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memorizing-crypto.pdf
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H Koba
March 20, 2025
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830
memorizing-crypto.pdf
H Koba
March 20, 2025
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Transcript
LPCBF ҉هͷ͢͢Ί ҉߸ٕज़Λཧղ͢ΔͨͬͨҰͭͷํ๏
͜ͷɾ֮ࣜ͑ͯ·͔͢ w ͋ͷ࠷ۙΓͷ44)ͷ伴λΠϓɺͳΜ͚ͩͬʜ w FE
͜ͷɾ֮ࣜ͑ͯ·͔͢ w ͋ͷ࠷ۙΓͷ44)ͷ伴λΠϓɺͳΜ͚ͩͬʜ w FE
͜ͷɾ֮ࣜ͑ͯ·͔͢ w FEͷݩͱͳ͍ͬͯΔ$VSWFͷࣜɺͳΜ͚ͩͬʜ w W?V? V? V
͜ͷɾ֮ࣜ͑ͯ·͔͢ w FEͷݩͱͳ͍ͬͯΔ$VSWFͷࣜɺͳΜ͚ͩͬʜ w W?V? V? V
͜ͷɾ֮ࣜ͑ͯ·͔͢ w ͰFEͷࣜɺͳΜ͚ͩͬʜ w Y? Z? Y?Z?
͜ͷɾ֮ࣜ͑ͯ·͔͢ w ͰFEͷࣜɺͳΜ͚ͩͬʜ w Y? Z? Y?Z?
͜ͷɾ֮ࣜ͑ͯ·͔͢ w ͱͷؔɺͲ͏͔ͩͬͨͳʜ w " Eͱ͢ΔͱE w
W?V? "V? VͱY? Z? EY?Z?ͷมɺͲ͏ͩͬͨ ͔ͳʜ w Y Z
͜ͷɾ֮ࣜ͑ͯ·͔͢ w ͱͷؔɺͲ͏͔ͩͬͨͳʜ w " Eͱ͢ΔͱE " "
w W?V? "V? VͱY? Z? EY?Z?ͷมɺͲ͏ͩͬͨ ͔ͳʜ w YTRSU " VW Z V V
Ͱɺ֮͑ͯԿͷʹཱͭͷ ϝϦοτ৭ʑ͋Δ͕ɺҎԼͷ͜ͱ͕େ͖͍ w ಄ͷதͰ࣮ݧɾߟͰ͖Δ w $5'Ͱมͳ࣮Λݟͨ࣌ɺᄿ͕֮ಇ͍͙ͯ͢ʹΘ͔Δ w ϫʔΩϯάϝϞϦʔ͕͑ΒΕΔ
಄ͷதͰ࣮ݧɾߟͰ͖Δ w &EXBSETͷجͷZͬͯਐͰͲ͏͚ͩͬͨͬʜ w Q?͔ͩΒQNPE w ͔ͩΒNPEQ Q
ͩ w Q ?Y ff ff ff Cͷͣ w ͜ΕΛͰׂΔͱYͩ w ࣮ࡍʹιʔεΛಡΈʹߦ͘ͱ߹ͬͯΔ
಄ͷதͰ࣮ݧɾߟͰ͖Δ w ҉ࢉͰΔҙຯ w ͜͏ͨۤ͠࿑ͨ͠Τϐιʔυ͕͋ΔͱZͱ͍͏ΛΕʹ͍͘ w $VSWFͰVͱ͍͏ͷ͓͚֮͑ͯɺZ V
V V Z Z Εʹ͍͘ w ୶͍هԱΛෳ͓͍࣋ͬͯͯఆظతʹݕࢉ͢Δ͜ͱͰɺهԱΛڧݻʹ͢ΔΠϝ ʔδ
$5'Ͱมͳ࣮Λݟͨ࣌ɺᄿ͕֮ಇ͍͙ͯ͢ʹΘ͔Δ Ͳ͜ʹ੬ऑੑ͕͋ΔͰ͠ΐ͏
$5'Ͱมͳ࣮Λݟͨ࣌ɺᄿ͕֮ಇ͍͙ͯ͢ʹΘ͔Δ Ͳ͜ʹ੬ऑੑ͕͋ΔͰ͠ΐ͏ˠೖྗͷWBMJEBUJPOΛ͍ͯ͠ͳ͍ IUUQTBMQBDBIBDLDPNDUGT[FSQUTDUGDIBMMFOHFTFEEI
ϫʔΩϯάϝϞϦʔ͕͑ΒΕΔ w $5'҉߸ϥΠϒϥϦʔͷಡղͰɺ͍ͦͦ͜͜ίʔυͷཧղ͕ඞཁ w ಄ͷதʹଟ͘ͷใ͕ࡌͤΒΕΔͱɺʹཱͭ w ಄ͷதͷใ͕ϦϯΫ͞Ε͍ͯΔͱɺ٫͠ʹ͍͘
هԱϝιου w ࣮ΛಡΉ Πϯϓοτ w จ3'$ΛಡΉ Πϯϓοτ w
ࣗͰίʔυΰϧϑͯ͠ΈΔ Ξτϓοτ w ҉ࢉ͢Δ ੳ
࣮ΛಡΉ Πϯϓοτ w ϏοτԋࢉΛۦͯ͠ذΛফͤΔͷ͔ʜ ڻ͖ w IUUQTHJUIVCDPNPQFOTTIPQFOTTIQPSUBCMFCMPC7@@@1 FED
จ3'$ΛಡΉ Πϯϓοτ w $VSWF w ͷV࠲ඪ͚ͩͰପԁۂઢͷεΧϥʔഒ͕ܭࢉͰ͖Δ w IUUQTDSZQUPFDEIDVSWFQEG
ࣗͰίʔυΰϧϑͯ͠ΈΔ Ξτϓοτ w $3$ͱ͔͔ͳΓ؆୯ චऀ࣮(PͰߦ w IUUQTTJ[VNFLPCB@FQPTUTVCBLNEPLOW
͜Ε͚ͩͰ͢
҉ࢉ͢Δ ੳ w ո͍͠ެࣜΛෳΈ߹Θͤͯɺໃ६Λݕग़ɾղফ͢Δ w ϫʔΩϯάϝϞϦʔΛϑϧՔಇͤͯ͞ɺ಄ͷதʹͯ͢ࡌͤΔ w ࡌΒͳ͔ͬͨΒؤுΖ͏
҉ࢉ͢Δ ੳ Γํ w Kෆมྔ͕Kͷପԁۂઢɺ ͚ͩͬͨͬʜ w ͷKෆมྔ ͚ͩͬͨͬʜ
w ͦ͏ࢥͬͯKෆมྔΛܭࢉ͢Δͱ ʹͳΓɺΕʹؾ͚Δ y2 = x3 − 3j j − 1728 x + 2j j − 1728 y2 = x3 + ax + b 4a3 4a3 + 27b2 j 1728
҉ࢉ͢Δ ੳ Γํ w ಄ͷதͰΔํ๏ w ͱͯ͠͠·͓͏ɻBK" CK" w
w ࢠͱΛK?"?ͰׂΔͱ j − 1728 = A 4a3 4a3 + 27b2 = −108j3 A3 −108j3 A3 + 108j2 A2 −j −j + A = j 1728
͓͢͢Ί w "4$**DPEF w ॳ w ପԁۂઢ
"4$**DPEF w "Yͱ͔ͩͬͨΓ͢ΔΞϨ w ӳޠͷΞϧϑΝϕοτʹϥϯμϜΞΫηεͰ͖Δͱڧ͍ w ͤΊ్ͯதͷϙΠϯτΛ֮͑Α͏ w % )
- 1 5 9 w ेਐͱेਐͷมͰ͖ΔΑ͏ʹͳ͓ͬͯ͜͏ w ·ͰͷͷഒͰ͍͍ͷͰָ
ॳ w ۩ମྫͷๅݿ w ฏํ༨ͱ͔๏܈ w ͳΒBฏํ༨ w ͳΒNPEQͰͷ͕ࠜ͋Δ w
ଘࡏఆཧͳͷͰ࣮ࡍͷߏஙͱͷؒʹΪϟοϓ͋Γɺ࣮ࡍʹௐΑ͏ w ϥϯμϜʹBΛͱΕ ͕֬Ͱඇࣗ໌ͳࠜ a(p−1)/2 ≡ 1 (mod p) p ≡ 1 (mod 3) a(p−1)/3
ପԁۂઢ w ͓͠Ζ͍ w ପԁۂઢେ͗ͯ͢໎͍͍͢ͷͰɺ$5'Ͱ͍ͦ͏ͳͱ͜Ζ͔Β w ͓͢͢Ίॳख༗ݶମ্ͷପԁۂઢ w खલຯḩIUUQTRJJUBDPNLPCBFJUFNT FCGCGGDBG
w 4BGF$VSWFTIUUQTTBGFDVSWFTDSZQUP
·ͱΊ w $5'Ͱ҉߸ཧͰɺָ͠ΉͨΊʹεϜʔζʹهԱ͢Δ͜ͱ͕ෆՄܽ w εϜʔζͳهԱͷͨΊʹΤϐιʔυ͕େࣄ w ҉ࢉεΩϧΛߴΊͯΤϐιʔυΛ಄ͷதͰ࿉͠Α͏
༧උεϥΠυ ༧උεϥΠυ
YFEC@ͬͯԿ w (' ͷଟ߲ࣜ w ͭ·ΓNPEͰ৭ʑΔͱ͍͏͜ͱ w ԼҐϏοτ͕࣍ͷߴ͍ଆ w
Y?লུ͞Ε͍ͯΔ w ͭ·Γ ͱ͍͏͜ͱ w YFC YECʹҙɻ (1 + x + x2) + (x4 + x5 + x7) + ⋯ + x26 + x32
&EXBSETͷՃ๏ެࣜ w IUUQTFEDSZQUPFEQEG w ΄ͱΜͲෳૉͷੵɺͨͩ͠ʜ w YͱZ͕ٯɺJ? w ͕͋Δ