memorizing-crypto.pdf

Transcript

1. LPCBF
    ҉هͷ͢͢Ί ҉߸ٕज़Λཧղ͢ΔͨͬͨҰͭͷํ๏
2. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w ͋ͷ࠷ۙ͸΍Γͷ
   44)
   ͷ伴λΠϓɺͳΜ͚ͩͬʜ
   w FE
3. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w ͋ͷ࠷ۙ͸΍Γͷ
   44)
   ͷ伴λΠϓɺͳΜ͚ͩͬʜ
   w FE
4. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w FE
   ͷݩͱͳ͍ͬͯΔ
   $VSWF
   ͷࣜɺͳΜ͚ͩͬʜ
   w W?
   
   V?
   
   V?
   
   V
5. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w FE
   ͷݩͱͳ͍ͬͯΔ
   $VSWF
   ͷࣜɺͳΜ͚ͩͬʜ
   w W?
   
   V?
   
   V?
   
   V
6. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w Ͱ͸
   FE
   ͷࣜɺͳΜ͚ͩͬʜ
   w Y?
   
   Z?
   
   
   
    Y?Z?
7. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w Ͱ͸
   FE
   ͷࣜɺͳΜ͚ͩͬʜ
   w Y?
   
   Z?
   
   
   
    Y?Z?
8. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w 
   ͱ
   
   ͷؔ܎ɺͲ͏͔ͩͬͨͳʜ
   w "
   
   
   E
   
   
   ͱ͢Δͱ
   E
   
    
   w W?
   
   V?
   
   "V?
   
   V
   ͱ
   Y?
   
   Z?
   
   
   
   EY?Z?
   ͷม׵ɺͲ͏ͩͬͨ ͔ͳʜ
   w Y
   
   
   Z
   
   
9. ͜ͷ஋ɾ֮ࣜ͑ͯ·͔͢ w 
   ͱ
   
   ͷؔ܎ɺͲ͏͔ͩͬͨͳʜ
   w "
   
   
   E
   
   
   ͱ͢Δͱ
   E
   
    "  " 
   w W?
   
   V?
   
   "V?
   
   V
   ͱ
   Y?
   
   Z?
   
   
   
   EY?Z?
   ͷม׵ɺͲ͏ͩͬͨ ͔ͳʜ
   w Y
   
   TRSU  "  VW
   Z
   
   V  V 
10. Ͱɺ֮͑ͯԿͷ໾ʹཱͭͷ ϝϦοτ͸৭ʑ͋Δ͕ɺҎԼͷ͜ͱ͕େ͖͍
    w ಄ͷதͰ࣮ݧɾߟ࡯Ͱ͖Δ
    w $5'
    Ͱมͳ࣮૷Λݟͨ࣌ɺᄿ͕֮ಇ͍͙ͯ͢ʹΘ͔Δ
    w ϫʔΩϯάϝϞϦʔ͕஁͑ΒΕΔ
11. ಄ͷதͰ࣮ݧɾߟ࡯Ͱ͖Δ w &EXBSET
    ͷج఺ͷ
    Z
    
    
    ͬͯ
    
    ਐͰͲ͏͚ͩͬͨͬʜ
    w Q
    
    ?
    
    
    ͔ͩΒ
    Q
    NPE
    
    
    
    
    
    
    
    w ͔ͩΒ
    
    NPE
    Q
    
    Q  
    ͩ
    w Q 
    
    ?
    
    
    
    Y
    ff
    ff
    
    ff
    C
    ͷ͸ͣ
    w ͜ΕΛ
    
    ͰׂΔͱ
    Y
    
    
    
    
    ͩ
    w ࣮ࡍʹιʔεΛಡΈʹߦ͘ͱ߹ͬͯΔ
12. ಄ͷதͰ࣮ݧɾߟ࡯Ͱ͖Δ w ҉ࢉͰ΍Δҙຯ͸
    w ͜͏ͨۤ͠࿑ͨ͠Τϐιʔυ͕͋Δͱ
    Z
    
    
    ͱ͍͏஋Λ๨Εʹ͍͘
    w $VSWF
    Ͱ
    V
    
    
    ͱ͍͏ͷ΋͓͚֮͑ͯ͹ɺZ
    
    V  V 
    ΍
    V
    
     Z  Z
    ΋๨Εʹ͍͘
    w ୶͍هԱΛෳ਺͓͍࣋ͬͯͯఆظతʹݕࢉ͢Δ͜ͱͰɺهԱΛڧݻʹ͢ΔΠϝ ʔδ
13. $5'
    Ͱมͳ࣮૷Λݟͨ࣌ɺᄿ͕֮ಇ͍͙ͯ͢ʹΘ͔Δ Ͳ͜ʹ੬ऑੑ͕͋ΔͰ͠ΐ͏
14. $5'
    Ͱมͳ࣮૷Λݟͨ࣌ɺᄿ͕֮ಇ͍͙ͯ͢ʹΘ͔Δ Ͳ͜ʹ੬ऑੑ͕͋ΔͰ͠ΐ͏ˠೖྗͷ
    WBMJEBUJPO
    Λ͍ͯ͠ͳ͍
    IUUQTBMQBDBIBDLDPNDUGT[FSQUTDUGDIBMMFOHFTFEEI
    
15. ϫʔΩϯάϝϞϦʔ͕஁͑ΒΕΔ w $5'
    ΍҉߸ϥΠϒϥϦʔͷಡղͰ͸ɺͦͦ͜͜௕͍ίʔυͷཧղ͕ඞཁ
    w ಄ͷதʹଟ͘ͷ৘ใ͕ࡌͤΒΕΔͱɺ໾ʹཱͭ
    w ಄ͷதͷ৘ใ͕ϦϯΫ͞Ε͍ͯΔͱɺ๨٫͠ʹ͍͘
16. هԱϝιου w ࣮૷ΛಡΉ
    Πϯϓοτ
    w ࿦จ3'$
    ΛಡΉ
    Πϯϓοτ
    w ࣗ෼Ͱίʔυΰϧϑͯ͠ΈΔ
    Ξ΢τϓοτ
    w ҉ࢉ͢Δ
    ෼ੳ
17. ࣮૷ΛಡΉ
    Πϯϓοτ w ϏοτԋࢉΛۦ࢖ͯ͠෼ذΛফͤΔͷ͔ʜ
    ڻ͖
    w IUUQTHJUIVCDPNPQFOTTIPQFOTTIQPSUBCMFCMPC7@@@1 FED
    
18. ࿦จ3'$
    ΛಡΉ
    Πϯϓοτ w $VSWF
    w ఺ͷ
    V
    ࠲ඪ͚ͩͰପԁۂઢͷεΧϥʔഒ͕ܭࢉͰ͖Δ
    w IUUQTDSZQUPFDEIDVSWFQEG
19. ࣗ෼Ͱίʔυΰϧϑͯ͠ΈΔ
    Ξ΢τϓοτ w $3$
    ͱ͔͸͔ͳΓ؆୯
    චऀ࣮૷͸
    (P
    Ͱ
    
    ߦ
    w IUUQTTJ[VNFLPCB@FQPTUTVCBLNEPLOW
20. ͜Ε͚ͩͰ͢
21. ҉ࢉ͢Δ
    ෼ੳ w ո͍͠ެࣜΛෳ਺૊Έ߹Θͤͯɺໃ६Λݕग़ɾղফ͢Δ
    w ϫʔΩϯάϝϞϦʔΛϑϧՔಇͤͯ͞ɺ಄ͷதʹ͢΂ͯࡌͤΔ
    w ࡌΒͳ͔ͬͨΒؤுΖ͏
22. ҉ࢉ͢Δ
    ෼ੳ ΍Γํ w Kෆมྔ͕
    K
    ͷପԁۂઢɺ
    ͚ͩͬͨͬʜ
    w
    ͷ
    Kෆมྔ͸
    
    ͚ͩͬͨͬʜ
    w ͦ͏ࢥͬͯ
    KෆมྔΛܭࢉ͢Δͱ
    
    ʹͳΓɺ
    ๨Εʹؾ෇͚Δ y2 = x3 − 3j j − 1728 x + 2j j − 1728 y2 = x3 + ax + b 4a3 4a3 + 27b2 j 1728
23. ҉ࢉ͢Δ
    ෼ੳ ΍Γํ w ಄ͷதͰ΍Δํ๏
    w
    ͱͯ͠͠·͓͏ɻB
    
    K"
    C
    
    K"
    w
    w ෼ࢠͱ෼฼Λ
    K?"?
    ͰׂΔͱ
    j − 1728 = A 4a3 4a3 + 27b2 = −108j3 A3 −108j3 A3 + 108j2 A2 −j −j + A = j 1728
24. ͓͢͢Ί෼໺ w "4$**
    DPEF
    w ॳ౳੔਺࿦
    w ପԁۂઢ࿦
25. "4$**
    DPEF w "
    
    Y
    ͱ͔ͩͬͨΓ͢ΔΞϨ
    w ӳޠͷΞϧϑΝϕοτʹϥϯμϜΞΫηεͰ͖Δͱڧ͍
    w ͤΊ్ͯதͷϙΠϯτΛ֮͑Α͏
    w %
    
    
    )
    
    
    -
    
    
    1
    
    
    5
    
    
    9
    
    
    w ेਐ਺ͱे࿡ਐ਺ͷม׵΋Ͱ͖ΔΑ͏ʹͳ͓ͬͯ͜͏
    w 
    
    
    
    
    
    
    
    
    
    
    ·Ͱͷ
    
    ͷഒ਺Ͱ͍͍ͷͰָ
26. ॳ౳੔਺࿦ w ۩ମྫͷๅݿ
    w ฏํ৒༨ͱ͔৐๏܈
    w
    ͳΒ
    B
    ͸ฏํ৒༨
    w
    ͳΒ
    NPE
    Q
    Ͱ
    
    ͷ
    
    ৐͕ࠜ͋Δ
    w ଘࡏఆཧͳͷͰ࣮ࡍͷߏஙͱͷؒʹ͸Ϊϟοϓ͋Γɺ࣮ࡍʹௐ΂Α͏
    w ϥϯμϜʹ
    B
    ΛͱΕ͹
    
    ͕֬཰
    
    Ͱඇࣗ໌ͳ
    
    ৐ࠜ a(p−1)/2 ≡ 1 (mod p) p ≡ 1 (mod 3) a(p−1)/3
27. ପԁۂઢ࿦ w ͓΋͠Ζ͍
    w ପԁۂઢ࿦͸޿େ͗ͯ͢໎͍΍͍͢ͷͰɺ$5'
    Ͱ࢖͍ͦ͏ͳͱ͜Ζ͔Β
    w ͓͢͢Ίॳख
    ༗ݶମ্ͷପԁۂઢ
    w खલຯḩ
    IUUQTRJJUBDPNLPCBFJUFNT FCGCGGDBG
    
    w 4BGF$VSWFT
    IUUQTTBGFDVSWFTDSZQUP
    
28. ·ͱΊ w $5'
    Ͱ΋҉߸ཧ࿦Ͱ΋ɺָ͠ΉͨΊʹ͸εϜʔζʹهԱ͢Δ͜ͱ͕ෆՄܽ
    w εϜʔζͳهԱͷͨΊʹ͸Τϐιʔυ͕େࣄ
    w ҉ࢉεΩϧΛߴΊͯΤϐιʔυΛ಄ͷதͰ࿉੒͠Α͏
29. ༧උεϥΠυ ༧උεϥΠυ
30. YFEC@
    ͬͯԿ w (' 
    ͷଟ߲ࣜ
    w ͭ·Γ
    NPE
    
    Ͱ৭ʑ΍Δͱ͍͏͜ͱ
    w ԼҐϏοτ͕࣍਺ͷߴ͍ଆ
    w Y?
    ͸লུ͞Ε͍ͯΔ
    w ͭ·Γ
    
    ͱ͍͏͜ͱ
    w YF
    
    C
    YE
    
    C
    ʹ஫ҙɻ (1 + x + x2) + (x4 + x5 + x7) + ⋯ + x26 + x32
31. &EXBSET
    ͷՃ๏ެࣜ͸ w IUUQTFEDSZQUPFEQEG
    w ΄ͱΜͲෳૉ਺ͷੵɺͨͩ͠ʜ
    w Y
    ͱ
    Z
    ͕ٯɺJ?
    
    
    w ෼฼͕͋Δ