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é##C#Ññh:q\Í ¼ªFhD#444##g#g:¼?/4ô#ggò|FÓ4Ô6ggô: ¯Üg#Ó4gѼC#g#\¼C|FÓªDÔ8gg4D4h 464:hª§CYh¼ åKggÒg:¼??Ü44h###h?ghÑ©#yhgâ?h#âÓªh##hg4Ñg`ϤY#©|h/gÐͧ #g#_Ü44##g:¼?/4:¼?:gY#gD:g#ÈëñæóÍ|_:êâæ¤#Rͧêâ#h æóͪ¼ªê 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\ 64Ѽ?#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
Ϥ/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#$IQ Ö6zÖý Ö
54ú¿h|_§q4ÒÝ#444##g#g:¼?/4P¯#gÔ8D/ªggª§§g#gñhÑ6YhÎh#×|_È# É h©#g##g:¯?gYh#ò64Ѽ?#4ñ_ÑѼ&Ô84# #ògg#¤¼P|hh\g
σ ##×g8F_ g4#/_4#g
4 Ì #Ýgg#.¼&h6g\gâ \î|#4#gâh gÑh<#444##Èg:¼?/4:¼?:FhD#DhÈh#|hÑѼq4Òg:¼?/4:¼?:Fg
Ïó©#g#g:¼?/4 hÑ44ï#E4\h # 4|hÑA##/y##q4 h h #ÝÔ6g
rhÐY\
Zâ6#Ýg#g:¼?/4ð#gF_ÑÑ G£ £CØÙý/) h#g:ÍFhÑÑ ¥ Ú
§/¨¦ ý/) t7g#gY§Pg:¼?/h¼?:g ò#4Ô|hÔ#Ñ:Ý#g×Ô6gÑ4#y¯ÓªghÑ åKggªg##g:¯?§Ý4#gh4 g6g #4Y#h 4ß?#ÝÔ6g
r ∈ Z Ú /Ò#h|_
F¯J¯J¯® Q³C²(®(§¯V h
hÐÔ6\q4# gg:¼?/4ChÑ44#!gYh# \g#:hâªHhA#g¾:#:©4#g.Üh4Ô6\Ñ4#4#È4ÓªgCg##g:¯?FëHù?gDÔ8gÑÍkÙA#h#Ñ 4 :/ P# íÝÖ #:#g4hâ?P#4Ô|hÔ#Ñ:hâK#444##Ýg:¼?/4:¼?:g #×hÑ44#
A#4#Ñ Ô6#4Ô|hÔ#Ñ:hâ
?&ð# Y#P##/g##g:¯?g Ö 4Ý#:h#ghâf#ÝFh(Ô8yFhÑ Ô¼ h##Ѽ?#YÔ#Ñ #YhÑ44ï#ëqgÝé\î|#4 4 :/ih í 4&ðë \g#: í \4#/_4Ûh`#g#g:¼?/4Qh7#ò# #\h hA#D#Qâ|g6g\Ó4gѼ4 LIKÏ Ö6zÖý=Ö Ï¤p#Yq4ÑÑRÍ#yͧYÍÑÑ<4ò#Cgg#¤¼p|hh\g
σh #
##Ô#ÑÈÎ4\¼ÛÛ# gg/4&h`#ÝhÑ44#×gûpÍÑÑ2ÍEke
(m)4E(m)
q4E(1σ, ke,m)
h Dk(c)
4D(c)
q4D(1σ, k, c)
Í#g×#D# g:h #`84Ô#ÑhÏó##gÐ#4Ô|_Ô#Ñ:ÒÍ©ÍÑÑ8hÑòÍ
Eke(m)
4E(m)
h<Í©gÑÑ6hEke
(m, r)4E(m, r)
q4E(1σ, ke,m, r)
Ö #:#g4hâ/Í©DÍÑÑKÍ
(E(m), E(m′)) = E(m m′)
g#hò|_Ò#hÑ44ï#¿¿ëqg#gÒ # 4çQ#Ñ í h##Ñ 4y¤Í©Yg#\¼?/h#hA#gh4g
m,m′ ∈ (M, )h 4/#/hÛg#g:¼?/4Ûh
m m′â8Hhâ8#4Ñ
|_D(
(1σ, ke, Eke(m), Eke
(m′))) = m m′
¯/gg/Íy#g4Ñ4Ô#Ñ#4Ô|hÔ#Ñ祿4D\yIL#$&(W0 Ö6zÖR-Ö 7gͧ×4Ó4×hô¯#h#Ñh§ \g#:hâÝ#Ñ#ÑF_ïÓ4gѼÛ#_Ü44##ð#g h hk¯#h#Ñðq4y(#4Ô|hÔ#Ñhâ # Ó4gѼ#444###gh
54 Ì #Ð\ÑñhFhÑöù? #g\ hG<ªÕhȯ#h#Ñfh# \g#hâR#Ñ#ÑFïÓ4gѼ#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#Ѽ4â`#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#ÑFhyFhѼCÔ8ò#Ñg×g hçÛ#Ñï
(Eke(m1), Eke
(m2)) := Eke(m1) · Eke
(m2).
ã|hÑѼYC#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 # Ó4gѼQ#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#ѯܤÎh\¼"#444##yg#g:¼?/4¯#gÛ4''ëqgïÜ í #
(M,+, ·)Fh'Ô8
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Aâ§Hhâ©YY # ïÜ
Ó4gѼYh Ý#Ñ#ÑF_ïÓ4gѼ×#444##_Ð#hh<ù?#P#gghFhÑÑ Ø Â V£ ØCØÙ_L¨B¨BÊL1 HJ0mÔ;C Ö6zÖý3Ö Q # Ó4gѼ#444##\#g:¼?/4y¯#gD4QyëqgïÜ í #(M,+, ·)
<FhÑÑ Ø ïL¨B¨BÊL1 6#g¯/:©È#4Ô|hÔ#Ñ:hâ4¯/6g\ 64ѼªÜ#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#Ѽ(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#Ñ祿4HJ0mÔ;C Ö6zÖ xgÖ £ØÙ ¥ 1Â$ Ø Â CC1¨ ×ô#4Ô|hÔ#Ñ:hâ<64Ѽ?#4ñ_Ñ©yhÑ4_Ü#QâKÍ#4ô##/
1σ, ke
âÕh c ∈ Eke
(m, r)Í#g
r ∈ Z_ 4Ѽۯ#4g
4/#/òh#h#gDg#g:¼?/4c′ ∈ Eke
(m, r′)hmÍ#g
r′ ∈ Z#4gô##q4Ѽ
_h 4Q
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Ö 4Ò#:h#ghâ|y×#4Ô|hÔ#Ñ:hâ/#444##g:¼?/4:¼?:g4(M, )
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ch h
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M
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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 # ÓªgѼC#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 /ªggѼ4 /
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:4Ϥð#Dg\4ðͧî|:§#ggª§gÑñ_ Í©4Î×g4# g#ò#Ð#4Ô#ÑgQ Ì #gF_ngÐͧ ggÔ8#gÑñÜ4##×Ô6\¤Í©ggPhÑ4gÔ#hFhÑÑï¼Ý#444##Ò#gg§h #444##Ò#ggÐ46ggñhѪ#4?Ü.hÔ6gÑhò44##g`çQ4<#gggѼ4âͧ<#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# 4Ó4gƼ##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#ÑgQÈϤÕ5 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
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#444##g:¼?/4:¼?:g FhðÔ6Ô#4Îhgô&#Ô?ܤ¯/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?ܤ¯/64#gªñhÑ7hâ8?# Ö #:#g4hâ6fÍ©hggÎ#Ûq4hÑ4gÔ#hFhÑÑï¼ð#444##×##Ô#ѯܤÎh\¼ô¯#g×gÝ4 YhÑÑ`î|gÑ 4È#4#&h
M = F2
â|4ÔªÓ?4#ѼC#&#gÈ|hÒ×Ô6ò#4Ô|hÔ#Ñ:Ô8ògg#hn #g©h/#4Ð|FÓ4 òî| gh _g<q4ÐhÑ4gÔ#hFhÑÑï¼Ý#444##Ò#gggϤA5 jGjGN â Ö gÑÑÍÒh ú4Ô#Ñ É #gg hQhÑ4gÔ#h##Ô#ѯܤÎh\¼Pg:¼?/4:¼?:gFhÑÑ Ø74ÑѼ ÷ hÎhgp Pm:HDÏóhÑ4gÔ#hFhÑÑï¼y#444##Èh #RÓhhÔ#ѼQgg#hÈã/q4:?Ü|_gѼ4â7#Y#gY|hQ#Ô8gh 6ªg#Ñgh È#hÝ/ªggªÝg4#gg##Û#g##g:¯?òÑg#h Ö :Ѽ4âØ74ÑѼ ÷ hÎhgDDC84Ѽ?#4ñhѹܤÔ|_ :¼?:\Q Ì #g\q4hâg4#/#hP\#g:¼?/4yh2#D# #\
E(m1 ·m2)hդͩgh4g
m1
h m2
Ô¼Ý#Ñ#Ñï¼?#C#g4g84 #Ûg##g:¯?84Ѽ?#4ñhÑ
E(m1)h
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x 7→
1 0 x
0 1 0
0 0 1
=: M(x).
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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
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∃ 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
],
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(SL(3, p), ·)
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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
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sh
t_¤Í§×6ggñhÑÓ4g:gghA#D4h#
ûp×YF¼ðh#|_V = s = 1, . . . , t = λ
h |_Dq4DFh 4(i, j) ∈ E
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w(i, j) =n∑
k=1
alkxk + bl, alk, bl ∈ Rh
1 ≤ l ≤ λ2.
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q4i1ikGâ_#\Á¤/  12h φ \î|# ×Ô6D#ò# #¯
w(i1, i2) · w(i2, i3) · · ·w(ik−1, ik)W (i, j)
g#hg#p§ ØÁ¤/  1hAhÑÑ g\ |_#©q4
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x = (x1, . . . , xn)í
Ì |_W (i, j) =
∑
φ(i,j)
k−1∏
m=1
w(im, im+1)
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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#$&(W0 =Ö)LÖý Ö ü2\RÔ8òÝg4Ý/_Ó4#Í]54fÒhy¯#h#ÑhAÝÔ#h#?Ü
#ô#44h h É <ð4(##/ x1, x2, x3, x4
g4# g#Pq4ÑÑRÍ#ð4h#â<Í#gx1, x2, x3, x4, a, b ∈ R
1
2 x 1
x 4
5
3
4
x 3
x 2 + bx 3
a+ x 4
ûpò|FÓª
w(1, 2) = x1, w(2, 3) = x3, w(3, 5) = a+ x4, w(1, 4) = x4, w(4, 5) = x2 + bx3.
7g#ghâ|#DÍ©g4ªh7#ò|_φ = (1, 4, 5)
\î|# ×Ô6
w(1, 4)w(4, 5) = x4 · (x2 + bx3).
Ì #ÝhhÑÕÍ©g4ªW (i, j)
gá|hÑx1x3(a + x4) + x4(x2 + bx3)
_ ?##q##\4g4#/ Ôª¼
BP
f(x) = x1x3(a+ x4) + x4(x2 + bx3).
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f(x)FhðÔ8ÝYh#8 Y#
g4#/_4Ph7# \g|hªh`Y_ïCP#Dq4ÑÑRÍ#ÝͧF¼pëqgòh#ú <P í ü2\
x = (x1, . . . , xn) ∈ Rn Ì #g &h* ¾hgg#\¼jY_ï
H(x)Ô6gÑ4#4#(
Fh'Ô#h#¯##p#44hBP = (G,w, s, t)
Í#gG = (V,E)
ÐÏóDÍ©ðh#Û|_V = s = 1, . . . , t = λ
h |_Dq4DFh 4(i, j) ∈ E
ò#hÑ D|_i < j
â2#gH(x)
Dhλ × λ
Y_ïQÍ#g\:¼(i, j)
gá|hÑF:C(i, j) /∈ E
h w(i, j)
h#g.ÜÍh©ù?#g
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H(x)hQ##8gñh#4#Ñh§Y_ïYÍ#
ò#Ñ6hgªFYÏórD##Ô8gÈh§ 4gh©#Ñ4#4g:È|_q4
s = 1t = λ#g
H(x)r+1 = 0û #gð#P# g:h #Y64Ô#ÑÈͧÍÑÑ2Í
H#:F
hH(x)
ü2\
k Ú C g#hYy|_|_g4#:Èh§¯#h\Ѽ k 4gg Ì #g:¼ (i, j)h©#
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H∗ :=∞∑
k=0
Hk =r∑
k=0
Hk
§#òY_ïYÍ#4(i, j)
gª:¼y\á|hÑW (i, j)
©ù?#gHr+1 = 0
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(I −H)H∗ = H∗(I −H) =r∑
k=0
Hk −
r∑
k=0
Hk+1 = H0 = I
⇒ H∗ = (I −H)−1.
Ö #:#g4hâA#Y##6gÝ4gª:¼(1, λ) = (s, t)
hH∗â g#h Ô¼
H∗1,λ
â`gá|hÑW (1, λ)(x)
Ì # \g#gÐ#4/#/f(x)
h2#q##\4f : Rn → R
g4#/ Ôª¼
BP Ì g4#/#Dg:¼
(1, λ)hH∗
Í©òFhÛh##Ѽ ÷ hgg #Ñh7g#g
H∗1,λ = W (1, λ) = ((I −H)−1)1,λ = (−1)λ+1 detM
det(I −H),
Í#gM g#hg©##Ô#Y_ïYh
I −H4Ô/h# Ôª¼ gÑ\#È#î|:Òg4Ñ#Ch
#ÈÑñh:ëλÜó í RÍòù?#g
HhÛ##8gñh#4#ÑñhÒY_ïCÍ©|FÓª
det(I −H) s 547g#gH∗
1,λ = (−1)λ+1 detM = f(x).
01#$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.ܤ É CÔ#h###ô#44hg Ö #:#g.Ü4hâ6#ÝÔ#h###C#44h É Ý#PÑñh4g|h&#g4g64 #Cq4Ý#Ññ× É hf 4 hHú7Rͧ\Ó4ggâK64Ѽ?#4ñ_ÑïÜó É òÔ#h###Y#44hhÈ#4Ô|hÔ#Ѽy#h6Rͧg:q#Ñ
g#4#4"/ªggªÑ¼(g4#/QhÑÑÒ84Ѽ?#4ñhѹÜó×Pg4#/hÔ#ÑPq##\4#g Ì #RÍ#yP#×Th\ò|_D#6RÍ©gÈh©64Ѽ?#4ñhѹÜó É Ô#h###Q#4h_òg4#g gÍP|_h åKggªÒÓhhñhªÒhAÑ4_ܤ|hgDg4#/_4b <G<H 01#$IQÏ=Ö)LÖý3Ö Ì #P##6gÔ64# q4#C É PhÔ#h###ð#44h×Ýg4#/#Ú ?4ÑFhòq##\4#
f : 0, 1n → 0, 17¯/64#gªñhÑH Ö 4Õ6ggñhÑgÑñhgAh#q##\4#
×YhÑÑg§Ô64# FhÛÔ8È#RÓªg©Q¯#h#Ñòh7q##\4#|_FhÛÔ8Èg4#/ Ô¼Ñ#Fh.ܤ É §Ô#h#¯##D#44hfh Ú ?4ÑFhÝq##\4# f(x) = 1+
∑J⊆1,...,n
∏i∈J xi
7g#ghâ
f(x) =∏n
i=1(1 + xi)h #gQ¯/:YôÔ#h####44h h É n + 1g4#/#
f
D\yIL#$&(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|hgѼ"#g4#:#\4&hf_ܤFhÑÑ h 4 É #64Ѽ?#4hÑÒܩͧhª #g 2 <G<?â< O `h q#:#gPhg#ghÑ 4 hH Ì #gôh 4 É #p6hѼ?#4ñhÑChð g#ggª_4h©q##¯4#òÔ¼ðÑR꤆ g4g×64Ѽ?#4ñ_ÑÍ#òg8ggñhÑѼ&#\q#Ñfq4D# g4hgg#ÈÝ#Ñ|h:¤¼Yg4#/_4P#h?g4Ñg
oA1I0 > ÂÉ<½CË · ÁPÇÃ · ËÆ|kÁy¾78A<PËÕÅÆ:~ ¶ Ë ¶
Ϥ#Yg\4Í©&g4# gCh64:hªYh##ÑF_hhD#444##yg:¼?/4:¼?.Ügg ¨\ ¢ ) 1 ÁqCC g § ¥B¥m `ûpî|:Ò4Ó4DhC #\4Y#§#4Ô#ÑgÔª¼ ggÔ##yÈh 4Ó?#P¯#h#Ñgg Ì #gÍ© \ÓªgÑ4py#hg4ÑA|_È4ÑÓ4gD## gѼ?#ô#4Ô#ÑgQ n ##h?g4ѧÝ#4?ܤgh\Ó4hâ</ªggª×q4Ý64Ѽ?#4ñ_ÑïÜó É Ô#h###Q#44hgâ7h #RÓhhÔ#Ñï¼ôgg#h Ì #gâ`ïÈÈy#\Íßh##ªhyî| Ü#RÓª 4Ñ/4#§q4©#D# gѼ?#×#4Ô#ÑgQ÷ 4# gÝ#Cq4ÑÑRÍ#ôù?gg|h 54Ïóͧh #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 ¨\ ¢ ) 1 ÁqCC] g § ¥ú¡¢1g ) \â/|_Ògá#Ó_hÑgÒYù?gg|h54dG01-IQLÞÑgô|hÛ q##\4
f : Rn → R|_Û# ?gQ#hQͧhªQ\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#Ѽ 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#AͧÒg4#Ñ ##ÎÝhÔ64/
fÔ8g#ÈD##Ô#Ñ©q##\4Ûëqgh#
k O âQPZh í Ì #È4Ñ/4P4Ó4gyÔ6gÑRÍ FhQhÑ×Ô6ò# 4ÑÓ4D#Ògg|h#4 ϤQ4DFhgÑgD#4#Ñ ÑFhP#ò4/#/Fâ/#N n #ѼCÑgD#4#Ñ ÑFhP#ò4/#/ f(x)
01#$IQ=Öý Ö)LÖ n ÔªÓ?4#Ѽ Ñgh ##/FhhÑðÔ6yggh&î// |h:×h#gq##\4
f7g#g
k = 0hÑ×64Ô#Ñh
D\yIL#$&(W0^=Ö6 Ö6 Ö ¿ÍjÍ©ò4Ó4ò¯#h#Ñgh7#ÈhÔ6RÓ4ògg|hYh §Óhhñhªg54Ýh ¯#h×#ÑYhù?gg|h$5Í©Cg4#Ñ ##ÎôhÔ64/Ñg|FÓ?#&h /ªg\hÑ44ï#Þq4DT_\4#P|_È##gÈQFh4#\¼&Íph #òÍÒhªòÎhgg&#Óh_h Ú 4ÔÛ|hhQ##/|#òͧh|RÓ4òTh\4 É ©ÑgòhåKgÚ 4ÔP×Th\4Ò#Ò##/Òq4Ò4#\¼4
4 D.¼?#FhÑ7¯#h#Ñhf#î|:ÓhhñhªÍ©Ýg4#Ñ ##ÎÛhÔ64/ÑgòÍÒhª#×gÑÑ2#gg4#/#6RÍ©gÒ Ú 4ÔPq4##Ô#ÑòhÑ44ï#Q
N Ïóf×##Ô#ÑYÜh×(ä<hñhJ5PÜh Ô6h"Ñgyh Ú 4Ô"#4#Ñ ÑFh #
4/#/ÜhÛäÐhñhª 4 ܧͧÝ|FÓª#Ýgg|hC#|hÑѼQg4# \ 4 ܤ|h:¤¼g4#/_4#h?g4ÑgQÝh ¯#h#Ñ×ͧYg4# g
fhÔ6g#Û#q##\4
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gg Í#Phf#gÞFh#D4È4#\¼4ÈçQ#Ñ|h:¤¼Qgh#/_4ÛhÑ/4#q4D#¯ Pgg|h åKgDq44#ò4Ñ/4ð#gð#q4gD4##4Ñgh<ÑgÝh ôÚ 4ÔôhÝ#|hÑѼÛ:¼?\hâKHhâ6#g4#/_4##\¼Û|RÓ4Ý6g:q4 hf#FhѼ#<hhAù?g©ù?g\4¸P# O q42#ÒÝ#Ñ|_:.¼g4#/_4#h?g4Ñ2h q#:#g \hÑg
P#ÑgÒͧhª#È ÝÈg4##g.ܤ4##4×64ÑÑ|ÐhY¯#h#ÑhäÐhñhª N q7ghâÑg×g4ÑÑg\È##/ÈÓhhÑ#gh Ú 4Ô'ëTh h#gÈgg í &\Ó_hÑ|_×#\#ïÜÓh_gÑï¼4
mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ 4 k
÷ ÑFhѼÈù?gg|hY5ÐFhÈÔ6Ð4ÑÓ4 ÍÈ4g#ghѪ#h?g4Ñ2q4Õg\#<q##\4È\ÓhhÑ|_4ëqgÒh#â% k O âPZh í ;7Rͧ\Ó4ggâªÍ©§Í©4#Ñ ÑÎhÐ|RÓ44Ñ/4|_``#4?ܤªgh\Ó4hâHhâ`Ñ\Yh Ú 4Ô #4#Ñ |FÓ4C4#Ѽô4#Y4# h _Û¯/|h#4Ph #4#Ñ #h|FÓª×Û\4##F_Ýq#:#g ##y#\ÓhhÑ|_4ôh©#q##\h Ö #:#g4hâÍ©4#Ñ Ô6Ý4/YhÑÕP#RÓ? ×C4Ñ/4&ÛÍ#¯&#ÝͧhÎѪ |_òÑgÝ|h6g:q4 g8g Ð4#Ѽ4#Ò/ªgg#\¼h8#Ò# gѼ?#Dg#g:¼?/4ׯ#ghâÔ#/<#h4y#æ¤ É ¯êDh f â|Hhâ/#ò É òh`gg#g4#/# f ù?4f#4?ܤªgh\Ó4`4Ñ/4#|RÓ4<Ô6gg ggÔ8 #fÑg_#hâRg<h#¤ hk/âG5hH7Rͧ\Ó4ggâ##4#hÕ#ghåKg§ g6g g#¯¼Ph2#æ¤ É ¯êh f Ì #DÔ8g:Î#RÍy#4?ܪgh\Óª4Ñ/4&y4#D\g|hQ ggÔ6 Ô¼ôé Ú FFÓªgòE O H×Ïóò#Ó? gÈ4Ñ/4×q4Ð#4 \g#:Ñ4_ܤ|hgq##\4#ë)¿ü n ì ù/Ø6 ÷ Ù í h ©ÐÔ|h 4Ûg4#:#\4 Ôª¼ ù/h g\hÑH$ hkH Ì #CÑñ_g ggÔ8 Û#h?g4Ñ<##&#RÓªgÑg4#:#\4#©Í#Pgá#g§4#Ѽ4#4# _Õgh\4Ý\Ó_hÑ|_
NC1gg#
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NC1gg#gq7Rͧ\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
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R<î|#
î|gÑ âªhÑ#4#4Ý#h\g©Í©Ò#|hÑѼYF¼Ýh#©|_`#Ò#Z NZ
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# #\hA¤Í©YÑñh4D#gÔ6g|FÓ4gÑÎhò×î|#Dî|gÑ ëqgÝögYhÎ N 4 N í Ì ?#ͧ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\ѼÛ#444##
g#g:¼?/4ô#ggÈh \î|# 4Z kZ
q44k ∈ N
DÍPg:\R
Ô6g#CY#òî|gÑ Ì #ͧÝR¼y#Ý#4Ô|hÔ#Ñ:hâ6 # Ó4gѼy#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
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f : Rn → RÍ#g
R = Z NZh N = pq
D## #\h©¤Í§QÑñh4×#gg ì Ó4gx ∈ Z NZ
##4Ô|hÔ#Ѥ¼|_
gcd(x,N) = 1 φ(N)
N=∏
p|N(1 − 1p) = (1 − 1
p)(1 − 1
q)Èù?#gÈ##4Ô|hÔ#Ѥ¼
§#g4Ñ4Ô#Ñï¼gÑ4J54â|hÑ4:§hÑÑ6gÑggÒhÓ4g:Ô#Ñh Ì ?#gâ|C#h\gͧDYF¼h#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 # ÓªgѼ#444##ׯ#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#ѧÎh\¼ke = B
Ì ##h_4EB(M)
Fh#A|_EB
h##Ñ 4ÛFh¯&gª:¼Û_Mh |_#g#ÑÍg&ÛYY_ïQhªh
û #gÛh##Ѽ?##ÈhÑ44ï#© # 4çQï/ Ü:çQ#ÑÒ4y#¯Q\##g:¯?Y_gggâ?ͧh##ѼY#g4yFhQg:¼Pg|h_gѼ4Ö 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:ͧ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 # Ó4gѼÛ#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#ѼDÔ8ÐF 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¯Ü5×64Ѽ?#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
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zh ∈ xi|1 ≤ i ≤ kâHhâ`#gª:¼&hf#ÝY_ï
H−
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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
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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#ק # 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#
#© # Ó4gѼòh 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#yϤ§ògá|hÑλ − 1
âÕ#×4/#/hf(x, y)gá|hѸ54â#h#g:ÍY:QëqgögYhÎ N 654 N í
mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ Nm4
Ïófͧh \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#$IQ=Öý Öý3Ö Ïó
EB(M)ͧh×#4##4#ÑñhÈ#g"4#Ph 4#4##4#ÑñhÝY_ï
Í©4#Ñ |FÓ4&Ô8gg'KªggªPùg 4 pÔ#Ñ #Ûgªg(EB(M))i,j
âÒÔ#/C#gÑg gD#hÎ#RÍ #Ýgg\òÎh\¼4â##hDhÔ#ÑÈ gg:¼?/òh Cg4#/#h#Î8
Ì #òq4ÑÑRÍ#ÝYh4#Yhg§#ò FhA4#Ò4Ñ/4
f , x = ( x 1 ,..., x k )
D B ( E B ( f (x, y ))) = f (x, y ) f (x, E B ( y )) = E B ( f (x, y ))
E B ( y )
Alice Bob
E B ( f (x, y ))
y = ( y 1 ,..., y m )
Ö 4# N 654 ÷ 4#/#Íyg#g:¼?/ Û _
Ϥph#gÈÍ©4 ò#Ô|h FyhÒh#È4Ñ/4D#q4ÑÑRÍ##òÑg×|hQ#Óh_q##\4
fh #Óh_D##/
x Ú 4Ôy|h#Ó__D##/
y|_#Dg#g:¼?/##
#ò##Ô#ÑÎh\¼Bh g È#g#g:¼?/4ôQÑghYù?#\ÓhhÑ|_gò#gòq##\4
f4(#g#Ó__P##/×h "Ú 4Ô g#g:¼?/ ##/F ù?#y î|g#yg#ÑÝô# #gÒq##\4Qh 4Ô/h#ÒhQg#g:¼?/ æóq##\4PÓ_hÑ#¯ê
EB(f(x, y))©ù?#òg #
ÓhhÑ#Ô|_¯Î Ú 4Ôq7<#Í hÔ#ÑÒ gg:¼?/§Ð###Ðgg\§Îh\¼Ch Èg4#/#ò4/#/
f(x, y)
01#$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\Ó4gѼpëqgt <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#ѯ/¤¼P#gò#|hÑѼP # 4#Òh Ý#Ñ#ÑF_4#ÒhAgÑFh:¯?Ó_hÑ#ghò4D/ªÜggª|hP#ò#h4òh7#ÈhÑ44#ק # 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 # Ó4gѼð#444##×g#g:¼?/4p#ghâfh g È#gg#g:¼?/4#
EB(yj)×Ñgh
4 Ì g4#/<hò4/#/f(x, y)
|_ÕFhò4#ѼÔ8<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#hgªgѼQg#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\Ѽ4 Ì #D4Ýh<#_¹K gªgDhÝg#g:¼?/ â4h|h:Ѽg#g:¼?/ h #gYh##DggÐhgÑFh:¯?<ÓhhÑ#gg
ë íPÌ DÔ#Ñ #g`Y_¹HÑgЯ#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:Ѽg#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#ѯ/.¼y4##hg4Ñ74/YhÑ2#g#4?ܪgh\ÓªhâHhâ_Õ#g %¾:#:74#Ð4# h#g4Ý##F_4 ÷ 4#\g###<g4#/Ü4|hÑg4#ѯ/¤¼È g8g <h#Ѽò4#© É §h|#©# gѼ?#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#/ªggѼ4 n ÔªÓ?4#Ѽ4âÕ#/ªgg#\¼h§##h?g4Ñ#`g:\ Dq##\4#`|_ÐhÒg4#/hÔ#ѧԪ¼64Ѽ?#4ñ_ÑïÜó É ©Ô#_#¯###44hg Ì #ÈgÑñhh`#Ûq##\4#Ò|_hÈg4#/hÔ#ÑDÔ¼yÔ#h####44hhK64Ѽ?#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_ܤ|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#gòÑgͧhªÒ4\|h q4g # Ú 4ÔP#òg#g:¼?/ 4/#/Fâ/#_hÎCÒ##ÑÎhgѼ4
mQ" ! ¯X(±Q)³ý¬ (wv ³Êx wyq¬²®(°^z&+KK+ N <
#h#g_h¯Îðͧ4#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# gѼ?#&g#g:¼?/4#g#gÈÑgD4#ѼP4Ô/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#Ѽy4\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#ѼyÓhhÑ#g|_ Ú 4Ôð\ggÓªg
q4ÞÑghF7g#ghâ2f#Y_ïQDh&g#g:¼?/4ôhÐCh 4ÞY_ïKâ Ú 4Ôð#hhÔ#ÑתhQ_¼P¯?×/q4Y4yhÔ64/Ñgh §q##\4CÜЯ/gg/Ò É h 01#2#$I8=Öý Ö xgÖ ü2\
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h Ñ\Sh
TÔ8§##ïq4ܤh 4
#4##4#Ññhn× n
Y_gg©Í \(T )·
\(S) = 1
RÓ4gFq
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64Ô#ÑDY_ïSMT
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\(T )·
\(S) = 1 ⊆ GL(n, q)×GL(n, q)
Ì #44#ð48g_gD4&#Ý\
M := M | M
n × nY_ïCRÓ4g
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M ∈ M ÷ 4# gò#×##44#
GM
hGÍ
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#ò#Ô#\G.M := SMT | (S, T ) ∈ G
hM Ì #gy#òYh###
G/GM → G.M, (S, T ) ·GM 7→ SMT
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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 ¢G _¥ /§ ¢ 4J [ ØÙ x ¢1 CL;g§ Ø - ÀmúCL C /¢ C ¡ f §_£ Y¢1 / Ø C /¢ C ¥t¢ g ¥ 1 ¡ 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\ѼYÔ8òh##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 # ÓªgѼ
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#Ѽ&ͧh#ÛÑFhphp##8gÔ64# h©# É ×#gͧFhô##Y_gg
Uh
N \î|# Ô6gÑRÍ!ögYhÎ N N 4 #:F hÐ#
Y_ggHh
M
4 Ú 4Ôy#g#ÈhÑ44#ק # 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# gѼ?#×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\ÓªgѼ4
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_ܤY¹K×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 ͧ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#Ýͧ4#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
ׯ|h#4D#ò É gh7#òY_gg S h T gá|_gѼhÌ #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#/#È6Rͧg©#g#Y_gg©hÑñh4gf|hCÔ6\q4h 01#$IQ$=Öý=Öý=Ö Ï¤ 4 h ÷ hgf\©hÑHA#464 DͧR¼ÝȪhÔ#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|hѼ?©h2#§#hg4ÑK©¯#h\Ѽ#Dhhh`#È4#Èðù?\\4 N 4 ¼7g#ghâKhÔ6\q4È4##h?g4ÑÕ#RÓhhÔ#Ѽygg#hâK#4?ܪgh\Óªòh /ªggªq4Ò84Ѽ?#4ñhѹÜó 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#Ѽ×#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_ܤÎ#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×à ¶ <^< · ÁPÇ
ϤÛ#g\4ÛÍ©ª #g#\Í#48g:.¼y|_Í©FhÑÑ(Àm/p/Á m 1 ©ûp#4:Ѽ g4#:_Dͪ¼P##48g:.¼Ph`g:¼?/4:¼?:gÒ#\q#ÑHâ|g6ggñhÑѼCP#òg4ª¯?hÐ#D|h/gg Ö #:#g4hâÍ©×#RÍ!|_D#ghÑF ¼&¯/:òÝ#Ñ#ÑF_ïÓªgÑï¼&hÍ©gÑÑÕh # Ó4gѼY#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 1 7#gD¯/:hy/ªgghÑ44#¿úD\¼ªÜ:ùͧhy|_4Ó4gQhg#g:¼?/4
Ea(Eb(m))g4#/ghyg#g:¼?/4
Eb(Ea(m))
k#4Ô|hÔ#Ñ:hâ # Ó4gѼh 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|_#ͧhª
È\ÓhhÑ|_h Ú 4ÔC_ Ñg#4#Ñ ÑFh#h##hÔ84/Ð#h#gÐ4#gÐ##/gâHhâfh
x#4#Ñ gYhògg\fh 4#Ѽ 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 # ÓªgѼQh 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 FhQFhѼCÔ8ÈhÓªg 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:ͧ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#ѼYäÐhñh N hfù?gg|h5ÈFhyÔ6ò4ÑÓ4 h|hÑ444#Ѽ4 01#$IQÕ=ÖR-Öý Ö #h#g§64Ô#Ñ祿×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)
ϤyhQ_|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©
Ϥô#òg\4ôÍ©×#ggȤͩQ¯#h#Ñgòh©#444##Ý#ggò|_È_:n¼ðÎh\¼:ͧh####
T+'-0^DFWÊÜJIL#$ILWïd'-01#$0
Ì #×ÙfÑ ì hYhÑÕ#g N kAͧhî|: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_ïÓ4gѼY#444####Ô#ѯܤÎh\¼C#gh Ö #:#g4hâ#Ò##64:Îh\¼:ͧ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_ïÓ4gѼY#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#Ѽ Ô6Û#Ñ\gª Ôª¼"#Ñ#Ѽ?#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\Ó4gѼ48¿h|_ò#×#p#4#Ñ Ô6×#×h×ôÔ6hÎh\¼?gâAHhâ2ͧh
Í©4Î# #&hÛ44#âÔ#/C#&4g#g_4gYF¼"Ô6g 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|FÓ4òׯ?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:ùͧhúD\¼ªÜ:ùͧh
(Ea(Eb(m, r′)), r) =
úD\¼ªÜ:ùͧh(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:ùͧh8¾:#:D:ͧh#D#î|:.Í©Pg?h ¹Ü
|_gh7#Ò#Ñ×4Ô/hEb(Ea(m, r), r
′) Ì #FhQhÔÓ?4#ѼCÔ8 h#ò¯?gѼ
/ªggªÑ¼4ì Ó4gÝhÝg#g:¼?/4
Eb(Ea(m, r), r′)#©g4g64 #gg\fÎh\¼È#©##Ô#ÑÐÎh\¼
bFhCÔ6# Ý4Ô/hEa(m, r)
Ì #©©:h4ªq4:ͧh `Ph|hѼ?Ðh2#gg#¤¼h7#ÈÙfÑ ì hYhÑ6#gÈh q#:#g \hѧFhQÔ6Dq4# A N kH
T+'-0]HJIL#$?«QB¬I-[d'-01#$0¿RÍ*ͧ4Ó4Ýh#h#g¯#h#ÑÈh<h&g#g:¼?/4&#g|_Îh\¼Û:ͧh#### Ì #f77#4Ô|hÔ#Ñ:hâ # ÓªgѼDh FhÑñhÕ#444##`g:¼?/4:¼?:F h #g#gFh Ô6P# &hÑÓªY#Cgg|h4#ggª p#PÑñh:Ýg\4#gôÏóͧhÝ\ggѼ##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 # Ó4gѼ#444###gC4Ó4g
Ea(m1, r1)h
Ea(m2, r2)ͧ
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##Ѽ?#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\Ó4gѼ4Ý¿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:ͧ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":ͧh(#Cî|:פͩôg_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
|~ ®
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N2Í#g
N## #\òh<¤Í§PÑñh4Ý#ggÈÏóDͧhò##Ô#Ñ# 5 jGjGj
h h|hѼ? 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#RÓªgQgYhªFhÑѼYgg#ÈgÑñ_Ó4Y#\Í \î|# #4Ô#ÑgQϤÛ#ÝhÈͧR¼ÛhÏKéhªõh 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 # Ó4gѼ"#444##Ûg:¼??Ü4:¼?:ggâ7Í#phhÑyFhÑñhò#44×4##hâ2Hhâ2#\¼ô|FÓª##48g:gò|_
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n #<#g#4Ñ Ó4g4ChÑÑRÍ<h¼×#Ô#\§h k 4/©h ` #ÑñF¼ªgf gg:¼?/ÒÈg##g:¯?g4g\Ѽh /ªggªÑ¼Í#ÑÒ hÑÑRÍÐhª¼×#Ô#\§hÑgf|hk#ÑñF¼ªgfÈ4Ô/h
hª¼Q/q4Y_4QhÔ64/#Ègh4h Ú h 4Û4#Ò#g#4Ñ Ó4g4QͧhhÔ#ÑD ggÔ6Óhh4#©h##ÑF_4#ÐCÑhfͧR¼Ch©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`ïܤ#\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'ͧô#ggª& Ý#Ñ|h:¤¼g4#/_4Y#h?g4Ñ6Í#yhÑÑRÍÐÝg4#/DhPg#g:¼?/4PhÕ## #\Òhդͩgh4g4Ó4gy#Èg4g64 #g##g:¯?Ô#/#h#ògh4g#ggÑÓªgg©
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ûpC#4:Ѽô #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#:Ó4gͧò\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#Dͧ#g yg4# gD#4
R = Z NsZÍ
NÔ6g#P## #\òhФͧy#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
Q"$# «§¬g)®¯Z°±Z²)³¯¬A§¯Ây ' ' ³()³C²ï±®À,F¯ÃÀ®Ä¼³ý¬ ( Z/NsZ
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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|_Òͧ 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
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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#Ѽ4â 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)
<ϤQ|h:g#Ññhgâͧò4\
|Ej(Z NsZ)| = N · |Ej+1(Z NsZ)|
q4ÒhÑÑ1 ≤ j ≤ s− 1
7g#g|E1(Z NsZ)| = N s−1
¿h|_§q4s = 2
ͧ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)
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E(Z NZ)
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|E(Z NsZ)| = |E1(Z NsZ)| · |E(Z NZ)|
= N s−1 · |E(Z NZ)|.
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4E(Z N2Z)
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mPi = Pmi
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mP1
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Z NZâHhâ
gcd(N, 6(4a3 + 27b2)) = 1
ü2\M = |E(Fp)| · |E(Fq)|
Ô6D#D4 ghE(Z NZ)
Ì #gÛÎ#RÍÑ 4òhM
64Ѽ?#4hѹÜó×gá#ÓhhÑgªÈQÎ#ÍÑ 4h§#×Th\4_hhN = pq
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•÷ #?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
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E(Z NZ)
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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
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m ∈ Z NZ#?4×h 4 g4g
1 ≤ r < Nh g4#/
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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 #xܤg?4 |_
mMNg#\
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N'4Ô/h
mM ∈ Z NZh #gÝ#Ñ#Ñï¼ Ôª¼(#QÓ4gÛh
M mod NpggRÓªg#Ûg.Ü
h4m ∈ Z NZ
yë)¿h|_ͧ×|FÓ4Z NZ ' E1(Z N2Z)
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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© #
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(E(m1), E(m2)) := E(m1) + E(m2) + r′Q = E(m1 +m2)
Q" !wº &y ' ' ³()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. <G<
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×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#ѧFhCÔ6
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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
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N − 1ûpFh&4#4ò#hÑ4#Chq4#ÝC4Ô/h# #\Í
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C 6= (mN : 1 : 0)âÍ#YFhYFhѼÝÔ6Ó4gî| n #g:ÍÍ©|FÓªÝÔ#Ñ #gh4g§Ôª¼C # #
r′QÍ#g
r′ ∈ 1, . . . , N − 1Ðh 4Ѽ
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Ì #Ègg#¤¼yh|hѼ?Ò§Ó4g:¼yÑhÐ×|_h`ØAhÑÑgg §44|hÑ6#gh¤¿hD|_#ÛgÑÑ/Pg#:Óª
E(#¯#gÛ#4g'h 4Ѽ4Ý7g#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)|
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Q
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N = pqh
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(Z N2Z)∗4Ch 4
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ë Ì ##ògh4D¯ÍÑÑKÔ6D#ò44#Z NsZ
í
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Z NZâ`Hhâ
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M = |E(Fp)| · |E(Fq)|
•÷ #?h64ª
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ord(Q)|ME(Z Ns+1Z)
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Q′ = (x′ : y′ : z′) ∈ E(Z Ns+1Z)h
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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)
ûpȯ_Î4ÈgÛ#zܤg?4 |_È#ªÑ2# g4gÝ4F\|h
s + 1ëqg
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|h4 gN sâ#Hhâ
N sP1 = O
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âK#g/ªgg(a, b)
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#84C = rQ +mP1
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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_Ó4gѼÛh _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#Ѽ
#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 #xܤg4 |_Dh7#Ò84
gá|hÑ ∑im
′iN
i ·N + · · · = m′0N +m′
1N2 + · · · + · · ·
fûÈFh \g##DÓhhÑ#hm′
0
hm′
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q4m′P1
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m′1 = x
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Q ∈ E(Z Ns+1Z)h2h \ Ó? #
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h`¤Í©YÑñh4ò#gh 4Ó4g&h 46hC ∈ E(Z Ns+1Z)
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kh
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t ≤ k − 1h |_C#
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ke
hÑ4#ôÍgg\PÎ4\¼|hgs1, . . . , s`
âh #Ó4gî|F_4(Îh\¼?
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•÷ #?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 hÑͧF¼?
Q"%»º Q®,O ¯ ' °½¼¾®O󳯬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. O 5
h#D|_gcd(M,N s) = 1.
í ÷ 4#:#\#ò84Ѽ?#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#ÑñF¼4ggâÐͧPq#:#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#ѧÓ_hÑ#g
Vh hÑÑ
Vi
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f_Ð#gk64ªÐ##á#gѼ \g##g?/ªggª
ai
q40 < i < k
hfh #g#g
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1, . . . , ` Ö #:#g4hâq4§hª¼Y#Ô#\Òh
k− 164ª1, . . . , `
# :Ô#/4#h`#ÈÓhhÑ#ghf_#gÝ84h##qh
h Ý/|hÑÑï¼C \6g gªF
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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®O󳯬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. Om4
Ì #g ##p ¨t£ 1 # 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<ͧgÑÑ6h©hª¼×¯?g|hÑ4Ô#g:Ó4gÈ|_È#ÑñR¼4g
i|hò g Ý#Ñ#Ñ Ôª¼ô#ògg\Ý|h
si
YϤ ù?g\4AP#i< #4?ܤªgh\Ó4ð#hg4Ñq4C#ChÎ"ÍÑÑÔ6ð#gg Í# PÔ|h 4'#g4g64 #×4#òA kG:HÖ 4hª¼P#Ô#\
Shk64ªÒ
1, . . . , `h q4
i ∈ Sͧ \î|#
λS0,i = ∆
∏
i′∈ S\i
−i′
i− i′∈ Z.
Ì #gÝÓ_hÑ#gh g\Ѽ 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#Ññݧq4ÑÑRÍ©|_
∆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
ͧò|FÓª
C ′ = ∆2M ′rQ︸ ︷︷ ︸O
+∆2M ′mP1 = ∆2M ′mP1 = ∆2mP1,
Q"%»º Q®,O ¯ ' °½¼¾®O󳯬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. OGN
Í#gm`# g #Ññh¯?F(7g#gͧFhg4×#/
mg_Ó4gѼÝh<Ô8\q4ëqg
ù?g\4P# P#65 í h #Ñ#Ñï¼Y#Èg#ÑÔª¼(∆2)−1 mod N s.
¿hò|_Í©ÈFh&hÑͧF¼?h#D|_
p, q `
ù?#g×#g#g:¼?/4 #g #YgYh#ò##¯|h#4 ÈF_ FhѼðÔ6ggp|_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
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k − 1#ÑñF¼4gq4#ÝÔ6g4####ûp#×C:h h
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Ø m/ ¡ ¨CL¦£  L 1 MY× ¢ L/¨B, ¢ ¨BYCLF ¥ ¦ ¡CL  / ØÙý ¥ /ØCØÙ ) _/¢ R CØCØÙ / ¢ ¨\) MG L CL £*ú L/¨B\ý q M M  ^ LL/§ONGqCL ¥ ) § Ø ¢ Ù ¢ C+CL Ø §ONGìON §ÁqCC  ØÙ Â )£Øý¤G£ £CØÙC) M
Q"%»º Q®,O ¯ ' °½¼¾®O󳯬^¯\¤x §y ' ' ³¿()³C²ï±®À&ZÁ+³ ' ' ³®YL²E ,. O P
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Ci
âÍ#¯òÍÑÑ4Ô6Ъ #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?g4Ñ6ͧ: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#ÑѼ \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 # 64Ѽ?#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#
64Ѽ?#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
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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#ѼÛ#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#Ѽpgá#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#:#\Σܤ#h?g4ÑKq4§#ògÑñ_4
R = ((ke, C), r) | C = Eke,r(0)Í#g
ke
§#ò##Ô#ÑDÎh\¼.
#hg4Ñq4§#ÒFhQFhѼYÔ6ò4Ô/h# q4#òg4g64 #4#òA <kHaYLGW[Ê%J'-01)-?b@B'-0m'-0ZBIL01W01#$01t).¦IL01G,&-E%[
0 -&(mÞke = (N,S, (a, b), Q), CaYC`QIK0 )-&(t[Ê%t'-0 &(1`0ZZÞ
rÍ
0 ≤ r < N s×|_
C = rQ = Eke,r(0)
54 Ì #ò#Ó4gP#?4g
r′Í
0 ≤ r′ < N s_h 4Q7òg4#/g
A = r′Q = Eke,r′(0)
h g A×#Óªgî|g
V
4 V|hÑÑg#4g
PÍQh 4
tܤÔ#Ò:#
e
N Pg4#/g
z = r′ + erh g
z
V
P#V¯#gΩÍ#\#g
A,C ∈ E(Z Ns+1Z)h |_
Eke,z(0) = A+ eC7hggg/
Ah 4#ѼCÕ#§#òFhh 01#2#$I¦R-Öý3Öý=Ö Ì #©hÔ6RÓ4Ð#h?\4Ñ4q4Õ#gÎ#Í#\#g`hÈgÑg×gÕ7hÈg#g:¼?/4h
0
Σܤ#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\ñhѪ#4#g:7Ó4gî|g É g_ܤÎ#RÍÑ 4fÍ©Ð|FÓ4fÝ#Ññ_`g4ªÓ4g_4 (A, e, z)Í(#yhy#4Ô|hÔ#Ñ祿 :Ô#/h Ì #g\q4y#QÝ#Ññ_hݯ#4gzh
eh 4Ѽ"#gCg6g\Ó4 4Yh#gÏ7ð\A = Eke,z(0) − eC
h 4/#/(A, e, z)
â/Í#¯y4ÔªÓ?4#ѼPg4ªÓªg_4PÍy#òhD#4Ô|hÔ#Ñ祿 :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#/
Ϥ&4#h##ÑF_4Qh`#h #qhÑÑRÍ##h?\4ÑÍ©h#È|_#Ý #Ñ#N = pq
|hݤͩð#Th\4Ýh4#4#Ѽ#ChC É hA7g#ghâ< |N |È#PÔ#
Ñg#h×hNâªÍ©Fh×\
t = |N |/2òg##|_f##4Ô|hÔ#Ѥ¼hD#F_#ò#RÓ4g
×YhÎ4#DÓ4gî|ghggg/≤ 2−t
ðï6[ ñèéTò6ò"Aò"n òFééT¨ªóák¦Ká大¦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#ÑÒׯ/#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
tܤÔ#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#ѼCÕ#Ò§#DFhh 01#2#$IpR-Öý3ÖR-Ö Ì #©hÔ6RÓ4Ð#h?g4Ñ4q4Õ#RÓ?#g4g\AÝ#Ñ#ÑF_4ò7
Σܤ#h?g4Ñ
#RÓ?#òÎ#RÍÑ 4hm2, r2, r3
#¯×|_Cm2
= r2Q+m2P1
h D = m2Cm1
+r3Qâ
Hhâ|#RÓ?#|_Dg#g:¼?/
m2m1
Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' hG<
R ¡ M ù?#Í#×g4#Ñ\g#gÒ:h4ªqhͧh #g
wP1 + zQ = (m+ em2)P1 + (u+ er2)Q = mP1 + uQ+ e(r2Q+m2P1) = B + eCm2
h
wCm1+yQ = (m+em2)Cm1
+(v+er3)Q = mCm1+vQ+e(m2Cm1
+r3Q) = A+eD.
Ö 46ggñhÑ#4#g:CÓ4gî|g É g_ܤÎ#RÍÑ 4hâ4Ó4g'hª¼"¯|hÑÑ\#4 e g4g\Ѽ .ÜÔ#/ g4ªÓªg_4yFhQÔ6òÝ#Ññ_ h§q4ÑÑRÍgA#òÝ#Ññ_4©#?4g#ÓhhÑ#gw, y, z
_òh 4ð#gDg8g\Óª 4Yh#g¦7Ý#\g4#/gòY_##CÓhhÑ#gA,B
##×#Dgá|_4#wP1 + zQ = B + eCm2
h wCm1
+ yQ = A+ eD
Ö 44# #gÍ©h#È|_q44òÓhhÑ#Èh(A,B)
##RÓ4gPFh&g4g\Ѽ
h#:Í©gC.Í© åKggªC¯|hÑÑg#4ge, e′
Í#g&Íò ÑH # #e > e′
7g#ghâ#Ûq4ÑÑRÍ#gá|_4#hD_:î|
wP1 + zQ = B + eCm2, wCm1
+ yQ = A+ eD
h w′P1 + z′Q = B + e′Cm2
, w′Cm1+ y′Q = A+ e′D.
Ì ##Ñg§|_(w − w′)P1 + (z − z′)Q = (e− e′)Cm2
h (w − w′)Cm1
+ (y − y′)Q = (e− e′)D.
7g#gCm2
= (w − w′)(e− e′)−1P1 + (z − z′)(e− e′)−1Q
h D = (w − w′)(e− e′)−1Cm1
+ (y − y′)(e− e′)−1Q.
ûpPFh"g4#gÑ P|_m2 = (w − w′)(e − e′)−1, r2 = (z − z′)(e − e′)−1
h |_r3 = (y − y′)(e− e′)−1
Ì #DÒ g hQg#g:¼?/4Qh
m2m1
Q"¿ëa% ±ìK³ ' ³+®.Zq®¯1§¯1²¯ ' h O
ðï6ðï ñèéTò6ò"Aò"¡ñu¦¢Öåßᨾ bái`ßòC ¦¨àÖ¨
ûpòͧh4Ô/hÛΣܤ#h?g4Ñ|_4Ó4g&hQg#g:¼?/4
C = Eke(m)
hm#Ó4g
|_PÎ#RÍÐ#g4g64 #gh4
m<ù?#PÈ#h?g4Ñ8<g4h# #ÑgÑï¼
(#ÛhÔ6RÓ4Q#h?g4ѧq4×#Ó?#ghg¯CÝ#Ñ#ÑF_47ghâ§#QgÑñ_hR
4ÔªÓ?4#Ѽ
R = (ke, Cm2), (m2, r2)) | m2 ∈ M, Cm2
= Eke,r2(m2).
aYLGW[Ê%8&1`-)-?$&(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
tܤÔ#Ò|hÑÑg#4
e
P
N Pg4#/g
w = m+ em2, z = u+ er2h g
(w, z)
V
P#V#gΩ|_
wP1 + zQ = B+ eCm2
h hggg/©h 4#ѼK#<f#Fhh 01#2#$I R-Öý3Öý3Ö Ì #hÔ6RÓ4#hg4Ñ8q4Ð#RÓ?##Ññhª¯?©Î#RÍÑ 4§
Σܤ#h?g4Ñ
#RÓ?#×Î#ÍÑ hÈhm2
h r2#y|
Cm2= r2Q+m2P1
Ì #È#h7q4ÑÑRͧ ñgѼq4 ü2gY_P#i<? P#
ðï6G ã*ò"Yáã*ò"Eã~@ñèéTòÁáò6ó¾òÖ¦
ûp#RÍ ggÔ6Σܤ#h?g4Ñ#|_§4Ó4g.Í©Ý\##g:¯?
C1, C2
â#h ###Ô#ÑÎh\¼ke
#RÓ4gA|_A#©#Ó4g`Î#RÍ`4#©h|#©g4g64 #gh4gAÍ#4/A\Ó4FhÑ#Í#ð4#g Ì ##46g:¤¼ÛFhÑÑ ÁqC _ ¥ ý/) 1¢ ý £CØÙC)_DÔ8\q4Ý#È\á#ÓhhÑgªÈÛ#RÓ?#y|_È##RÓªgÈÎ#RÍÝ4#g4g84 #Û_ 4 ÓhhÑ#
r&.Í©ðg#g:¼?/4#×h\:/âfHhâ<C1 = Eke,r1
(0) = r1Qh
C2 = Eke,r2(0) = r2Qhg#g:¼?/4## gÈ#Yh##Ô#Ñ×Î4\¼
ke
h #Yh 4 ?#ÝÔ6gr1h
r2â
Q"e;% T ' ³²-+K)³¯¬]§¯$± ' )³¿B+®V$ ¯X(±QK+K)³¯¬ hGh
g6g\Ó4gѼ4âAQ#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_gѼÔ##Ñ #"&#h?g4ÑHÛûpCÍÑÑÐh#CÍ#4/ÝÑ4Ýh4g#ghѤ¼ô|_Ý#C#Ó4gÎ#RÍ
r1#|_
C1 = Eke,r1(0)
h Í#gS g#hg<##4#g::ÜóÓ4gî|g<#Ññ4
q4<##hg4Ñ8qhЯ#gÎ#Í#\#gÒhYgÑggª©©hYg#g:¼?/4Ch0h7ù?#Ô#g\4
P#i<? N ¬4¬%[x¬ $&LGW -&(mÞ
ke = (N,S, (a, b), Q), C1, C2aYC`QIK0B)-&(Y[Ê%Y'-0B&1`0ZZÞr1Í
0 ≤ r1 < N sÈ|_
C1 = r1Q = Eke,r1(0)
54 Ì #×#RÓ4gP#4gÈh 4Ñï¼
rÍ
0 ≤ r < N s+7תÓ44ÎhgD##4#g::Ü
Ó4gî|gÈÝ#Ññ_4S4p##/
ke, C2
Û4Ô/hyg4ªÓªg_4(A2, e2, z2)
Pg4#/g
A1 = rQ = Eke,r(0)h \
A1
h A2
#DÓ4gî|gV
4 V#?4g_ 4ѼP
tܤÔ#Ò¯|hÑÑg#4
eh \ §
P
N Pg4#/g
e1 = e − e2
âKÍ#\ÝÍò ÑH # #e > e2
âh z1 = r + e1r1
¸7g e1, z1, e2, z2
V
P#V¯#gΩÍ#\#g
A1, A2, C1, C2 ∈ E(Z Ns+1Z)h |_
Eke,z1(0) = A1 + e1C1h
Eke,z2(0) = A2 + e2C2
7Èhggg/Ah 4#ѼCÕ#§#òFhh
¿RÍ*ü2gY_P#i<? N h #ò#?h7g##á#gq4 4O 2#Ѽ#Dq4ÑÑRÍ#×ü2gY/ 01#2#$IbR-Öý3Ö Ö Ì #hÔ8Ó4p5\ܤ4/:ܤhñÜ 4 #h?g4Ñ
Σܤ#hg4ÑH
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E(b)h ð#Ññh¯?
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Pi
âRq41 ≤ i ≤ n
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E(c) = E(a · b)4Ó4gQg#g:¼?/4#hE(a)
h E(b)
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E(a), E(b)ot&(mÞE(c) = E(a · b)
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E(a)h
E(r1), . . . , E(rn) Ì #ò
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h
ù?#gYͧCÔ|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
hÑѼ?#Ph/#4gAi
q41 ≤ i ≤ n
ÍÑÑ`h\h gg:¼?/4g:Óªgg Ì Q4|hhªg#òg4g\#gh7#òî||hÑhÑѼͩhÑ×|FÓ4òh#D|_
N s > mLâ#Í#g
m
##ÝÔ6gÐhÓ4hg§h L##ÝÔ6gÐh2Fh _ggq¿h|_<#ÐFhPhÑͧF¼?ÐÔ6
q#Ñî|ÑÑ Ôª¼P#?4#s4NÑñh4g#4#4
ϤD#`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\Ó4gѼÝ#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#:ͧgzY¿h
|_Í©|RÓ4È#gÑ ID(Vi)
y###/hhY#\Ó4gÓ4h ##ÑF_4 Ö #.Ü
#g4hâ#hÒ|_ÐYhÑÑ|h?/Ñh:¼Ý#hg4Ñ<<FhYhÑÍÒF¼?<Ô6FhѼg4#/ 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\ܤ4/:ܤhñÜ 4 #hg4Ñ`_Òù?#Ô#g\4bP#i<? O Y764:D#g#g:¼?/
Ó4hD44\#gÒÍP#ò#?h7#òÔ##ÑÑ\PÔ8ªh B
4 Ù<hyhÑѼCh/#4.¼Aj
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C = O
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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
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E Ö 4Ð#Ñg祿ÝÍ©Dh##Í|_§hÑÑ|Ó4hg§h
ÓhhÑ Ì #gyͧò4Ô/h
C =∑
i
E(vi, ri) = E(∑
i
vi mod N s,∑
ri mod N s).
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Í# ∑
i vi
#gDÍ©ò|RÓ4N s > m
|hÑ444#Ѽ 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#Ñ祿4âFg4#/_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?ܤªgh\ÓªgѼhâh#\¼hÈÓªgî8hÔ#ÑÔª¼&hª¼Û4Ô#g:Ó4gg Ö #:#g4hâKhª¼Û4Ô#g:ÓªgFhð#g¯ÎQ#Èî||hÑ7hÑѼÍyg6g\hÑÑKÓ_hÑ Ô|hÑÑhg Ì #4Ó4g###Ó4ghÑÓ4gî8hÔ#Ѥ¼4##&#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
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CÏó©Í§Yh#×|__4:k − 1
hÑѼ?#Qh/#4gòhg4#/ â#ͧ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
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