File size: 2,420 Bytes
bff4ec7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | /*@ 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;
/*@ Inv Assert
exists values,
p == p@pre && inverse == inverse@pre &&
PrimeForLinearInverse(p@pre) &&
2 <= p@pre && p@pre <= 46340 &&
2 <= i && i <= p@pre &&
ModularInversePrefix(p@pre, i, values) &&
IntArray::seg(inverse@pre, 1, i, values) *
IntArray::undef_seg(inverse@pre, i, p@pre)
*/
for (int i = 2; i < p; ++i) {
int quotient = p / i;
int remainder = p % i;
/*@ Assert
exists values,
p == p@pre && inverse == inverse@pre &&
PrimeForLinearInverse(p@pre) &&
2 <= p@pre && p@pre <= 46340 &&
2 <= i && i < p@pre &&
quotient == p@pre / i &&
remainder == p@pre % i &&
p@pre == quotient * i + remainder &&
1 <= quotient &&
1 <= remainder && remainder < i &&
0 < p@pre - quotient && p@pre - quotient < p@pre &&
0 < values[remainder - 1] &&
values[remainder - 1] < p@pre &&
0 < (p@pre - quotient) * values[remainder - 1] &&
(p@pre - quotient) * values[remainder - 1] <= INT_MAX &&
ModularInversePrefix(p@pre, i, values) &&
IntArray::seg(inverse@pre, 1, i, values) *
IntArray::undef_seg(inverse@pre, i, p@pre)
*/
inverse[i] = ((p - quotient) * inverse[remainder]) % p;
}
}
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