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Unknown Evolution of the Built-in Function pow
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HayaoSuzuki
October 15, 2021
Technology
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Unknown Evolution of the Built-in Function pow
PyCon JP 2021
HayaoSuzuki
October 15, 2021
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Transcript
ΈࠐΈؔ pow ͷΒΕ͟ΔਐԽ Unknown Evolution of the Built-in Function pow
Hayao Suzuki PyCon JP 2021 October 15, 2021
ൃදʹࡍͯ͠ GitHub › https://github.com/HayaoSuzuki/pyconjp2021 Twitter ϋογϡλά › #pyconjp #pyconjp_3 PyCon
JP 2021 Discord › #hayao-suzuki-ΈࠐΈؔ pow ͷΒΕ͟ΔਐԽ › #pyconjp_3 ʢ1 17:30ʙ18:15ʣ 2 / 33
Who am I ? ͓લ୭Α ໊લ Hayao Suzukiʢླɹॣʣ Twitter @CardinalXaro
ࣄ Software Developer @ BeProud Inc. › גࣜձࣾϏʔϓϥυ › IT ษڧձࢧԉαʔϏε connpass › ΦϯϥΠϯֶशαʔϏε PyQ › γεςϜ։ൃͷͨΊͷυΩϡϝϯταʔϏε Tracery 3 / 33
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 Ͱ͡ΊΔσΟʔϓϥʔχϯά (O’Reilly Japan) › ࣮ફ ࣌ܥྻղੳ (O’Reilly Japan) New! › ػցֶशσβΠϯύλʔϯ (O’Reilly Japan) New! https://xaro.hatenablog.jp/ ʹϦετ͕͋Γ·͢ɻ 4 / 33
Who am I ? ൃදϦετʢൈਮʣ › ϨΨγʔ Django ΞϓϦέʔγϣϯͷݱԽ (DjangoCongress
JP 2018) › SymPy ʹΑΔࣜॲཧ (PyCon JP 2018) › Python ͱָ͠Ήॳ (PyCon mini Hiroshima 2019) › ܅ cmath Λ͍ͬͯΔ͔ (PyCon mini Shizuoka 2020) › ΠϯϝϞϦʔετϦʔϜ׆༻ज़ (PyCon JP 2020) https://xaro.hatenablog.jp/ ʹϦετ͕͋Γ·͢ɻ 5 / 33
ࠓͷඪ ΈࠐΈؔ pow › pow ؔͷႈΛฦؔ͢ › Python ʹݶΒͣɺେͷݴޠʹ pow
͕ؔଘࡏ͢Δ Python 3.8 ͰػೳՃ › m Λ๏ͱ͢Δ༨ྨʹ͓͚Δ๏ٯݩ͕ܭࢉͰ͖Δ › Α͘Θ͔Βͳ͍୯ޠΛฒΔͳʂ 6 / 33
ࠓͷඪ ΈࠐΈؔ pow ͷΒΕ͟ΔਐԽ › Python 3.8 ͰՃ͞Εͨ pow ؔͷ৽ػೳΛཧղ͢Δ
› ʮ m Λ๏ͱ͢Δ༨ྨʹ͓͚ΔٯݩʯͷҙຯΛཧղ͢Δ › ʮ m Λ๏ͱ͢Δ༨ྨʹ͓͚ΔٯݩʯΛܭࢉ͢ΔΞϧΰ ϦζϜΛཧղ͢Δ 7 / 33
ࠓ·Ͱͷ pow ؔ Python 3.7 ·Ͱͷ pow ؔΛ෮श͠Α͏ 8 /
33
ͷႈ ఆٛ (ͷႈ) b ͱࣗવ n ʹରͯ͠ɺႈ bn Λ
bn ≜ n ݸ z }| { b ˆ b ˆ ´ ´ ´ ˆ b ͱఆٛ͢Δɻb Λఈɺn ΛࢦͱݺͿɻ ͷႈͷྫ 232 = 4294967296: 9 / 33
ͷႈ Python ʹ͓͚Δႈ ΈࠐΈؔ pow ·ͨ**ԋࢉࢠΛ͏ɻ ႈͷ࣮ߦྫ >>> pow(2, 32)
4294967296 >>> 2 ** 32 4294967296 10 / 33
ႈ༨ ఆٛ (ႈ༨) ࣗવͷఈ b ͱࣗવ n; m ʹରͯ͠ɺ bn
mod m Λ m Λ๏ͱ͢Δႈ༨ͱఆٛ͢Δɻ ႈ༨ͷྫ 232 mod 65535 = 1: 11 / 33
ႈ༨ Python ʹ͓͚Δႈ༨ › ΈࠐΈؔ pow ͰޮతʹܭࢉͰ͖Δɻ › **ԋࢉࢠ͓Αͼ%ԋࢉࢠͰܭࢉՄೳ͕ͩޮ͕ѱ͍ɻ ›
Python 1.5 ͔Βར༻Մೳʢint ܕͷൣғͳͲͷ੍ݶ ͋ͬͨʣ ɻ ႈ༨ͷ࣮ߦྫ >>> pow(2, 262144, 65535) 1 >>> (2 ** 262144) % 65535 1 12 / 33
ႈ༨ ͲΕ͚ͩޮత͔ >>> import timeit >>> timeit.timeit("pow(2, 262144, 65535)", number=1000)
0.0007324999999999693 >>> timeit.timeit("(2 ** 262144) % 65535", number=1000) 0.868453 ݁ՌΛ࣮ߦճͰׂΕฏۉ͕࣌ؒΘ͔Δɻ 13 / 33
ႈ༨ ԋࢉࢠͱؔʹ͓͚Δܭࢉ࣌ؒͷൺֱ 14 / 33
͜Ε͔Βͷ pow ؔ Python 3.8 ͔Βͷ pow ؔΛཧղ͢ΔͨΊʹ 15 /
33
ͷ߹ಉ ఆٛ (ͷ߹ಉ) a ͕ m Λ๏ͱͯ͠ b ͱ߹ಉͰ͋Δͱ
m ͕ a ` b ΛׂΓΔ ͜ͱΛ͍͍ɺ a ” b (mod m) ͱද͢ɻ ͷ߹ಉͷྫ 47 ” 35 (mod 6) 47 ` 35 = 12 6 ͰׂΓΕΔɻ 16 / 33
༨ྨ ఆٛ (༨ྨ) ू߹ Z ͱ m Λ๏ͱ͢Δ߹ಉؔʹΑΔಉྨ͔ΒͳΔू ߹Λ༨ྨͱݺͼɺZm ͱද͢ɻ
༨ྨͷΠϝʔδ Λ m Ͱׂͬͨ༨Ͱྨͯ͠ɺ༨͕ಉ͡ಉ༷͡ͳͷͱ ͯ͠ߟ͑Δɻͭ·ΓɺΛ 0; 1; : : : ; m ` 1 ͷ͍ͣΕ͔ʹྨͯ͠ ී௨ͷͷΘΓʹ 0; 1; : : : ; m ` 1 ͚ͩͷੈքΛߟ͍͑ͯΔɻ m = 2 ͳΒɺΛۮ͔حͷ 2 ͭʹྨ͢Δ͜ͱͱಉ͡ɻ 17 / 33
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
༨ྨʹ͓͚Δ๏ٯݩ Python ʹ͓͚Δ༨ྨʹ͓͚Δ๏ٯݩ › ΈࠐΈؔ pow ͷୈ 2 Ҿʹ `1
ΛͤܭࢉՄೳ › ͜Ε͕ Python 3.8 ͷ৽ػೳ ༨ྨʹ͓͚Δ๏ٯݩͷ࣮ߦྫ >>> pow(38, -1, 97) 23 >>> (38 * 23) % 97 == 1 True 19 / 33
༨ྨʹ͓͚Δ๏ٯݩ ඞͣ͠๏ٯݩ͕ଘࡏ͢ΔͱݶΒͳ͍ >>> pow(2, -1, 6) Traceback (most recent call
last): File "<stdin>", line 1, in <module> ValueError: base is not invertible for the given modulus ༩͑ΒΕͨ๏ʹରͯ͠ఈ͕๏ٯݩΛ࣋ͨͳ͍ʢԿނʁʣ ɻ 20 / 33
๏ٯݩΛٻΊͯ ๏ٯݩͷҙຯ a ʹରͯ͠ɺm Λ๏ͱ͢Δ߹ಉํఔࣜ ax ” 1 (mod
m) Λղ͘͜ͱʹଞͳΒͳ͍ɻ 21 / 33
߹ಉͷఆٛʹཱͪฦΔ ๏ٯݩͷҙຯ ax ” 1 (mod m) Λมܗ͢Δͱɺ1 ࣍ෆఆํఔࣜ ax
` my = 1 ͕ղ x; y Λ࣋ͭ͜ͱʹଞͳΒͳ͍ɻ 22 / 33
߹ಉํఔࣜͷղ ఆཧ a; c ʹରͯ͠ɺm Λ๏ͱ͢Δ߹ಉํఔࣜ ax ” c
(mod m) ɺc ͕ gcd(a; m) ͰׂΓΕΔͱ͖ͷΈɺͪΐ͏Ͳ gcd(a; m) ݸͷޓ͍ʹ߹ಉͰͳ͍ղΛ࣋ͭɻͨͩ͠ɺgcd(a; m) a ͱ m ͷ ࠷େެͰ͋Δɻ ূ໌ɺదͳॳͷڭՊॻΛࢀর͍ͯͩ͘͠͞ɻ 23 / 33
ࠓճͷέʔε ܥ a; ʹରͯ͠ɺm Λ๏ͱ͢Δ߹ಉํఔࣜ ax ” 1 (mod
m) ɺgcd(a; m) = 1 ͷ߹ͷΈɺ1 ݸͷޓ͍ʹ߹ಉͰͳ͍ղΛ ࣋ͭɻ ๏ٯݩ͕ଘࡏ͢Δ͔Ͳ͏ֶ͔తͳཪ͚͕͋Δɻ 24 / 33
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
༨ྨʹ͓͚Δ๏ٯݩ ๏ٯݩ͕ଘࡏ͠ͳ͍έʔε >>> import math >>> math.gcd(2, 6) 2 >>>
pow(2, -1, 6) Traceback (most recent call last): File "<stdin>", line 1, in <module> ValueError: base is not invertible for the given modulus gcd(2; 6) 6= 1 ͳͷͰɺ6 Λ๏ͱ͢Δ 2 ͷ๏ٯݩଘࡏ͠ͳ͍ɻ 26 / 33
༨ྨʹ͓͚Δ๏ٯݩ ఆཧ a ʹରͯ͠ɺm Λ๏ͱ͢Δ๏ٯݩ͕ଘࡏ͢ΔͨΊͷඞཁे ݅ gcd(a; m) =
1 Ͱ͋Δɻ ެࣜυΩϡϝϯτʹʮIf mod is present and exp is negative, base must be relatively prime to mod.ʯͱ͋Δɻ 27 / 33
Euclid ͷޓআ๏ ఆཧ a; b Λ a – b Ͱ͋Δɺr
Λ a Λ b Ͱׂͬͨ༨Γͱ͢Δɻ͜ͷ ͱ͖ɺ gcd(a; b) = gcd(b; r) ͕Γཱͭɻ 28 / 33
Euclid ͷޓআ๏ Euclid ͷޓআ๏ͱ 1 ࣍ෆఆํఔࣜ a Λ b Ͱׂͬͨͱ༨ΛͦΕͧΕ
q; r ͱ͢Δɻ1 ࣍ෆఆํఔࣜ ax + by = 1 ʹ a = qb + r Λೖ͢Δͱɺ (qb + r)x + by = 1 )b(qx + y) + rx = 1 ͱͳΔɻ ͭ·Γɺax + by = 1 ͔Β bs + rt = 1 ͱ͢Δ͜ͱ͕Ͱ͖Δɻ x = t; y = s ` qt ͱ͍͏ؔɻ 29 / 33
pow ͷத longobject.c ͷίϝϯτʹ͋Δ࣮ʢҰ෦վมʣ 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
༨ྨʹ͓͚Δ๏ٯݩ ΞϧΰϦζϜ a ͱ m ͷ࠷େެΛܭࢉ͢ΔաఔͰ๏ٯݩΛܭࢉ͢Δ͜ͱ͕Ͱ ͖Δɻ Remark a
ʹରͯ͠ɺm Λ๏ͱ͢Δ๏ٯݩΛܭࢉ͢ΔΞϧΰϦζϜ֦ ு Euclid ͷޓআ๏ͱݺΕΔɻ 31 / 33
ԿނՃ͞Εͨͷ͔ bpo-36027 ʹॻ͔Ε͍ͯΔཧ༝ Here is another number theory basic that
I’ve needed every now... ͷجຊతͳ͔ؔͩΒඞཁͩΑͶʂ 32 / 33
·ͱΊ ·ͱΊ › pow ؔͷႈΛฦؔ͢Ͱ͋Δɻ › ႈ༨Λܭࢉ͢Δ߹ඞͣ pow ؔΛ͏ɻ ›
Python 3.8 Ͱ༨ྨͷ๏ٯݩ͕ܭࢉͰ͖ΔΑ͏ʹͳͬͨɻ › ๏ٯݩ͕ܭࢉͰ͖ΔΈ Euclid ͷޓআ๏ʹ͋Δɻ PyCon JP 2021 Discord ΑΖ͘͠ › #hayao-suzuki-ΈࠐΈؔ pow ͷΒΕ͟ΔਐԽ › #pyconjp_3 ʢ1 17:30ʙ18:15ʣ 33 / 33