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memorizing-crypto.pdf

H Koba
March 20, 2025
790

 memorizing-crypto.pdf

H Koba

March 20, 2025
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Transcript

  1. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w ͱͷؔ܎ɺͲ͏͔ͩͬͨͳʜ  w " Eͱ͢ΔͱE   w

    W?V? "V? VͱY? Z? EY?Z?ͷม׵ɺͲ͏ͩͬͨ ͔ͳʜ  w Y  Z 
  2. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w ͱͷؔ܎ɺͲ͏͔ͩͬͨͳʜ  w " Eͱ͢ΔͱE "  "

      w W?V? "V? VͱY? Z? EY?Z?ͷม׵ɺͲ͏ͩͬͨ ͔ͳʜ  w YTRSU  "  VW Z V  V 
  3. ಄ͷதͰ࣮ݧɾߟ࡯Ͱ͖Δ w &EXBSETͷج఺ͷZͬͯਐͰͲ͏͚ͩͬͨͬʜ  w Q?͔ͩΒQNPE w ͔ͩΒNPEQ Q 

    ͩ w Q ?Y ff  ff  ff Cͷ͸ͣ w ͜ΕΛͰׂΔͱYͩ w ࣮ࡍʹιʔεΛಡΈʹߦ͘ͱ߹ͬͯΔ
  4. ಄ͷதͰ࣮ݧɾߟ࡯Ͱ͖Δ w ҉ࢉͰ΍Δҙຯ͸  w ͜͏ͨۤ͠࿑ͨ͠Τϐιʔυ͕͋ΔͱZͱ͍͏஋Λ๨Εʹ͍͘ w $VSWFͰVͱ͍͏ͷ΋͓͚֮͑ͯ͹ɺZ V 

    V  ΍V  Z  Z ΋๨Εʹ͍͘ w ୶͍هԱΛෳ਺͓͍࣋ͬͯͯఆظతʹݕࢉ͢Δ͜ͱͰɺهԱΛڧݻʹ͢ΔΠϝ ʔδ
  5. هԱϝιου w ࣮૷ΛಡΉ Πϯϓοτ  w ࿦จ3'$ΛಡΉ Πϯϓοτ  w

    ࣗ෼Ͱίʔυΰϧϑͯ͠ΈΔ Ξ΢τϓοτ  w ҉ࢉ͢Δ ෼ੳ
  6. ҉ࢉ͢Δ ෼ੳ ΍Γํ w Kෆมྔ͕Kͷପԁۂઢɺ ͚ͩͬͨͬʜ  w ͷKෆมྔ͸ ͚ͩͬͨͬʜ

     w ͦ͏ࢥͬͯKෆมྔΛܭࢉ͢Δͱ ʹͳΓɺ๨Εʹؾ෇͚Δ y2 = x3 − 3j j − 1728 x + 2j j − 1728 y2 = x3 + ax + b 4a3 4a3 + 27b2 j 1728
  7. ҉ࢉ͢Δ ෼ੳ ΍Γํ w ಄ͷதͰ΍Δํ๏ w ͱͯ͠͠·͓͏ɻBK" CK" w 

    w ෼ࢠͱ෼฼ΛK?"?ͰׂΔͱ j − 1728 = A 4a3 4a3 + 27b2 = −108j3 A3 −108j3 A3 + 108j2 A2 −j −j + A = j 1728
  8. "4$**DPEF w "Yͱ͔ͩͬͨΓ͢ΔΞϨ w ӳޠͷΞϧϑΝϕοτʹϥϯμϜΞΫηεͰ͖Δͱڧ͍ w ͤΊ్ͯதͷϙΠϯτΛ֮͑Α͏ w % )

    - 1 5 9 w ेਐ਺ͱे࿡ਐ਺ͷม׵΋Ͱ͖ΔΑ͏ʹͳ͓ͬͯ͜͏ w        ·Ͱͷͷഒ਺Ͱ͍͍ͷͰָ
  9. ॳ౳੔਺࿦ w ۩ମྫͷๅݿ w ฏํ৒༨ͱ͔৐๏܈ w ͳΒB͸ฏํ৒༨ w ͳΒNPEQͰͷ৐͕ࠜ͋Δ w

    ଘࡏఆཧͳͷͰ࣮ࡍͷߏஙͱͷؒʹ͸Ϊϟοϓ͋Γɺ࣮ࡍʹௐ΂Α͏ w ϥϯμϜʹBΛͱΕ͹ ͕֬཰Ͱඇࣗ໌ͳ৐ࠜ a(p−1)/2 ≡ 1 (mod p) p ≡ 1 (mod 3) a(p−1)/3
  10. YFEC@ͬͯԿ w ('  ͷଟ߲ࣜ w ͭ·ΓNPEͰ৭ʑ΍Δͱ͍͏͜ͱ w ԼҐϏοτ͕࣍਺ͷߴ͍ଆ w

    Y?͸লུ͞Ε͍ͯΔ w ͭ·Γ ͱ͍͏͜ͱ w YFC YECʹ஫ҙɻ (1 + x + x2) + (x4 + x5 + x7) + ⋯ + x26 + x32