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Coin Locker - Time Operation

Coin Locker - Time Operation

現代暗号理論の輪講講義用のスライド.
公開鍵暗号・RSAを理解するための準備.
教科書:暗号技術入門 第3版(著:結城 浩)

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Kensuke Kosaka

October 20, 2015

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  1. Ճࢉ • ӈਤͷঢ়ଶ͔Β13Λ଍͢ͱɼ ݁Ռతʹ਑͸1Λࢦ͢ • (0 + 13) mod 12

    = 1 • modͱ͸ɼׂΓࢉΛͯ͠༨Γ ΛٻΊΔܭࢉΛද͢ • ී௨ʹ଍͠ࢉΛ͠ɼͦͷ݁Ռ Λ12Ͱׂͬͨ༨Γ͕਑ͷҐஔ ʹͳΔ 20
  2. ݮࢉ • ਑͸ӈճΓʹ͔͠ճΒͳ͍ • ͲͷΑ͏ʹͯ͠਑Λ0ʹ
 ࣋ͬͯߦ͔͘ • (3 + 9)

    mod 12 = 0 • ࣌ܭԋࢉʹ͓͍ͯɼ9͸–3ͱ
 ಉ͡໾ׂΛՌͨ͢ • ݮࢉΛՃࢉʹม׵͢Δ 21
  3. আࢉ • ৐ࢉͷٯԋࢉΛߟ͑Δ • (0 + 7 × A) mod

    12 = 1 • ͜Ε͸ɼmod12ͷੈքʹ͓͚ ΔҎԼͷࣜͱಉ͡ • 0 + A = 1 ÷ 7 • Ұͭ໨ͷࣜΑΓɼAʹ0͔Βॱ ʹ౰ͯ͸Ί͍ͯ͘ͱ7͕ٻ·Δ • mod12ͷੈքͰ͸ɼ1 ÷ 7ͷ౴ ͑͸7ʹͳΔ 23
  4. ٯ਺ • mod12ͷੈքʹ͓͚Δɼm × n = 1 • mͱn͸ɼٯ਺ͷؔ܎ʹ͋Δʢֻ͚Δͱ1ʹͳΔʣ •

    0͔Β11ͷதͰɼͲΜͳ਺ʹ΋ٯ਺͕͋Δͷ͔ʁ • RSAͰެ։伴ͱϖΞʹͳΔϓϥΠϕʔτ伴͕ଘࡏ͢Δ ͔Ͳ͏͔ͱ͍͏໰୊ʹͭͳ͕Δ 24
  5. ٯ਺ • mod12ͷੈքʹ͓͚Δٯ਺Λ࣋ͭ਺ • ͦͷ਺ͱ12ͷ࠷େެ໿਺͕1Ͱ͋Ε͹ɼٯ਺Λ࣋ͭ • 5 × 5 mod

    12 = 1 • 7 × 7 mod 12 = 1 • 12ͱޓ͍ʹૉͳ਺ʢ12ʹͱͬͯͷૉ਺ʣ 25
  6. ྦྷ৐ʢ΂͖৐ʣ • ྦྷ৐͸ɼ৐ࢉΛ܁Γฦͨ͠΋ͷ • 74 = 7 × 7 ×

    7 × 7 • ࣌ܭԋࢉͰ͸ɼྦྷ৐ͷܭࢉΛ͔ͯ͠ΒmodΛऔΕ͹͍͍ • 74 mod 12 = 7 × 7 × 7 × 7 mod 12
 = 2401 mod 12
 = 1 • ·ͨɼྦྷ৐ͷܭࢉ్தʹmodͷܭࢉΛڬΜͰ΋ྑ͍ • ܭࢉաఔͰѻ͏਺͕େ͖͘ͳΔͨΊʢRSAͷॲཧͰ΋࢖ΘΕΔʣ 26
  7. ର਺ • ྦྷ৐ͷٯԋࢉ • 7k = 49 • kʹ͸2͕ద౰Ͱ͋ΔͱΘ͔Δ •

    ී௨ͷ਺ֶͰ͸ɼ͞΄Ͳ೉͍͠ܭࢉͰ͸ͳ͍ 27