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fp_inv.cu
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// bls12_381: Arithmetic for BLS12-381
// Copyright 2022-2023 Dag Arne Osvik
// Copyright 2022-2023 Luan Cardoso dos Santos
#include "fp.cuh"
/**
* @brief Calculates the multiplicative inverse of x and stores in z
*
* * This function calculates the multiplicative inverse of the argument.
* An integer a is the inverse of z if a*z mod r == 1
*
* Normally, the inverse is found by using the Extended Euclidean Algorithm, to
* find integers (z,y) to satisfy the Bézout's identity:
* a*z + r*y == gcd(a, r) == 1
* which can be rewritten as:
* az-1 == (-y)*m which follows that a*z mod r == 1. This approach has complexity
* in the order of O(log2(r)).
*
* This implementation uses Euler's theorem, calculating the inverse as z^(phi(r)-1).
* where phi is Euler's totient function. For the special case where r is prime,
* phi(r) = r-1. Therefore, the inverse here is calculated as z^(r-2).
* Although this is asymptotically worse than EEA, this implementation avoid flow
* divergence and uses 279 squarings and 128 multiplications.
* Furthermore, since curve operations are done in projective coordinates, inversions
* are needed only at the very end when projective coordinates are translated into
* affine coordinates.
*
* @param[out] z
* @param[in] x
* @return void
*/
__device__ void fp_inv(fp_t &z, const fp_t &x) {
// p-2 = 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559785
// = 5 * 5683 * 8572387 * 60565473370971138564533387 * 271298243114458311795492131869447332014235303245210008608904867493862992995991
uint64_t x1[6], x3[6];
// 5: 101
fp_cpy(x1, x);
fp_sqr(z, x);
fp_sqr(z, z);
fp_mul(x1, z, x1);
// 5683: 1011000110011
fp_sqr(z, x1);
fp_mul(x3, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(x1, z, x3);
// 8572387: 100000101100110111100011
// 11 1 11 11 11 11
// 11
fp_sqr(z, x1);
fp_mul(x3, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(x1, z, x3);
// 60565473370971138564533387: 11001000011001001111001110001111101100000100111001111101010010111100111010110010001011
// 11 1 11 1 11 11 11 1 11 1 11 11 1 1 1 1 11 11 1 11 1 1 11
// 11 1 11 1 11 11 1
fp_sqr(z, x1);
fp_mul(z, z, x1);
fp_cpy(x3, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(x1, z, x3);
// 271298243114458311795492131869447332014235303245210008608904867493862992995991
fp_sqr(z, x1);
fp_mul(x3, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x1);
fp_sqr(z, z);
fp_sqr(z, z);
fp_sqr(z, z);
fp_mul(z, z, x3);
fp_sqr(z, z);
fp_mul(z, z, x1);
}
// vim: ts=4 et sw=4 si