/*********************************************************************** * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra * * Distributed under the MIT software license, see the accompanying * * file COPYING or https://www.opensource.org/licenses/mit-license.php.* ***********************************************************************/ #ifndef SECP256K1_ECMULT_CONST_IMPL_H #define SECP256K1_ECMULT_CONST_IMPL_H #include "scalar.h" #include "group.h" #include "ecmult_const.h" #include "ecmult_impl.h" /** Fill a table 'pre' with precomputed odd multiples of a. * * The resulting point set is brought to a single constant Z denominator, stores the X and Y * coordinates as ge_storage points in pre, and stores the global Z in globalz. * It only operates on tables sized for WINDOW_A wnaf multiples. */ static void rustsecp256k1_v0_9_1_ecmult_odd_multiples_table_globalz_windowa(rustsecp256k1_v0_9_1_ge *pre, rustsecp256k1_v0_9_1_fe *globalz, const rustsecp256k1_v0_9_1_gej *a) { rustsecp256k1_v0_9_1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)]; rustsecp256k1_v0_9_1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr, globalz, a); rustsecp256k1_v0_9_1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr); } /* This is like `ECMULT_TABLE_GET_GE` but is constant time */ #define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \ int m = 0; \ /* Extract the sign-bit for a constant time absolute-value. */ \ int volatile mask = (n) >> (sizeof(n) * CHAR_BIT - 1); \ int abs_n = ((n) + mask) ^ mask; \ int idx_n = abs_n >> 1; \ rustsecp256k1_v0_9_1_fe neg_y; \ VERIFY_CHECK(((n) & 1) == 1); \ VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \ VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \ VERIFY_SETUP(rustsecp256k1_v0_9_1_fe_clear(&(r)->x)); \ VERIFY_SETUP(rustsecp256k1_v0_9_1_fe_clear(&(r)->y)); \ /* Unconditionally set r->x = (pre)[m].x. r->y = (pre)[m].y. because it's either the correct one \ * or will get replaced in the later iterations, this is needed to make sure `r` is initialized. */ \ (r)->x = (pre)[m].x; \ (r)->y = (pre)[m].y; \ for (m = 1; m < ECMULT_TABLE_SIZE(w); m++) { \ /* This loop is used to avoid secret data in array indices. See * the comment in ecmult_gen_impl.h for rationale. */ \ rustsecp256k1_v0_9_1_fe_cmov(&(r)->x, &(pre)[m].x, m == idx_n); \ rustsecp256k1_v0_9_1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \ } \ (r)->infinity = 0; \ rustsecp256k1_v0_9_1_fe_negate(&neg_y, &(r)->y, 1); \ rustsecp256k1_v0_9_1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \ } while(0) /** Convert a number to WNAF notation. * The number becomes represented by sum(2^{wi} * wnaf[i], i=0..WNAF_SIZE(w)+1) - return_val. * It has the following guarantees: * - each wnaf[i] an odd integer between -(1 << w) and (1 << w) * - each wnaf[i] is nonzero * - the number of words set is always WNAF_SIZE(w) + 1 * * Adapted from `The Width-w NAF Method Provides Small Memory and Fast Elliptic Scalar * Multiplications Secure against Side Channel Attacks`, Okeya and Tagaki. M. Joye (Ed.) * CT-RSA 2003, LNCS 2612, pp. 328-443, 2003. Springer-Verlag Berlin Heidelberg 2003 * * Numbers reference steps of `Algorithm SPA-resistant Width-w NAF with Odd Scalar` on pp. 335 */ static int rustsecp256k1_v0_9_1_wnaf_const(int *wnaf, const rustsecp256k1_v0_9_1_scalar *scalar, int w, int size) { int global_sign; int skew; int word = 0; /* 1 2 3 */ int u_last; int u; int flip; rustsecp256k1_v0_9_1_scalar s = *scalar; VERIFY_CHECK(w > 0); VERIFY_CHECK(size > 0); /* Note that we cannot handle even numbers by negating them to be odd, as is * done in other implementations, since if our scalars were specified to have * width < 256 for performance reasons, their negations would have width 256 * and we'd lose any performance benefit. Instead, we use a variation of a * technique from Section 4.2 of the Okeya/Tagaki paper, which is to add 1 to the * number we are encoding when it is even, returning a skew value indicating * this, and having the caller compensate after doing the multiplication. * * In fact, we _do_ want to negate numbers to minimize their bit-lengths (and in * particular, to ensure that the outputs from the endomorphism-split fit into * 128 bits). If we negate, the parity of our number flips, affecting whether * we want to add to the scalar to ensure that it's odd. */ flip = rustsecp256k1_v0_9_1_scalar_is_high(&s); skew = flip ^ rustsecp256k1_v0_9_1_scalar_is_even(&s); rustsecp256k1_v0_9_1_scalar_cadd_bit(&s, 0, skew); global_sign = rustsecp256k1_v0_9_1_scalar_cond_negate(&s, flip); /* 4 */ u_last = rustsecp256k1_v0_9_1_scalar_shr_int(&s, w); do { int even; /* 4.1 4.4 */ u = rustsecp256k1_v0_9_1_scalar_shr_int(&s, w); /* 4.2 */ even = ((u & 1) == 0); /* In contrast to the original algorithm, u_last is always > 0 and * therefore we do not need to check its sign. In particular, it's easy * to see that u_last is never < 0 because u is never < 0. Moreover, * u_last is never = 0 because u is never even after a loop * iteration. The same holds analogously for the initial value of * u_last (in the first loop iteration). */ VERIFY_CHECK(u_last > 0); VERIFY_CHECK((u_last & 1) == 1); u += even; u_last -= even * (1 << w); /* 4.3, adapted for global sign change */ wnaf[word++] = u_last * global_sign; u_last = u; } while (word * w < size); wnaf[word] = u * global_sign; VERIFY_CHECK(rustsecp256k1_v0_9_1_scalar_is_zero(&s)); VERIFY_CHECK(word == WNAF_SIZE_BITS(size, w)); return skew; } static void rustsecp256k1_v0_9_1_ecmult_const(rustsecp256k1_v0_9_1_gej *r, const rustsecp256k1_v0_9_1_ge *a, const rustsecp256k1_v0_9_1_scalar *scalar) { rustsecp256k1_v0_9_1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)]; rustsecp256k1_v0_9_1_ge tmpa; rustsecp256k1_v0_9_1_fe Z; int skew_1; rustsecp256k1_v0_9_1_ge pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)]; int wnaf_lam[1 + WNAF_SIZE(WINDOW_A - 1)]; int skew_lam; rustsecp256k1_v0_9_1_scalar q_1, q_lam; int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)]; int i; if (rustsecp256k1_v0_9_1_ge_is_infinity(a)) { rustsecp256k1_v0_9_1_gej_set_infinity(r); return; } /* build wnaf representation for q. */ /* split q into q_1 and q_lam (where q = q_1 + q_lam*lambda, and q_1 and q_lam are ~128 bit) */ rustsecp256k1_v0_9_1_scalar_split_lambda(&q_1, &q_lam, scalar); skew_1 = rustsecp256k1_v0_9_1_wnaf_const(wnaf_1, &q_1, WINDOW_A - 1, 128); skew_lam = rustsecp256k1_v0_9_1_wnaf_const(wnaf_lam, &q_lam, WINDOW_A - 1, 128); /* Calculate odd multiples of a. * All multiples are brought to the same Z 'denominator', which is stored * in Z. Due to secp256k1' isomorphism we can do all operations pretending * that the Z coordinate was 1, use affine addition formulae, and correct * the Z coordinate of the result once at the end. */ VERIFY_CHECK(!a->infinity); rustsecp256k1_v0_9_1_gej_set_ge(r, a); rustsecp256k1_v0_9_1_ecmult_odd_multiples_table_globalz_windowa(pre_a, &Z, r); for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) { rustsecp256k1_v0_9_1_fe_normalize_weak(&pre_a[i].y); } for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) { rustsecp256k1_v0_9_1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]); } /* first loop iteration (separated out so we can directly set r, rather * than having it start at infinity, get doubled several times, then have * its new value added to it) */ i = wnaf_1[WNAF_SIZE_BITS(128, WINDOW_A - 1)]; VERIFY_CHECK(i != 0); ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A); rustsecp256k1_v0_9_1_gej_set_ge(r, &tmpa); i = wnaf_lam[WNAF_SIZE_BITS(128, WINDOW_A - 1)]; VERIFY_CHECK(i != 0); ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, i, WINDOW_A); rustsecp256k1_v0_9_1_gej_add_ge(r, r, &tmpa); /* remaining loop iterations */ for (i = WNAF_SIZE_BITS(128, WINDOW_A - 1) - 1; i >= 0; i--) { int n; int j; for (j = 0; j < WINDOW_A - 1; ++j) { rustsecp256k1_v0_9_1_gej_double(r, r); } n = wnaf_1[i]; ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A); VERIFY_CHECK(n != 0); rustsecp256k1_v0_9_1_gej_add_ge(r, r, &tmpa); n = wnaf_lam[i]; ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A); VERIFY_CHECK(n != 0); rustsecp256k1_v0_9_1_gej_add_ge(r, r, &tmpa); } { /* Correct for wNAF skew */ rustsecp256k1_v0_9_1_gej tmpj; rustsecp256k1_v0_9_1_ge_neg(&tmpa, &pre_a[0]); rustsecp256k1_v0_9_1_gej_add_ge(&tmpj, r, &tmpa); rustsecp256k1_v0_9_1_gej_cmov(r, &tmpj, skew_1); rustsecp256k1_v0_9_1_ge_neg(&tmpa, &pre_a_lam[0]); rustsecp256k1_v0_9_1_gej_add_ge(&tmpj, r, &tmpa); rustsecp256k1_v0_9_1_gej_cmov(r, &tmpj, skew_lam); } rustsecp256k1_v0_9_1_fe_mul(&r->z, &r->z, &Z); } static int rustsecp256k1_v0_9_1_ecmult_const_xonly(rustsecp256k1_v0_9_1_fe* r, const rustsecp256k1_v0_9_1_fe *n, const rustsecp256k1_v0_9_1_fe *d, const rustsecp256k1_v0_9_1_scalar *q, int known_on_curve) { /* This algorithm is a generalization of Peter Dettman's technique for * avoiding the square root in a random-basepoint x-only multiplication * on a Weierstrass curve: * https://mailarchive.ietf.org/arch/msg/cfrg/7DyYY6gg32wDgHAhgSb6XxMDlJA/ * * * === Background: the effective affine technique === * * Let phi_u be the isomorphism that maps (x, y) on secp256k1 curve y^2 = x^3 + 7 to * x' = u^2*x, y' = u^3*y on curve y'^2 = x'^3 + u^6*7. This new curve has the same order as * the original (it is isomorphic), but moreover, has the same addition/doubling formulas, as * the curve b=7 coefficient does not appear in those formulas (or at least does not appear in * the formulas implemented in this codebase, both affine and Jacobian). See also Example 9.5.2 * in https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf. * * This means any linear combination of secp256k1 points can be computed by applying phi_u * (with non-zero u) on all input points (including the generator, if used), computing the * linear combination on the isomorphic curve (using the same group laws), and then applying * phi_u^{-1} to get back to secp256k1. * * Switching to Jacobian coordinates, note that phi_u applied to (X, Y, Z) is simply * (X, Y, Z/u). Thus, if we want to compute (X1, Y1, Z) + (X2, Y2, Z), with identical Z * coordinates, we can use phi_Z to transform it to (X1, Y1, 1) + (X2, Y2, 1) on an isomorphic * curve where the affine addition formula can be used instead. * If (X3, Y3, Z3) = (X1, Y1) + (X2, Y2) on that curve, then our answer on secp256k1 is * (X3, Y3, Z3*Z). * * This is the effective affine technique: if we have a linear combination of group elements * to compute, and all those group elements have the same Z coordinate, we can simply pretend * that all those Z coordinates are 1, perform the computation that way, and then multiply the * original Z coordinate back in. * * The technique works on any a=0 short Weierstrass curve. It is possible to generalize it to * other curves too, but there the isomorphic curves will have different 'a' coefficients, * which typically does affect the group laws. * * * === Avoiding the square root for x-only point multiplication === * * In this function, we want to compute the X coordinate of q*(n/d, y), for * y = sqrt((n/d)^3 + 7). Its negation would also be a valid Y coordinate, but by convention * we pick whatever sqrt returns (which we assume to be a deterministic function). * * Let g = y^2*d^3 = n^3 + 7*d^3. This also means y = sqrt(g/d^3). * Further let v = sqrt(d*g), which must exist as d*g = y^2*d^4 = (y*d^2)^2. * * The input point (n/d, y) also has Jacobian coordinates: * * (n/d, y, 1) * = (n/d * v^2, y * v^3, v) * = (n/d * d*g, y * sqrt(d^3*g^3), v) * = (n/d * d*g, sqrt(y^2 * d^3*g^3), v) * = (n*g, sqrt(g/d^3 * d^3*g^3), v) * = (n*g, sqrt(g^4), v) * = (n*g, g^2, v) * * It is easy to verify that both (n*g, g^2, v) and its negation (n*g, -g^2, v) have affine X * coordinate n/d, and this holds even when the square root function doesn't have a * deterministic sign. We choose the (n*g, g^2, v) version. * * Now switch to the effective affine curve using phi_v, where the input point has coordinates * (n*g, g^2). Compute (X, Y, Z) = q * (n*g, g^2) there. * * Back on secp256k1, that means q * (n*g, g^2, v) = (X, Y, v*Z). This last point has affine X * coordinate X / (v^2*Z^2) = X / (d*g*Z^2). Determining the affine Y coordinate would involve * a square root, but as long as we only care about the resulting X coordinate, no square root * is needed anywhere in this computation. */ rustsecp256k1_v0_9_1_fe g, i; rustsecp256k1_v0_9_1_ge p; rustsecp256k1_v0_9_1_gej rj; /* Compute g = (n^3 + B*d^3). */ rustsecp256k1_v0_9_1_fe_sqr(&g, n); rustsecp256k1_v0_9_1_fe_mul(&g, &g, n); if (d) { rustsecp256k1_v0_9_1_fe b; #ifdef VERIFY VERIFY_CHECK(!rustsecp256k1_v0_9_1_fe_normalizes_to_zero(d)); #endif rustsecp256k1_v0_9_1_fe_sqr(&b, d); VERIFY_CHECK(SECP256K1_B <= 8); /* magnitude of b will be <= 8 after the next call */ rustsecp256k1_v0_9_1_fe_mul_int(&b, SECP256K1_B); rustsecp256k1_v0_9_1_fe_mul(&b, &b, d); rustsecp256k1_v0_9_1_fe_add(&g, &b); if (!known_on_curve) { /* We need to determine whether (n/d)^3 + 7 is square. * * is_square((n/d)^3 + 7) * <=> is_square(((n/d)^3 + 7) * d^4) * <=> is_square((n^3 + 7*d^3) * d) * <=> is_square(g * d) */ rustsecp256k1_v0_9_1_fe c; rustsecp256k1_v0_9_1_fe_mul(&c, &g, d); if (!rustsecp256k1_v0_9_1_fe_is_square_var(&c)) return 0; } } else { rustsecp256k1_v0_9_1_fe_add_int(&g, SECP256K1_B); if (!known_on_curve) { /* g at this point equals x^3 + 7. Test if it is square. */ if (!rustsecp256k1_v0_9_1_fe_is_square_var(&g)) return 0; } } /* Compute base point P = (n*g, g^2), the effective affine version of (n*g, g^2, v), which has * corresponding affine X coordinate n/d. */ rustsecp256k1_v0_9_1_fe_mul(&p.x, &g, n); rustsecp256k1_v0_9_1_fe_sqr(&p.y, &g); p.infinity = 0; /* Perform x-only EC multiplication of P with q. */ #ifdef VERIFY VERIFY_CHECK(!rustsecp256k1_v0_9_1_scalar_is_zero(q)); #endif rustsecp256k1_v0_9_1_ecmult_const(&rj, &p, q); #ifdef VERIFY VERIFY_CHECK(!rustsecp256k1_v0_9_1_gej_is_infinity(&rj)); #endif /* The resulting (X, Y, Z) point on the effective-affine isomorphic curve corresponds to * (X, Y, Z*v) on the secp256k1 curve. The affine version of that has X coordinate * (X / (Z^2*d*g)). */ rustsecp256k1_v0_9_1_fe_sqr(&i, &rj.z); rustsecp256k1_v0_9_1_fe_mul(&i, &i, &g); if (d) rustsecp256k1_v0_9_1_fe_mul(&i, &i, d); rustsecp256k1_v0_9_1_fe_inv(&i, &i); rustsecp256k1_v0_9_1_fe_mul(r, &rj.x, &i); return 1; } #endif /* SECP256K1_ECMULT_CONST_IMPL_H */