/* 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_grpc/base.h>
#include <openssl_grpc/bn.h>
#include <openssl_grpc/ec.h>
#include <openssl_grpc/err.h>
#include <openssl_grpc/mem.h>
#include <openssl_grpc/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];
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:
// <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);
}
#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