| /*@ Import Lean | |
| import Algorithms.linear_modular_inverse.lean.spec_lib | |
| open scoped SimpleC | |
| */ | |
| /* | |
| * Compute the modular inverses of 1, ..., p - 1 in linear time. | |
| * | |
| * For a prime p and 2 <= i < p, write p = (p / i) * i + p % i. | |
| * Since 1 <= p % i < i, its inverse has already been computed. Therefore | |
| * | |
| * inverse[i] = (p - p / i) * inverse[p % i] (mod p). | |
| * | |
| * The verification contract supplies the primality and writable-array | |
| * assumptions. The upper bound p <= 46340 keeps the product in signed | |
| * 32-bit range. | |
| */ | |
| /*@ Extern Coq | |
| (PrimeForLinearInverse : Z -> Prop) | |
| (ModularInversePrefix : Z -> Z -> list Z -> Prop) | |
| */ | |
| void linear_modular_inverse(int p, int *inverse) | |
| /*@ Require | |
| PrimeForLinearInverse(p) && | |
| 2 <= p && p <= 46340 && | |
| IntArray::undef_seg(inverse, 1, p) | |
| Ensure | |
| exists values, | |
| ModularInversePrefix(p, p, values) && | |
| IntArray::seg(inverse, 1, p, values) | |
| */ | |
| { | |
| inverse[1] = 1; | |
| for (int i = 2; i < p; ++i) { | |
| int quotient = p / i; | |
| int remainder = p % i; | |
| inverse[i] = ((p - quotient) * inverse[remainder]) % p; | |
| } | |
| } | |