ref: 2cdc6577ef8a251bf1740439cf5bfc47050dab41
dir: /third_party/boringssl/src/crypto/fipsmodule/ec/ec_montgomery.c/
/* Originally written by Bodo Moeller and Nils Larsch for the OpenSSL project.
* ====================================================================
* Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
*
* 3. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* openssl-core@openssl.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
* OF THE POSSIBILITY OF SUCH DAMAGE.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com).
*
*/
/* ====================================================================
* Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
*
* Portions of the attached software ("Contribution") are developed by
* SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
*
* The Contribution is licensed pursuant to the OpenSSL open source
* license provided above.
*
* The elliptic curve binary polynomial software is originally written by
* Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
* Laboratories. */
#include <openssl/ec.h>
#include <openssl/bn.h>
#include <openssl/err.h>
#include <openssl/mem.h>
#include "../bn/internal.h"
#include "../delocate.h"
#include "internal.h"
int ec_GFp_mont_group_init(EC_GROUP *group) {
int ok;
ok = ec_GFp_simple_group_init(group);
group->mont = NULL;
return ok;
}
void ec_GFp_mont_group_finish(EC_GROUP *group) {
BN_MONT_CTX_free(group->mont);
group->mont = NULL;
ec_GFp_simple_group_finish(group);
}
int ec_GFp_mont_group_set_curve(EC_GROUP *group, const BIGNUM *p,
const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) {
BN_MONT_CTX_free(group->mont);
group->mont = BN_MONT_CTX_new_for_modulus(p, ctx);
if (group->mont == NULL) {
OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
return 0;
}
if (!ec_GFp_simple_group_set_curve(group, p, a, b, ctx)) {
BN_MONT_CTX_free(group->mont);
group->mont = NULL;
return 0;
}
return 1;
}
static void ec_GFp_mont_felem_to_montgomery(const EC_GROUP *group,
EC_FELEM *out, const EC_FELEM *in) {
bn_to_montgomery_small(out->words, in->words, group->field.width,
group->mont);
}
static void ec_GFp_mont_felem_from_montgomery(const EC_GROUP *group,
EC_FELEM *out,
const EC_FELEM *in) {
bn_from_montgomery_small(out->words, group->field.width, in->words,
group->field.width, group->mont);
}
static void ec_GFp_mont_felem_inv0(const EC_GROUP *group, EC_FELEM *out,
const EC_FELEM *a) {
bn_mod_inverse0_prime_mont_small(out->words, a->words, group->field.width,
group->mont);
}
void ec_GFp_mont_felem_mul(const EC_GROUP *group, EC_FELEM *r,
const EC_FELEM *a, const EC_FELEM *b) {
bn_mod_mul_montgomery_small(r->words, a->words, b->words, group->field.width,
group->mont);
}
void ec_GFp_mont_felem_sqr(const EC_GROUP *group, EC_FELEM *r,
const EC_FELEM *a) {
bn_mod_mul_montgomery_small(r->words, a->words, a->words, group->field.width,
group->mont);
}
void ec_GFp_mont_felem_to_bytes(const EC_GROUP *group, uint8_t *out,
size_t *out_len, const EC_FELEM *in) {
EC_FELEM tmp;
ec_GFp_mont_felem_from_montgomery(group, &tmp, in);
ec_GFp_simple_felem_to_bytes(group, out, out_len, &tmp);
}
int ec_GFp_mont_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out,
const uint8_t *in, size_t len) {
if (!ec_GFp_simple_felem_from_bytes(group, out, in, len)) {
return 0;
}
ec_GFp_mont_felem_to_montgomery(group, out, out);
return 1;
}
static void ec_GFp_mont_felem_reduce(const EC_GROUP *group, EC_FELEM *out,
const BN_ULONG *words, size_t num) {
// Convert "from" Montgomery form so the value is reduced mod p.
bn_from_montgomery_small(out->words, group->field.width, words, num,
group->mont);
// Convert "to" Montgomery form to remove the R^-1 factor added.
ec_GFp_mont_felem_to_montgomery(group, out, out);
// Convert to Montgomery form to match this implementation's representation.
ec_GFp_mont_felem_to_montgomery(group, out, out);
}
static void ec_GFp_mont_felem_exp(const EC_GROUP *group, EC_FELEM *out,
const EC_FELEM *a, const BN_ULONG *exp,
size_t num_exp) {
bn_mod_exp_mont_small(out->words, a->words, group->field.width, exp, num_exp,
group->mont);
}
static int ec_GFp_mont_point_get_affine_coordinates(const EC_GROUP *group,
const EC_RAW_POINT *point,
EC_FELEM *x, EC_FELEM *y) {
if (ec_GFp_simple_is_at_infinity(group, point)) {
OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
return 0;
}
// Transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3). Note the check above
// ensures |point->Z| is non-zero, so the inverse always exists.
EC_FELEM z1, z2;
ec_GFp_mont_felem_inv0(group, &z2, &point->Z);
ec_GFp_mont_felem_sqr(group, &z1, &z2);
if (x != NULL) {
ec_GFp_mont_felem_mul(group, x, &point->X, &z1);
}
if (y != NULL) {
ec_GFp_mont_felem_mul(group, &z1, &z1, &z2);
ec_GFp_mont_felem_mul(group, y, &point->Y, &z1);
}
return 1;
}
static int ec_GFp_mont_jacobian_to_affine_batch(const EC_GROUP *group,
EC_AFFINE *out,
const EC_RAW_POINT *in,
size_t num) {
if (num == 0) {
return 1;
}
// Compute prefix products of all Zs. Use |out[i].X| as scratch space
// to store these values.
out[0].X = in[0].Z;
for (size_t i = 1; i < num; i++) {
ec_GFp_mont_felem_mul(group, &out[i].X, &out[i - 1].X, &in[i].Z);
}
// Some input was infinity iff the product of all Zs is zero.
if (ec_felem_non_zero_mask(group, &out[num - 1].X) == 0) {
OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
return 0;
}
// Invert the product of all Zs.
EC_FELEM zinvprod;
ec_GFp_mont_felem_inv0(group, &zinvprod, &out[num - 1].X);
for (size_t i = num - 1; i < num; i--) {
// Our loop invariant is that |zinvprod| is Z0^-1 * Z1^-1 * ... * Zi^-1.
// Recover Zi^-1 by multiplying by the previous product.
EC_FELEM zinv, zinv2;
if (i == 0) {
zinv = zinvprod;
} else {
ec_GFp_mont_felem_mul(group, &zinv, &zinvprod, &out[i - 1].X);
// Maintain the loop invariant for the next iteration.
ec_GFp_mont_felem_mul(group, &zinvprod, &zinvprod, &in[i].Z);
}
// Compute affine coordinates: x = X * Z^-2 and y = Y * Z^-3.
ec_GFp_mont_felem_sqr(group, &zinv2, &zinv);
ec_GFp_mont_felem_mul(group, &out[i].X, &in[i].X, &zinv2);
ec_GFp_mont_felem_mul(group, &out[i].Y, &in[i].Y, &zinv2);
ec_GFp_mont_felem_mul(group, &out[i].Y, &out[i].Y, &zinv);
}
return 1;
}
void ec_GFp_mont_add(const EC_GROUP *group, EC_RAW_POINT *out,
const EC_RAW_POINT *a, const EC_RAW_POINT *b) {
if (a == b) {
ec_GFp_mont_dbl(group, out, a);
return;
}
// The method is taken from:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-add-2007-bl
//
// Coq transcription and correctness proof:
// <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L467>
// <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L544>
EC_FELEM x_out, y_out, z_out;
BN_ULONG z1nz = ec_felem_non_zero_mask(group, &a->Z);
BN_ULONG z2nz = ec_felem_non_zero_mask(group, &b->Z);
// z1z1 = z1z1 = z1**2
EC_FELEM z1z1;
ec_GFp_mont_felem_sqr(group, &z1z1, &a->Z);
// z2z2 = z2**2
EC_FELEM z2z2;
ec_GFp_mont_felem_sqr(group, &z2z2, &b->Z);
// u1 = x1*z2z2
EC_FELEM u1;
ec_GFp_mont_felem_mul(group, &u1, &a->X, &z2z2);
// two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2
EC_FELEM two_z1z2;
ec_felem_add(group, &two_z1z2, &a->Z, &b->Z);
ec_GFp_mont_felem_sqr(group, &two_z1z2, &two_z1z2);
ec_felem_sub(group, &two_z1z2, &two_z1z2, &z1z1);
ec_felem_sub(group, &two_z1z2, &two_z1z2, &z2z2);
// s1 = y1 * z2**3
EC_FELEM s1;
ec_GFp_mont_felem_mul(group, &s1, &b->Z, &z2z2);
ec_GFp_mont_felem_mul(group, &s1, &s1, &a->Y);
// u2 = x2*z1z1
EC_FELEM u2;
ec_GFp_mont_felem_mul(group, &u2, &b->X, &z1z1);
// h = u2 - u1
EC_FELEM h;
ec_felem_sub(group, &h, &u2, &u1);
BN_ULONG xneq = ec_felem_non_zero_mask(group, &h);
// z_out = two_z1z2 * h
ec_GFp_mont_felem_mul(group, &z_out, &h, &two_z1z2);
// z1z1z1 = z1 * z1z1
EC_FELEM z1z1z1;
ec_GFp_mont_felem_mul(group, &z1z1z1, &a->Z, &z1z1);
// s2 = y2 * z1**3
EC_FELEM s2;
ec_GFp_mont_felem_mul(group, &s2, &b->Y, &z1z1z1);
// r = (s2 - s1)*2
EC_FELEM r;
ec_felem_sub(group, &r, &s2, &s1);
ec_felem_add(group, &r, &r, &r);
BN_ULONG yneq = ec_felem_non_zero_mask(group, &r);
// This case will never occur in the constant-time |ec_GFp_mont_mul|.
BN_ULONG is_nontrivial_double = ~xneq & ~yneq & z1nz & z2nz;
if (is_nontrivial_double) {
ec_GFp_mont_dbl(group, out, a);
return;
}
// I = (2h)**2
EC_FELEM i;
ec_felem_add(group, &i, &h, &h);
ec_GFp_mont_felem_sqr(group, &i, &i);
// J = h * I
EC_FELEM j;
ec_GFp_mont_felem_mul(group, &j, &h, &i);
// V = U1 * I
EC_FELEM v;
ec_GFp_mont_felem_mul(group, &v, &u1, &i);
// x_out = r**2 - J - 2V
ec_GFp_mont_felem_sqr(group, &x_out, &r);
ec_felem_sub(group, &x_out, &x_out, &j);
ec_felem_sub(group, &x_out, &x_out, &v);
ec_felem_sub(group, &x_out, &x_out, &v);
// y_out = r(V-x_out) - 2 * s1 * J
ec_felem_sub(group, &y_out, &v, &x_out);
ec_GFp_mont_felem_mul(group, &y_out, &y_out, &r);
EC_FELEM s1j;
ec_GFp_mont_felem_mul(group, &s1j, &s1, &j);
ec_felem_sub(group, &y_out, &y_out, &s1j);
ec_felem_sub(group, &y_out, &y_out, &s1j);
ec_felem_select(group, &x_out, z1nz, &x_out, &b->X);
ec_felem_select(group, &out->X, z2nz, &x_out, &a->X);
ec_felem_select(group, &y_out, z1nz, &y_out, &b->Y);
ec_felem_select(group, &out->Y, z2nz, &y_out, &a->Y);
ec_felem_select(group, &z_out, z1nz, &z_out, &b->Z);
ec_felem_select(group, &out->Z, z2nz, &z_out, &a->Z);
}
void ec_GFp_mont_dbl(const EC_GROUP *group, EC_RAW_POINT *r,
const EC_RAW_POINT *a) {
if (group->a_is_minus3) {
// 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>
EC_FELEM delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta;
// delta = z^2
ec_GFp_mont_felem_sqr(group, &delta, &a->Z);
// gamma = y^2
ec_GFp_mont_felem_sqr(group, &gamma, &a->Y);
// beta = x*gamma
ec_GFp_mont_felem_mul(group, &beta, &a->X, &gamma);
// alpha = 3*(x-delta)*(x+delta)
ec_felem_sub(group, &ftmp, &a->X, &delta);
ec_felem_add(group, &ftmp2, &a->X, &delta);
ec_felem_add(group, &tmptmp, &ftmp2, &ftmp2);
ec_felem_add(group, &ftmp2, &ftmp2, &tmptmp);
ec_GFp_mont_felem_mul(group, &alpha, &ftmp, &ftmp2);
// x' = alpha^2 - 8*beta
ec_GFp_mont_felem_sqr(group, &r->X, &alpha);
ec_felem_add(group, &fourbeta, &beta, &beta);
ec_felem_add(group, &fourbeta, &fourbeta, &fourbeta);
ec_felem_add(group, &tmptmp, &fourbeta, &fourbeta);
ec_felem_sub(group, &r->X, &r->X, &tmptmp);
// z' = (y + z)^2 - gamma - delta
ec_felem_add(group, &delta, &gamma, &delta);
ec_felem_add(group, &ftmp, &a->Y, &a->Z);
ec_GFp_mont_felem_sqr(group, &r->Z, &ftmp);
ec_felem_sub(group, &r->Z, &r->Z, &delta);
// y' = alpha*(4*beta - x') - 8*gamma^2
ec_felem_sub(group, &r->Y, &fourbeta, &r->X);
ec_felem_add(group, &gamma, &gamma, &gamma);
ec_GFp_mont_felem_sqr(group, &gamma, &gamma);
ec_GFp_mont_felem_mul(group, &r->Y, &alpha, &r->Y);
ec_felem_add(group, &gamma, &gamma, &gamma);
ec_felem_sub(group, &r->Y, &r->Y, &gamma);
} else {
// The method is taken from:
// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl
//
// Coq transcription and correctness proof:
// <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L102>
// <https://github.com/davidben/fiat-crypto/blob/c7b95f62b2a54b559522573310e9b487327d219a/src/Curves/Weierstrass/Jacobian.v#L534>
EC_FELEM xx, yy, yyyy, zz;
ec_GFp_mont_felem_sqr(group, &xx, &a->X);
ec_GFp_mont_felem_sqr(group, &yy, &a->Y);
ec_GFp_mont_felem_sqr(group, &yyyy, &yy);
ec_GFp_mont_felem_sqr(group, &zz, &a->Z);
// s = 2*((x_in + yy)^2 - xx - yyyy)
EC_FELEM s;
ec_felem_add(group, &s, &a->X, &yy);
ec_GFp_mont_felem_sqr(group, &s, &s);
ec_felem_sub(group, &s, &s, &xx);
ec_felem_sub(group, &s, &s, &yyyy);
ec_felem_add(group, &s, &s, &s);
// m = 3*xx + a*zz^2
EC_FELEM m;
ec_GFp_mont_felem_sqr(group, &m, &zz);
ec_GFp_mont_felem_mul(group, &m, &group->a, &m);
ec_felem_add(group, &m, &m, &xx);
ec_felem_add(group, &m, &m, &xx);
ec_felem_add(group, &m, &m, &xx);
// x_out = m^2 - 2*s
ec_GFp_mont_felem_sqr(group, &r->X, &m);
ec_felem_sub(group, &r->X, &r->X, &s);
ec_felem_sub(group, &r->X, &r->X, &s);
// z_out = (y_in + z_in)^2 - yy - zz
ec_felem_add(group, &r->Z, &a->Y, &a->Z);
ec_GFp_mont_felem_sqr(group, &r->Z, &r->Z);
ec_felem_sub(group, &r->Z, &r->Z, &yy);
ec_felem_sub(group, &r->Z, &r->Z, &zz);
// y_out = m*(s-x_out) - 8*yyyy
ec_felem_add(group, &yyyy, &yyyy, &yyyy);
ec_felem_add(group, &yyyy, &yyyy, &yyyy);
ec_felem_add(group, &yyyy, &yyyy, &yyyy);
ec_felem_sub(group, &r->Y, &s, &r->X);
ec_GFp_mont_felem_mul(group, &r->Y, &r->Y, &m);
ec_felem_sub(group, &r->Y, &r->Y, &yyyy);
}
}
static int ec_GFp_mont_cmp_x_coordinate(const EC_GROUP *group,
const EC_RAW_POINT *p,
const EC_SCALAR *r) {
if (!group->field_greater_than_order ||
group->field.width != group->order.width) {
// Do not bother optimizing this case. p > order in all commonly-used
// curves.
return ec_GFp_simple_cmp_x_coordinate(group, p, 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.
EC_FELEM r_Z2, Z2_mont, X;
ec_GFp_mont_felem_mul(group, &Z2_mont, &p->Z, &p->Z);
// r < order < p, so this is valid.
OPENSSL_memcpy(r_Z2.words, r->words, group->field.width * sizeof(BN_ULONG));
ec_GFp_mont_felem_mul(group, &r_Z2, &r_Z2, &Z2_mont);
ec_GFp_mont_felem_from_montgomery(group, &X, &p->X);
if (ec_felem_equal(group, &r_Z2, &X)) {
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.
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.
bn_add_words(r_Z2.words, r->words, group->order.d, group->field.width);
ec_GFp_mont_felem_mul(group, &r_Z2, &r_Z2, &Z2_mont);
if (ec_felem_equal(group, &r_Z2, &X)) {
return 1;
}
}
return 0;
}
DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_mont_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_mont_point_get_affine_coordinates;
out->jacobian_to_affine_batch = ec_GFp_mont_jacobian_to_affine_batch;
out->add = ec_GFp_mont_add;
out->dbl = ec_GFp_mont_dbl;
out->mul = ec_GFp_mont_mul;
out->mul_base = ec_GFp_mont_mul_base;
out->mul_batch = ec_GFp_mont_mul_batch;
out->mul_public_batch = ec_GFp_mont_mul_public_batch;
out->init_precomp = ec_GFp_mont_init_precomp;
out->mul_precomp = ec_GFp_mont_mul_precomp;
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->felem_reduce = ec_GFp_mont_felem_reduce;
out->felem_exp = ec_GFp_mont_felem_exp;
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_mont_cmp_x_coordinate;
}