/* Copyright (c) 2020, Google Inc. * * Permission to use, copy, modify, and/or distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ // An implementation of the NIST P-256 elliptic curve point multiplication. // 256-bit Montgomery form for 64 and 32-bit. Field operations are generated by // Fiat, which lives in //third_party/fiat. #include #include #include #include #include #include #include #include #include "../../internal.h" #include "../delocate.h" #include "./internal.h" // MSVC does not implement uint128_t, and crashes with intrinsics #if defined(BORINGSSL_HAS_UINT128) #define BORINGSSL_NISTP256_64BIT 1 #include "../../../third_party/fiat/p256_64.h" #else #include "../../../third_party/fiat/p256_32.h" #endif // utility functions, handwritten #if defined(BORINGSSL_NISTP256_64BIT) #define FIAT_P256_NLIMBS 4 typedef uint64_t fiat_p256_limb_t; typedef uint64_t fiat_p256_felem[FIAT_P256_NLIMBS]; static const fiat_p256_felem fiat_p256_one = {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}; #else // 64BIT; else 32BIT #define FIAT_P256_NLIMBS 8 typedef uint32_t fiat_p256_limb_t; typedef uint32_t fiat_p256_felem[FIAT_P256_NLIMBS]; static const fiat_p256_felem fiat_p256_one = { 0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}; #endif // 64BIT static fiat_p256_limb_t fiat_p256_nz( const fiat_p256_limb_t in1[FIAT_P256_NLIMBS]) { fiat_p256_limb_t ret; fiat_p256_nonzero(&ret, in1); return ret; } static void fiat_p256_copy(fiat_p256_limb_t out[FIAT_P256_NLIMBS], const fiat_p256_limb_t in1[FIAT_P256_NLIMBS]) { for (size_t i = 0; i < FIAT_P256_NLIMBS; i++) { out[i] = in1[i]; } } static void fiat_p256_cmovznz(fiat_p256_limb_t out[FIAT_P256_NLIMBS], fiat_p256_limb_t t, const fiat_p256_limb_t z[FIAT_P256_NLIMBS], const fiat_p256_limb_t nz[FIAT_P256_NLIMBS]) { fiat_p256_selectznz(out, !!t, z, nz); } static void fiat_p256_from_generic(fiat_p256_felem out, const EC_FELEM *in) { fiat_p256_from_bytes(out, in->bytes); } static void fiat_p256_to_generic(EC_FELEM *out, const fiat_p256_felem in) { // This works because 256 is a multiple of 64, so there are no excess bytes to // zero when rounding up to |BN_ULONG|s. OPENSSL_STATIC_ASSERT( 256 / 8 == sizeof(BN_ULONG) * ((256 + BN_BITS2 - 1) / BN_BITS2), "fiat_p256_to_bytes leaves bytes uninitialized"); fiat_p256_to_bytes(out->bytes, in); } // fiat_p256_inv_square calculates |out| = |in|^{-2} // // Based on Fermat's Little Theorem: // a^p = a (mod p) // a^{p-1} = 1 (mod p) // a^{p-3} = a^{-2} (mod p) static void fiat_p256_inv_square(fiat_p256_felem out, const fiat_p256_felem in) { // This implements the addition chain described in // https://briansmith.org/ecc-inversion-addition-chains-01#p256_field_inversion fiat_p256_felem x2, x3, x6, x12, x15, x30, x32; fiat_p256_square(x2, in); // 2^2 - 2^1 fiat_p256_mul(x2, x2, in); // 2^2 - 2^0 fiat_p256_square(x3, x2); // 2^3 - 2^1 fiat_p256_mul(x3, x3, in); // 2^3 - 2^0 fiat_p256_square(x6, x3); for (int i = 1; i < 3; i++) { fiat_p256_square(x6, x6); } // 2^6 - 2^3 fiat_p256_mul(x6, x6, x3); // 2^6 - 2^0 fiat_p256_square(x12, x6); for (int i = 1; i < 6; i++) { fiat_p256_square(x12, x12); } // 2^12 - 2^6 fiat_p256_mul(x12, x12, x6); // 2^12 - 2^0 fiat_p256_square(x15, x12); for (int i = 1; i < 3; i++) { fiat_p256_square(x15, x15); } // 2^15 - 2^3 fiat_p256_mul(x15, x15, x3); // 2^15 - 2^0 fiat_p256_square(x30, x15); for (int i = 1; i < 15; i++) { fiat_p256_square(x30, x30); } // 2^30 - 2^15 fiat_p256_mul(x30, x30, x15); // 2^30 - 2^0 fiat_p256_square(x32, x30); fiat_p256_square(x32, x32); // 2^32 - 2^2 fiat_p256_mul(x32, x32, x2); // 2^32 - 2^0 fiat_p256_felem ret; fiat_p256_square(ret, x32); for (int i = 1; i < 31 + 1; i++) { fiat_p256_square(ret, ret); } // 2^64 - 2^32 fiat_p256_mul(ret, ret, in); // 2^64 - 2^32 + 2^0 for (int i = 0; i < 96 + 32; i++) { fiat_p256_square(ret, ret); } // 2^192 - 2^160 + 2^128 fiat_p256_mul(ret, ret, x32); // 2^192 - 2^160 + 2^128 + 2^32 - 2^0 for (int i = 0; i < 32; i++) { fiat_p256_square(ret, ret); } // 2^224 - 2^192 + 2^160 + 2^64 - 2^32 fiat_p256_mul(ret, ret, x32); // 2^224 - 2^192 + 2^160 + 2^64 - 2^0 for (int i = 0; i < 30; i++) { fiat_p256_square(ret, ret); } // 2^254 - 2^222 + 2^190 + 2^94 - 2^30 fiat_p256_mul(ret, ret, x30); // 2^254 - 2^222 + 2^190 + 2^94 - 2^0 fiat_p256_square(ret, ret); fiat_p256_square(out, ret); // 2^256 - 2^224 + 2^192 + 2^96 - 2^2 } // Group operations // ---------------- // // Building on top of the field operations we have the operations on the // elliptic curve group itself. Points on the curve are represented in Jacobian // coordinates. // // Both operations were transcribed to Coq and proven to correspond to naive // implementations using Affine coordinates, for all suitable fields. In the // Coq proofs, issues of constant-time execution and memory layout (aliasing) // conventions were not considered. Specification of affine coordinates: // // As a sanity check, a proof that these points form a commutative group: // // fiat_p256_point_double calculates 2*(x_in, y_in, z_in) // // The method is taken from: // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b // // Coq transcription and correctness proof: // // // // Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. // while x_out == y_in is not (maybe this works, but it's not tested). static void fiat_p256_point_double(fiat_p256_felem x_out, fiat_p256_felem y_out, fiat_p256_felem z_out, const fiat_p256_felem x_in, const fiat_p256_felem y_in, const fiat_p256_felem z_in) { fiat_p256_felem delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta; // delta = z^2 fiat_p256_square(delta, z_in); // gamma = y^2 fiat_p256_square(gamma, y_in); // beta = x*gamma fiat_p256_mul(beta, x_in, gamma); // alpha = 3*(x-delta)*(x+delta) fiat_p256_sub(ftmp, x_in, delta); fiat_p256_add(ftmp2, x_in, delta); fiat_p256_add(tmptmp, ftmp2, ftmp2); fiat_p256_add(ftmp2, ftmp2, tmptmp); fiat_p256_mul(alpha, ftmp, ftmp2); // x' = alpha^2 - 8*beta fiat_p256_square(x_out, alpha); fiat_p256_add(fourbeta, beta, beta); fiat_p256_add(fourbeta, fourbeta, fourbeta); fiat_p256_add(tmptmp, fourbeta, fourbeta); fiat_p256_sub(x_out, x_out, tmptmp); // z' = (y + z)^2 - gamma - delta fiat_p256_add(delta, gamma, delta); fiat_p256_add(ftmp, y_in, z_in); fiat_p256_square(z_out, ftmp); fiat_p256_sub(z_out, z_out, delta); // y' = alpha*(4*beta - x') - 8*gamma^2 fiat_p256_sub(y_out, fourbeta, x_out); fiat_p256_add(gamma, gamma, gamma); fiat_p256_square(gamma, gamma); fiat_p256_mul(y_out, alpha, y_out); fiat_p256_add(gamma, gamma, gamma); fiat_p256_sub(y_out, y_out, gamma); } // fiat_p256_point_add calculates (x1, y1, z1) + (x2, y2, z2) // // The method is taken from: // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, // adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). // // Coq transcription and correctness proof: // // // // This function includes a branch for checking whether the two input points // are equal, (while not equal to the point at infinity). This case never // happens during single point multiplication, so there is no timing leak for // ECDH or ECDSA signing. static void fiat_p256_point_add(fiat_p256_felem x3, fiat_p256_felem y3, fiat_p256_felem z3, const fiat_p256_felem x1, const fiat_p256_felem y1, const fiat_p256_felem z1, const int mixed, const fiat_p256_felem x2, const fiat_p256_felem y2, const fiat_p256_felem z2) { fiat_p256_felem x_out, y_out, z_out; fiat_p256_limb_t z1nz = fiat_p256_nz(z1); fiat_p256_limb_t z2nz = fiat_p256_nz(z2); // z1z1 = z1z1 = z1**2 fiat_p256_felem z1z1; fiat_p256_square(z1z1, z1); fiat_p256_felem u1, s1, two_z1z2; if (!mixed) { // z2z2 = z2**2 fiat_p256_felem z2z2; fiat_p256_square(z2z2, z2); // u1 = x1*z2z2 fiat_p256_mul(u1, x1, z2z2); // two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 fiat_p256_add(two_z1z2, z1, z2); fiat_p256_square(two_z1z2, two_z1z2); fiat_p256_sub(two_z1z2, two_z1z2, z1z1); fiat_p256_sub(two_z1z2, two_z1z2, z2z2); // s1 = y1 * z2**3 fiat_p256_mul(s1, z2, z2z2); fiat_p256_mul(s1, s1, y1); } else { // We'll assume z2 = 1 (special case z2 = 0 is handled later). // u1 = x1*z2z2 fiat_p256_copy(u1, x1); // two_z1z2 = 2z1z2 fiat_p256_add(two_z1z2, z1, z1); // s1 = y1 * z2**3 fiat_p256_copy(s1, y1); } // u2 = x2*z1z1 fiat_p256_felem u2; fiat_p256_mul(u2, x2, z1z1); // h = u2 - u1 fiat_p256_felem h; fiat_p256_sub(h, u2, u1); fiat_p256_limb_t xneq = fiat_p256_nz(h); // z_out = two_z1z2 * h fiat_p256_mul(z_out, h, two_z1z2); // z1z1z1 = z1 * z1z1 fiat_p256_felem z1z1z1; fiat_p256_mul(z1z1z1, z1, z1z1); // s2 = y2 * z1**3 fiat_p256_felem s2; fiat_p256_mul(s2, y2, z1z1z1); // r = (s2 - s1)*2 fiat_p256_felem r; fiat_p256_sub(r, s2, s1); fiat_p256_add(r, r, r); fiat_p256_limb_t yneq = fiat_p256_nz(r); fiat_p256_limb_t is_nontrivial_double = constant_time_is_zero_w(xneq | yneq) & ~constant_time_is_zero_w(z1nz) & ~constant_time_is_zero_w(z2nz); if (is_nontrivial_double) { fiat_p256_point_double(x3, y3, z3, x1, y1, z1); return; } // I = (2h)**2 fiat_p256_felem i; fiat_p256_add(i, h, h); fiat_p256_square(i, i); // J = h * I fiat_p256_felem j; fiat_p256_mul(j, h, i); // V = U1 * I fiat_p256_felem v; fiat_p256_mul(v, u1, i); // x_out = r**2 - J - 2V fiat_p256_square(x_out, r); fiat_p256_sub(x_out, x_out, j); fiat_p256_sub(x_out, x_out, v); fiat_p256_sub(x_out, x_out, v); // y_out = r(V-x_out) - 2 * s1 * J fiat_p256_sub(y_out, v, x_out); fiat_p256_mul(y_out, y_out, r); fiat_p256_felem s1j; fiat_p256_mul(s1j, s1, j); fiat_p256_sub(y_out, y_out, s1j); fiat_p256_sub(y_out, y_out, s1j); fiat_p256_cmovznz(x_out, z1nz, x2, x_out); fiat_p256_cmovznz(x3, z2nz, x1, x_out); fiat_p256_cmovznz(y_out, z1nz, y2, y_out); fiat_p256_cmovznz(y3, z2nz, y1, y_out); fiat_p256_cmovznz(z_out, z1nz, z2, z_out); fiat_p256_cmovznz(z3, z2nz, z1, z_out); } #include "./p256_table.h" // fiat_p256_select_point_affine selects the |idx-1|th point from a // precomputation table and copies it to out. If |idx| is zero, the output is // the point at infinity. static void fiat_p256_select_point_affine( const fiat_p256_limb_t idx, size_t size, const fiat_p256_felem pre_comp[/*size*/][2], fiat_p256_felem out[3]) { OPENSSL_memset(out, 0, sizeof(fiat_p256_felem) * 3); for (size_t i = 0; i < size; i++) { fiat_p256_limb_t mismatch = i ^ (idx - 1); fiat_p256_cmovznz(out[0], mismatch, pre_comp[i][0], out[0]); fiat_p256_cmovznz(out[1], mismatch, pre_comp[i][1], out[1]); } fiat_p256_cmovznz(out[2], idx, out[2], fiat_p256_one); } // fiat_p256_select_point selects the |idx|th point from a precomputation table // and copies it to out. static void fiat_p256_select_point(const fiat_p256_limb_t idx, size_t size, const fiat_p256_felem pre_comp[/*size*/][3], fiat_p256_felem out[3]) { OPENSSL_memset(out, 0, sizeof(fiat_p256_felem) * 3); for (size_t i = 0; i < size; i++) { fiat_p256_limb_t mismatch = i ^ idx; fiat_p256_cmovznz(out[0], mismatch, pre_comp[i][0], out[0]); fiat_p256_cmovznz(out[1], mismatch, pre_comp[i][1], out[1]); fiat_p256_cmovznz(out[2], mismatch, pre_comp[i][2], out[2]); } } // fiat_p256_get_bit returns the |i|th bit in |in| static crypto_word_t fiat_p256_get_bit(const uint8_t *in, int i) { if (i < 0 || i >= 256) { return 0; } return (in[i >> 3] >> (i & 7)) & 1; } // OPENSSL EC_METHOD FUNCTIONS // Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = // (X/Z^2, Y/Z^3). static int ec_GFp_nistp256_point_get_affine_coordinates( const EC_GROUP *group, const EC_RAW_POINT *point, EC_FELEM *x_out, EC_FELEM *y_out) { if (ec_GFp_simple_is_at_infinity(group, point)) { OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); return 0; } fiat_p256_felem z1, z2; fiat_p256_from_generic(z1, &point->Z); fiat_p256_inv_square(z2, z1); if (x_out != NULL) { fiat_p256_felem x; fiat_p256_from_generic(x, &point->X); fiat_p256_mul(x, x, z2); fiat_p256_to_generic(x_out, x); } if (y_out != NULL) { fiat_p256_felem y; fiat_p256_from_generic(y, &point->Y); fiat_p256_square(z2, z2); // z^-4 fiat_p256_mul(y, y, z1); // y * z fiat_p256_mul(y, y, z2); // y * z^-3 fiat_p256_to_generic(y_out, y); } return 1; } static void ec_GFp_nistp256_add(const EC_GROUP *group, EC_RAW_POINT *r, const EC_RAW_POINT *a, const EC_RAW_POINT *b) { fiat_p256_felem x1, y1, z1, x2, y2, z2; fiat_p256_from_generic(x1, &a->X); fiat_p256_from_generic(y1, &a->Y); fiat_p256_from_generic(z1, &a->Z); fiat_p256_from_generic(x2, &b->X); fiat_p256_from_generic(y2, &b->Y); fiat_p256_from_generic(z2, &b->Z); fiat_p256_point_add(x1, y1, z1, x1, y1, z1, 0 /* both Jacobian */, x2, y2, z2); fiat_p256_to_generic(&r->X, x1); fiat_p256_to_generic(&r->Y, y1); fiat_p256_to_generic(&r->Z, z1); } static void ec_GFp_nistp256_dbl(const EC_GROUP *group, EC_RAW_POINT *r, const EC_RAW_POINT *a) { fiat_p256_felem x, y, z; fiat_p256_from_generic(x, &a->X); fiat_p256_from_generic(y, &a->Y); fiat_p256_from_generic(z, &a->Z); fiat_p256_point_double(x, y, z, x, y, z); fiat_p256_to_generic(&r->X, x); fiat_p256_to_generic(&r->Y, y); fiat_p256_to_generic(&r->Z, z); } static void ec_GFp_nistp256_point_mul(const EC_GROUP *group, EC_RAW_POINT *r, const EC_RAW_POINT *p, const EC_SCALAR *scalar) { fiat_p256_felem p_pre_comp[17][3]; OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp)); // Precompute multiples. fiat_p256_from_generic(p_pre_comp[1][0], &p->X); fiat_p256_from_generic(p_pre_comp[1][1], &p->Y); fiat_p256_from_generic(p_pre_comp[1][2], &p->Z); for (size_t j = 2; j <= 16; ++j) { if (j & 1) { fiat_p256_point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2], p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2], 0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1], p_pre_comp[j - 1][2]); } else { fiat_p256_point_double(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2], p_pre_comp[j / 2][0], p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]); } } // Set nq to the point at infinity. fiat_p256_felem nq[3] = {{0}, {0}, {0}}, ftmp, tmp[3]; // Loop over |scalar| msb-to-lsb, incorporating |p_pre_comp| every 5th round. int skip = 1; // Save two point operations in the first round. for (size_t i = 255; i < 256; i--) { // double if (!skip) { fiat_p256_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); } // do other additions every 5 doublings if (i % 5 == 0) { crypto_word_t bits = fiat_p256_get_bit(scalar->bytes, i + 4) << 5; bits |= fiat_p256_get_bit(scalar->bytes, i + 3) << 4; bits |= fiat_p256_get_bit(scalar->bytes, i + 2) << 3; bits |= fiat_p256_get_bit(scalar->bytes, i + 1) << 2; bits |= fiat_p256_get_bit(scalar->bytes, i) << 1; bits |= fiat_p256_get_bit(scalar->bytes, i - 1); crypto_word_t sign, digit; ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); // select the point to add or subtract, in constant time. fiat_p256_select_point((fiat_p256_limb_t)digit, 17, (const fiat_p256_felem(*)[3])p_pre_comp, tmp); fiat_p256_opp(ftmp, tmp[1]); // (X, -Y, Z) is the negative point. fiat_p256_cmovznz(tmp[1], (fiat_p256_limb_t)sign, tmp[1], ftmp); if (!skip) { fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, tmp[0], tmp[1], tmp[2]); } else { fiat_p256_copy(nq[0], tmp[0]); fiat_p256_copy(nq[1], tmp[1]); fiat_p256_copy(nq[2], tmp[2]); skip = 0; } } } fiat_p256_to_generic(&r->X, nq[0]); fiat_p256_to_generic(&r->Y, nq[1]); fiat_p256_to_generic(&r->Z, nq[2]); } static void ec_GFp_nistp256_point_mul_base(const EC_GROUP *group, EC_RAW_POINT *r, const EC_SCALAR *scalar) { // Set nq to the point at infinity. fiat_p256_felem nq[3] = {{0}, {0}, {0}}, tmp[3]; int skip = 1; // Save two point operations in the first round. for (size_t i = 31; i < 32; i--) { if (!skip) { fiat_p256_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); } // First, look 32 bits upwards. crypto_word_t bits = fiat_p256_get_bit(scalar->bytes, i + 224) << 3; bits |= fiat_p256_get_bit(scalar->bytes, i + 160) << 2; bits |= fiat_p256_get_bit(scalar->bytes, i + 96) << 1; bits |= fiat_p256_get_bit(scalar->bytes, i + 32); // Select the point to add, in constant time. fiat_p256_select_point_affine((fiat_p256_limb_t)bits, 15, fiat_p256_g_pre_comp[1], tmp); if (!skip) { fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], tmp[1], tmp[2]); } else { fiat_p256_copy(nq[0], tmp[0]); fiat_p256_copy(nq[1], tmp[1]); fiat_p256_copy(nq[2], tmp[2]); skip = 0; } // Second, look at the current position. bits = fiat_p256_get_bit(scalar->bytes, i + 192) << 3; bits |= fiat_p256_get_bit(scalar->bytes, i + 128) << 2; bits |= fiat_p256_get_bit(scalar->bytes, i + 64) << 1; bits |= fiat_p256_get_bit(scalar->bytes, i); // Select the point to add, in constant time. fiat_p256_select_point_affine((fiat_p256_limb_t)bits, 15, fiat_p256_g_pre_comp[0], tmp); fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], tmp[1], tmp[2]); } fiat_p256_to_generic(&r->X, nq[0]); fiat_p256_to_generic(&r->Y, nq[1]); fiat_p256_to_generic(&r->Z, nq[2]); } static void ec_GFp_nistp256_point_mul_public(const EC_GROUP *group, EC_RAW_POINT *r, const EC_SCALAR *g_scalar, const EC_RAW_POINT *p, const EC_SCALAR *p_scalar) { #define P256_WSIZE_PUBLIC 4 // Precompute multiples of |p|. p_pre_comp[i] is (2*i+1) * |p|. fiat_p256_felem p_pre_comp[1 << (P256_WSIZE_PUBLIC - 1)][3]; fiat_p256_from_generic(p_pre_comp[0][0], &p->X); fiat_p256_from_generic(p_pre_comp[0][1], &p->Y); fiat_p256_from_generic(p_pre_comp[0][2], &p->Z); fiat_p256_felem p2[3]; fiat_p256_point_double(p2[0], p2[1], p2[2], p_pre_comp[0][0], p_pre_comp[0][1], p_pre_comp[0][2]); for (size_t i = 1; i < OPENSSL_ARRAY_SIZE(p_pre_comp); i++) { fiat_p256_point_add(p_pre_comp[i][0], p_pre_comp[i][1], p_pre_comp[i][2], p_pre_comp[i - 1][0], p_pre_comp[i - 1][1], p_pre_comp[i - 1][2], 0 /* not mixed */, p2[0], p2[1], p2[2]); } // Set up the coefficients for |p_scalar|. int8_t p_wNAF[257]; ec_compute_wNAF(group, p_wNAF, p_scalar, 256, P256_WSIZE_PUBLIC); // Set |ret| to the point at infinity. int skip = 1; // Save some point operations. fiat_p256_felem ret[3] = {{0}, {0}, {0}}; for (int i = 256; i >= 0; i--) { if (!skip) { fiat_p256_point_double(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2]); } // For the |g_scalar|, we use the precomputed table without the // constant-time lookup. if (i <= 31) { // First, look 32 bits upwards. crypto_word_t bits = fiat_p256_get_bit(g_scalar->bytes, i + 224) << 3; bits |= fiat_p256_get_bit(g_scalar->bytes, i + 160) << 2; bits |= fiat_p256_get_bit(g_scalar->bytes, i + 96) << 1; bits |= fiat_p256_get_bit(g_scalar->bytes, i + 32); if (bits != 0) { size_t index = (size_t)(bits - 1); fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], 1 /* mixed */, fiat_p256_g_pre_comp[1][index][0], fiat_p256_g_pre_comp[1][index][1], fiat_p256_one); skip = 0; } // Second, look at the current position. bits = fiat_p256_get_bit(g_scalar->bytes, i + 192) << 3; bits |= fiat_p256_get_bit(g_scalar->bytes, i + 128) << 2; bits |= fiat_p256_get_bit(g_scalar->bytes, i + 64) << 1; bits |= fiat_p256_get_bit(g_scalar->bytes, i); if (bits != 0) { size_t index = (size_t)(bits - 1); fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], 1 /* mixed */, fiat_p256_g_pre_comp[0][index][0], fiat_p256_g_pre_comp[0][index][1], fiat_p256_one); skip = 0; } } int digit = p_wNAF[i]; if (digit != 0) { assert(digit & 1); size_t idx = (size_t)(digit < 0 ? (-digit) >> 1 : digit >> 1); fiat_p256_felem *y = &p_pre_comp[idx][1], tmp; if (digit < 0) { fiat_p256_opp(tmp, p_pre_comp[idx][1]); y = &tmp; } if (!skip) { fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], 0 /* not mixed */, p_pre_comp[idx][0], *y, p_pre_comp[idx][2]); } else { fiat_p256_copy(ret[0], p_pre_comp[idx][0]); fiat_p256_copy(ret[1], *y); fiat_p256_copy(ret[2], p_pre_comp[idx][2]); skip = 0; } } } fiat_p256_to_generic(&r->X, ret[0]); fiat_p256_to_generic(&r->Y, ret[1]); fiat_p256_to_generic(&r->Z, ret[2]); } static int ec_GFp_nistp256_cmp_x_coordinate(const EC_GROUP *group, const EC_RAW_POINT *p, const EC_SCALAR *r) { if (ec_GFp_simple_is_at_infinity(group, p)) { return 0; } // We wish to compare X/Z^2 with r. This is equivalent to comparing X with // r*Z^2. Note that X and Z are represented in Montgomery form, while r is // not. fiat_p256_felem Z2_mont; fiat_p256_from_generic(Z2_mont, &p->Z); fiat_p256_mul(Z2_mont, Z2_mont, Z2_mont); fiat_p256_felem r_Z2; fiat_p256_from_bytes(r_Z2, r->bytes); // r < order < p, so this is valid. fiat_p256_mul(r_Z2, r_Z2, Z2_mont); fiat_p256_felem X; fiat_p256_from_generic(X, &p->X); fiat_p256_from_montgomery(X, X); if (OPENSSL_memcmp(&r_Z2, &X, sizeof(r_Z2)) == 0) { return 1; } // During signing the x coefficient is reduced modulo the group order. // Therefore there is a small possibility, less than 1/2^128, that group_order // < p.x < P. in that case we need not only to compare against |r| but also to // compare against r+group_order. assert(group->field.width == group->order.width); if (bn_less_than_words(r->words, group->field_minus_order.words, group->field.width)) { // We can ignore the carry because: r + group_order < p < 2^256. EC_FELEM tmp; bn_add_words(tmp.words, r->words, group->order.d, group->order.width); fiat_p256_from_generic(r_Z2, &tmp); fiat_p256_mul(r_Z2, r_Z2, Z2_mont); if (OPENSSL_memcmp(&r_Z2, &X, sizeof(r_Z2)) == 0) { return 1; } } return 0; } DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp256_method) { out->group_init = ec_GFp_mont_group_init; out->group_finish = ec_GFp_mont_group_finish; out->group_set_curve = ec_GFp_mont_group_set_curve; out->point_get_affine_coordinates = ec_GFp_nistp256_point_get_affine_coordinates; out->add = ec_GFp_nistp256_add; out->dbl = ec_GFp_nistp256_dbl; out->mul = ec_GFp_nistp256_point_mul; out->mul_base = ec_GFp_nistp256_point_mul_base; out->mul_public = ec_GFp_nistp256_point_mul_public; out->felem_mul = ec_GFp_mont_felem_mul; out->felem_sqr = ec_GFp_mont_felem_sqr; out->felem_to_bytes = ec_GFp_mont_felem_to_bytes; out->felem_from_bytes = ec_GFp_mont_felem_from_bytes; out->scalar_inv0_montgomery = ec_simple_scalar_inv0_montgomery; out->scalar_to_montgomery_inv_vartime = ec_simple_scalar_to_montgomery_inv_vartime; out->cmp_x_coordinate = ec_GFp_nistp256_cmp_x_coordinate; } #undef BORINGSSL_NISTP256_64BIT