðúñ õæñ

t = 4îiäëòðø/åö/õZü

Q"% T ' ³²-+K)³¯¬]§¯^¯t³ x.¬FL²E ,., j P

Ñ\Ý#Ph 4tÔ8P#P44#(4 g

M4gá#Ó_hÑgªÑï¼p#P#YTh\4h

N

ûp \î|#b = q = N

(4Ô/h 4g#g_4CÍ ##Ô#Ñð4 gg!ûpô|FÓ4ð|_4

(g) = N = |E(Z N2Z)/G2|^7g#gC4 ë

gí hÈÍ§gÑÑ©_

b′ = 2σ/2−1#6g84Ñ¼ªÜ

#4ñhÑ<σ n ÔÓ?4#Ñ¼p4Óª\ (G,H, f, g, q, b, b′)

Í§PFh( # 64ªÝ4E(Z N2Z)ëqgQù?g\4 P#65 í â`Ý#Ñ#Ñï¼ô#gÔª¼ ªg4ggâ`hÎhPªÓ4ggh64E(Z N2Z)&#4Ô|hÔ#ÑÈ64Ñ¼?#4ñ_ÑÕ_D # 4|hÑÑ¼hâ6Í©Fhðh#Ñh 4ÞgÑggª

E(Z N2Z)hÐ#h hÔ6RÓªh Ö #:#g4hâª#gÑÑ/g#:Ó4ØAhÑÑg<h#/44|h.Ü

hªgg #h#|hÔ#Ñï¤¼4Ïó§Í©Yh×4Ó4g(G,H, f, g, q, b, b′)

h # 4|hÑÑï¼Û#h ?h

t = MÍ§Fhôg4#/

i mod N4Óªg

ig + f(R) = iP1 + NR##P#

gg:¼?/4ChÑ44ï#yâªHhâÝ#Ñ#Ñ¼?#DÔ¼M Ì #Ð#RÍ<|_<#4g#g_4<fq#ÑÑ¼

¯?h\hÔ#Ñh Ì #Ññh:©#48g:.¼×|_©|hÐÔ8Óªgî| qÜ¤ªÓªg:Ô#Ñï¤¼ÝÍ#Yq4ÑÑRÍ

_gÑï¼qh¿# \î|#47g#ghâ##D#Òg#Ñ<héhªõ_ h ¿gÑg N P8h ggÔ8 ù?#Ô#g\4tP# k/ 4Í©È4Ô/hC#Dq4ÑÑRÍ#Ý#g4gQT+'-0Z%01#R-ÖýfÖý=Ö5 ¥ CL Û RÜ CcýYL/)£ØýB§A£ ¢ CØ ¥A §L Ø¨¦ N1 ¥A ¨ Ú¨¦C)¨B t L/¨B$ÁqCC×¨B Â b§ Z Z® NZ

ÁqCCE ¡ Õ ¡ /Ø ) E ¡ CL ¡ ¨m = f(m1,m2, . . . ,ml)

Á(L/fý Øý ¥ § ¥ l = O(log(k)) ¥ ¡ ¡Ym¥m¥ C) ¥ ¨ ¢ ØÙ) ØÙ )¸/Ø ) ¸¨B ¥¢ Øý N M

|~ ¯

±J |Ï ° ~ ± t

Ì #`#g FhÑfÍÒ×#444##©g:¼?/4:¼?:FÐh #gfh##ÑF_h#gAÏ¤ ÷ |h?Üg 4 Í©Q|RÓ4Q/ªggªÑ¼(g4#:#\ h_Ñ4gÔ#hFhÑÑï¼ #444##Pg:¼?/4:¼?:g4Ó4gyÝ#444##g:¼?/4:¼?g 4y6ggñhÑK#4?Ü.hÔ6gÑñhY44#<ég4##Ý#hhÑ4gÔ#hFhÑÑï¼ #444##Pg:¼?/4:¼?:F |hYÔ8gg h46g#4Ô#Ñg q44|h 4 :ò¼4Fh©#gò64Ó4g#Ñ©ÑF <È/ªgg§h #Ñ4Ñ/4#f\Ó4ghÑg:¼?/44h##C#h?g4ÑgðûpCÍ©gyhÔ#ÑYð|h:ñhÑÑï¼4ÑÓ4Y#ÝÑ4#&:h #ð46g#4Ô#Ñg Ôª¼ #g#&ÈQ#YFh q4È6ggñhÑÐ/ªggªÝ#444##×g:¼?/4:¼?.ÜggÌ #Ð4Ñ/4#2#gg ÷ |h/g N Í©gÐÔ|h 4ò#Ð FÒh#g4# g#Ô#_#¯###44hf#:F hKgg#<hÐDg4#/_4|hÑ/ gÑ#q4`q##\4#g Ú ¼#¯Fh#Í©CÍ©gyhÔ#Ñð#RÓ? C#4?Ü¤ªgh\Ó4Ch #RÓhhÔ#Ñ¼gg#y#h?g4Ñ&g#g:¼?/q##\4#§4ÓªgPÔª¼C64Ñ¼?#4hÑ6Ô#h####44h©P#ÛÍ§F¼Y|_§#\¼PhD:ÑÑ¯/gg/hÔ#ÑhË7g#ghâ/Í© ggÔ6 #\Íjh##ªhCg#Ññh4#gÑñh©hÕg#g:¼?/hÔ#Ñq##\4# 84Ñ¼?#4ñhÑÒÔ#h### #44h×Í#'|h:g#Ññh×6RÍ§g:q#Ñ#gQÜP#×Ô8g:hÐ4#òÎ#RÍÑ 4ÝÜÒ#\×h×#yg4#g\q##\4#òÎ#RÍô|_òFh##hÔ6&g#ggª Ô¼64Ñ¼?#4hÑÔ#h### #44hg Ö /#&gFh'ÍÑÑ#RÍÞ#gP¯/:4Ñ/4#ÈÔ|h 4 #444##Y#ggÝ|_×hC g8g g×hÒ# É YhÒ#C# gÑ¼?#Qq##\4 Ö |hÑÑï¼4â7#Yá#g:4(hgÈÍ#\#g×hÑ4gÔ#hFhÑÑï¼#444##Èg:¼?/4:¼?:gDh:ÑÑ2#gggh:¼Qq48ggî|h##ÑF_4#4Í#\#g#444##g:¼?/4:¼?:FhDKªggF

¯¬² ' ±ó³¯¬T+¬°K²Ö=¬m³F± ,V)³¯¬T jGO

Ï¤ ÷ |_/\BPÍ§Qg4#:#\ ô#g#4Ñ gg:¼?/4Ó4g4h#ÛgÑÑ/yg#:Ó4ØAhÑÑgf#gh Ì #<Óª\hCÐg6gghÑÑ¼# q4ÐYhª¼×h##ÑF_4#f#gÓhh4#6RÍ©g:q#Ñ§h?/Ññh:¼ð#h?g4ÑÈFh Ô8CÔ##ÑÔ|h 4 F n ##h?g4Ñh#Yî|:gÑÑ/Ðg#:Ó4§Ó4g4#`h8#AÎ AÏó``ªgg:###g§¯/:`q#:#g`gÑÑ/Ðg#:Ó4Ó4g4#Ýh§#CØAhÑÑgD¯#g|_hYh6RÍ©g:q#Ñ©h4##ghâ`HhP|_ÑF Ý#Dh#hg4Ñg Ö #:#g4hâ/4#Dg4#Ñ _|hÑ¼?D4#§g#Ñ© \g##6ggî|È#48g:ggá# h7#ò# gÑ¼?#×g:¼?/4:¼?g×4Ô/hQhÑÑgª4# h?/Ññh.¼Y#h?g4Ñg

´ Eµ¹±¥gÏ |~ÐÑ

5ÝçÔ| Hâ/à/ Ö g4gÔ|h#Qâ Ü /¢ C /¢ C Û Ø ¢Q ) Ú¤ R § Øg£ ¥ Ú ¥Ú 1Â ¡ ¨ ) c¡ ¨ q¶ Z Øý\â?àh4#|hÑ|h ÷ :¼?/hÑ4h¼4âªä<4ÑHQ54â1¿# 4 â%5 jGj :/âª#5\·%5 4

4 ÈF gÑÔ|hâ4ù6úò_ É gÔ6gggâhù/ g4#Hâ § Â L1I¸A/Cº¹ §/¨ À 1Â J» /§ ) 1Â ¹ §/¨ À¤ÁqCCy¸^ C¨ ØLT¼;/2½ ÁìØý ¥Â » ý Øý ¢ \âÏ.ÝØf?gg Ü#4Õh/#©Ùf#48Fhù?4|hÑØf?gg# ÷ 4/qgg#g 4 :G: 4 â Ì 4#Ñ4#hâ Ö h#ghâ 4 :G: 4

N Ö Ú h#â ØÙ/ýY ¡c R * £Øý Ü /¢ \¥m¥ C) ¥ ¸ ¢ ØÙ) ØÙ ) R Z Ú ¨B¨BÊL1ý/¨âûp4Î#4Û4 ÷ #h &÷ :¼?/44h#ª¼4â 4 :G: N â/ä<ghÑÑg

PéF Ú h#h4â ¢1¥ ¥Ú ÁqCC R ØÙ ¨¦ Ø Ú /¾ g 1 1Â R Â ¨+¿¼ Â ¾Û N Z )ØÙ_CL(À 1Â¢QÂ \ NC1

â?Ï¤PØf?gg #4©h2#+5k_Y ÷ ç ù¼?6h#4P# Ì #g4:¼yh ÷ 4#/##âg5 j k O â##5\·K<

<éòç Ú h#h4â_7Èòùh#Ô###âÝé Ì #ggâÔY LÚ§¢1 ¡ ¨ \¢ §¨ */Ø ¢ L\â#Ï¤/q4Y_4yh Û÷ 4#/_4â?ä<4ÑHLk j âL¿# 4 â5 jGj :/â##5: j ·%5 Nm4

O é Ú FRÓ4ggâ¸^ C¨ Ø Ú À § g Ü /¢ × ¢1g ) AÛ Ø ¢Q ) â Óhh#gg© ÷ :¼?/4ÑïÜ4h¼Ü©Ù<ãö n ÷ ö Ø Ì 4 :G:G:/â|ü-¿ ÷ ù]5kG:mh?â|ù?##4g.Üóä<gÑñh#â 4 :G:G:/â## NGN <.· N <:

hÝç Ú gÑÑñhhâ<Ø7`ö4ªRÍ§F¼4â¿ ¥ ¨ ¶ Z Øý R Z ) Ø$J R m¥ Â ¨ ¡ » Ú/ Â 1Â2Û Ù R § Ø \â Ö : ÷ ç ÷ 4/qgg#g4 ÷ 4#/gh ô÷ 4Ý?Ü#F_h#ù?gg#¤¼4â5 jGjGN â## Om4 ·Kh N

kÝà/ ÷ Ú g|hÑ4â-ÁL/ £Øý Ü / Ú Ø Øý Û Øý ) \â2Øfé Ì #ggâ Ì g##FhÑfö¯Ü84:c< O 5hâ hÑòã#Ó4g.¼4â|ég|h:g_ ÷ 4#/gù?gg#ghâ-5 j kk

n « nÂ «\² Ì Ä % ZÖÃ j k

j Ýà/ Ú g|hÑ4â » R G£ £CØÙý/) bÛ¤g ) âªÏ¤YØfgg #4<h6#Òû4Î#44ù?gÑg¯ Fhh ÷ :¼?/44h#ª¼4â-5 jGj P#â/#5 4 :1·%5 4 k

5:Ýà/ Ú g|hÑ4â8ç ###â » ý/))£ ¢ ) 1Â CL R Á¤/+ ¡¸ Ø */ ¨B § Û¤ g FCLR Z t ¡ ÁL§/\â/Ï¤PØf?gg #h§h#¸<Y ÷ ç ù¼?64#4PØf#g#ÑgÐhé:Ô#/ Û÷ 4#/#QëTØ n é ÷ k O í â%¿\Í 4Î8â5 j k O â##< 4 · Om4

5G5Ýç Ú g?Ü n gâªöò ÷ Ñ\Ó4hâ ¨\ ¢ ) 1Â Ø Â V£ × ¨ ¢ Ø / 1Â+ / ¾Ô ¢ ¨t£/ ¡ ¿¼ Â ý/§/\â8ù?Ï¤Dç àh4#|hÑ4 ÷ 4#/##â?ä<4ÑH 4 54â%¿#54â5 jGjm4 â##<P·K<k

5 4 ÈÏ Ú ÑñhÎhhâ ì 4ù?g4#HâZ¿Èªù?Yh:Fâ Û ØCØÙ ) ¢ ¼ § Â L1_âªü24 4çÛ_?ÜgY_ghÑÕù?g\.¼hâ8ü2g\#+¿hÈù?gg 4O <?â ÷ hÝÔ# 4òã#Ó4g.¼PØfggâ-5 jGjGj

5 N Ýç Ú Ñ#Qâ × ØÙ m 1Â £ G /Øý§LL \â<Ï.Ù<Ù<Ùù?## ÷ n çÛØ ÷ n çâú5 j k 4 â<#5 NGN ·L5 N h5VP Ú 4#gâÕü2/4â Ü 1 1ÂF¡ Û Øý/¨B C¦ Ø Z À 4N × /Ø ¥ ¥A mØÙ ) \â

Ï¤QØf?gg #4h ÷ ¼?/Q jGO â|ü-¿ ÷ ù]5G5: j â8ù?##4g.Üóä<gÑñh#â5 jGjGO â## 4 k N · 4j h5< ì Ú hh âhé ÷ |h#Qâ ÷ â ÷ ýg6Fhâ0¸^ C¨ ¢ ¨ » ý Øý ¢ R ¡ ì ¡ ½ ÁìØý ¥Â \â

àh4#|hÑh ÷ 4#/gh ù¼?:gù?gg#gggâ|ä<4ÑH N h?âL¿# 4 â|#5< O Ü5k j â5 j kGk5 O ÙÒ Ö Ú Î4gÑÑHâ hg4Ô#Hâ ¶ï R Z Ú ¨B¨BÊL1ý/¨+\âR Ó_h#gg` ÷ :¼?/4Ñ4h¼

Ü©Ù<ãö n ÷ ö Ø Ì 5 j kmh?â|ü-¿ ÷ ù N :P#â8ù?##4g.Üóä<gÑñh#âg5 j kmh?â/#5G5h.·%5 4O5h ÷ ÷ h¯#â6à/ ÷ h\#¯âKà/8úDÑhâ|à/Kç

¨#ÑÑggâ ¶ï Ú ¢1%¥ Ü /¢ ¨\ ¢ )

¥ Ü /¢ ú¢ § ¨B ¢ "¸AG£CØý ËÂ C\âÏ¤Øf?gg #4Õh?# 4 hòÏ¤ªg|_4|hÑ÷ 4ÑÑ?á##4×/4Y_/âªüÕh#4|h4g<h Øf44h#×ëqÏ ÷ üÕØ í âü-¿ ÷ ù85km< N âù?##4g.ÜóäfgÑñh#â 4 :G:G:/â##<15 4 ·K< 4N

5köò ÷ h#\Hâ©ç Ö #Ñâ / ØCØÙ ¨\L £Øý ¨¦¨¦C)¨B C\âf Óh_#gg÷ :¼?/4Ñ4h¼Ü ÷ ö Ø Ì n :K54â|ü-¿ ÷ ù 4 5 NGj â|ù?##4g.Üóä<gÑñh#â 4 :G:K54â##5 j ·ZPm:

5 j é ÷ _hÑñh##âföò ì g#|h#â¿È;7RÍ4FÓª¯Ü ì h|hQâ G L C Ü /¢ C) ¥ R CØ ÚØÙ /TË Û¤g Ü L/¨B ¥ CC mØÙ ) \â Ó_h#ggÈ ÷ :¼?/4Ñ4h¼QÜÙ<ãÒÜö n ÷ ö Ø Ì 4 :G:K54â|ü-¿ ÷ ù 4 :PZ<?â|ù?##4g.Üóä<gÑñh#â 4 :G:K54â## 4G4j · 4 P N

4 :é ÷ _hÑñh##â6öò ì g#|h#â¿È7ÍhFÓ4¯Ü ì h|hQâ R CØCØÙ /TË G ¥ × ¢1g ) Ú ¥ ¢ A§ O(n)

CC\â8àh4#|hÑ2h ÷ :¼?/4Ñ4h¼4â/ä<4ÑH5<?â 4 :G: 4 â## 4 <15\· 4OGj

n « nÂ «\² Ì Ä % ZÖÃ jGj

4 5é ÷ _hÑñh##âhöò ì g#|h#âG¿È7RÍ4RÓ4¯Ü ì h|hQâ_Ø7ÄÝ¿4/¼4gâ R CØCØÙ /TË Ú§//§/¨Å¿¼-ý/C§ ¥ â7Ï¤ Øfgg #4h©#Bk_p ÷ ç ÷ 4/qgg#gC4 ÷ 4#/gh Û÷ 4Ý##F_4yù?gg#.¼hâ 4 ::K54â## 4 : O · 4 5VP

4G4 Ýà/ ÷ 4#gâAçp Ö #ggâ ¿¼G£ ¢ / ¥ ÁL/ £Øý § Â L1 ØCØÙ Ü /¢ Û Øý /Ú) Ü L/¨B\âKÏ¤ôØf?gg #4h<# 4O ðÏ.Ù<Ù<Ù ù¼?84#Þ4 Ö 4# _4#h÷ 4#/gù?gg#ghâ8Ï.Ù<Ù<Ù ÷ 4#/gù??\\.¼4â5 j km<?â## N h 4 · N k 4

4N öò ÷ hggâ#Ï|éh×ªõh â^¼;/ Ú ½ ÁìØý ¥Â ¡ × C§ × /Ø ¥X CC1¨B/) M ¶ 4J ¼;/ Ú ½ ÁìØý ¥Â ]£ ¡ × © âÏ¤kØf?\g #4yh ÷ ö Ø Ì n j k/âü-¿ ÷ ù×5VP Om4 âù?##4g.ÜóäfgÑñh#âg5 jGj k/â##LP 4 P·ZPGPQ5

4 Pöò ÷ hggâfÏ`éhªõh âfãÈfçÛh#ggâ Ø / Ø Ü /¢ U¸ ¢ ØÙ) /) ¨\ ¢ ) ¡ ¨ #À( Ü / Ú 1Â Ü L/¨B\âÏ¤ Óhh#ggÈ ÷ :¼?/4Ñ4h¼QÜÙ<ãö n Ü÷ ö Ø Ì 4 :G:G:/â|ü-¿ ÷ ù]5kG:mh?â8ù?##4g.ÜóäfgÑñh#â 4 :G:G:/â## N 5 O · NGN P

4 <öò ÷ h\FâÐÏ<éhªõh â§à/Ë¿gÑgâI¸ ¢ ØÙ) ) ¨\ ¢ ) ^¡ ¨ G 1LØ ¥Ú ¨B¨BÊL1 Û¤g ) â7Ï¤ Øfgg #4ÈhÒÙ<ãö n ÷ ö Ø Ì :K54â7ü-¿ ÷ ù 4 :PZ<?â# 4 kG:Ü 4jGj â 4 :G:K5

4O öò ÷ hggâÏÒéhªõh â Ú ù?#?g#YhÎhggâ R ¡ A ¡ R ) ØI½ ÁìØý ¥Â \âÏ¤Øfgg #4h ÷ ö Ø Ì n j P#â|ü-¿ ÷ ùk NGj â8ù?##4g.Üóä<gÑñh#âg5 jGj P#â/#65hP·%5kmh

4 höò ÷ hggâAù6 Ö g#gâ 2Ï¤|hHâAÙÒÕúD##Ñ\Ó? É â Û Ù I¸ ¢ ØÙ) Ú R ) ¨\ ¢ Ú) /Z¿¼ 1Â \â#Ï¤ÛØf?gg #4hfÙ<ãö n ÷ ö Ø Ì 4 :G: N â|ü-¿ ÷ ù 4O < O â8ù?##4g.ÜäfgÑñh#â 4 :G: N

4 köò ÷ hggâföò ì g#|h#â Ú `ù?#?g#YhÎhggâ Ü /¢ ¥o¶ )C¨ ØCØÙ Û Ù ¸ ¢ ØÙ) Úxú¢ CLC) Û Øý ) Ü L/¨B\â8Ï¤ðÙf#46Fh Ì h#h\4#4 Ì gÑgg4Ý##ïÜF_4#gâ/ä<4ÑHQk/âL¿#<?â5 jGj h?â##LPmkK5\Ü§P j :

4j ÈÏhéD_ªõh â ¶ï CL Û NGý/§ g \ ¡ C ¨p¨¦C)¨B Ü L/¨B ¥ ¼;/ Ú À ÁìØý ¥Â R ¡ \âA Ó_h#gg ÷ :¼?/4Ñ4h¼ðÜ ÷ ö Ø Ì n k j âfü-¿ ÷ ù$P N <?âfù?##4g.Üóä<gÑñh#â5 j k j â##5h.· 4j

N :ÈÏ|éhªõh â ¶ï Σ Ú R § Ø \â ÷ :¼?/4Ñ44DØfh?g4Ñ Ì #g4:¼ ÷ 4#hâ 4 :G: 4 âãöüfÆÇ0Ç,ÈIÉÊ0ÊË0Ë0ËIÌaÍ+ÎÏ.Ð^ÏÑÌÒÎÓIÌaÍ,Ô¬Ê˜Ï.ÕÎÖ¬Ê×Ø0ÙÌGÆ+Ç,Ð¬Ú

n « nÂ «\² Ì Ä % ZÖÃ 5:G:

N 5ÈÏéhªõh âhç_àh#Î8â Ø / ØÙý ) Ü C¨\ØÙ ) t¥ ¨B mØÙ ) ¡ R CØCØÙ /TË R G£ £CØÙý/) R ¢ £ØÙ /Ú ½¦/ Ü /§/¨âKØf#Ô#ÑúD\¼ ÷ :¼?/44h#ª¼ ëTØ<ú ÷4 :G:K5 í â#ü-¿ ÷ ù]5 jGjm4 â8ù?##4g.Üóä<gÑñh#â 4 ::K54â##5G5 j Ü5 NGO

Nm4 ÈÏÕéhªõh âAç7àh#Î8â À- 1Â C Ú × ØýONG)£Øý G 1LØ ¥ §//§/¨ÁqCC mØÙ Ú ) \â#Ï¤ÛØfgg #4h7#Yk_Q#:hÑñhñh ÷ 4/qgg#gÈ4QÏ¤/q4Y_hQù?¯Üg#¤¼Ph ØfÓhh\¼ëq ÷ Ï:ù/Ø 4 :G: N í â|ü-¿ ÷ ù 4 h 4 h?â8ù?##4g.Üóä<gÑñh#â 4 :G: N

NGN ÈÏ_éhªõh â4ç4àh#Î8âªà/¿gÑgâ Ø / ØÙý ) ¡ R CØCØÙ /TË R ¢ £ØÙ /Ú ½¦/ Ü Ú§/¨ ÁCC mØÙ ) ¤§ Û Øý ) Á%) 1Â âªù?8ggñhÑ#gA4 Ö |h#gñhÑ ÷ :¼?/4_Üh#ª¼4âfÏ¤ªg|_4|hÑÒàh4#|hÑÐ4 Ï¤/q4Y_4 ù?g\#ï.¼'ëqÏ:àhÏ:ù í âfù?##4g.Üóä<gÑñh#â4 :G: N

N PÈÏKéhªõh â2à/g¿gÑgâ R / ¡ Ú ¥ 1Â¥ R / ¡ ¥ 1Â / ØCØÙ ¨ ÚL £Øý ¨¦¨¦C)¨B Ü L/¨BÁqCC / Û N/ / × Z §¯âÐ Ó_h#ggY÷ :¼?/4Ñ4h¼Ü ÷ ö Ø Ì n 4 :G: 4 â|ü-¿ ÷ ù 4 PGP 4 â8ù?##4g.Üóä<gÑñh#â 4 :G: 4 â##<kK5\·K< jGO

N < ì é ÷ ggg É #â?à/_úò_ É âhöò n :Ó?Îª¼4â4hù? Û Ù ¥ R / ¡ )ØÙ Ú Ú ¥ 1ÂÔY LÚ §/ Z ),&ÔY LÚ ¨ ØCØý £Øý ¨¦¨¦C)¨B Tâ< Óhh#gg ÷ :¼?/4Ñhh¼pÜÙ<ãÒÜö n ÷ ö Ø Ì 4 :G:K54â|ü-¿ ÷ ù 4 :PZ<?â 4 :G:K54â##LPm:1·K< j

NGO é6é4Ñ\Ó6â ÷ KéÍ©4Î8â2çg¿h4gâBÔY LÚ ¨ ØCØý £Øý § Â L1_âKù?Ï¤Dç àh4#|hÑÕ4÷ 4#/#/â/ä<4ÑH N :/â%¿# 4 â 4 :G:G:/â## NGj 5\Ü§P N h

N hÝà/#é4#4_Ü Ö gggâ R * £ØÙ Ü /¢ ú¥m¥ C) ¥ ¸ ¢ ØÙ) ØÙ ) R Z Ú Ú¨B¨BÊL1ý/¨â8Ï¤ôØf?gg #4hf#p<ðÏ¤ªg|_4|hÑ ÷ 4/qgg#g4ðÏ¤/q4YÜ4Qù?gg#¤¼ëqÏ:ù ÷ 4 :G: 4 í â#ü-¿ ÷ ù 4 P NGN â8ù?##4g.Üóä<gÑñh#â 4 :: 4 â##LPZh15\·ZPmk N

N k Ì §ÙfÑ ì hYhÑTâ R ¢ £ØÙ ½¦/ §//§/¨ ¥E Ü Â ¢ Ü L/¨B]£ ¥ » ý /§ºÀ- Âm CC1¨+\âÕ Ó_h#ggÝ ÷ :¼?/4Ñ4h¼&ÜØf?gg #4h ÷ ö Ø Ì n kP#âü-¿ ÷ ù]5 jGO â5 j km<?â/#5:1·%5k

NGj Ýà/ Ö g4gÔ|h#Qâç2çQgFâ ¶ L iÛï¢ /) , G Ø À £/) Z C, ¥ Ü ¢ ¨¦¨ ¡» ý /¢ / ¯â<éÏ:çQ ÷ ù"ù?gg ég\ÛçÛ_#gY_gÝh Ì #g4\FhÑ ÷ 4Ü#/gù?gg#ghâ#ä<4ÑH 4 â5 jGj 54â##5\·ZPZ<

n « nÂ «\² Ì Ä % ZÖÃ 5:K5

Pm:Ýç Ö gÑÑRÍgâ1¿È/ú4Ô#Ñ É â ¨B£ % § Ø §//§/¨+ Ø Ø6VÜâÏ¤ Ö # Ö gÑ g#g4:¼hâ/h##ÑF_4#fh hÑ44ï#×gâ ÷ 4g84h:¼çÛ_#gY_ggâ4ä<4ÑHK5 O k/âüÕhäfgªhgâ5 jGjGN â##<15\· O 5

PQ5È Ö ñ_FâRù?|hgâ Ú Á G R *ÞÝ ¢ /Ø ¡ J R Z ) Ø Ü Ø ¢ ) §ì ¥ ) ) ¥ Ü Â% ¢ R G£Øý/¨+\â2 Óhh#ggÈ ÷ :¼?/4Ñ4h¼QÜ ÷ ö Ø Ì n k O â7ü-¿ ÷ ù 4OGN âù?##4g.ÜóäfgÑñh#âg5 j kmh?â##5k O ·%5 j P

P 4 ØÕ Ö 4#á#hâ ì 8ØAh#|h âKà/Kùgâ Ü 1Â » ) CL §ONG¼ ¡ ÁL) 1ÂZÀ-)§/ý\â Ö |h#gñhÑ ÷ :¼?/44h#ª¼ 4 :G:G:/â#ü-¿ ÷ ù]5 jGOm4 â8ù?##4g.Üóä<gÑñh#â 4 :G:G:

P N Ýù6Õé ì hÑÔ#hâ Û ØCØÙ ) ¢ R CØCØÙ / Ü L/¨B\â7àh4#|hÑ<h ÷ :¼?/4Ñ4h¼4â2ä<4ÑH5<?âL¿# 4 â 4 :G: 4

PGP e ì hÑÑTâù6m7hÔ6ggâ/ç ###â § Â L1 ¨\ ¢ ) J Ü /¢ × ¢ ØÙ Ú §Øý/ R § Ø ¥ CL R ¢ £ØÙ /Ú Àm/)¸A ¥ /Ø¹âA Óhh#ggÝ ÷ :¼?/4Ñ4h¼&Ü ÷ ö Ø Ì n kmh?âü-¿ ÷ ù 4jGN â8ù?##4g.ÜóäfgÑñh#âg5 j kGk/â##5 N <RÜ5<G<

PZ<Ýù6 ì 4Ñ Í§hggâ?ù6çQFhÑHâ R G£ £CØÙý/) bÛ¤g ) ß Ú Á2§ R Ø 5¸A Ø R Àm/½¦§ 1Â Ü / ØCØ R ) Ø% ¡ ¨ ) â#Øfgg #4hÕ#p5VPhP ÷ çù¼?6_Ü#¿4y# Ì #g4:¼yh ÷ 4#/##â5 j k 4 â/# 4 h:Ü 4jGj

P O Ýù6 ì 4Ñ ÍÒhggâÐù6<çQFhÑHâ R G£ £CØÙý/) Û¤g ) âÐàh4#|hÑ§h ÷ 4#/gh ù¼?:gÞù?gg#gggâ|ä<4ÑH 4 k/âL¿# 4 â5 j kP#â## 4 h:1· 4jGj

PZh n ì 4Ñ gâ2ù6çQFhÑHâ66û \4â Ú Ás§ R Ø U¸A Ø Ø ¨B\âK ÷ çù¼?84# 4 Ì #g4:¼yh ÷ 4#/#QëHù Ì n ÷ í â-5 j kmh?â## 4 5k1· 4G4j

Pmk n ì 4Ñ gâù6çQFhÑHâ§û g4â R ¡ ]C qÝ- /Ø ¥ ÔYC1 1Â £ ¢ pCL/CÁ ØÙ ¥ C) ØCØEÀ 1Â¢QÂ Y Ô R -¼;/ Ú ½ ÁìØý ¥Â R ¡ Ü /§/¨+\â6àh4#|hÑh7#ò ÷ çâ#ä<4ÑH N k/âL¿#54â5 jGj 54â/# OGj 5\Üòh 4j

P j ØÕ ì 4ÑÑhâçà4hÎ44Ô#4âàh#gÑgâØ7ù¼Óªg4âK / ØI¿¼ Ú g ) A¡ ¸^ N /C\âÏ¤ Øf?gg #4Phò#ðöù? ÷ 4/qgg#g ÷ :¼?/44h##gg Ì h¯Îj :P#âü-¿ ÷ ù 4jGO P#â|ù?##4g.Üóä<gÑñh#â 4 :G:P

<:é ì 44\Ó6âÏØ74#4Yhg#Îh#â Ú ¨B¨BÊ%1 R ¢ £ØÙ /Ú Àm/ §//§/¨+ ¥Û¤g ) 1Â Øý C /¢ CC\â2ûp4Î#4p4 ÷ gh ÷ :¼?/44h#¼hâAÏã¿öÏ¤âäfghÑÑggâ 4 :G: N

n « nÂ «\² Ì Ä % ZÖÃ 5: 4

<15é ì 44\Ó6â©Ï©Ø74#4Yhg#Î4#â Ú ¨B¨BÊL1 R ¢ £ØÙ /Ú Àm/ §//§/¨+b*/Ø ¢ L ¥ ¿ì 1Â \â#ißGßhh./Ó6 4ZßhhÔ#ß_gg ÷ ö\ß: N : j :K5:

< 4 Ýà/ ì hâ¿¼/ ¥ ¨¦¾ £Øý ¥ ¿ú§Ø £Øý \¥m ) L LL/§ONG ) Z ÀÜ /¢ §//§/¨+\âÏ¤jØfgg #4QhÈ# Ö : Ì #g4:¼ h ÷ :¼?/44h#ª¼ë Ì ÷Ò÷ 4 :G:P í â|ü-¿ ÷ ù 4j <154â8ù?##4g.Üóä<gÑñh/â 4 :G:P#â##5< 4 ·%5h:

< N Ýà/KûÕØ7-7/qgÑ â R à ) Ø ¨B/) 8*/ × C§ × /Ø ¥ \â n ?q4 çÛ_#¯ÜY_F_Ñ2çQ4#44h##gâ ÷ Ññhg 4ÛØfggâ n ?q4 â# N :/â5 j h j <P 6Ï¤|hHâÙÒKúD##Ñ\Ó? É â R § Ü C¨ ¢ ØÙ ¢ -¸A Â R § Ø _ÁqCC mØÙ Ú

) \âªÏ¤YØfgg #4Ðh6# Ö n×Ï¤hgÑ6ù¼?64#)4 Ì #g4:¼×h ÷ 4#/#h ù¼?:gÝëqÏ:ù Ì ÷ ùb5 jGj h í â/Ï.Ù<Ù<Ù ÷ 4#/gù?g\.¼hâ5 jGj h?â##5hP·%5k N

<G< 4Ï¤|hHâÙÒªúD##Ñ\Ó? É â¿ G%¥ ¨¦¾ 1Â R ØÙ ¨¦ Ø J /Áq¿¼§ ) ÁqCC mØÙ ) ¼§5¿¼ ¢1¥Ú xÙ Ü /¢ ¨\ ¢ ) âªÏ¤Øfgg #4<h6#ìPQ5F:#|hÑfù¼?84#Þ4 Ö 4# _4#h ÷ 4#/gDù?gg#gQë Ö n ÷ ù 4 :G:G: í â 4 :G:G:/â# 4j P· N :P

< O ?Ï¤|hHâ#ÙÒ#úD##Ñ\Ó?ï É â R / ¡ / Ú ¿¼ ¢1¥ Ü /¢ ¨\ ¢ ) R / Ú¡ :¿ ¥ ¨¦¾ 1Â R ØÙ ¨¦ Ø \â/Ï¤QØf?gg #4h7# 4j yÏ¤ªg|_4|hÑ ÷ 4ÑÑ_Üá##!4×/4Y_/â4üÕh#4|h4gÐh Øf44h#×ëqÏ ÷ üÕØ 4 :G: 4 í âü-¿ ÷ ù 4N k 4 âù?##4g.ÜóäfgÑñh#â 4 :G: 4 â## 4 PGP· 4 < O

<GhÝù62à4hgÎHâü¼?:¼h#ÎR¼ª/â ú¥m )/ØÙ Ü /¢ G KLØ ¥ § Â L1ÃJ¸ ) Ú¥¢L 1Â g/¢ g º¿¼/¨B* 1ÂÛ ¢ \â Óhh#ggy ÷ :¼?/4Ñ4h¼(ÜYÙ<ãö n Ü÷ ö Ø Ì 4 :G:G:/â|ü-¿ ÷ ù]5kG:mh?â8ù?##4g.ÜóäfgÑñh#â 4 :G:G:/â## 4G4 5\· 4 P 4

<kÝç7àh#Î8â Û NG§ / p§ CL R CØ ØÙ / §//§/¨ÁqCC mØÙ ) ¦§ Ú§Øý Â Ø R § Ø \â/Øf éÝ Ì #ggâ|ég|h:gÒh ÷ 4#/gù?gg#ghâ8ã#Ó4g¤¼hAh#gâ8ég#YhÎ8â 4 :G: N

< j úK7È#ú#â|ù6|à//ü2ghâ8à/Q7È ÷ #g4â8à/ûQ7hâ|à/|ù6#úòh##â ÷ #ØAhÎ8âÔY/Á R ¢ £ØÙ /ÚÀm/ §//§/¨á/ 1Â ¥ Ø ¢ L\â Óhh#ggP ÷ :¼?/4Ñ4_¼(Ü ÷ ö Ø Ì n4 :G:G:?â|ü-¿ ÷ ù]5kGkG:/â8ù?##4g.Üóä<gÑñh#â 4 :G:G:/â/#5 OGO ·%5k N

O :+¿Èªú4Ô#Ñ É â Ø Â V£ ê §L c ¡ § Â L1_âªÑ44ï#`h C÷ 4×#/¯4çÛ_#gY_ggâ/ä<hÑH N â8ù?##4g.Üóä<gÑñh#âQ¿\Í 4Î8â5 jGj k/

n « nÂ «\² Ì Ä % ZÖÃ 5: N

O 5+7gÑhgÐü2#Y4/â"ÁL/ £Øý Ú ¨B¨BÊL1 y¶ £ØÙ ¢ G ¡ / ¥ R § Û H ¢Q Ø ÚC) G /Tâ_ Óhh#\gA ÷ :¼?/4Ñ4h¼Ü6ñhg:¼?/ 4 :G: N â_ü¿ ÷ ù 4 k j P#â_ù?##4g.Üóä<gÑñh#â4 :G: N

Om4 üfÐäÝÒü¼4â R ØCØÙ G Á¤ ÚÂ R ¢ £ØÙ /Ú ½¦/ §//§/¨ £ ¥ R ØCØÙ Z Àm/¯âØf é Ì #ggâ6ö##.Ü.ã#Ó4g Ú ¯#Qâ 4 :G: 4

OGN é¸¿hgFh#hâCà/Èùg ÔY/Á R ¢ £ØÙ ½¦/ §//§/¨ ¥ Ú Â L/¿¼/ ¥¢ \â8Ï¤&Øf?gg #4hA#¦<Û ÷ ç ÷ 4/qgg#gÝ4 ÷ 4#/gh ð÷ 4ÜÝ##F_h#ù?gg#¤¼4â5 jGj k/â##%< j · OGO

O PÝç¿h4gâ C ¨¦¨¦C)¨B Þ/ ZÂ R ¢Q¥ Ú ¿ ¥ ¨ \â8àh4#|hÑ2h ÷ :¼?/4Ñ4h¼hâäf4ÑHQP#â5 jGj 54â##5<15\·%5<k

O <Ýç¿h4gâKöò n :RÓ?Îª¼4â2öò|äfg#Î__ghâ2ç ###â R / ¡ T¼;/ Ú ½ ÁìØý ¥Â ÚÂ¢ ¨B C ¡ §Ô R £ ¥ Ø / Ø ¨\ØýONGC) ¢ ¨\) \â| Óh_#gg ÷ :¼?/4Ñ4h¼YÜ ÷ ö Ø Ì n jG4 â6ü-¿ ÷ ùhPm:/â5 jGjm4 â/#5 jGO · 4 5VP

OGO Ì n Î_hh#â|ù6|ã#¼hY/â ÔY/Á R ¢ £ØÙ /Ú ½¦/ §//§/¨ Ü /¢ × Z/Ú§ 1Â â7 Ó_h#ggÝ ÷ :¼?/4Ñ4h¼ÛÜÙ<ãö n ÷ ö Ø Ì 5 jGj k/âAü-¿ ÷ ùÕ5VPm: N âAù?##4g.ÜäfgÑñh#â5 jGj k/â## N :GkÜ N 5k

O hÝù6 Üã7È7Ø`hg##âfú Ü ÷ (7?â<à/(7ÈAúDQâ`ù6 ÷ #ghâ ÷ `ØAhÎ8âÔY/Á R ¢ £ØÙ ½¦/ Ú§//§/¨â/ 1Â × C§ÖÔY £/ØÙ Ø ¢ L\â_ Ó_h#ggA ÷ :¼?/4Ñ4h¼Ü ÷ ö Ø Ì n4 :G:K5hâ|ü-¿ ÷ ù 4 5 NGj â8ù?##4g.Üóä<gÑñh#â 4 :G:K54â/#LPZh:1·ZPmkm<

O kØÕØAhÑÑggâ R ¢ £ØÙ /Ú ½¦/ §ã/¨+ ¥ ¨\L/C§ » Â -¿ú/ ¥¢ /C) Ø ¯â# Ó_h#gg ÷ :¼?/4Ñ4h¼YÜ©Ù<ãö n ÷ ö Ø Ì 5 jGjGj â8ü-¿ ÷ ù]5< jm4 â8ù?##4g.ÜäfgÑñh#â5 jGjGj â## 4G4N · 4N k

OGj ØÕ?ØAhÑÑggâ G ¥ 1Â » ý /§IÀ- Âm CC1¨+c AÛ ØCØÙ ) ¢ ¸*/¿ì 1Â \âª ÜÓ_h#gg ÷ :¼?/4Ñ4h¼Ü<Dù?Ï¤ ÷ ö Ø Ì 4 :G:G:/â#ü-¿ ÷ ù]5 j h O â|ù?##4g.Üóä<gÑñh#â 4 :G:G:/â#<h N ·K<kP

h:ØÕ?ØAhÑÑggâªé?Ø74ª¯#\Ó_hÑHâ Û Ù R ¢ £ØÙ /Ú ½¦/ §//§/¨+ R * £ØÙ Ü /¢ ËÂm / \ ) ú¥ / \âª Ó_h#gg§ ÷ :¼?/4Ñ4h¼ÝÜAù?Ï. ÷ ö Ø Ì jGj â?ü-¿ ÷ ù5h15 O â8ù?##4g.Üóä<gÑñh#âg5 jGjGj â/#5 O <.·%5h j

n « nÂ «\² Ì Ä % ZÖÃ 5:P

h15 Ú Ø`î# É Yh#â7çKû h #ggâ ú ¨B ¢ × 1Â /Ê ) 1Â âK Ó_h#ggò ÷ :¼?/4ÑïÜ4h¼Ü©Ù<ãö n ÷ ö Ø Ì 5 jGj h?â|ü-¿ ÷ ù]5 4NGN â|ù?##4g.Üóä<gÑñh#âg5 jGj h?â##%kGk1·%5: 4

h 4 ì ?Ø74#|h â|à/#ùgâ × C Û¤g g) ¡ ¿ Ü ½¦/\â/ Ó_h#ggÒ ÷ :¼?/4Ñ4h¼ÜÙ<ãö n ÷ ö Ø Ì 4 :G:G:/â|ü-¿ ÷ ù]5kG:mh?â8ù?##4g.Üóä<gÑñh#â 4 :G:G:

h N éÕú7öh#8hâ Ø Â V£ ý ^L¨B¨BÊLLÞ½Y§//§/¨B\âAé#Ñ4Y Ì #ggâ`ã#ïÜÓ4g¤¼Ph`é4:# â ì gYh¼hâ 4 :G:G:

hPöò§öÓ4g:Fâüf§ ÑgYhâçÒég:4 É 4gâ ¶ï » À ¥ R Z Ú ¨B Ú¨BÊL1ý/¨+\â Ö 4# _4#hÈù?gg# ÷ 4#/_4â©ÒF gÛØfggâ¸5 j hk/â§#5 OGj ·L5hGh

hG<öò8öÓ4g:Fâ8Kù?|hgâ8üf| ÑgYhâ ¸A/CL ¥c¡ ¶ £ 1Â » Â C Ø Ü Â ¢ ¥ R ¢ £ØÙ /Ú ½¦/ g§//§/¨+\â ÷ 4Ý##F_h#DhÐ#× ÷ çâä<4ÑH 4 54â-¿# 4 â5 j hk?â##5 4 :1·%5 4O

h O Ì fù/h ggâ ÷ Ö Ì ? â G Á ¥ ¸AG£CØý § Â L1hâ`Ï.Ù<Ù<Ù)ù¼?6h#4Ûù?gg#¤¼ ß ØfÓ_h\¼ð j k/â n hÎÑñh â ÷ hÑq4#/âg5 jGj k/â/# 4 5<.· 4G4 P

hGh Ì 6ù/h ggâ ÷ Ö Ì ¯ â R § ) 1Â ¸AG£CØý ¤Â ËÂm /¸ ØÙ ¢ Ú /C\âQçQ4Ô#ÑD4gªh ù?gg#¤¼4â8ü-¿ ÷ ù]5VPQ5 j â5 jGj k

hk Ì <ù/h ggâ© 4###â§ç ###â6ÔY LÚ §/ Z ) § ¨\ ¢ ) 1Âp¡ NC1â

Ï¤(Øfgg #4hÒ#8Pm:_ #|hÑÒù¼?64#4 Ö 4# _4#h ÷ 4#/gù?gg#\hâ8Ï.Ù<Ù<ÙÒâ-5 jGjGj â##%<G<P·K< O h

h j È2ù?|hgâ Ú Ás§ Ü Ü /Hâ ÷ 4Ý##F_4#h<# ÷ çâÓ44ÑH 4G4 â##5G54â5 j h j â/# O 5 4 · O 5 N

kG:òäÝ<ù?#4#â R Z ) Ø G 1LØ ¥ Ü Â ¢ \âÐ Óhh#gg ÷ :¼?/4Ñhh¼pÜÙ<ãö n Ü÷ ö Ø Ì 4 :G:G:/â|ü-¿ ÷ ù]5kG:mh?â8ù?##4g.ÜóäfgÑñh#â 4 :G:G:/â## 4 :mh.· 4G4 : kK5Ýà/L7È6ù?ÑÓ4gYhâ G L CC1¨B/) ¡Û Ø ØÙ ¢ \â ì |_ Ì ¯?ÛçÛ_#¯Ü

Y_g©ä<4ÑHg5: O â8ù?##4g.Üóä<gÑñh#âg5 j k O k 4 éhûph4#ggâ G% Øiýc ¡¼2 Ø Â V£ R Z Ú ¨B¨BÊL1ý/¨âhÏ¤Øf?gg Ü

#4òh©# O pÏ¤ªg|_4|hÑ ÷ 4/qgg#gY4pÏ¤/q4Y_4ù?gg#.¼ëqÏ:ù ÷ 4 :G: N í â

n « nÂ «\² Ì Ä % ZÖÃ 5:m<

ü-¿ ÷ ù 4 km<154â|ù?##4g.Üóä<gÑñh#â 4 :G: N âãöüfÆÇ0Ç,ÈIÉÊ0ÊË0Ë0ËIÌã0ä:Ìcåæç,ÔæÚ,æèÌÒæÍÓÊ

˜Í+ÎËÊ1ÈÎÈæçä0Ê&ëq\Ó?: Ó4g4 í

k N +¿È4öòhû h4#ggâ?çªöòªçÛhh¼ªhÎ8â R ¢ £ØÙ ½¦/ §//§/¨ ¥ CL$¹¦ ¥R G£Øý/¨â/ Óhh#gg ÷ :¼?/4Ñ4h¼Ü ÷ ö Ø Ì n kP#â|ü-¿ ÷ ù]5 jGO â|ù?##4g.Üóä<gÑñh#â5 j km<â##5 j · NGO

kPÈÏ§ûg4g#ggâ G L ¨\ØýONGC) ¡ Øý × ¢1g ) \âû Ñ\¼ªÜ Ì g#Ô##gÛù?ggy÷ 4#/gù?gg#ghâ-5 j kGh

km<È h/â R § Ø ¡ Ü /¢ ¨\ ¢ ) ëq¯?g hÔ#:h\ í â?Ï¤ 4N #|hÑù¼?84# 4 Ö 4# _4#Òh ÷ 4#/gù?gg#ghâ8Ï.Ù<Ù<ÙÒâ-5 j k 4 â##5 O :1·%5 O P

k O È h#â Ú Á § Ø / § ¥^Û N 1Â Ü /C\âAÏ¤ 4 hp#|hÑ§ù¼?6h#4 Ö 4# _4#Òh ÷ 4#/gù?gg#gCë Ö n ÷ ù^5 j k O í â#Ï.Ù<Ù<ÙÒâ-5 j k O â/#5 Om4 Ü5 O h

mé

NC1gg#Fâ 4 5

ΣÜ¤#h?g4ÑHâ O h

kÜ¤|_â 4Nq4#¯ÜóÍÒF¼P#444##¯?h\hÔ#Ñhâ j :

q4#¯ÜóÍÒF¼j#444## 4g#g_4gâ

j :qÜ¤ªÓªg:Ô#Ñï.¼4â j 5 Ó4gh:¼

h\Ó4hâhk h/Ó4hâhk|hÓ4hâghk:_hâ%< j â OGN âhk

Ô# ##46g:¤¼4â%k OÔ#Ñ ##â jÔ#h####4h_Qâ 4G4Ô##ÑÑ\PÔ8ªh â%kK5g4×gªÒ¯#ghâ5k/âLkm<

gá#Ó4?FhÔ#Ñhâ%kGk¯?h\hÔ#Ñhâ%kGkï/ âLkmh?âLkGk6ggñhÑ2ï/ âLk jh ?4gâLkGk

g4#Ñ\g#ggâ O kg4#/_4|hÑÑï¼Y :#4#|hÔ#Ñhâ j 5

g4#/#"Í*g#g:¼?/ q##\4#gâ5 O

g4#/#ÍÝ\#g:¼?/ × _/âQ5 O â 4Og4#/#"Í*g#g:¼?/ q##\4#gâ

NGOg:¼?/4:¼?:g

# Ó4gÑ¼Y#444##hâ OhÑ4gÔ#hFhÑÑï¼Y#444##hâ j \g#:hâ O#444##hâL<Ý#Ñ#ÑF_ïÓªgÑ¼Y#44h##hâ O#4Ô|hÔ#Ñ:hâ OFhÑñhÒ#444##hâ j

g#ñhÑhAg:Ó?ghâ N PÙ ÷ ØÐù6âg< NgÑg\4y#ghâ5hgÑg\4#Ó4h##âLkG:gÑ4Ô#Ñï¤¼4âQk 4gÑÑ/g#:Ó4ØAhÑÑg§h#/4â%<G<gÑÑ/g#:Ó4ØAhÑÑg©¯#ghâ< Ngá#Ó4?FhÔ#Ñ¤¼4âQk j¯?h\hÔ#Ñ¤¼4â j 5¯?h\hÔ#Ñ

q#ÑÑ¼4â j 5¯?h\4âLk jTh§g#g:¼?/4#gâ 4 :

«Òêz§y:ë 5:mh

Ö ñ_:Ü:ù?|h×§#g#:hâ%h15î|#4g#ª##â5k4g#ghÑ gÑÑ/Ag#:Ó4ÐØ`hÑÑgh#?Ü

4â%<Gh?â OGN4g#ghÑ gÑÑ/Ag#:Ó4ÐØ`hÑÑgK#ghâ

< Oì 4Ñ Í§hg.Ü:çQFhÑK#ghâk/â 4j# ##46g:¤¼4â%kmh#4#g::ÜóÓ4gî|g É g_Ü¤Î#RÍÑ 4f#?h.â OGj :#4#|hÔ#Ñï¤¼4â j 5Îh\¼y :#4#|hÔ#Ñï¤¼4âQk jÑg#h?ÜÊÈ|¯/Ô#Ñhâ O <Ñhg:¼4â5kïÜ¤#\Fâg5 j â OGOï/ g4×gªÒ#ghâkmh?âLkGk4Ô#ÑDh4gFâ5 O â N hÝ#Ñ|h:¤¼Yg4#/_4âg5 O â%hGh#g4Ñ4Ô#Ñhâ O¿ü n ì ù/Ø6 ÷ ÙÒâ 4 k#4?Üóq44FhÔ#Ñ¤¼4âL<k#4?Ü¤ªgh\Óªhâ 4 k4Ô#ÑÓ?4#©h#:qggâ5kn Î_hh_Ü.ã#¼ªhY#ghâ 4jØAhÑÑgg Òh#/4â%<G<6g:qg\Fâ%kmhØ74ÑÑ¼ ÷ hÎhggâ-5G5#Óhh\¼4âLk 4

g4#/_4|hÑTâQk 4

/q4Y_4?ÜH#g4\hâQk 4#h?g4Ñ

Σâ O k5\Ü¤4/:Ü¤hñÜ 4 âLhGh#g¯ÎPhAg##g:¯?#ggâh 4g4g\Ý#Ñ#ÑF_4âLhPgá|hÑ¤¼Ch g\òÑ4ªh#×gâ OGjÝ#Ñ|h:¤¼Yg4#/_4âLh j#Ññhª¯?Î#RÍÑ 4hâh OÓ4h##âLk N

##Ô#ÑD4 ggâ j :h 4 4hgÑ gÑHâ OGN â%h15h 4 gÑñÜ¤ #gÔ#Ñï.¼hâ5 jh 4 É #Ý64Ñ¼?#4ñ_Ñgâ 4O¯Ü¤g#g:¼?/4â5 j¯Ü¤h 4 É hÔ#Ñg#g:¼?/4â5 j4Ô##:#ggâ<k?â%k 4öù?âhgg\|h#×#ghâ-5hgg#Ègg#\Ó_hÑ|_4â N hgg#¤¼y|hh¯\Fâ%<|hòg4ÝÔ####â Om4|h gg:¼?/4â O 56ggñhÑÕ#4#g::ÜóÓ4gî|g É g_Ü¤Î#RÍÑ 4hâ

O k6ggñhÑï/ g4×gª7#ghâmkm<?â

k j6ggñhÑ24# #\gâ OGjÌ Ù ÷ ØÐù6âg<Gh#g#4Ñ gÑÑ/Ag#:Ó4ÐØAhÑÑgK#ghâ

<Gh#g#4Ñ #ghâ5h?â<k

«Òêz§y:ë 5:Gk

hhÑKÍ©g4Fâ 4G4##Ó4ghÑ2g464hÔ#Ñï¤¼4âQkm<?âLkmh##Ó4ghÑÓªgî8hÔ#Ñï¤¼4âQk 4Ó4gî8hÔ#ÑDg#g:¼?/4â 4 :Ó4gî8hÔ#ÑDgg\|h#×#ghâghkÓ4h ##ÑF_4âQk 4Í§_gYhÎ##â5kÍ©g4Fâ 4G4Í#ggâ O kÉ g_Ü¤Î#RÍÑ 4ò#?h.â5h

Date post:	25-Dec-2020
Category:	Documents
Upload:	others
View:	6 times
Download:	0 times

¼C | h g ª#$ #$ % &(' ) *+ , &- ./, ./ 01#2. 3 4 65 Ñ 4 gÔ# h F hÑ Ñï¼ 87 4 4 4 # # ÷ :¼?...

Documents