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Unknown Evolution of the Built-in Function pow

HayaoSuzuki
October 15, 2021

Unknown Evolution of the Built-in Function pow

PyCon JP 2021

HayaoSuzuki

October 15, 2021
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  1. ૊ΈࠐΈؔ਺ pow ͷ஌ΒΕ͟ΔਐԽ
    Unknown Evolution of the Built-in Function pow
    Hayao Suzuki
    PyCon JP 2021
    October 15, 2021

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  2. ൃදʹࡍͯ͠
    GitHub
    › https://github.com/HayaoSuzuki/pyconjp2021
    Twitter ϋογϡλά
    › #pyconjp #pyconjp_3
    PyCon JP 2021 Discord
    › #hayao-suzuki-૊ΈࠐΈؔ਺ pow ͷ஌ΒΕ͟ΔਐԽ
    › #pyconjp_3 ʢ1 ೔໨ 17:30ʙ18:15ʣ
    2 / 33

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  3. Who am I ?
    ͓લ୭Α
    ໊લ Hayao Suzukiʢླ໦ɹॣʣ
    Twitter @CardinalXaro
    ࢓ࣄ Software Developer @ BeProud Inc.
    › גࣜձࣾϏʔϓϥ΢υ
    › IT ษڧձࢧԉαʔϏε connpass
    › ΦϯϥΠϯֶशαʔϏε PyQ
    › γεςϜ։ൃͷͨΊͷυΩϡϝϯταʔϏε Tracery
    3 / 33

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  4. Who am I ?
    ؂༁ɾࠪಡٕͨ͠ज़ॻʢൈਮʣ
    › ೖ໳ Python 3 ୈ 2 ൛ (O’Reilly Japan)
    › Effective Python ୈ 2 ൛ (O’Reilly Japan)
    › ػցֶशʹΑΔ࣮༻ΞϓϦέʔγϣϯߏங (O’Reilly Japan)
    › PyTorch ͱ fastai Ͱ͸͡ΊΔσΟʔϓϥʔχϯά (O’Reilly
    Japan)
    › ࣮ફ ࣌ܥྻղੳ (O’Reilly Japan) New!
    › ػցֶशσβΠϯύλʔϯ (O’Reilly Japan) New!
    https://xaro.hatenablog.jp/ ʹϦετ͕͋Γ·͢ɻ
    4 / 33

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  5. Who am I ?
    ൃදϦετʢൈਮʣ
    › ϨΨγʔ Django ΞϓϦέʔγϣϯͷݱ୅Խ
    (DjangoCongress JP 2018)
    › SymPy ʹΑΔ਺ࣜॲཧ (PyCon JP 2018)
    › Python ͱָ͠Ήॳ౳੔਺࿦ (PyCon mini Hiroshima
    2019)
    › ܅͸ cmath Λ஌͍ͬͯΔ͔ (PyCon mini Shizuoka 2020)
    › ΠϯϝϞϦʔετϦʔϜ׆༻ज़ (PyCon JP 2020)
    https://xaro.hatenablog.jp/ ʹϦετ͕͋Γ·͢ɻ
    5 / 33

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  6. ࠓ೔ͷ໨ඪ
    ૊ΈࠐΈؔ਺ pow
    › pow ؔ਺͸਺ͷႈ৐Λฦؔ͢਺
    › Python ʹݶΒͣɺେ఍ͷݴޠʹ͸ pow ؔ਺͕ଘࡏ͢Δ
    Python 3.8 Ͱػೳ௥Ճ
    › ੔਺ m Λ๏ͱ͢Δ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ͕ܭࢉͰ͖Δ
    › Α͘Θ͔Βͳ͍୯ޠΛฒ΂Δͳʂ
    6 / 33

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  7. ࠓ೔ͷ໨ඪ
    ૊ΈࠐΈؔ਺ pow ͷ஌ΒΕ͟ΔਐԽ
    › Python 3.8 Ͱ௥Ճ͞Εͨ pow ؔ਺ͷ৽ػೳΛཧղ͢Δ
    › ʮ੔਺ m Λ๏ͱ͢Δ੔਺৒༨ྨʹ͓͚ΔٯݩʯͷҙຯΛཧղ͢Δ
    › ʮ੔਺ m Λ๏ͱ͢Δ੔਺৒༨ྨʹ͓͚ΔٯݩʯΛܭࢉ͢ΔΞϧΰ
    ϦζϜΛཧղ͢Δ
    7 / 33

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  8. ࠓ·Ͱͷ pow ؔ਺
    Python 3.7 ·Ͱͷ pow ؔ਺Λ෮श͠Α͏
    8 / 33

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  9. ੔਺ͷႈ৐
    ఆٛ (੔਺ͷႈ৐)
    ੔਺ b ͱࣗવ਺ n ʹରͯ͠ɺႈ৐ bn Λ
    bn ≜
    n ݸ
    z }| {
    b ˆ b ˆ ´ ´ ´ ˆ b
    ͱఆٛ͢Δɻb Λఈɺn Λࢦ਺ͱݺͿɻ
    ੔਺ͷႈ৐ͷྫ
    232 = 4294967296:
    9 / 33

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  10. ੔਺ͷႈ৐
    Python ʹ͓͚Δႈ৐
    ૊ΈࠐΈؔ਺ pow ·ͨ͸**ԋࢉࢠΛ࢖͏ɻ
    ႈ৐ͷ࣮ߦྫ
    >>> pow(2, 32)
    4294967296
    >>> 2 ** 32
    4294967296
    10 / 33

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  11. ႈ৐৒༨
    ఆٛ (ႈ৐৒༨)
    ࣗવ਺ͷఈ b ͱࣗવ਺ n; m ʹରͯ͠ɺ
    bn mod m
    Λ m Λ๏ͱ͢Δႈ৐৒༨ͱఆٛ͢Δɻ
    ႈ৐৒༨ͷྫ
    232 mod 65535 = 1:
    11 / 33

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  12. ႈ৐৒༨
    Python ʹ͓͚Δႈ৐৒༨
    › ૊ΈࠐΈؔ਺ pow Ͱޮ཰తʹܭࢉͰ͖Δɻ
    › **ԋࢉࢠ͓Αͼ%ԋࢉࢠͰ΋ܭࢉՄೳ͕ͩޮ཰͕ѱ͍ɻ
    › Python 1.5 ͔Βར༻Մೳʢint ܕͷൣғ಺ͳͲͷ੍ݶ͸
    ͋ͬͨʣ
    ɻ
    ႈ৐৒༨ͷ࣮ߦྫ
    >>> pow(2, 262144, 65535)
    1
    >>> (2 ** 262144) % 65535
    1
    12 / 33

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  13. ႈ৐৒༨
    ͲΕ͚ͩޮ཰త͔
    >>> import timeit
    >>> timeit.timeit("pow(2, 262144, 65535)", number=1000)
    0.0007324999999999693
    >>> timeit.timeit("(2 ** 262144) % 65535", number=1000)
    0.868453
    ݁ՌΛ࣮ߦճ਺ͰׂΕ͹ฏۉ͕࣌ؒΘ͔Δɻ
    13 / 33

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  14. ႈ৐৒༨
    ԋࢉࢠͱؔ਺ʹ͓͚Δܭࢉ࣌ؒͷൺֱ
    14 / 33

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  15. ͜Ε͔Βͷ pow ؔ਺
    Python 3.8 ͔Βͷ pow ؔ਺Λཧղ͢ΔͨΊʹ
    15 / 33

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  16. ੔਺ͷ߹ಉ
    ఆٛ (੔਺ͷ߹ಉ)
    ੔਺ a ͕ m Λ๏ͱͯ͠ b ͱ߹ಉͰ͋Δͱ͸ m ͕ a ` b ΛׂΓ੾Δ
    ͜ͱΛ͍͍ɺ
    a ” b (mod m)
    ͱද͢ɻ
    ੔਺ͷ߹ಉͷྫ
    47 ” 35 (mod 6)
    47 ` 35 = 12 ͸ 6 ͰׂΓ੾ΕΔɻ
    16 / 33

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  17. ੔਺৒༨ྨ
    ఆٛ (੔਺৒༨ྨ)
    ੔਺ू߹ Z ͱ੔਺ m Λ๏ͱ͢Δ߹ಉؔ܎ʹΑΔಉ஋ྨ͔ΒͳΔ঎ू
    ߹Λ੔਺৒༨ྨͱݺͼɺZm
    ͱද͢ɻ
    ੔਺৒༨ྨͷΠϝʔδ
    ੔਺Λ m Ͱׂͬͨ৒༨Ͱ෼ྨͯ͠ɺ৒༨͕ಉ͡੔਺͸ಉ༷͡ͳ΋ͷͱ
    ͯ͠ߟ͑Δɻͭ·Γɺ੔਺Λ 0; 1; : : : ; m ` 1 ͷ͍ͣΕ͔ʹ෼ྨͯ͠
    ී௨ͷ੔਺ͷ୅ΘΓʹ 0; 1; : : : ; m ` 1 ͚ͩͷੈքΛߟ͍͑ͯΔɻ
    m = 2 ͳΒ͹ɺ੔਺Λۮ਺͔ح਺ͷ 2 ͭʹ෼ྨ͢Δ͜ͱͱಉ͡ɻ
    17 / 33

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  18. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    ఆٛ (Zm
    ʹ͓͚Δ৐๏ٯݩ)
    ੔਺ a; b ͱࣗવ਺ m ʹରͯ͠ɺ
    ab ” 1 (mod m)
    ͱͳΔͱ͖ɺb Λ a ͷ৐๏ٯݩͱݺͼɺa`1 ͱද͢ɻ
    ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩͷྫ
    38 ˜ 23 ” 1 (mod 97)
    38 ͷ 97 Λ๏ͱ͢Δ৐๏ٯݩ͸ 23
    18 / 33

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  19. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    Python ʹ͓͚Δ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    › ૊ΈࠐΈؔ਺ pow ͷୈ 2 Ҿ਺ʹ `1 Λ౉ͤ͹ܭࢉՄೳ
    › ͜Ε͕ Python 3.8 ͷ৽ػೳ
    ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩͷ࣮ߦྫ
    >>> pow(38, -1, 97)
    23
    >>> (38 * 23) % 97 == 1
    True
    19 / 33

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  20. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    ඞͣ͠΋৐๏ٯݩ͕ଘࡏ͢Δͱ͸ݶΒͳ͍
    >>> pow(2, -1, 6)
    Traceback (most recent call last):
    File "", line 1, in
    ValueError: base is not invertible for the given modulus
    ༩͑ΒΕͨ๏ʹରͯ͠ఈ͕৐๏ٯݩΛ࣋ͨͳ͍ʢԿނʁʣ
    ɻ
    20 / 33

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  21. ৐๏ٯݩΛٻΊͯ
    ৐๏ٯݩͷҙຯ
    ੔਺ a ʹରͯ͠ɺm Λ๏ͱ͢Δ߹ಉํఔࣜ
    ax ” 1 (mod m)
    Λղ͘͜ͱʹଞͳΒͳ͍ɻ
    21 / 33

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  22. ߹ಉͷఆٛʹཱͪฦΔ
    ৐๏ٯݩͷҙຯ
    ax ” 1 (mod m) Λมܗ͢Δͱɺ1 ࣍ෆఆํఔࣜ
    ax ` my = 1
    ͕੔਺ղ x; y Λ࣋ͭ͜ͱʹଞͳΒͳ͍ɻ
    22 / 33

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  23. ߹ಉํఔࣜͷղ
    ఆཧ
    ੔਺ a; c ʹରͯ͠ɺm Λ๏ͱ͢Δ߹ಉํఔࣜ
    ax ” c (mod m)
    ͸ɺc ͕ gcd(a; m) ͰׂΓ੾ΕΔͱ͖ͷΈɺͪΐ͏Ͳ gcd(a; m)
    ݸͷޓ͍ʹ߹ಉͰ͸ͳ͍ղΛ࣋ͭɻͨͩ͠ɺgcd(a; m) ͸ a ͱ m ͷ
    ࠷େެ໿਺Ͱ͋Δɻ
    ূ໌͸ɺద౰ͳॳ౳੔਺࿦ͷڭՊॻΛࢀর͍ͯͩ͘͠͞ɻ
    23 / 33

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  24. ࠓճͷέʔε
    ܥ
    ੔਺ a; ʹରͯ͠ɺm Λ๏ͱ͢Δ߹ಉํఔࣜ
    ax ” 1 (mod m)
    ͸ɺgcd(a; m) = 1 ͷ৔߹ͷΈɺ1 ݸͷޓ͍ʹ߹ಉͰ͸ͳ͍ղΛ
    ࣋ͭɻ
    ৐๏ٯݩ͕ଘࡏ͢Δ͔Ͳ͏͔͸਺ֶతͳཪ෇͚͕͋Δɻ
    24 / 33

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  25. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    ৐๏ٯݩ͕ଘࡏ͢Δέʔε
    >>> import math
    >>> math.gcd(38, 97)
    1
    >>> pow(38, -1, 97)
    23
    gcd(38; 97) = 1 ͳͷͰɺ97 Λ๏ͱ͢Δ 38 ͷ৐๏ٯݩ͕ଘࡏ͢Δɻ
    25 / 33

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  26. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    ৐๏ٯݩ͕ଘࡏ͠ͳ͍έʔε
    >>> import math
    >>> math.gcd(2, 6)
    2
    >>> pow(2, -1, 6)
    Traceback (most recent call last):
    File "", line 1, in
    ValueError: base is not invertible for the given modulus
    gcd(2; 6) 6= 1 ͳͷͰɺ6 Λ๏ͱ͢Δ 2 ͷ৐๏ٯݩ͸ଘࡏ͠ͳ͍ɻ
    26 / 33

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  27. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    ఆཧ
    ੔਺ a ʹରͯ͠ɺm Λ๏ͱ͢Δ৐๏ٯݩ͕ଘࡏ͢ΔͨΊͷඞཁे෼৚
    ݅͸ gcd(a; m) = 1 Ͱ͋Δɻ
    ެࣜυΩϡϝϯτʹ΋ʮIf mod is present and exp is negative,
    base must be relatively prime to mod.ʯͱ͋Δɻ
    27 / 33

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  28. Euclid ͷޓআ๏
    ఆཧ
    a; b Λ a – b Ͱ͋Δ੔਺ɺr Λ a Λ b Ͱׂͬͨ༨Γͱ͢Δɻ͜ͷ
    ͱ͖ɺ
    gcd(a; b) = gcd(b; r)
    ͕੒Γཱͭɻ
    28 / 33

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  29. Euclid ͷޓআ๏
    Euclid ͷޓআ๏ͱ 1 ࣍ෆఆํఔࣜ
    a Λ b Ͱׂͬͨ঎ͱ৒༨ΛͦΕͧΕ q; r ͱ͢Δɻ1 ࣍ෆఆํఔࣜ
    ax + by = 1 ʹ a = qb + r Λ୅ೖ͢Δͱɺ
    (qb + r)x + by = 1
    )b(qx + y) + rx = 1
    ͱͳΔɻ
    ͭ·Γɺax + by = 1 ͔Β bs + rt = 1 ͱ͢Δ͜ͱ͕Ͱ͖Δɻ
    x = t; y = s ` qt ͱ͍͏ؔ܎ɻ
    29 / 33

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  30. pow ͷத਎
    longobject.c ͷίϝϯτʹ͋Δ࣮૷ʢҰ෦վมʣ
    def invmod(a, m):
    x, y = 1, 0
    while m:
    q, r = divmod(a, m)
    a, m = m, r
    x, y = y, x - q * y
    if a == 1:
    return x
    raise ValueError("Not invertible")
    30 / 33

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  31. ੔਺৒༨ྨʹ͓͚Δ৐๏ٯݩ
    ΞϧΰϦζϜ
    a ͱ m ͷ࠷େެ໿਺Λܭࢉ͢ΔաఔͰ৐๏ٯݩΛܭࢉ͢Δ͜ͱ͕Ͱ
    ͖Δɻ
    Remark
    ੔਺ a ʹରͯ͠ɺm Λ๏ͱ͢Δ৐๏ٯݩΛܭࢉ͢ΔΞϧΰϦζϜ͸֦
    ு Euclid ͷޓআ๏ͱݺ͹ΕΔɻ
    31 / 33

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  32. Կނ௥Ճ͞Εͨͷ͔
    bpo-36027 ʹॻ͔Ε͍ͯΔཧ༝
    Here is another number theory basic that I’ve needed
    every now...
    ਺࿦ͷجຊతͳؔ਺͔ͩΒඞཁͩΑͶʂ
    32 / 33

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  33. ·ͱΊ
    ·ͱΊ
    › pow ؔ਺͸਺ͷႈ৐Λฦؔ͢਺Ͱ͋Δɻ
    › ႈ৐৒༨Λܭࢉ͢Δ৔߹͸ඞͣ pow ؔ਺Λ࢖͏ɻ
    › Python 3.8 Ͱ੔਺৒༨ྨͷ৐๏ٯݩ͕ܭࢉͰ͖ΔΑ͏ʹͳͬͨɻ
    › ৐๏ٯݩ͕ܭࢉͰ͖Δ࢓૊Έ͸ Euclid ͷޓআ๏ʹ͋Δɻ
    PyCon JP 2021 Discord ΋ΑΖ͘͠
    › #hayao-suzuki-૊ΈࠐΈؔ਺ pow ͷ஌ΒΕ͟ΔਐԽ
    › #pyconjp_3 ʢ1 ೔໨ 17:30ʙ18:15ʣ
    33 / 33

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