/*********************************************************************** * Copyright (c) 2020 Peter Dettman * * Distributed under the MIT software license, see the accompanying * * file COPYING or https://www.opensource.org/licenses/mit-license.php.* **********************************************************************/ #ifndef SECP256K1_MODINV64_IMPL_H #define SECP256K1_MODINV64_IMPL_H #include "int128.h" #include "modinv64.h" /* This file implements modular inversion based on the paper "Fast constant-time gcd computation and * modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. * * For an explanation of the algorithm, see doc/safegcd_implementation.md. This file contains an * implementation for N=62, using 62-bit signed limbs represented as int64_t. */ /* Data type for transition matrices (see section 3 of explanation). * * t = [ u v ] * [ q r ] */ typedef struct { int64_t u, v, q, r; } secp256k1_modinv64_trans2x2; #ifdef VERIFY /* Helper function to compute the absolute value of an int64_t. * (we don't use abs/labs/llabs as it depends on the int sizes). */ static int64_t secp256k1_modinv64_abs(int64_t v) { VERIFY_CHECK(v > INT64_MIN); if (v < 0) return -v; return v; } static const secp256k1_modinv64_signed62 SECP256K1_SIGNED62_ONE = {{1}}; /* Compute a*factor and put it in r. All but the top limb in r will be in range [0,2^62). */ static void secp256k1_modinv64_mul_62(secp256k1_modinv64_signed62 *r, const secp256k1_modinv64_signed62 *a, int alen, int64_t factor) { const uint64_t M62 = UINT64_MAX >> 2; secp256k1_int128 c, d; int i; secp256k1_i128_from_i64(&c, 0); for (i = 0; i < 4; ++i) { if (i < alen) secp256k1_i128_accum_mul(&c, a->v[i], factor); r->v[i] = secp256k1_i128_to_u64(&c) & M62; secp256k1_i128_rshift(&c, 62); } if (4 < alen) secp256k1_i128_accum_mul(&c, a->v[4], factor); secp256k1_i128_from_i64(&d, secp256k1_i128_to_i64(&c)); VERIFY_CHECK(secp256k1_i128_eq_var(&c, &d)); r->v[4] = secp256k1_i128_to_i64(&c); } /* Return -1 for ab*factor. A has alen limbs; b has 5. */ static int secp256k1_modinv64_mul_cmp_62(const secp256k1_modinv64_signed62 *a, int alen, const secp256k1_modinv64_signed62 *b, int64_t factor) { int i; secp256k1_modinv64_signed62 am, bm; secp256k1_modinv64_mul_62(&am, a, alen, 1); /* Normalize all but the top limb of a. */ secp256k1_modinv64_mul_62(&bm, b, 5, factor); for (i = 0; i < 4; ++i) { /* Verify that all but the top limb of a and b are normalized. */ VERIFY_CHECK(am.v[i] >> 62 == 0); VERIFY_CHECK(bm.v[i] >> 62 == 0); } for (i = 4; i >= 0; --i) { if (am.v[i] < bm.v[i]) return -1; if (am.v[i] > bm.v[i]) return 1; } return 0; } /* Check if the determinant of t is equal to 1 << n. If abs, check if |det t| == 1 << n. */ static int secp256k1_modinv64_det_check_pow2(const secp256k1_modinv64_trans2x2 *t, unsigned int n, int abs) { secp256k1_int128 a; secp256k1_i128_det(&a, t->u, t->v, t->q, t->r); if (secp256k1_i128_check_pow2(&a, n, 1)) return 1; if (abs && secp256k1_i128_check_pow2(&a, n, -1)) return 1; return 0; } #endif /* Take as input a signed62 number in range (-2*modulus,modulus), and add a multiple of the modulus * to it to bring it to range [0,modulus). If sign < 0, the input will also be negated in the * process. The input must have limbs in range (-2^62,2^62). The output will have limbs in range * [0,2^62). */ static void secp256k1_modinv64_normalize_62(secp256k1_modinv64_signed62 *r, int64_t sign, const secp256k1_modinv64_modinfo *modinfo) { const int64_t M62 = (int64_t)(UINT64_MAX >> 2); int64_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4]; int64_t cond_add, cond_negate; #ifdef VERIFY /* Verify that all limbs are in range (-2^62,2^62). */ int i; for (i = 0; i < 5; ++i) { VERIFY_CHECK(r->v[i] >= -M62); VERIFY_CHECK(r->v[i] <= M62); } VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, -2) > 0); /* r > -2*modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, 1) < 0); /* r < modulus */ #endif /* In a first step, add the modulus if the input is negative, and then negate if requested. * This brings r from range (-2*modulus,modulus) to range (-modulus,modulus). As all input * limbs are in range (-2^62,2^62), this cannot overflow an int64_t. Note that the right * shifts below are signed sign-extending shifts (see assumptions.h for tests that that is * indeed the behavior of the right shift operator). */ cond_add = r4 >> 63; r0 += modinfo->modulus.v[0] & cond_add; r1 += modinfo->modulus.v[1] & cond_add; r2 += modinfo->modulus.v[2] & cond_add; r3 += modinfo->modulus.v[3] & cond_add; r4 += modinfo->modulus.v[4] & cond_add; cond_negate = sign >> 63; r0 = (r0 ^ cond_negate) - cond_negate; r1 = (r1 ^ cond_negate) - cond_negate; r2 = (r2 ^ cond_negate) - cond_negate; r3 = (r3 ^ cond_negate) - cond_negate; r4 = (r4 ^ cond_negate) - cond_negate; /* Propagate the top bits, to bring limbs back to range (-2^62,2^62). */ r1 += r0 >> 62; r0 &= M62; r2 += r1 >> 62; r1 &= M62; r3 += r2 >> 62; r2 &= M62; r4 += r3 >> 62; r3 &= M62; /* In a second step add the modulus again if the result is still negative, bringing * r to range [0,modulus). */ cond_add = r4 >> 63; r0 += modinfo->modulus.v[0] & cond_add; r1 += modinfo->modulus.v[1] & cond_add; r2 += modinfo->modulus.v[2] & cond_add; r3 += modinfo->modulus.v[3] & cond_add; r4 += modinfo->modulus.v[4] & cond_add; /* And propagate again. */ r1 += r0 >> 62; r0 &= M62; r2 += r1 >> 62; r1 &= M62; r3 += r2 >> 62; r2 &= M62; r4 += r3 >> 62; r3 &= M62; r->v[0] = r0; r->v[1] = r1; r->v[2] = r2; r->v[3] = r3; r->v[4] = r4; #ifdef VERIFY VERIFY_CHECK(r0 >> 62 == 0); VERIFY_CHECK(r1 >> 62 == 0); VERIFY_CHECK(r2 >> 62 == 0); VERIFY_CHECK(r3 >> 62 == 0); VERIFY_CHECK(r4 >> 62 == 0); VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, 0) >= 0); /* r >= 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, 1) < 0); /* r < modulus */ #endif } #if 0 /* Compute the transition matrix and eta for 59 divsteps (where zeta=-(delta+1/2)). * Note that the transformation matrix is scaled by 2^62 and not 2^59. * * Input: zeta: initial zeta * f0: bottom limb of initial f * g0: bottom limb of initial g * Output: t: transition matrix * Return: final zeta * * Implements the divsteps_n_matrix function from the explanation. */ static int64_t secp256k1_modinv64_divsteps_59(int64_t zeta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) { /* u,v,q,r are the elements of the transformation matrix being built up, * starting with the identity matrix times 8 (because the caller expects * a result scaled by 2^62). Semantically they are signed integers * in range [-2^62,2^62], but here represented as unsigned mod 2^64. This * permits left shifting (which is UB for negative numbers). The range * being inside [-2^63,2^63) means that casting to signed works correctly. */ uint64_t u = 8, v = 0, q = 0, r = 8; uint64_t c1, c2, f = f0, g = g0, x, y, z; int i; for (i = 3; i < 62; ++i) { VERIFY_CHECK((f & 1) == 1); /* f must always be odd */ VERIFY_CHECK((u * f0 + v * g0) == f << i); VERIFY_CHECK((q * f0 + r * g0) == g << i); /* Compute conditional masks for (zeta < 0) and for (g & 1). */ c1 = zeta >> 63; c2 = -(g & 1); /* Compute x,y,z, conditionally negated versions of f,u,v. */ x = (f ^ c1) - c1; y = (u ^ c1) - c1; z = (v ^ c1) - c1; /* Conditionally add x,y,z to g,q,r. */ g += x & c2; q += y & c2; r += z & c2; /* In what follows, c1 is a condition mask for (zeta < 0) and (g & 1). */ c1 &= c2; /* Conditionally change zeta into -zeta-2 or zeta-1. */ zeta = (zeta ^ c1) - 1; /* Conditionally add g,q,r to f,u,v. */ f += g & c1; u += q & c1; v += r & c1; /* Shifts */ g >>= 1; u <<= 1; v <<= 1; /* Bounds on zeta that follow from the bounds on iteration count (max 10*59 divsteps). */ VERIFY_CHECK(zeta >= -591 && zeta <= 591); } /* Return data in t and return value. */ t->u = (int64_t)u; t->v = (int64_t)v; t->q = (int64_t)q; t->r = (int64_t)r; #ifdef VERIFY /* The determinant of t must be a power of two. This guarantees that multiplication with t * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which * will be divided out again). As each divstep's individual matrix has determinant 2, the * aggregate of 59 of them will have determinant 2^59. Multiplying with the initial * 8*identity (which has determinant 2^6) means the overall outputs has determinant * 2^65. */ VERIFY_CHECK(secp256k1_modinv64_det_check_pow2(t, 65, 0)); #endif return zeta; } #endif /* Compute the transition matrix and eta for 62 divsteps (variable time, eta=-delta). * * Input: eta: initial eta * f0: bottom limb of initial f * g0: bottom limb of initial g * Output: t: transition matrix * Return: final eta * * Implements the divsteps_n_matrix_var function from the explanation. */ static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) { /* Transformation matrix; see comments in secp256k1_modinv64_divsteps_62. */ uint64_t u = 1, v = 0, q = 0, r = 1; uint64_t f = f0, g = g0, m; uint32_t w; int i = 62, limit, zeros; for (;;) { /* Use a sentinel bit to count zeros only up to i. */ zeros = secp256k1_ctz64_var(g | (UINT64_MAX << i)); /* Perform zeros divsteps at once; they all just divide g by two. */ g >>= zeros; u <<= zeros; v <<= zeros; eta -= zeros; i -= zeros; /* We're done once we've done 62 divsteps. */ if (i == 0) break; VERIFY_CHECK((f & 1) == 1); VERIFY_CHECK((g & 1) == 1); VERIFY_CHECK((u * f0 + v * g0) == f << (62 - i)); VERIFY_CHECK((q * f0 + r * g0) == g << (62 - i)); /* Bounds on eta that follow from the bounds on iteration count (max 12*62 divsteps). */ VERIFY_CHECK(eta >= -745 && eta <= 745); /* If eta is negative, negate it and replace f,g with g,-f. */ if (eta < 0) { uint64_t tmp; eta = -eta; tmp = f; f = g; g = -tmp; tmp = u; u = q; q = -tmp; tmp = v; v = r; r = -tmp; /* Use a formula to cancel out up to 6 bits of g. Also, no more than i can be cancelled * out (as we'd be done before that point), and no more than eta+1 can be done as its * sign will flip again once that happens. */ limit = ((int)eta + 1) > i ? i : ((int)eta + 1); VERIFY_CHECK(limit > 0 && limit <= 62); /* m is a mask for the bottom min(limit, 6) bits. */ m = (UINT64_MAX >> (64 - limit)) & 63U; /* Find what multiple of f must be added to g to cancel its bottom min(limit, 6) * bits. */ w = (f * g * (f * f - 2)) & m; } else { /* In this branch, use a simpler formula that only lets us cancel up to 4 bits of g, as * eta tends to be smaller here. */ limit = ((int)eta + 1) > i ? i : ((int)eta + 1); VERIFY_CHECK(limit > 0 && limit <= 62); /* m is a mask for the bottom min(limit, 4) bits. */ m = (UINT64_MAX >> (64 - limit)) & 15U; /* Find what multiple of f must be added to g to cancel its bottom min(limit, 4) * bits. */ w = f + (((f + 1) & 4) << 1); w = (-w * g) & m; } g += f * w; q += u * w; r += v * w; VERIFY_CHECK((g & m) == 0); } /* Return data in t and return value. */ t->u = (int64_t)u; t->v = (int64_t)v; t->q = (int64_t)q; t->r = (int64_t)r; #ifdef VERIFY /* The determinant of t must be a power of two. This guarantees that multiplication with t * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which * will be divided out again). As each divstep's individual matrix has determinant 2, the * aggregate of 62 of them will have determinant 2^62. */ VERIFY_CHECK(secp256k1_modinv64_det_check_pow2(t, 62, 0)); #endif return eta; } #if 0 /* Compute the transition matrix and eta for 62 posdivsteps (variable time, eta=-delta), and keeps track * of the Jacobi symbol along the way. f0 and g0 must be f and g mod 2^64 rather than 2^62, because * Jacobi tracking requires knowing (f mod 8) rather than just (f mod 2). * * Input: eta: initial eta * f0: bottom limb of initial f * g0: bottom limb of initial g * Output: t: transition matrix * Input/Output: (*jacp & 1) is bitflipped if and only if the Jacobi symbol of (f | g) changes sign * by applying the returned transformation matrix to it. The other bits of *jacp may * change, but are meaningless. * Return: final eta */ static int64_t secp256k1_modinv64_posdivsteps_62_var(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t, int *jacp) { /* Transformation matrix; see comments in secp256k1_modinv64_divsteps_62. */ uint64_t u = 1, v = 0, q = 0, r = 1; uint64_t f = f0, g = g0, m; uint32_t w; int i = 62, limit, zeros; int jac = *jacp; for (;;) { /* Use a sentinel bit to count zeros only up to i. */ zeros = secp256k1_ctz64_var(g | (UINT64_MAX << i)); /* Perform zeros divsteps at once; they all just divide g by two. */ g >>= zeros; u <<= zeros; v <<= zeros; eta -= zeros; i -= zeros; /* Update the bottom bit of jac: when dividing g by an odd power of 2, * if (f mod 8) is 3 or 5, the Jacobi symbol changes sign. */ jac ^= (zeros & ((f >> 1) ^ (f >> 2))); /* We're done once we've done 62 posdivsteps. */ if (i == 0) break; VERIFY_CHECK((f & 1) == 1); VERIFY_CHECK((g & 1) == 1); VERIFY_CHECK((u * f0 + v * g0) == f << (62 - i)); VERIFY_CHECK((q * f0 + r * g0) == g << (62 - i)); /* If eta is negative, negate it and replace f,g with g,f. */ if (eta < 0) { uint64_t tmp; eta = -eta; tmp = f; f = g; g = tmp; tmp = u; u = q; q = tmp; tmp = v; v = r; r = tmp; /* Update bottom bit of jac: when swapping f and g, the Jacobi symbol changes sign * if both f and g are 3 mod 4. */ jac ^= ((f & g) >> 1); /* Use a formula to cancel out up to 6 bits of g. Also, no more than i can be cancelled * out (as we'd be done before that point), and no more than eta+1 can be done as its * sign will flip again once that happens. */ limit = ((int)eta + 1) > i ? i : ((int)eta + 1); VERIFY_CHECK(limit > 0 && limit <= 62); /* m is a mask for the bottom min(limit, 6) bits. */ m = (UINT64_MAX >> (64 - limit)) & 63U; /* Find what multiple of f must be added to g to cancel its bottom min(limit, 6) * bits. */ w = (f * g * (f * f - 2)) & m; } else { /* In this branch, use a simpler formula that only lets us cancel up to 4 bits of g, as * eta tends to be smaller here. */ limit = ((int)eta + 1) > i ? i : ((int)eta + 1); VERIFY_CHECK(limit > 0 && limit <= 62); /* m is a mask for the bottom min(limit, 4) bits. */ m = (UINT64_MAX >> (64 - limit)) & 15U; /* Find what multiple of f must be added to g to cancel its bottom min(limit, 4) * bits. */ w = f + (((f + 1) & 4) << 1); w = (-w * g) & m; } g += f * w; q += u * w; r += v * w; VERIFY_CHECK((g & m) == 0); } /* Return data in t and return value. */ t->u = (int64_t)u; t->v = (int64_t)v; t->q = (int64_t)q; t->r = (int64_t)r; #ifdef VERIFY /* The determinant of t must be a power of two. This guarantees that multiplication with t * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which * will be divided out again). As each divstep's individual matrix has determinant 2 or -2, * the aggregate of 62 of them will have determinant 2^62 or -2^62. */ VERIFY_CHECK(secp256k1_modinv64_det_check_pow2(t, 62, 1)); #endif *jacp = jac; return eta; } #endif /* Compute (t/2^62) * [d, e] mod modulus, where t is a transition matrix scaled by 2^62. * * On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range * (-2^62,2^62). * * This implements the update_de function from the explanation. */ static void secp256k1_modinv64_update_de_62(secp256k1_modinv64_signed62 *d, secp256k1_modinv64_signed62 *e, const secp256k1_modinv64_trans2x2 *t, const secp256k1_modinv64_modinfo* modinfo) { const uint64_t M62 = UINT64_MAX >> 2; const int64_t d0 = d->v[0], d1 = d->v[1], d2 = d->v[2], d3 = d->v[3], d4 = d->v[4]; const int64_t e0 = e->v[0], e1 = e->v[1], e2 = e->v[2], e3 = e->v[3], e4 = e->v[4]; const int64_t u = t->u, v = t->v, q = t->q, r = t->r; int64_t md, me, sd, se; secp256k1_int128 cd, ce; #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, -2) > 0); /* d > -2*modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, 1) < 0); /* d < modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, -2) > 0); /* e > -2*modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, 1) < 0); /* e < modulus */ VERIFY_CHECK(secp256k1_modinv64_abs(u) <= (((int64_t)1 << 62) - secp256k1_modinv64_abs(v))); /* |u|+|v| <= 2^62 */ VERIFY_CHECK(secp256k1_modinv64_abs(q) <= (((int64_t)1 << 62) - secp256k1_modinv64_abs(r))); /* |q|+|r| <= 2^62 */ #endif /* [md,me] start as zero; plus [u,q] if d is negative; plus [v,r] if e is negative. */ sd = d4 >> 63; se = e4 >> 63; md = (u & sd) + (v & se); me = (q & sd) + (r & se); /* Begin computing t*[d,e]. */ secp256k1_i128_mul(&cd, u, d0); secp256k1_i128_accum_mul(&cd, v, e0); secp256k1_i128_mul(&ce, q, d0); secp256k1_i128_accum_mul(&ce, r, e0); /* Correct md,me so that t*[d,e]+modulus*[md,me] has 62 zero bottom bits. */ md -= (modinfo->modulus_inv62 * secp256k1_i128_to_u64(&cd) + md) & M62; me -= (modinfo->modulus_inv62 * secp256k1_i128_to_u64(&ce) + me) & M62; /* Update the beginning of computation for t*[d,e]+modulus*[md,me] now md,me are known. */ secp256k1_i128_accum_mul(&cd, modinfo->modulus.v[0], md); secp256k1_i128_accum_mul(&ce, modinfo->modulus.v[0], me); /* Verify that the low 62 bits of the computation are indeed zero, and then throw them away. */ VERIFY_CHECK((secp256k1_i128_to_u64(&cd) & M62) == 0); secp256k1_i128_rshift(&cd, 62); VERIFY_CHECK((secp256k1_i128_to_u64(&ce) & M62) == 0); secp256k1_i128_rshift(&ce, 62); /* Compute limb 1 of t*[d,e]+modulus*[md,me], and store it as output limb 0 (= down shift). */ secp256k1_i128_accum_mul(&cd, u, d1); secp256k1_i128_accum_mul(&cd, v, e1); secp256k1_i128_accum_mul(&ce, q, d1); secp256k1_i128_accum_mul(&ce, r, e1); if (modinfo->modulus.v[1]) { /* Optimize for the case where limb of modulus is zero. */ secp256k1_i128_accum_mul(&cd, modinfo->modulus.v[1], md); secp256k1_i128_accum_mul(&ce, modinfo->modulus.v[1], me); } d->v[0] = secp256k1_i128_to_u64(&cd) & M62; secp256k1_i128_rshift(&cd, 62); e->v[0] = secp256k1_i128_to_u64(&ce) & M62; secp256k1_i128_rshift(&ce, 62); /* Compute limb 2 of t*[d,e]+modulus*[md,me], and store it as output limb 1. */ secp256k1_i128_accum_mul(&cd, u, d2); secp256k1_i128_accum_mul(&cd, v, e2); secp256k1_i128_accum_mul(&ce, q, d2); secp256k1_i128_accum_mul(&ce, r, e2); if (modinfo->modulus.v[2]) { /* Optimize for the case where limb of modulus is zero. */ secp256k1_i128_accum_mul(&cd, modinfo->modulus.v[2], md); secp256k1_i128_accum_mul(&ce, modinfo->modulus.v[2], me); } d->v[1] = secp256k1_i128_to_u64(&cd) & M62; secp256k1_i128_rshift(&cd, 62); e->v[1] = secp256k1_i128_to_u64(&ce) & M62; secp256k1_i128_rshift(&ce, 62); /* Compute limb 3 of t*[d,e]+modulus*[md,me], and store it as output limb 2. */ secp256k1_i128_accum_mul(&cd, u, d3); secp256k1_i128_accum_mul(&cd, v, e3); secp256k1_i128_accum_mul(&ce, q, d3); secp256k1_i128_accum_mul(&ce, r, e3); if (modinfo->modulus.v[3]) { /* Optimize for the case where limb of modulus is zero. */ secp256k1_i128_accum_mul(&cd, modinfo->modulus.v[3], md); secp256k1_i128_accum_mul(&ce, modinfo->modulus.v[3], me); } d->v[2] = secp256k1_i128_to_u64(&cd) & M62; secp256k1_i128_rshift(&cd, 62); e->v[2] = secp256k1_i128_to_u64(&ce) & M62; secp256k1_i128_rshift(&ce, 62); /* Compute limb 4 of t*[d,e]+modulus*[md,me], and store it as output limb 3. */ secp256k1_i128_accum_mul(&cd, u, d4); secp256k1_i128_accum_mul(&cd, v, e4); secp256k1_i128_accum_mul(&ce, q, d4); secp256k1_i128_accum_mul(&ce, r, e4); secp256k1_i128_accum_mul(&cd, modinfo->modulus.v[4], md); secp256k1_i128_accum_mul(&ce, modinfo->modulus.v[4], me); d->v[3] = secp256k1_i128_to_u64(&cd) & M62; secp256k1_i128_rshift(&cd, 62); e->v[3] = secp256k1_i128_to_u64(&ce) & M62; secp256k1_i128_rshift(&ce, 62); /* What remains is limb 5 of t*[d,e]+modulus*[md,me]; store it as output limb 4. */ d->v[4] = secp256k1_i128_to_i64(&cd); e->v[4] = secp256k1_i128_to_i64(&ce); #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, -2) > 0); /* d > -2*modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, 1) < 0); /* d < modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, -2) > 0); /* e > -2*modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, 1) < 0); /* e < modulus */ #endif } #if 0 /* Compute (t/2^62) * [f, g], where t is a transition matrix scaled by 2^62. * * This implements the update_fg function from the explanation. */ static void secp256k1_modinv64_update_fg_62(secp256k1_modinv64_signed62 *f, secp256k1_modinv64_signed62 *g, const secp256k1_modinv64_trans2x2 *t) { const uint64_t M62 = UINT64_MAX >> 2; const int64_t f0 = f->v[0], f1 = f->v[1], f2 = f->v[2], f3 = f->v[3], f4 = f->v[4]; const int64_t g0 = g->v[0], g1 = g->v[1], g2 = g->v[2], g3 = g->v[3], g4 = g->v[4]; const int64_t u = t->u, v = t->v, q = t->q, r = t->r; secp256k1_int128 cf, cg; /* Start computing t*[f,g]. */ secp256k1_i128_mul(&cf, u, f0); secp256k1_i128_accum_mul(&cf, v, g0); secp256k1_i128_mul(&cg, q, f0); secp256k1_i128_accum_mul(&cg, r, g0); /* Verify that the bottom 62 bits of the result are zero, and then throw them away. */ VERIFY_CHECK((secp256k1_i128_to_u64(&cf) & M62) == 0); secp256k1_i128_rshift(&cf, 62); VERIFY_CHECK((secp256k1_i128_to_u64(&cg) & M62) == 0); secp256k1_i128_rshift(&cg, 62); /* Compute limb 1 of t*[f,g], and store it as output limb 0 (= down shift). */ secp256k1_i128_accum_mul(&cf, u, f1); secp256k1_i128_accum_mul(&cf, v, g1); secp256k1_i128_accum_mul(&cg, q, f1); secp256k1_i128_accum_mul(&cg, r, g1); f->v[0] = secp256k1_i128_to_u64(&cf) & M62; secp256k1_i128_rshift(&cf, 62); g->v[0] = secp256k1_i128_to_u64(&cg) & M62; secp256k1_i128_rshift(&cg, 62); /* Compute limb 2 of t*[f,g], and store it as output limb 1. */ secp256k1_i128_accum_mul(&cf, u, f2); secp256k1_i128_accum_mul(&cf, v, g2); secp256k1_i128_accum_mul(&cg, q, f2); secp256k1_i128_accum_mul(&cg, r, g2); f->v[1] = secp256k1_i128_to_u64(&cf) & M62; secp256k1_i128_rshift(&cf, 62); g->v[1] = secp256k1_i128_to_u64(&cg) & M62; secp256k1_i128_rshift(&cg, 62); /* Compute limb 3 of t*[f,g], and store it as output limb 2. */ secp256k1_i128_accum_mul(&cf, u, f3); secp256k1_i128_accum_mul(&cf, v, g3); secp256k1_i128_accum_mul(&cg, q, f3); secp256k1_i128_accum_mul(&cg, r, g3); f->v[2] = secp256k1_i128_to_u64(&cf) & M62; secp256k1_i128_rshift(&cf, 62); g->v[2] = secp256k1_i128_to_u64(&cg) & M62; secp256k1_i128_rshift(&cg, 62); /* Compute limb 4 of t*[f,g], and store it as output limb 3. */ secp256k1_i128_accum_mul(&cf, u, f4); secp256k1_i128_accum_mul(&cf, v, g4); secp256k1_i128_accum_mul(&cg, q, f4); secp256k1_i128_accum_mul(&cg, r, g4); f->v[3] = secp256k1_i128_to_u64(&cf) & M62; secp256k1_i128_rshift(&cf, 62); g->v[3] = secp256k1_i128_to_u64(&cg) & M62; secp256k1_i128_rshift(&cg, 62); /* What remains is limb 5 of t*[f,g]; store it as output limb 4. */ f->v[4] = secp256k1_i128_to_i64(&cf); g->v[4] = secp256k1_i128_to_i64(&cg); } #endif /* Compute (t/2^62) * [f, g], where t is a transition matrix for 62 divsteps. * * Version that operates on a variable number of limbs in f and g. * * This implements the update_fg function from the explanation. */ static void secp256k1_modinv64_update_fg_62_var(int len, secp256k1_modinv64_signed62 *f, secp256k1_modinv64_signed62 *g, const secp256k1_modinv64_trans2x2 *t) { const uint64_t M62 = UINT64_MAX >> 2; const int64_t u = t->u, v = t->v, q = t->q, r = t->r; int64_t fi, gi; secp256k1_int128 cf, cg; int i; VERIFY_CHECK(len > 0); /* Start computing t*[f,g]. */ fi = f->v[0]; gi = g->v[0]; secp256k1_i128_mul(&cf, u, fi); secp256k1_i128_accum_mul(&cf, v, gi); secp256k1_i128_mul(&cg, q, fi); secp256k1_i128_accum_mul(&cg, r, gi); /* Verify that the bottom 62 bits of the result are zero, and then throw them away. */ VERIFY_CHECK((secp256k1_i128_to_u64(&cf) & M62) == 0); secp256k1_i128_rshift(&cf, 62); VERIFY_CHECK((secp256k1_i128_to_u64(&cg) & M62) == 0); secp256k1_i128_rshift(&cg, 62); /* Now iteratively compute limb i=1..len of t*[f,g], and store them in output limb i-1 (shifting * down by 62 bits). */ for (i = 1; i < len; ++i) { fi = f->v[i]; gi = g->v[i]; secp256k1_i128_accum_mul(&cf, u, fi); secp256k1_i128_accum_mul(&cf, v, gi); secp256k1_i128_accum_mul(&cg, q, fi); secp256k1_i128_accum_mul(&cg, r, gi); f->v[i - 1] = secp256k1_i128_to_u64(&cf) & M62; secp256k1_i128_rshift(&cf, 62); g->v[i - 1] = secp256k1_i128_to_u64(&cg) & M62; secp256k1_i128_rshift(&cg, 62); } /* What remains is limb (len) of t*[f,g]; store it as output limb (len-1). */ f->v[len - 1] = secp256k1_i128_to_i64(&cf); g->v[len - 1] = secp256k1_i128_to_i64(&cg); } #if 0 /* Compute the inverse of x modulo modinfo->modulus, and replace x with it (constant time in x). */ static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) { /* Start with d=0, e=1, f=modulus, g=x, zeta=-1. */ secp256k1_modinv64_signed62 d = {{0, 0, 0, 0, 0}}; secp256k1_modinv64_signed62 e = {{1, 0, 0, 0, 0}}; secp256k1_modinv64_signed62 f = modinfo->modulus; secp256k1_modinv64_signed62 g = *x; int i; int64_t zeta = -1; /* zeta = -(delta+1/2); delta starts at 1/2. */ /* Do 10 iterations of 59 divsteps each = 590 divsteps. This suffices for 256-bit inputs. */ for (i = 0; i < 10; ++i) { /* Compute transition matrix and new zeta after 59 divsteps. */ secp256k1_modinv64_trans2x2 t; zeta = secp256k1_modinv64_divsteps_59(zeta, f.v[0], g.v[0], &t); /* Update d,e using that transition matrix. */ secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo); /* Update f,g using that transition matrix. */ #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, -1) > 0); /* f > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, 1) <= 0); /* f <= modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, -1) > 0); /* g > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, 1) < 0); /* g < modulus */ #endif secp256k1_modinv64_update_fg_62(&f, &g, &t); #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, -1) > 0); /* f > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, 1) <= 0); /* f <= modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, -1) > 0); /* g > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, 1) < 0); /* g < modulus */ #endif } /* At this point sufficient iterations have been performed that g must have reached 0 * and (if g was not originally 0) f must now equal +/- GCD of the initial f, g * values i.e. +/- 1, and d now contains +/- the modular inverse. */ #ifdef VERIFY /* g == 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &SECP256K1_SIGNED62_ONE, 0) == 0); /* |f| == 1, or (x == 0 and d == 0 and |f|=modulus) */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &SECP256K1_SIGNED62_ONE, -1) == 0 || secp256k1_modinv64_mul_cmp_62(&f, 5, &SECP256K1_SIGNED62_ONE, 1) == 0 || (secp256k1_modinv64_mul_cmp_62(x, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && secp256k1_modinv64_mul_cmp_62(&d, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && (secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, 1) == 0 || secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, -1) == 0))); #endif /* Optionally negate d, normalize to [0,modulus), and return it. */ secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo); *x = d; } #endif /* Compute the inverse of x modulo modinfo->modulus, and replace x with it (variable time). */ static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) { /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ secp256k1_modinv64_signed62 d = {{0, 0, 0, 0, 0}}; secp256k1_modinv64_signed62 e = {{1, 0, 0, 0, 0}}; secp256k1_modinv64_signed62 f = modinfo->modulus; secp256k1_modinv64_signed62 g = *x; #ifdef VERIFY int i = 0; #endif int j, len = 5; int64_t eta = -1; /* eta = -delta; delta is initially 1 */ int64_t cond, fn, gn; /* Do iterations of 62 divsteps each until g=0. */ while (1) { /* Compute transition matrix and new eta after 62 divsteps. */ secp256k1_modinv64_trans2x2 t; eta = secp256k1_modinv64_divsteps_62_var(eta, f.v[0], g.v[0], &t); /* Update d,e using that transition matrix. */ secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo); /* Update f,g using that transition matrix. */ #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, -1) > 0); /* f > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, -1) > 0); /* g > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ #endif secp256k1_modinv64_update_fg_62_var(len, &f, &g, &t); /* If the bottom limb of g is zero, there is a chance that g=0. */ if (g.v[0] == 0) { cond = 0; /* Check if the other limbs are also 0. */ for (j = 1; j < len; ++j) { cond |= g.v[j]; } /* If so, we're done. */ if (cond == 0) break; } /* Determine if len>1 and limb (len-1) of both f and g is 0 or -1. */ fn = f.v[len - 1]; gn = g.v[len - 1]; cond = ((int64_t)len - 2) >> 63; cond |= fn ^ (fn >> 63); cond |= gn ^ (gn >> 63); /* If so, reduce length, propagating the sign of f and g's top limb into the one below. */ if (cond == 0) { f.v[len - 2] |= (uint64_t)fn << 62; g.v[len - 2] |= (uint64_t)gn << 62; --len; } #ifdef VERIFY VERIFY_CHECK(++i < 12); /* We should never need more than 12*62 = 744 divsteps */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, -1) > 0); /* f > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, -1) > 0); /* g > -modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ #endif } /* At this point g is 0 and (if g was not originally 0) f must now equal +/- GCD of * the initial f, g values i.e. +/- 1, and d now contains +/- the modular inverse. */ #ifdef VERIFY /* g == 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &SECP256K1_SIGNED62_ONE, 0) == 0); /* |f| == 1, or (x == 0 and d == 0 and |f|=modulus) */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &SECP256K1_SIGNED62_ONE, -1) == 0 || secp256k1_modinv64_mul_cmp_62(&f, len, &SECP256K1_SIGNED62_ONE, 1) == 0 || (secp256k1_modinv64_mul_cmp_62(x, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && secp256k1_modinv64_mul_cmp_62(&d, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && (secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) == 0 || secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, -1) == 0))); #endif /* Optionally negate d, normalize to [0,modulus), and return it. */ secp256k1_modinv64_normalize_62(&d, f.v[len - 1], modinfo); *x = d; } #if 0 /* Do up to 25 iterations of 62 posdivsteps (up to 1550 steps; more is extremely rare) each until f=1. * In VERIFY mode use a lower number of iterations (744, close to the median 756), so failure actually occurs. */ #ifdef VERIFY #define JACOBI64_ITERATIONS 12 #else #define JACOBI64_ITERATIONS 25 #endif /* Compute the Jacobi symbol of x modulo modinfo->modulus (variable time). gcd(x,modulus) must be 1. */ static int secp256k1_jacobi64_maybe_var(const secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) { /* Start with f=modulus, g=x, eta=-1. */ secp256k1_modinv64_signed62 f = modinfo->modulus; secp256k1_modinv64_signed62 g = *x; int j, len = 5; int64_t eta = -1; /* eta = -delta; delta is initially 1 */ int64_t cond, fn, gn; int jac = 0; int count; /* The input limbs must all be non-negative. */ VERIFY_CHECK(g.v[0] >= 0 && g.v[1] >= 0 && g.v[2] >= 0 && g.v[3] >= 0 && g.v[4] >= 0); /* If x > 0, then if the loop below converges, it converges to f=g=gcd(x,modulus). Since we * require that gcd(x,modulus)=1 and modulus>=3, x cannot be 0. Thus, we must reach f=1 (or * time out). */ VERIFY_CHECK((g.v[0] | g.v[1] | g.v[2] | g.v[3] | g.v[4]) != 0); for (count = 0; count < JACOBI64_ITERATIONS; ++count) { /* Compute transition matrix and new eta after 62 posdivsteps. */ secp256k1_modinv64_trans2x2 t; eta = secp256k1_modinv64_posdivsteps_62_var(eta, f.v[0] | ((uint64_t)f.v[1] << 62), g.v[0] | ((uint64_t)g.v[1] << 62), &t, &jac); /* Update f,g using that transition matrix. */ #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 0) > 0); /* f > 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 0) > 0); /* g > 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ #endif secp256k1_modinv64_update_fg_62_var(len, &f, &g, &t); /* If the bottom limb of f is 1, there is a chance that f=1. */ if (f.v[0] == 1) { cond = 0; /* Check if the other limbs are also 0. */ for (j = 1; j < len; ++j) { cond |= f.v[j]; } /* If so, we're done. When f=1, the Jacobi symbol (g | f)=1. */ if (cond == 0) return 1 - 2*(jac & 1); } /* Determine if len>1 and limb (len-1) of both f and g is 0. */ fn = f.v[len - 1]; gn = g.v[len - 1]; cond = ((int64_t)len - 2) >> 63; cond |= fn; cond |= gn; /* If so, reduce length. */ if (cond == 0) --len; #ifdef VERIFY VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 0) > 0); /* f > 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 0) > 0); /* g > 0 */ VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ #endif } /* The loop failed to converge to f=g after 1550 iterations. Return 0, indicating unknown result. */ return 0; } #endif #endif /* SECP256K1_MODINV64_IMPL_H */