| /* 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 <openssl/base.h> |
| |
| #include <openssl/bn.h> |
| #include <openssl/ec.h> |
| #include <openssl/err.h> |
| #include <openssl/mem.h> |
| #include <openssl/type_check.h> |
| |
| #include <assert.h> |
| #include <string.h> |
| |
| #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]; |
| #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]; |
| |
| #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 (int 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: |
| // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Spec/WeierstrassCurve.v#L28> |
| // As a sanity check, a proof that these points form a commutative group: |
| // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/AffineProofs.v#L33> |
| |
| // 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: |
| // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L93> |
| // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L201> |
| // |
| // 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: |
| // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L135> |
| // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L205> |
| // |
| // 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); |
| } |
| |
| // Base point pre computation |
| // -------------------------- |
| // |
| // Two different sorts of precomputed tables are used in the following code. |
| // Each contain various points on the curve, where each point is three field |
| // elements (x, y, z). |
| // |
| // For the base point table, z is usually 1 (0 for the point at infinity). |
| // This table has 2 * 16 elements, starting with the following: |
| // index | bits | point |
| // ------+---------+------------------------------ |
| // 0 | 0 0 0 0 | 0G |
| // 1 | 0 0 0 1 | 1G |
| // 2 | 0 0 1 0 | 2^64G |
| // 3 | 0 0 1 1 | (2^64 + 1)G |
| // 4 | 0 1 0 0 | 2^128G |
| // 5 | 0 1 0 1 | (2^128 + 1)G |
| // 6 | 0 1 1 0 | (2^128 + 2^64)G |
| // 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G |
| // 8 | 1 0 0 0 | 2^192G |
| // 9 | 1 0 0 1 | (2^192 + 1)G |
| // 10 | 1 0 1 0 | (2^192 + 2^64)G |
| // 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G |
| // 12 | 1 1 0 0 | (2^192 + 2^128)G |
| // 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G |
| // 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G |
| // 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G |
| // followed by a copy of this with each element multiplied by 2^32. |
| // |
| // The reason for this is so that we can clock bits into four different |
| // locations when doing simple scalar multiplies against the base point, |
| // and then another four locations using the second 16 elements. |
| // |
| // Tables for other points have table[i] = iG for i in 0 .. 16. |
| |
| // fiat_p256_g_pre_comp is the table of precomputed base points |
| #if defined(BORINGSSL_NISTP256_64BIT) |
| static const fiat_p256_felem fiat_p256_g_pre_comp[2][16][3] = { |
| {{{0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}}, |
| {{0x79e730d418a9143c, 0x75ba95fc5fedb601, 0x79fb732b77622510, |
| 0x18905f76a53755c6}, |
| {0xddf25357ce95560a, 0x8b4ab8e4ba19e45c, 0xd2e88688dd21f325, |
| 0x8571ff1825885d85}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x4f922fc516a0d2bb, 0xd5cc16c1a623499, 0x9241cf3a57c62c8b, |
| 0x2f5e6961fd1b667f}, |
| {0x5c15c70bf5a01797, 0x3d20b44d60956192, 0x4911b37071fdb52, |
| 0xf648f9168d6f0f7b}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x9e566847e137bbbc, 0xe434469e8a6a0bec, 0xb1c4276179d73463, |
| 0x5abe0285133d0015}, |
| {0x92aa837cc04c7dab, 0x573d9f4c43260c07, 0xc93156278e6cc37, |
| 0x94bb725b6b6f7383}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x62a8c244bfe20925, 0x91c19ac38fdce867, 0x5a96a5d5dd387063, |
| 0x61d587d421d324f6}, |
| {0xe87673a2a37173ea, 0x2384800853778b65, 0x10f8441e05bab43e, |
| 0xfa11fe124621efbe}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x1c891f2b2cb19ffd, 0x1ba8d5bb1923c23, 0xb6d03d678ac5ca8e, |
| 0x586eb04c1f13bedc}, |
| {0xc35c6e527e8ed09, 0x1e81a33c1819ede2, 0x278fd6c056c652fa, |
| 0x19d5ac0870864f11}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x62577734d2b533d5, 0x673b8af6a1bdddc0, 0x577e7c9aa79ec293, |
| 0xbb6de651c3b266b1}, |
| {0xe7e9303ab65259b3, 0xd6a0afd3d03a7480, 0xc5ac83d19b3cfc27, |
| 0x60b4619a5d18b99b}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xbd6a38e11ae5aa1c, 0xb8b7652b49e73658, 0xb130014ee5f87ed, |
| 0x9d0f27b2aeebffcd}, |
| {0xca9246317a730a55, 0x9c955b2fddbbc83a, 0x7c1dfe0ac019a71, |
| 0x244a566d356ec48d}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x56f8410ef4f8b16a, 0x97241afec47b266a, 0xa406b8e6d9c87c1, |
| 0x803f3e02cd42ab1b}, |
| {0x7f0309a804dbec69, 0xa83b85f73bbad05f, 0xc6097273ad8e197f, |
| 0xc097440e5067adc1}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x846a56f2c379ab34, 0xa8ee068b841df8d1, 0x20314459176c68ef, |
| 0xf1af32d5915f1f30}, |
| {0x99c375315d75bd50, 0x837cffbaf72f67bc, 0x613a41848d7723f, |
| 0x23d0f130e2d41c8b}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xed93e225d5be5a2b, 0x6fe799835934f3c6, 0x4314092622626ffc, |
| 0x50bbb4d97990216a}, |
| {0x378191c6e57ec63e, 0x65422c40181dcdb2, 0x41a8099b0236e0f6, |
| 0x2b10011801fe49c3}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xfc68b5c59b391593, 0xc385f5a2598270fc, 0x7144f3aad19adcbb, |
| 0xdd55899983fbae0c}, |
| {0x93b88b8e74b82ff4, 0xd2e03c4071e734c9, 0x9a7a9eaf43c0322a, |
| 0xe6e4c551149d6041}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x5fe14bfe80ec21fe, 0xf6ce116ac255be82, 0x98bc5a072f4a5d67, |
| 0xfad27148db7e63af}, |
| {0x90c0b6ac29ab05b3, 0x37a9a83c4e251ae6, 0xa7dc875c2aade7d, |
| 0x77387de39f0e1a84}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x1e9ecc49a56c0dd7, 0xa5cffcd846086c74, 0x8f7a1408f505aece, |
| 0xb37b85c0bef0c47e}, |
| {0x3596b6e4cc0e6a8f, 0xfd6d4bbf6b388f23, 0xaba453fac39cef4e, |
| 0x9c135ac8f9f628d5}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xa1c729495c8f8be, 0x2961c4803bf362bf, 0x9e418403df63d4ac, |
| 0xc109f9cb91ece900}, |
| {0xc2d095d058945705, 0xb9083d96ddeb85c0, 0x84692b8d7a40449b, |
| 0x9bc3344f2eee1ee1}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xd5ae35642913074, 0x55491b2748a542b1, 0x469ca665b310732a, |
| 0x29591d525f1a4cc1}, |
| {0xe76f5b6bb84f983f, 0xbe7eef419f5f84e1, 0x1200d49680baa189, |
| 0x6376551f18ef332c}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}}, |
| {{{0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}}, |
| {{0x202886024147519a, 0xd0981eac26b372f0, 0xa9d4a7caa785ebc8, |
| 0xd953c50ddbdf58e9}, |
| {0x9d6361ccfd590f8f, 0x72e9626b44e6c917, 0x7fd9611022eb64cf, |
| 0x863ebb7e9eb288f3}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x4fe7ee31b0e63d34, 0xf4600572a9e54fab, 0xc0493334d5e7b5a4, |
| 0x8589fb9206d54831}, |
| {0xaa70f5cc6583553a, 0x879094ae25649e5, 0xcc90450710044652, |
| 0xebb0696d02541c4f}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xabbaa0c03b89da99, 0xa6f2d79eb8284022, 0x27847862b81c05e8, |
| 0x337a4b5905e54d63}, |
| {0x3c67500d21f7794a, 0x207005b77d6d7f61, 0xa5a378104cfd6e8, |
| 0xd65e0d5f4c2fbd6}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xd433e50f6d3549cf, 0x6f33696ffacd665e, 0x695bfdacce11fcb4, |
| 0x810ee252af7c9860}, |
| {0x65450fe17159bb2c, 0xf7dfbebe758b357b, 0x2b057e74d69fea72, |
| 0xd485717a92731745}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xce1f69bbe83f7669, 0x9f8ae8272877d6b, 0x9548ae543244278d, |
| 0x207755dee3c2c19c}, |
| {0x87bd61d96fef1945, 0x18813cefb12d28c3, 0x9fbcd1d672df64aa, |
| 0x48dc5ee57154b00d}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xef0f469ef49a3154, 0x3e85a5956e2b2e9a, 0x45aaec1eaa924a9c, |
| 0xaa12dfc8a09e4719}, |
| {0x26f272274df69f1d, 0xe0e4c82ca2ff5e73, 0xb9d8ce73b7a9dd44, |
| 0x6c036e73e48ca901}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xe1e421e1a47153f0, 0xb86c3b79920418c9, 0x93bdce87705d7672, |
| 0xf25ae793cab79a77}, |
| {0x1f3194a36d869d0c, 0x9d55c8824986c264, 0x49fb5ea3096e945e, |
| 0x39b8e65313db0a3e}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xe3417bc035d0b34a, 0x440b386b8327c0a7, 0x8fb7262dac0362d1, |
| 0x2c41114ce0cdf943}, |
| {0x2ba5cef1ad95a0b1, 0xc09b37a867d54362, 0x26d6cdd201e486c9, |
| 0x20477abf42ff9297}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xf121b41bc0a67d2, 0x62d4760a444d248a, 0xe044f1d659b4737, |
| 0x8fde365250bb4a8}, |
| {0xaceec3da848bf287, 0xc2a62182d3369d6e, 0x3582dfdc92449482, |
| 0x2f7e2fd2565d6cd7}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xa0122b5178a876b, 0x51ff96ff085104b4, 0x50b31ab14f29f76, |
| 0x84abb28b5f87d4e6}, |
| {0xd5ed439f8270790a, 0x2d6cb59d85e3f46b, 0x75f55c1b6c1e2212, |
| 0xe5436f6717655640}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xc2965ecc9aeb596d, 0x1ea03e7023c92b4, 0x4704b4b62e013961, |
| 0xca8fd3f905ea367}, |
| {0x92523a42551b2b61, 0x1eb7a89c390fcd06, 0xe7f1d2be0392a63e, |
| 0x96dca2644ddb0c33}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x231c210e15339848, 0xe87a28e870778c8d, 0x9d1de6616956e170, |
| 0x4ac3c9382bb09c0b}, |
| {0x19be05516998987d, 0x8b2376c4ae09f4d6, 0x1de0b7651a3f933d, |
| 0x380d94c7e39705f4}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x3685954b8c31c31d, 0x68533d005bf21a0c, 0xbd7626e75c79ec9, |
| 0xca17754742c69d54}, |
| {0xcc6edafff6d2dbb2, 0xfd0d8cbd174a9d18, 0x875e8793aa4578e8, |
| 0xa976a7139cab2ce6}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0xce37ab11b43ea1db, 0xa7ff1a95259d292, 0x851b02218f84f186, |
| 0xa7222beadefaad13}, |
| {0xa2ac78ec2b0a9144, 0x5a024051f2fa59c5, 0x91d1eca56147ce38, |
| 0xbe94d523bc2ac690}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, |
| {{0x2d8daefd79ec1a0f, 0x3bbcd6fdceb39c97, 0xf5575ffc58f61a95, |
| 0xdbd986c4adf7b420}, |
| {0x81aa881415f39eb7, 0x6ee2fcf5b98d976c, 0x5465475dcf2f717d, |
| 0x8e24d3c46860bbd0}, |
| {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}}}; |
| #else |
| static const fiat_p256_felem fiat_p256_g_pre_comp[2][16][3] = { |
| {{{0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}, |
| {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}, |
| {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}}, |
| {{0x18a9143c, 0x79e730d4, 0x5fedb601, 0x75ba95fc, 0x77622510, 0x79fb732b, |
| 0xa53755c6, 0x18905f76}, |
| {0xce95560a, 0xddf25357, 0xba19e45c, 0x8b4ab8e4, 0xdd21f325, 0xd2e88688, |
| 0x25885d85, 0x8571ff18}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x16a0d2bb, 0x4f922fc5, 0x1a623499, 0xd5cc16c, 0x57c62c8b, 0x9241cf3a, |
| 0xfd1b667f, 0x2f5e6961}, |
| {0xf5a01797, 0x5c15c70b, 0x60956192, 0x3d20b44d, 0x71fdb52, 0x4911b37, |
| 0x8d6f0f7b, 0xf648f916}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xe137bbbc, 0x9e566847, 0x8a6a0bec, 0xe434469e, 0x79d73463, 0xb1c42761, |
| 0x133d0015, 0x5abe0285}, |
| {0xc04c7dab, 0x92aa837c, 0x43260c07, 0x573d9f4c, 0x78e6cc37, 0xc931562, |
| 0x6b6f7383, 0x94bb725b}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xbfe20925, 0x62a8c244, 0x8fdce867, 0x91c19ac3, 0xdd387063, 0x5a96a5d5, |
| 0x21d324f6, 0x61d587d4}, |
| {0xa37173ea, 0xe87673a2, 0x53778b65, 0x23848008, 0x5bab43e, 0x10f8441e, |
| 0x4621efbe, 0xfa11fe12}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x2cb19ffd, 0x1c891f2b, 0xb1923c23, 0x1ba8d5b, 0x8ac5ca8e, 0xb6d03d67, |
| 0x1f13bedc, 0x586eb04c}, |
| {0x27e8ed09, 0xc35c6e5, 0x1819ede2, 0x1e81a33c, 0x56c652fa, 0x278fd6c0, |
| 0x70864f11, 0x19d5ac08}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xd2b533d5, 0x62577734, 0xa1bdddc0, 0x673b8af6, 0xa79ec293, 0x577e7c9a, |
| 0xc3b266b1, 0xbb6de651}, |
| {0xb65259b3, 0xe7e9303a, 0xd03a7480, 0xd6a0afd3, 0x9b3cfc27, 0xc5ac83d1, |
| 0x5d18b99b, 0x60b4619a}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x1ae5aa1c, 0xbd6a38e1, 0x49e73658, 0xb8b7652b, 0xee5f87ed, 0xb130014, |
| 0xaeebffcd, 0x9d0f27b2}, |
| {0x7a730a55, 0xca924631, 0xddbbc83a, 0x9c955b2f, 0xac019a71, 0x7c1dfe0, |
| 0x356ec48d, 0x244a566d}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xf4f8b16a, 0x56f8410e, 0xc47b266a, 0x97241afe, 0x6d9c87c1, 0xa406b8e, |
| 0xcd42ab1b, 0x803f3e02}, |
| {0x4dbec69, 0x7f0309a8, 0x3bbad05f, 0xa83b85f7, 0xad8e197f, 0xc6097273, |
| 0x5067adc1, 0xc097440e}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xc379ab34, 0x846a56f2, 0x841df8d1, 0xa8ee068b, 0x176c68ef, 0x20314459, |
| 0x915f1f30, 0xf1af32d5}, |
| {0x5d75bd50, 0x99c37531, 0xf72f67bc, 0x837cffba, 0x48d7723f, 0x613a418, |
| 0xe2d41c8b, 0x23d0f130}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xd5be5a2b, 0xed93e225, 0x5934f3c6, 0x6fe79983, 0x22626ffc, 0x43140926, |
| 0x7990216a, 0x50bbb4d9}, |
| {0xe57ec63e, 0x378191c6, 0x181dcdb2, 0x65422c40, 0x236e0f6, 0x41a8099b, |
| 0x1fe49c3, 0x2b100118}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x9b391593, 0xfc68b5c5, 0x598270fc, 0xc385f5a2, 0xd19adcbb, 0x7144f3aa, |
| 0x83fbae0c, 0xdd558999}, |
| {0x74b82ff4, 0x93b88b8e, 0x71e734c9, 0xd2e03c40, 0x43c0322a, 0x9a7a9eaf, |
| 0x149d6041, 0xe6e4c551}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x80ec21fe, 0x5fe14bfe, 0xc255be82, 0xf6ce116a, 0x2f4a5d67, 0x98bc5a07, |
| 0xdb7e63af, 0xfad27148}, |
| {0x29ab05b3, 0x90c0b6ac, 0x4e251ae6, 0x37a9a83c, 0xc2aade7d, 0xa7dc875, |
| 0x9f0e1a84, 0x77387de3}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xa56c0dd7, 0x1e9ecc49, 0x46086c74, 0xa5cffcd8, 0xf505aece, 0x8f7a1408, |
| 0xbef0c47e, 0xb37b85c0}, |
| {0xcc0e6a8f, 0x3596b6e4, 0x6b388f23, 0xfd6d4bbf, 0xc39cef4e, 0xaba453fa, |
| 0xf9f628d5, 0x9c135ac8}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x95c8f8be, 0xa1c7294, 0x3bf362bf, 0x2961c480, 0xdf63d4ac, 0x9e418403, |
| 0x91ece900, 0xc109f9cb}, |
| {0x58945705, 0xc2d095d0, 0xddeb85c0, 0xb9083d96, 0x7a40449b, 0x84692b8d, |
| 0x2eee1ee1, 0x9bc3344f}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x42913074, 0xd5ae356, 0x48a542b1, 0x55491b27, 0xb310732a, 0x469ca665, |
| 0x5f1a4cc1, 0x29591d52}, |
| {0xb84f983f, 0xe76f5b6b, 0x9f5f84e1, 0xbe7eef41, 0x80baa189, 0x1200d496, |
| 0x18ef332c, 0x6376551f}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}}, |
| {{{0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}, |
| {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}, |
| {0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0}}, |
| {{0x4147519a, 0x20288602, 0x26b372f0, 0xd0981eac, 0xa785ebc8, 0xa9d4a7ca, |
| 0xdbdf58e9, 0xd953c50d}, |
| {0xfd590f8f, 0x9d6361cc, 0x44e6c917, 0x72e9626b, 0x22eb64cf, 0x7fd96110, |
| 0x9eb288f3, 0x863ebb7e}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xb0e63d34, 0x4fe7ee31, 0xa9e54fab, 0xf4600572, 0xd5e7b5a4, 0xc0493334, |
| 0x6d54831, 0x8589fb92}, |
| {0x6583553a, 0xaa70f5cc, 0xe25649e5, 0x879094a, 0x10044652, 0xcc904507, |
| 0x2541c4f, 0xebb0696d}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x3b89da99, 0xabbaa0c0, 0xb8284022, 0xa6f2d79e, 0xb81c05e8, 0x27847862, |
| 0x5e54d63, 0x337a4b59}, |
| {0x21f7794a, 0x3c67500d, 0x7d6d7f61, 0x207005b7, 0x4cfd6e8, 0xa5a3781, |
| 0xf4c2fbd6, 0xd65e0d5}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x6d3549cf, 0xd433e50f, 0xfacd665e, 0x6f33696f, 0xce11fcb4, 0x695bfdac, |
| 0xaf7c9860, 0x810ee252}, |
| {0x7159bb2c, 0x65450fe1, 0x758b357b, 0xf7dfbebe, 0xd69fea72, 0x2b057e74, |
| 0x92731745, 0xd485717a}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xe83f7669, 0xce1f69bb, 0x72877d6b, 0x9f8ae82, 0x3244278d, 0x9548ae54, |
| 0xe3c2c19c, 0x207755de}, |
| {0x6fef1945, 0x87bd61d9, 0xb12d28c3, 0x18813cef, 0x72df64aa, 0x9fbcd1d6, |
| 0x7154b00d, 0x48dc5ee5}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xf49a3154, 0xef0f469e, 0x6e2b2e9a, 0x3e85a595, 0xaa924a9c, 0x45aaec1e, |
| 0xa09e4719, 0xaa12dfc8}, |
| {0x4df69f1d, 0x26f27227, 0xa2ff5e73, 0xe0e4c82c, 0xb7a9dd44, 0xb9d8ce73, |
| 0xe48ca901, 0x6c036e73}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xa47153f0, 0xe1e421e1, 0x920418c9, 0xb86c3b79, 0x705d7672, 0x93bdce87, |
| 0xcab79a77, 0xf25ae793}, |
| {0x6d869d0c, 0x1f3194a3, 0x4986c264, 0x9d55c882, 0x96e945e, 0x49fb5ea3, |
| 0x13db0a3e, 0x39b8e653}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x35d0b34a, 0xe3417bc0, 0x8327c0a7, 0x440b386b, 0xac0362d1, 0x8fb7262d, |
| 0xe0cdf943, 0x2c41114c}, |
| {0xad95a0b1, 0x2ba5cef1, 0x67d54362, 0xc09b37a8, 0x1e486c9, 0x26d6cdd2, |
| 0x42ff9297, 0x20477abf}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xbc0a67d2, 0xf121b41, 0x444d248a, 0x62d4760a, 0x659b4737, 0xe044f1d, |
| 0x250bb4a8, 0x8fde365}, |
| {0x848bf287, 0xaceec3da, 0xd3369d6e, 0xc2a62182, 0x92449482, 0x3582dfdc, |
| 0x565d6cd7, 0x2f7e2fd2}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x178a876b, 0xa0122b5, 0x85104b4, 0x51ff96ff, 0x14f29f76, 0x50b31ab, |
| 0x5f87d4e6, 0x84abb28b}, |
| {0x8270790a, 0xd5ed439f, 0x85e3f46b, 0x2d6cb59d, 0x6c1e2212, 0x75f55c1b, |
| 0x17655640, 0xe5436f67}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x9aeb596d, 0xc2965ecc, 0x23c92b4, 0x1ea03e7, 0x2e013961, 0x4704b4b6, |
| 0x905ea367, 0xca8fd3f}, |
| {0x551b2b61, 0x92523a42, 0x390fcd06, 0x1eb7a89c, 0x392a63e, 0xe7f1d2be, |
| 0x4ddb0c33, 0x96dca264}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x15339848, 0x231c210e, 0x70778c8d, 0xe87a28e8, 0x6956e170, 0x9d1de661, |
| 0x2bb09c0b, 0x4ac3c938}, |
| {0x6998987d, 0x19be0551, 0xae09f4d6, 0x8b2376c4, 0x1a3f933d, 0x1de0b765, |
| 0xe39705f4, 0x380d94c7}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x8c31c31d, 0x3685954b, 0x5bf21a0c, 0x68533d00, 0x75c79ec9, 0xbd7626e, |
| 0x42c69d54, 0xca177547}, |
| {0xf6d2dbb2, 0xcc6edaff, 0x174a9d18, 0xfd0d8cbd, 0xaa4578e8, 0x875e8793, |
| 0x9cab2ce6, 0xa976a713}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0xb43ea1db, 0xce37ab11, 0x5259d292, 0xa7ff1a9, 0x8f84f186, 0x851b0221, |
| 0xdefaad13, 0xa7222bea}, |
| {0x2b0a9144, 0xa2ac78ec, 0xf2fa59c5, 0x5a024051, 0x6147ce38, 0x91d1eca5, |
| 0xbc2ac690, 0xbe94d523}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}, |
| {{0x79ec1a0f, 0x2d8daefd, 0xceb39c97, 0x3bbcd6fd, 0x58f61a95, 0xf5575ffc, |
| 0xadf7b420, 0xdbd986c4}, |
| {0x15f39eb7, 0x81aa8814, 0xb98d976c, 0x6ee2fcf5, 0xcf2f717d, 0x5465475d, |
| 0x6860bbd0, 0x8e24d3c4}, |
| {0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0}}}}; |
| #endif |
| |
| // 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 char 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) { |
| uint64_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); |
| uint8_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(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], 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. |
| uint64_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(bits, 16, 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(bits, 16, 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. |
| uint64_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); |
| fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], |
| 1 /* mixed */, fiat_p256_g_pre_comp[1][bits][0], |
| fiat_p256_g_pre_comp[1][bits][1], |
| fiat_p256_g_pre_comp[1][bits][2]); |
| 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); |
| fiat_p256_point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], |
| 1 /* mixed */, fiat_p256_g_pre_comp[0][bits][0], |
| fiat_p256_g_pre_comp[0][bits][1], |
| fiat_p256_g_pre_comp[0][bits][2]); |
| } |
| |
| int digit = p_wNAF[i]; |
| if (digit != 0) { |
| assert(digit & 1); |
| int idx = 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 |