暗号技術を支える素数

by Shohei Yamasaki

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҉߸ٕज़Λࢧ͑Δૉ਺ ϓϩάϥϚͷͨΊͷ਺ֶษڧձ@෱Ԭ  2015.09.04 @shoyan

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4IPIFJ:BNBTBLJ !TIPZBO • ϓϩάϥϚʔ • 2ࣇͷ෕ • GMOϖύϘ

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໨࣍ • ҉߸ٕज़ʹ͍ͭͯ • ެ։伴҉߸ʹ͍ͭͯ • RSA҉߸ͷ҆શੑʹ͍ͭͯ • ҉߸ֶతͳ਺ʹ͍ͭͯ • ·ͱΊ

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҉߸ٕज़ʹ͍ͭͯ

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Πϯλʔωοτͷ໰୊ • ৘ใͷ౪ௌ • ৘ใͷվ͟Μ • ͳΓ͢·͠

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͜ΕΒͷ໰୊Λղܾ͢ΔͨΊʹ ҉߸ٕज़͕࢖ΘΕ͍ͯΔ

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҉߸ٕज़͸ͭʹ෼ྨ͞ΕΔ

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৘ใΛಡΉ͜ͱ͕Ͱ͖ͳ͍Α͏ʹ  ͢ΔͨΊͷ҉߸ٕज़ • ౪ௌऀ͕৘ใΛݟͯ΋ɺ಺༰͕Θ͔Βͳ͍Α ͏ʹ͢Δ • SSL / TLS • SSH • IPsec

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৘ใ͕վ͟Μ͞Ε͍ͯͳ͍͜ͱΛ֬ ೝ͢ΔͨΊͷ҉߸ٕज़ • ૹΒΕ͖ͯͨ৘ใ͕ਖ਼͍͠৘ใ͔Ͳ͏͔Λ  νΣοΫ͢Δʹ͸ɺϋογϡνΣοΫΛ༻͍ Δ • SHA-1 • MD5

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৘ใΛૹͬͨ૬खΛ֬ೝ͢ΔͨΊͷ ҉߸ٕज़ • ৘ใΛૹ৴ͨ͠૬ख͕ਖ਼͍͠૬ख͔Ͳ͏͔Λ νΣοΫ͢Δɻిࢠॺ໊ͱݺ͹ΕΔํ๏͕͋ Δ • ೝূہʹిࢠূ໌ॻΛൃߦͯ͠΋Β͏ • https௨৴Ͱར༻͞Ε͍ͯΔ

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ެ։ݤ҉߸ʹ͍ͭͯ

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ެ։ݤ҉߸ • ΠϯλʔωοτͰ࠷΋࢖ΘΕ͍ͯΔ҉߸ٕज़ • ৘ใͷ౪ௌΛ๷͙҉߸ٕज़΍ిࢠূ໌ॻͷੜ ੒ʹ࢖ΘΕ͍ͯΔ

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ެ։ݤ҉߸ • ެ։伴Λ࢖ͬͯ҉߸Խ͢Δ • ൿີ伴Λ࢖ͬͯ෮߸Խ͢Δ • ड৴ऀ͸͋Β͔͡Ίެ։伴Λૹ৴ऀʹ౉͓ͯ͠ ͘ • ެ։伴Ͱ͸෮߸ԽͰ͖ͳ͍ͷͰ౪ௌ͞Εͯ΋໰ ୊ͳ͍

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ެ։ݤ҉߸ํࣜͰ࠷΋ීٴ͍ͯ͠Δ 34" • Ϛανϡʔηοπ޻ՊେֶͷRon Rivestɺ  Adi ShamirɺLeonard Adlemanͷ3ਓ͕։ൃ • 3ਓͷ։ൃऀͷ಄จࣈ͔ΒRSA҉߸ํࣜͱ໋໊ ͞Εͨ

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ެ։ݤͱൿີݤͷੜ੒ • ൿີ伴ͱެ։伴͸ϖΞʹͳ͍ͬͯΔඞཁ͕͋ Γɺ͜ͷ2ͭͷΩʔͷ͜ͱΛΩʔϖΞͱݺͿ

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ΩʔϖΞͷੜ੒खॱ PͱQΛ࢖ͬͯNΛܭࢉ͢Δ PͱQΛ࢖ͬͯLΛܭࢉ͢Δ LΛ࢖ͬͯEΛܭࢉ͢Δ EͱLΛ࢖ͬͯDΛܭࢉ͢Δ

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1ͱ2Λ࢖ͬͯ/Λܭࢉ͢Δ • ʮPʯͱʮQʯͷ2ͭͷૉ਺Λ४උ͢Δ • ஋͕খ͗͢͞Δͱղಡ͕༰қʹͳͬͯ͠·͏ • ஋͕େ͖͗͢Δͱॲཧʹ͕͔͔࣌ؒΔ • 1024 - 4096Ϗοτ(10ਐ਺Ͱ300-1000ܻఔ౓) ͕ਪ঑͞Ε͍ͯΔ

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ૉ਺ͱ͸ • ૉ਺ͱ͸1ͱࣗ෼ࣗ਎Ҏ֎ʹਖ਼ͷ໿਺Λ࣋ͨͳ ͍ࣗવ਺Ͱɺ1Ͱͳ͍਺ͷ͜ͱ ҎԼͷૉ਺ • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

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Τϥτεςωεͷ;Δ͍ • ࢦఆ͞Εͨ੔਺xҎԼͷશͯͷૉ਺Λൃݟ͢Δ ΞϧΰϦζϜ • ݹ୅ΪϦγϟͷՊֶऀɺΤϥτεςωε͕  ߟҊͨ͠ • ͍·΋ݱ໾Ͱ࢖ΘΕ͍ͯΔํ๏

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/ /VNCFS ͷܭࢉࣜ N = P ✖ Q N = 13 ✖ 17 N= 221 • ʮPʯΛʮ13ʯɺʮQʯΛʮ17ʯͱͯ͠ʮNʯ Λܭࢉ͢Δ

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1ͱ2Λ࢖ͬͯ- -$.-FBTU $PNNPO.VMUJQMF Λܭࢉ͢Δ L = lcm(P - 1, Q - 1) • ʮP - 1ʯͱʮQ - 1ʯͷ࠷খެഒ਺ΛٻΊΔ • LͷܭࢉͰ͸ɺNͷܭࢉͰ࢖ͬͨPͱQΛ࢖͏

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-ͷܭࢉ݁Ռ L = lcm(13 - 1, 17 - 1) L = 48 • 12(13 - 1) ͱ16(17 - 1) ͷ࠷খެഒ਺͸ʮ48ʯ

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-Λ࢖ͬͯ& &ODSZQUJPO Λ  ܭࢉ͢Δ Eͷܭࢉࣜ 1 < E < L god(E, L) = 1 • ʮEʯ͸ʮ1ʯΑΓେ͖͘ɺLΑΓখ͍͞ • ʮEʯͱʮLʯͷ࠷େެ໿਺͕1ͱͳΔ(ޓ͍ʹૉ)

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&ͷܭࢉ݁Ռ • 48ͱૉʹͳΔ਺ʮEʯΛٻΊΔ • ʮEʯͷ஋͸ෳ਺ಘΒΕΔͷͰɺ͜ͷத͔Βૉ਺Λ1ͭ  બ୒͢Δ god(E, 48) = 1 E = 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47

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&ͱ-Λ࢖ͬͯ% %FDSZQUJPO Λ  ܭࢉ͢Δ • ʮEʯͱʮDʯΛֻ͚߹Θͤɺʮmod LʯΛ  ܭࢉͨ͠஋͕1ʹͳΔΑ͏ͳʮDʯͷ஋ΛٻΊ Δ E ✖ D mod L = 1 Dͷܭࢉࣜ

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%ͷܭࢉ݁Ռ • ʮEʯͷ஋Λʮ5ʯɺʮLʯͷ஋Λʮ48ʯͱ͠ ͯܭࢉͨ͠৔߹ 5 ✖ D mod 48 = 1 D = 29

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࡞੒͞ΕͨΩʔϖΞ ެ։伴͸ʮEʯͱʮNʯ E = 5 N = 221 ൿີ伴͸ʮDʯͱʮNʯ D = 29 N = 221

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34"ͷ҉߸Խॲཧ • ฏจΛʮEʯ৐͠ɺͦͷ݁Ռͷ஋ΛʮNʯͰׂ Γɺ༨Γͷ஋ΛٻΊΔɻ • ʮEʯͱʮNʯ͕ެ։伴 ҉߸จ = ฏจ mod N E

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҉߸Խॲཧͷܭࢉ • ฏจͷ஋͕ʮ12ʯɺʮEʯͷ஋͕ʮ5ʯɺ ʮNʯͷ஋͕ʮ221ʯͷ৔߹ ҉߸จ = 12 mod 221 ҉߸จ = 207 5

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34"ͷ෮߸Խॲཧ • ҉߸จʮDʯ৐ͯ͠ɺͦͷ݁Ռͷ஋ΛʮNʯͰ ׂΓɺ༨Γͷ஋ΛٻΊΔɻ • ʮDʯͱʮNʯ͕ൿີ伴 ฏจ = ҉߸จ mod N D

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෮߸Խॲཧͷܭࢉ • ҉߸จͷ஋͕ʮ207ʯɺʮDʯͷ஋͕ʮ29ʯɺ ʮNʯͷ஋͕ʮ221ʯͷ৔߹ ฏจ = 207 mod 221 ҉߸จ = 12 29

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34"҉߸ͷ҆શੑʹ͍ͭͯ

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҉߸จΛղಡ͢Δͧʂ • ҉߸จ͸ʮDʯͱʮNʯ͕Θ͔Ε͹෮߸Ͱ͖Δ • ʮNʯͱʮEʯ͸౪ௌ͢Δ͜ͱ͕Ͱ͖Δ • N = P ✖ Q • ΩʔϖΞ͸ ʮPʯͱʮ Qʯ͔Β࡞੒͞Ε͍ͯ Δ

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ʮ/ʯΛʮ1ʯͱʮ2ʯʹ෼ղ ͢Ε͹҉߸ΛղಡͰ͖Δʂ

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ʮ/ʯΛʮ1ʯͱʮ2ʯʹ෼ղ ʮૉҼ਺෼ղʯ

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࠷΋؆୯ͳૉҼ਺෼ղ 6 = 2 ✖ 3 • ָউʂ

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ΛૉҼ਺෼ղ 221 = 13 ✖ 17 • ͜Ε͘Β͍ͳΒͰ͖Δʂ

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70000000013390000000171

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͜ͷ22ܻͷେ͖ͳ਺͸ 100000000019 ✖ 700000000009 ͱ͍͏2ͭͷૉ਺ͷੵʹ෼ղͰ͖Δ

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ࢴͱԖච͕͋Ε͹ೋ਺Λֻ͚ࢉͯ͠ɺ΋ͱͷ ਺ʹ໭͢͜ͱ͸༰қʹߦ͑Δ ͚ΕͲ΋ɺେ͖ͳ਺ΛૉҼ਺෼ղ͢Δͷ͸೉ ͍͠

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ཱͪ͸͔ͩΔૉҼ਺෼ղͷน

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਺ͷܭࢉͱ҉߸ • ૉҼ਺෼ղͷ೉͠͞͸ܭࢉػΛ༻͍ͯ΋Ұ޲ ʹղܾ͠ͳ͍ • ҰඵؒʹԿઍԯճ΋ͷܭࢉ͕Ͱ͖ΔܭࢉػΛ ࣋ͬͯͯ͠΋·ͬͨ͘ࣃ͕ͨͨͳ͍

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ޮ཰తͳૉҼ਺෼ղͷํ๏ • ۃΊͯߴ౓ͳ਺ֶͷཧ࿦Λ༻͍ͯૉҼ਺෼ղ ΛٻΊΔํ๏͕ੈքதͷ਺ֶऀͷؒͰ೔໷ݚ ڀ͞Ε͍ͯΔ͕… • ͦ͏ͨ͠ཧ࿦Λ༻͍ͯܭࢉػͷϓϩάϥϜΛ ࡞ͬͯ΋ɺ೉͍͠໰୊Ͱ͋Δ͜ͱʹ͸มΘΓ ͸ͳ͍

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34"҉߸ͷ҆શੑ • ૉҼ਺෼ղ͕ࠔ೉ͳ͜ͱʹΑΓ୲อ͞Ε͍ͯΔ • ͨ͘͞Μͷܭࢉػͱ్ํ΋ͳ͍೥݄Λ͔͚ͯ ΍ͬͱD͕ٻ·ͬͨͱͯ͠΋ͦͷࠒʹ͸౪ௌ͠ Α͏ͱ͍ͯͨ͠৘ใͷՁ஋͕ͳ͘ͳ͍ͬͯΔ • ౪ௌ͠Α͏ͱ͍ͯ͠Δ৘ใʹରͯ࣌ؒ͠ͱίε τ͕ݟ߹Θͳ͍

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҉߸ֶతͳ਺ʹ͍ͭͯ

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େ͖ͳ਺ • ެ։伴҉߸Ͱ͸ͱͯ΋େ͖ͳ਺Λ࢖͏ • RSA҉߸Ͱ͸1000Ϗοτఔ౓ • 10ਐ਺ʹ͢Δͱ300ܻҎ্ͷ਺ࣈ

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ఱจֶతͳ਺ • ෺ཧֶऀͷEddingtonʢΤσΟϯτϯʣ͸ɺશ Ӊ஦ͷཅࢠͷਖ਼֬ͳ਺͸136✖2 = 1.575 ✖ 10 (Eddingtonఆ਺)ͩͱ͍ͯ͠Δ • ೔ຊޠͰ͸ɺҰ൪େ͖͍਺ͷҐ͕ແྔେ਺ (10 ) ͳͷͰ͜ͷ਺ΛಡΈ্͛Δ୯Ґ͸ଘࡏ ͠ͳ͍ 256 79 68

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ਖ߷هʢ͡Μ͜͏͖ʣʹΑΔ਺ͷ਺ ͑ํ

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҉߸ֶతͳ਺ • RSA҉߸Ͱ͸1000Ϗοτఔ౓ • 1000Ϗοτ͸໿1.80 ✖ 10 • ਺্͑͛Δ͜ͱ΋Ͱ͖ͳ͍͠ɺ૝૾͢ΒͰ͖ ͳ͍ 308

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·ͱΊ

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·ͱΊ • ҉߸ٕज़͸େ͖͘3ͭʹΘ͚ΒΕΔ • ৘ใΛಡΉ͜ͱ͕Ͱ͖ͳ͍Α͏ʹ͢Δ • ৘ใͷվ͟ΜΛνΣοΫ͢Δ • ৘ใΛૹͬͨ૬खΛ֬ೝ͢ΔͨΊͷ҉߸ٕज़

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·ͱΊ • ެ։伴҉߸ • ެ։伴ͱൿີ伴ͷ࡞੒ํ๏ • ҉߸Խͱ෮߸Խʹ͍ͭͯ

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·ͱΊ • RSA҉߸ͷ҆શੑʹ͍ͭͯ • ૉҼ਺෼ղ͕ࠔ೉ͳ͜ͱʹΑͬͯ୲อ͞Εͯ ͍Δ • ҉߸ֶతͳ਺͸్ํ΋ͳ͘େ͖͍

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͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠