| /*********************************************************************** | |
| * Copyright (c) 2015, 2022 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" | |
| | |
| #if defined(EXHAUSTIVE_TEST_ORDER) | |
| /* We need 2^ECMULT_CONST_GROUP_SIZE - 1 to be less than EXHAUSTIVE_TEST_ORDER, because | |
| * the tables cannot have infinities in them (this breaks the effective-affine technique's | |
| * z-ratio tracking) */ | |
| # if EXHAUSTIVE_TEST_ORDER == 199 | |
| # define ECMULT_CONST_GROUP_SIZE 4 | |
| # elif EXHAUSTIVE_TEST_ORDER == 13 | |
| # define ECMULT_CONST_GROUP_SIZE 3 | |
| # elif EXHAUSTIVE_TEST_ORDER == 7 | |
| # define ECMULT_CONST_GROUP_SIZE 2 | |
| # else | |
| # error "Unknown EXHAUSTIVE_TEST_ORDER" | |
| # endif | |
| #else | |
| /* Group size 4 or 5 appears optimal. */ | |
| # define ECMULT_CONST_GROUP_SIZE 5 | |
| #endif | |
| | |
| #define ECMULT_CONST_TABLE_SIZE (1L << (ECMULT_CONST_GROUP_SIZE - 1)) | |
| #define ECMULT_CONST_GROUPS ((129 + ECMULT_CONST_GROUP_SIZE - 1) / ECMULT_CONST_GROUP_SIZE) | |
| #define ECMULT_CONST_BITS (ECMULT_CONST_GROUPS * ECMULT_CONST_GROUP_SIZE) | |
| | |
| /** 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 points in pre, and stores the global Z in globalz. | |
| * | |
| * 'pre' must be an array of size ECMULT_CONST_TABLE_SIZE. | |
| */ | |
| static void secp256k1_ecmult_const_odd_multiples_table_globalz(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) { | |
| secp256k1_fe zr[ECMULT_CONST_TABLE_SIZE]; | |
| | |
| secp256k1_ecmult_odd_multiples_table(ECMULT_CONST_TABLE_SIZE, pre, zr, globalz, a); | |
| secp256k1_ge_table_set_globalz(ECMULT_CONST_TABLE_SIZE, pre, zr); | |
| } | |
| | |
| /* Given a table 'pre' with odd multiples of a point, put in r the signed-bit multiplication of n with that point. | |
| * | |
| * For example, if ECMULT_CONST_GROUP_SIZE is 4, then pre is expected to contain 8 entries: | |
| * [1*P, 3*P, 5*P, 7*P, 9*P, 11*P, 13*P, 15*P]. n is then expected to be a 4-bit integer (range 0-15), and its | |
| * bits are interpreted as signs of powers of two to look up. | |
| * | |
| * For example, if n=4, which is 0100 in binary, which is interpreted as [- + - -], so the looked up value is | |
| * [ -(2^3) + (2^2) - (2^1) - (2^0) ]*P = -7*P. Every valid n translates to an odd number in range [-15,15], | |
| * which means we just need to look up one of the precomputed values, and optionally negate it. | |
| */ | |
| #define ECMULT_CONST_TABLE_GET_GE(r,pre,n) do { \ | |
| unsigned int m = 0; \ | |
| /* If the top bit of n is 0, we want the negation. */ \ | |
| volatile unsigned int negative = ((n) >> (ECMULT_CONST_GROUP_SIZE - 1)) ^ 1; \ | |
| /* Let n[i] be the i-th bit of n, then the index is | |
| * sum(cnot(n[i]) * 2^i, i=0..l-2) | |
| * where cnot(b) = b if n[l-1] = 1 and 1 - b otherwise. | |
| * For example, if n = 4, in binary 0100, the index is 3, in binary 011. | |
| * | |
| * Proof: | |
| * Let | |
| * x = sum((2*n[i] - 1)*2^i, i=0..l-1) | |
| * = 2*sum(n[i] * 2^i, i=0..l-1) - 2^l + 1 | |
| * be the value represented by n. | |
| * The index is (x - 1)/2 if x > 0 and -(x + 1)/2 otherwise. | |
| * Case x > 0: | |
| * n[l-1] = 1 | |
| * index = sum(n[i] * 2^i, i=0..l-1) - 2^(l-1) | |
| * = sum(n[i] * 2^i, i=0..l-2) | |
| * Case x <= 0: | |
| * n[l-1] = 0 | |
| * index = -(2*sum(n[i] * 2^i, i=0..l-1) - 2^l + 2)/2 | |
| * = 2^(l-1) - 1 - sum(n[i] * 2^i, i=0..l-1) | |
| * = sum((1 - n[i]) * 2^i, i=0..l-2) | |
| */ \ | |
| unsigned int index = ((unsigned int)(-negative) ^ n) & ((1U << (ECMULT_CONST_GROUP_SIZE - 1)) - 1U); \ | |
| secp256k1_fe neg_y; \ | |
| VERIFY_CHECK((n) < (1U << ECMULT_CONST_GROUP_SIZE)); \ | |
| VERIFY_CHECK(index < (1U << (ECMULT_CONST_GROUP_SIZE - 1))); \ | |
| /* 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_CONST_TABLE_SIZE; m++) { \ | |
| /* This loop is used to avoid secret data in array indices. See | |
| * the comment in ecmult_gen_impl.h for rationale. */ \ | |
| secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == index); \ | |
| secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == index); \ | |
| } \ | |
| (r)->infinity = 0; \ | |
| secp256k1_fe_negate(&neg_y, &(r)->y, 1); \ | |
| secp256k1_fe_cmov(&(r)->y, &neg_y, negative); \ | |
| } while(0) | |
| | |
| /* For K as defined in the comment of secp256k1_ecmult_const, we have several precomputed | |
| * formulas/constants. | |
| * - in exhaustive test mode, we give an explicit expression to compute it at compile time: */ | |
| #ifdef EXHAUSTIVE_TEST_ORDER | |
| static const secp256k1_scalar secp256k1_ecmult_const_K = ((SECP256K1_SCALAR_CONST(0, 0, 0, (1U << (ECMULT_CONST_BITS - 128)) - 2U, 0, 0, 0, 0) + EXHAUSTIVE_TEST_ORDER - 1U) * (1U + EXHAUSTIVE_TEST_LAMBDA)) % EXHAUSTIVE_TEST_ORDER; | |
| /* - for the real secp256k1 group we have constants for various ECMULT_CONST_BITS values. */ | |
| #elif ECMULT_CONST_BITS == 129 | |
| /* For GROUP_SIZE = 1,3. */ | |
| static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0xac9c52b3ul, 0x3fa3cf1ful, 0x5ad9e3fdul, 0x77ed9ba4ul, 0xa880b9fcul, 0x8ec739c2ul, 0xe0cfc810ul, 0xb51283ceul); | |
| #elif ECMULT_CONST_BITS == 130 | |
| /* For GROUP_SIZE = 2,5. */ | |
| static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0xa4e88a7dul, 0xcb13034eul, 0xc2bdd6bful, 0x7c118d6bul, 0x589ae848ul, 0x26ba29e4ul, 0xb5c2c1dcul, 0xde9798d9ul); | |
| #elif ECMULT_CONST_BITS == 132 | |
| /* For GROUP_SIZE = 4,6 */ | |
| static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0x76b1d93dul, 0x0fae3c6bul, 0x3215874bul, 0x94e93813ul, 0x7937fe0dul, 0xb66bcaaful, 0xb3749ca5ul, 0xd7b6171bul); | |
| #else | |
| # error "Unknown ECMULT_CONST_BITS" | |
| #endif | |
| | |
| static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q) { | |
| /* The approach below combines the signed-digit logic from Mike Hamburg's | |
| * "Fast and compact elliptic-curve cryptography" (https://eprint.iacr.org/2012/309) | |
| * Section 3.3, with the GLV endomorphism. | |
| * | |
| * The idea there is to interpret the bits of a scalar as signs (1 = +, 0 = -), and compute a | |
| * point multiplication in that fashion. Let v be an n-bit non-negative integer (0 <= v < 2^n), | |
| * and v[i] its i'th bit (so v = sum(v[i] * 2^i, i=0..n-1)). Then define: | |
| * | |
| * C_l(v, A) = sum((2*v[i] - 1) * 2^i*A, i=0..l-1) | |
| * | |
| * Then it holds that C_l(v, A) = sum((2*v[i] - 1) * 2^i*A, i=0..l-1) | |
| * = (2*sum(v[i] * 2^i, i=0..l-1) + 1 - 2^l) * A | |
| * = (2*v + 1 - 2^l) * A | |
| * | |
| * Thus, one can compute q*A as C_256((q + 2^256 - 1) / 2, A). This is the basis for the | |
| * paper's signed-digit multi-comb algorithm for multiplication using a precomputed table. | |
| * | |
| * It is appealing to try to combine this with the GLV optimization: the idea that a scalar | |
| * s can be written as s1 + lambda*s2, where lambda is a curve-specific constant such that | |
| * lambda*A is easy to compute, and where s1 and s2 are small. In particular we have the | |
| * secp256k1_scalar_split_lambda function which performs such a split with the resulting s1 | |
| * and s2 in range (-2^128, 2^128) mod n. This does work, but is uninteresting: | |
| * | |
| * To compute q*A: | |
| * - Let s1, s2 = split_lambda(q) | |
| * - Let R1 = C_256((s1 + 2^256 - 1) / 2, A) | |
| * - Let R2 = C_256((s2 + 2^256 - 1) / 2, lambda*A) | |
| * - Return R1 + R2 | |
| * | |
| * The issue is that while s1 and s2 are small-range numbers, (s1 + 2^256 - 1) / 2 (mod n) | |
| * and (s2 + 2^256 - 1) / 2 (mod n) are not, undoing the benefit of the splitting. | |
| * | |
| * To make it work, we want to modify the input scalar q first, before splitting, and then only | |
| * add a 2^128 offset of the split results (so that they end up in the single 129-bit range | |
| * [0,2^129]). A slightly smaller offset would work due to the bounds on the split, but we pick | |
| * 2^128 for simplicity. Let s be the scalar fed to split_lambda, and f(q) the function to | |
| * compute it from q: | |
| * | |
| * To compute q*A: | |
| * - Compute s = f(q) | |
| * - Let s1, s2 = split_lambda(s) | |
| * - Let v1 = s1 + 2^128 (mod n) | |
| * - Let v2 = s2 + 2^128 (mod n) | |
| * - Let R1 = C_l(v1, A) | |
| * - Let R2 = C_l(v2, lambda*A) | |
| * - Return R1 + R2 | |
| * | |
| * l will thus need to be at least 129, but we may overshoot by a few bits (see | |
| * further), so keep it as a variable. | |
| * | |
| * To solve for s, we reason: | |
| * q*A = R1 + R2 | |
| * <=> q*A = C_l(s1 + 2^128, A) + C_l(s2 + 2^128, lambda*A) | |
| * <=> q*A = (2*(s1 + 2^128) + 1 - 2^l) * A + (2*(s2 + 2^128) + 1 - 2^l) * lambda*A | |
| * <=> q*A = (2*(s1 + s2*lambda) + (2^129 + 1 - 2^l) * (1 + lambda)) * A | |
| * <=> q = 2*(s1 + s2*lambda) + (2^129 + 1 - 2^l) * (1 + lambda) (mod n) | |
| * <=> q = 2*s + (2^129 + 1 - 2^l) * (1 + lambda) (mod n) | |
| * <=> s = (q + (2^l - 2^129 - 1) * (1 + lambda)) / 2 (mod n) | |
| * <=> f(q) = (q + K) / 2 (mod n) | |
| * where K = (2^l - 2^129 - 1)*(1 + lambda) (mod n) | |
| * | |
| * We will process the computation of C_l(v1, A) and C_l(v2, lambda*A) in groups of | |
| * ECMULT_CONST_GROUP_SIZE, so we set l to the smallest multiple of ECMULT_CONST_GROUP_SIZE | |
| * that is not less than 129; this equals ECMULT_CONST_BITS. | |
| */ | |
| | |
| /* The offset to add to s1 and s2 to make them non-negative. Equal to 2^128. */ | |
| static const secp256k1_scalar S_OFFSET = SECP256K1_SCALAR_CONST(0, 0, 0, 1, 0, 0, 0, 0); | |
| secp256k1_scalar s, v1, v2; | |
| secp256k1_ge pre_a[ECMULT_CONST_TABLE_SIZE]; | |
| secp256k1_ge pre_a_lam[ECMULT_CONST_TABLE_SIZE]; | |
| secp256k1_fe global_z; | |
| int group, i; | |
| | |
| /* We're allowed to be non-constant time in the point, and the code below (in particular, | |
| * secp256k1_ecmult_const_odd_multiples_table_globalz) cannot deal with infinity in a | |
| * constant-time manner anyway. */ | |
| if (secp256k1_ge_is_infinity(a)) { | |
| secp256k1_gej_set_infinity(r); | |
| return; | |
| } | |
| | |
| /* Compute v1 and v2. */ | |
| secp256k1_scalar_add(&s, q, &secp256k1_ecmult_const_K); | |
| secp256k1_scalar_half(&s, &s); | |
| secp256k1_scalar_split_lambda(&v1, &v2, &s); | |
| secp256k1_scalar_add(&v1, &v1, &S_OFFSET); | |
| secp256k1_scalar_add(&v2, &v2, &S_OFFSET); | |
| | |
| #ifdef VERIFY | |
| /* Verify that v1 and v2 are in range [0, 2^129-1]. */ | |
| for (i = 129; i < 256; ++i) { | |
| VERIFY_CHECK(secp256k1_scalar_get_bits_limb32(&v1, i, 1) == 0); | |
| VERIFY_CHECK(secp256k1_scalar_get_bits_limb32(&v2, i, 1) == 0); | |
| } | |
| #endif | |
| | |
| /* Calculate odd multiples of A and A*lambda. | |
| * All multiples are brought to the same Z 'denominator', which is stored | |
| * in global_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. | |
| */ | |
| secp256k1_gej_set_ge(r, a); | |
| secp256k1_ecmult_const_odd_multiples_table_globalz(pre_a, &global_z, r); | |
| for (i = 0; i < ECMULT_CONST_TABLE_SIZE; i++) { | |
| secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]); | |
| } | |
| | |
| /* Next, we compute r = C_l(v1, A) + C_l(v2, lambda*A). | |
| * | |
| * We proceed in groups of ECMULT_CONST_GROUP_SIZE bits, operating on that many bits | |
| * at a time, from high in v1, v2 to low. Call these bits1 (from v1) and bits2 (from v2). | |
| * | |
| * Now note that ECMULT_CONST_TABLE_GET_GE(&t, pre_a, bits1) loads into t a point equal | |
| * to C_{ECMULT_CONST_GROUP_SIZE}(bits1, A), and analogously for pre_lam_a / bits2. | |
| * This means that all we need to do is add these looked up values together, multiplied | |
| * by 2^(ECMULT_GROUP_SIZE * group). | |
| */ | |
| for (group = ECMULT_CONST_GROUPS - 1; group >= 0; --group) { | |
| /* Using the _var get_bits function is ok here, since it's only variable in offset and count, not in the scalar. */ | |
| unsigned int bits1 = secp256k1_scalar_get_bits_var(&v1, group * ECMULT_CONST_GROUP_SIZE, ECMULT_CONST_GROUP_SIZE); | |
| unsigned int bits2 = secp256k1_scalar_get_bits_var(&v2, group * ECMULT_CONST_GROUP_SIZE, ECMULT_CONST_GROUP_SIZE); | |
| secp256k1_ge t; | |
| int j; | |
| | |
| ECMULT_CONST_TABLE_GET_GE(&t, pre_a, bits1); | |
| if (group == ECMULT_CONST_GROUPS - 1) { | |
| /* Directly set r in the first iteration. */ | |
| secp256k1_gej_set_ge(r, &t); | |
| } else { | |
| /* Shift the result so far up. */ | |
| for (j = 0; j < ECMULT_CONST_GROUP_SIZE; ++j) { | |
| secp256k1_gej_double(r, r); | |
| } | |
| secp256k1_gej_add_ge(r, r, &t); | |
| } | |
| ECMULT_CONST_TABLE_GET_GE(&t, pre_a_lam, bits2); | |
| secp256k1_gej_add_ge(r, r, &t); | |
| } | |
| | |
| /* Map the result back to the secp256k1 curve from the isomorphic curve. */ | |
| secp256k1_fe_mul(&r->z, &r->z, &global_z); | |
| } | |
| | |
| static int secp256k1_ecmult_const_xonly(secp256k1_fe* r, const secp256k1_fe *n, const secp256k1_fe *d, const secp256k1_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. | |
| */ | |
| | |
| secp256k1_fe g, i; | |
| secp256k1_ge p; | |
| secp256k1_gej rj; | |
| | |
| /* Compute g = (n^3 + B*d^3). */ | |
| secp256k1_fe_sqr(&g, n); | |
| secp256k1_fe_mul(&g, &g, n); | |
| if (d) { | |
| secp256k1_fe b; | |
| VERIFY_CHECK(!secp256k1_fe_normalizes_to_zero(d)); | |
| secp256k1_fe_sqr(&b, d); | |
| VERIFY_CHECK(SECP256K1_B <= 8); /* magnitude of b will be <= 8 after the next call */ | |
| secp256k1_fe_mul_int(&b, SECP256K1_B); | |
| secp256k1_fe_mul(&b, &b, d); | |
| secp256k1_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) | |
| */ | |
| secp256k1_fe c; | |
| secp256k1_fe_mul(&c, &g, d); | |
| if (!secp256k1_fe_is_square_var(&c)) return 0; | |
| } | |
| } else { | |
| secp256k1_fe_add_int(&g, SECP256K1_B); | |
| if (!known_on_curve) { | |
| /* g at this point equals x^3 + 7. Test if it is square. */ | |
| if (!secp256k1_fe_is_square_var(&g)) return 0; | |
| } | |
| } | |
| | |
| SECP256K1_FE_VERIFY_MAGNITUDE(&g, 2); | |
| | |
| /* 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. */ | |
| secp256k1_fe_mul(&p.x, &g, n); | |
| secp256k1_fe_sqr(&p.y, &g); | |
| p.infinity = 0; | |
| | |
| /* Perform x-only EC multiplication of P with q. */ | |
| VERIFY_CHECK(!secp256k1_scalar_is_zero(q)); | |
| secp256k1_ecmult_const(&rj, &p, q); | |
| VERIFY_CHECK(!secp256k1_gej_is_infinity(&rj)); | |
| | |
| /* 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)). */ | |
| secp256k1_fe_sqr(&i, &rj.z); | |
| secp256k1_fe_mul(&i, &i, &g); | |
| if (d) secp256k1_fe_mul(&i, &i, d); | |
| secp256k1_fe_inv(&i, &i); | |
| secp256k1_fe_mul(r, &rj.x, &i); | |
| | |
| return 1; | |
| } | |
| | |
| #endif /* SECP256K1_ECMULT_CONST_IMPL_H */ | |