༨ྨʹ͓͚Δ๏ٯݩ ඞͣ͠๏ٯݩ͕ଘࡏ͢ΔͱݶΒͳ͍ >>> 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
߹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
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๏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
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