/*********************************************************************** * 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_MODINV32_IMPL_H #define SECP256K1_MODINV32_IMPL_H #include "modinv32.h" #include "util.h" #include /* 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=30, using 30-bit signed limbs represented as int32_t. */ /* Take as input a signed30 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 (-2^30,2^30). The output will have limbs in range * [0,2^30). */ static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int32_t sign, const secp256k1_modinv32_modinfo *modinfo) { const int32_t M30 = (int32_t)(UINT32_MAX >> 2); int32_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4], r5 = r->v[5], r6 = r->v[6], r7 = r->v[7], r8 = r->v[8]; int32_t cond_add, cond_negate; /* 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 (-2^30,2^30), this cannot overflow an int32_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 = r8 >> 31; 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; r5 += modinfo->modulus.v[5] & cond_add; r6 += modinfo->modulus.v[6] & cond_add; r7 += modinfo->modulus.v[7] & cond_add; r8 += modinfo->modulus.v[8] & cond_add; cond_negate = sign >> 31; 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; r5 = (r5 ^ cond_negate) - cond_negate; r6 = (r6 ^ cond_negate) - cond_negate; r7 = (r7 ^ cond_negate) - cond_negate; r8 = (r8 ^ cond_negate) - cond_negate; /* Propagate the top bits, to bring limbs back to range (-2^30,2^30). */ r1 += r0 >> 30; r0 &= M30; r2 += r1 >> 30; r1 &= M30; r3 += r2 >> 30; r2 &= M30; r4 += r3 >> 30; r3 &= M30; r5 += r4 >> 30; r4 &= M30; r6 += r5 >> 30; r5 &= M30; r7 += r6 >> 30; r6 &= M30; r8 += r7 >> 30; r7 &= M30; /* In a second step add the modulus again if the result is still negative, bringing r to range * [0,modulus). */ cond_add = r8 >> 31; 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; r5 += modinfo->modulus.v[5] & cond_add; r6 += modinfo->modulus.v[6] & cond_add; r7 += modinfo->modulus.v[7] & cond_add; r8 += modinfo->modulus.v[8] & cond_add; /* And propagate again. */ r1 += r0 >> 30; r0 &= M30; r2 += r1 >> 30; r1 &= M30; r3 += r2 >> 30; r2 &= M30; r4 += r3 >> 30; r3 &= M30; r5 += r4 >> 30; r4 &= M30; r6 += r5 >> 30; r5 &= M30; r7 += r6 >> 30; r6 &= M30; r8 += r7 >> 30; r7 &= M30; r->v[0] = r0; r->v[1] = r1; r->v[2] = r2; r->v[3] = r3; r->v[4] = r4; r->v[5] = r5; r->v[6] = r6; r->v[7] = r7; r->v[8] = r8; } /* Data type for transition matrices (see section 3 of explanation). * * t = [ u v ] * [ q r ] */ typedef struct { int32_t u, v, q, r; } secp256k1_modinv32_trans2x2; /* Compute the transition matrix and eta for 30 divsteps. * * 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 function from the explanation. */ static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) { /* u,v,q,r are the elements of the transformation matrix being built up, * starting with the identity matrix. Semantically they are signed integers * in range [-2^30,2^30], but here represented as unsigned mod 2^32. This * permits left shifting (which is UB for negative numbers). The range * being inside [-2^31,2^31) means that casting to signed works correctly. */ uint32_t u = 1, v = 0, q = 0, r = 1; uint32_t c1, c2, f = f0, g = g0, x, y, z; int i; for (i = 0; i < 30; ++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 (eta < 0) and for (g & 1). */ c1 = eta >> 31; 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 (eta < 0) and (g & 1). */ c1 &= c2; /* Conditionally negate eta, and unconditionally subtract 1. */ eta = (eta ^ c1) - (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; } /* Return data in t and return value. */ t->u = (int32_t)u; t->v = (int32_t)v; t->q = (int32_t)q; t->r = (int32_t)r; return eta; } /* Compute the transition matrix and eta for 30 divsteps (variable time). * * 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 int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) { /* inv256[i] = -(2*i+1)^-1 (mod 256) */ static const uint8_t inv256[128] = { 0xFF, 0x55, 0x33, 0x49, 0xC7, 0x5D, 0x3B, 0x11, 0x0F, 0xE5, 0xC3, 0x59, 0xD7, 0xED, 0xCB, 0x21, 0x1F, 0x75, 0x53, 0x69, 0xE7, 0x7D, 0x5B, 0x31, 0x2F, 0x05, 0xE3, 0x79, 0xF7, 0x0D, 0xEB, 0x41, 0x3F, 0x95, 0x73, 0x89, 0x07, 0x9D, 0x7B, 0x51, 0x4F, 0x25, 0x03, 0x99, 0x17, 0x2D, 0x0B, 0x61, 0x5F, 0xB5, 0x93, 0xA9, 0x27, 0xBD, 0x9B, 0x71, 0x6F, 0x45, 0x23, 0xB9, 0x37, 0x4D, 0x2B, 0x81, 0x7F, 0xD5, 0xB3, 0xC9, 0x47, 0xDD, 0xBB, 0x91, 0x8F, 0x65, 0x43, 0xD9, 0x57, 0x6D, 0x4B, 0xA1, 0x9F, 0xF5, 0xD3, 0xE9, 0x67, 0xFD, 0xDB, 0xB1, 0xAF, 0x85, 0x63, 0xF9, 0x77, 0x8D, 0x6B, 0xC1, 0xBF, 0x15, 0xF3, 0x09, 0x87, 0x1D, 0xFB, 0xD1, 0xCF, 0xA5, 0x83, 0x19, 0x97, 0xAD, 0x8B, 0xE1, 0xDF, 0x35, 0x13, 0x29, 0xA7, 0x3D, 0x1B, 0xF1, 0xEF, 0xC5, 0xA3, 0x39, 0xB7, 0xCD, 0xAB, 0x01 }; /* Transformation matrix; see comments in secp256k1_modinv32_divsteps_30. */ uint32_t u = 1, v = 0, q = 0, r = 1; uint32_t f = f0, g = g0, m; uint16_t w; int i = 30, limit, zeros; for (;;) { /* Use a sentinel bit to count zeros only up to i. */ zeros = secp256k1_ctz32_var(g | (UINT32_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 30 divsteps. */ if (i == 0) break; VERIFY_CHECK((f & 1) == 1); VERIFY_CHECK((g & 1) == 1); VERIFY_CHECK((u * f0 + v * g0) == f << (30 - i)); VERIFY_CHECK((q * f0 + r * g0) == g << (30 - i)); /* If eta is negative, negate it and replace f,g with g,-f. */ if (eta < 0) { uint32_t tmp; eta = -eta; tmp = f; f = g; g = -tmp; tmp = u; u = q; q = -tmp; tmp = v; v = r; r = -tmp; } /* eta is now >= 0. In what follows we're going to cancel out the bottom bits of g. 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 once that happens. */ limit = ((int)eta + 1) > i ? i : ((int)eta + 1); /* m is a mask for the bottom min(limit, 8) bits (our table only supports 8 bits). */ m = (UINT32_MAX >> (32 - limit)) & 255U; /* Find what multiple of f must be added to g to cancel its bottom min(limit, 8) bits. */ w = (g * inv256[(f >> 1) & 127]) & m; /* Do so. */ g += f * w; q += u * w; r += v * w; VERIFY_CHECK((g & m) == 0); } /* Return data in t and return value. */ t->u = (int32_t)u; t->v = (int32_t)v; t->q = (int32_t)q; t->r = (int32_t)r; return eta; } /* Compute (t/2^30) * [d, e] mod modulus, where t is a transition matrix for 30 divsteps. * * On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range * (-2^30,2^30). * * This implements the update_de function from the explanation. */ static void secp256k1_modinv32_update_de_30(secp256k1_modinv32_signed30 *d, secp256k1_modinv32_signed30 *e, const secp256k1_modinv32_trans2x2 *t, const secp256k1_modinv32_modinfo* modinfo) { const int32_t M30 = (int32_t)(UINT32_MAX >> 2); const int32_t u = t->u, v = t->v, q = t->q, r = t->r; int32_t di, ei, md, me, sd, se; int64_t cd, ce; int i; /* [md,me] start as zero; plus [u,q] if d is negative; plus [v,r] if e is negative. */ sd = d->v[8] >> 31; se = e->v[8] >> 31; md = (u & sd) + (v & se); me = (q & sd) + (r & se); /* Begin computing t*[d,e]. */ di = d->v[0]; ei = e->v[0]; cd = (int64_t)u * di + (int64_t)v * ei; ce = (int64_t)q * di + (int64_t)r * ei; /* Correct md,me so that t*[d,e]+modulus*[md,me] has 30 zero bottom bits. */ md -= (modinfo->modulus_inv30 * (uint32_t)cd + md) & M30; me -= (modinfo->modulus_inv30 * (uint32_t)ce + me) & M30; /* Update the beginning of computation for t*[d,e]+modulus*[md,me] now md,me are known. */ cd += (int64_t)modinfo->modulus.v[0] * md; ce += (int64_t)modinfo->modulus.v[0] * me; /* Verify that the low 30 bits of the computation are indeed zero, and then throw them away. */ VERIFY_CHECK(((int32_t)cd & M30) == 0); cd >>= 30; VERIFY_CHECK(((int32_t)ce & M30) == 0); ce >>= 30; /* Now iteratively compute limb i=1..8 of t*[d,e]+modulus*[md,me], and store them in output * limb i-1 (shifting down by 30 bits). */ for (i = 1; i < 9; ++i) { di = d->v[i]; ei = e->v[i]; cd += (int64_t)u * di + (int64_t)v * ei; ce += (int64_t)q * di + (int64_t)r * ei; cd += (int64_t)modinfo->modulus.v[i] * md; ce += (int64_t)modinfo->modulus.v[i] * me; d->v[i - 1] = (int32_t)cd & M30; cd >>= 30; e->v[i - 1] = (int32_t)ce & M30; ce >>= 30; } /* What remains is limb 9 of t*[d,e]+modulus*[md,me]; store it as output limb 8. */ d->v[8] = (int32_t)cd; e->v[8] = (int32_t)ce; } /* Compute (t/2^30) * [f, g], where t is a transition matrix for 30 divsteps. * * This implements the update_fg function from the explanation. */ static void secp256k1_modinv32_update_fg_30(secp256k1_modinv32_signed30 *f, secp256k1_modinv32_signed30 *g, const secp256k1_modinv32_trans2x2 *t) { const int32_t M30 = (int32_t)(UINT32_MAX >> 2); const int32_t u = t->u, v = t->v, q = t->q, r = t->r; int32_t fi, gi; int64_t cf, cg; int i; /* Start computing t*[f,g]. */ fi = f->v[0]; gi = g->v[0]; cf = (int64_t)u * fi + (int64_t)v * gi; cg = (int64_t)q * fi + (int64_t)r * gi; /* Verify that the bottom 30 bits of the result are zero, and then throw them away. */ VERIFY_CHECK(((int32_t)cf & M30) == 0); cf >>= 30; VERIFY_CHECK(((int32_t)cg & M30) == 0); cg >>= 30; /* Now iteratively compute limb i=1..8 of t*[f,g], and store them in output limb i-1 (shifting * down by 30 bits). */ for (i = 1; i < 9; ++i) { fi = f->v[i]; gi = g->v[i]; cf += (int64_t)u * fi + (int64_t)v * gi; cg += (int64_t)q * fi + (int64_t)r * gi; f->v[i - 1] = (int32_t)cf & M30; cf >>= 30; g->v[i - 1] = (int32_t)cg & M30; cg >>= 30; } /* What remains is limb 9 of t*[f,g]; store it as output limb 8. */ f->v[8] = (int32_t)cf; g->v[8] = (int32_t)cg; } /* Compute the inverse of x modulo modinfo->modulus, and replace x with it (constant time in x). */ static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) { /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ secp256k1_modinv32_signed30 d = {{0}}; secp256k1_modinv32_signed30 e = {{1}}; secp256k1_modinv32_signed30 f = modinfo->modulus; secp256k1_modinv32_signed30 g = *x; int i; int32_t eta = -1; /* Do 25 iterations of 30 divsteps each = 750 divsteps. 724 suffices for 256-bit inputs. */ for (i = 0; i < 25; ++i) { /* Compute transition matrix and new eta after 30 divsteps. */ secp256k1_modinv32_trans2x2 t; eta = secp256k1_modinv32_divsteps_30(eta, f.v[0], g.v[0], &t); /* Update d,e using that transition matrix. */ secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo); /* Update f,g using that transition matrix. */ secp256k1_modinv32_update_fg_30(&f, &g, &t); } /* 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. */ VERIFY_CHECK((g.v[0] | g.v[1] | g.v[2] | g.v[3] | g.v[4] | g.v[5] | g.v[6] | g.v[7] | g.v[8]) == 0); /* Optionally negate d, normalize to [0,modulus), and return it. */ secp256k1_modinv32_normalize_30(&d, f.v[8], modinfo); *x = d; } /* Compute the inverse of x modulo modinfo->modulus, and replace x with it (variable time). */ static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) { /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ secp256k1_modinv32_signed30 d = {{0, 0, 0, 0, 0, 0, 0, 0, 0}}; secp256k1_modinv32_signed30 e = {{1, 0, 0, 0, 0, 0, 0, 0, 0}}; secp256k1_modinv32_signed30 f = modinfo->modulus; secp256k1_modinv32_signed30 g = *x; int j; int32_t eta = -1; int32_t cond; /* Do iterations of 30 divsteps each until g=0. */ while (1) { /* Compute transition matrix and new eta after 30 divsteps. */ secp256k1_modinv32_trans2x2 t; eta = secp256k1_modinv32_divsteps_30_var(eta, f.v[0], g.v[0], &t); /* Update d,e using that transition matrix. */ secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo); /* Update f,g using that transition matrix. */ secp256k1_modinv32_update_fg_30(&f, &g, &t); /* If the bottom limb of g is 0, there is a chance g=0. */ if (g.v[0] == 0) { cond = 0; /* Check if the other limbs are also 0. */ for (j = 1; j < 9; ++j) { cond |= g.v[j]; } /* If so, we're done. */ if (cond == 0) break; } } /* 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. */ /* Optionally negate d, normalize to [0,modulus), and return it. */ secp256k1_modinv32_normalize_30(&d, f.v[8], modinfo); *x = d; } #endif /* SECP256K1_MODINV32_IMPL_H */