/* Originally written by Bodo Moeller 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_grpc/ec.h> #include <string.h> #include <openssl_grpc/bn.h> #include <openssl_grpc/err.h> #include <openssl_grpc/mem.h> #include "internal.h" #include "../../internal.h" // Most method functions in this file are designed to work with non-trivial // representations of field elements if necessary (see ecp_mont.c): while // standard modular addition and subtraction are used, the field_mul and // field_sqr methods will be used for multiplication, and field_encode and // field_decode (if defined) will be used for converting between // representations. // // Functions here specifically assume that if a non-trivial representation is // used, it is a Montgomery representation (i.e. 'encoding' means multiplying // by some factor R). int ec_GFp_simple_group_init(EC_GROUP *group) { BN_init(&group->field); group->a_is_minus3 = 0; return 1; } void ec_GFp_simple_group_finish(EC_GROUP *group) { BN_free(&group->field); } int ec_GFp_simple_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { // p must be a prime > 3 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) { OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD); return 0; } int ret = 0; BN_CTX_start(ctx); BIGNUM *tmp = BN_CTX_get(ctx); if (tmp == NULL) { goto err; } // group->field if (!BN_copy(&group->field, p)) { goto err; } BN_set_negative(&group->field, 0); // Store the field in minimal form, so it can be used with |BN_ULONG| arrays. bn_set_minimal_width(&group->field); if (!ec_bignum_to_felem(group, &group->a, a) || !ec_bignum_to_felem(group, &group->b, b) || !ec_bignum_to_felem(group, &group->one, BN_value_one())) { goto err; } // group->a_is_minus3 if (!BN_copy(tmp, a) || !BN_add_word(tmp, 3)) { goto err; } group->a_is_minus3 = (0 == BN_cmp(tmp, &group->field)); ret = 1; err: BN_CTX_end(ctx); return ret; } int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b) { if ((p != NULL && !BN_copy(p, &group->field)) || (a != NULL && !ec_felem_to_bignum(group, a, &group->a)) || (b != NULL && !ec_felem_to_bignum(group, b, &group->b))) { return 0; } return 1; } void ec_GFp_simple_point_init(EC_RAW_POINT *point) { OPENSSL_memset(&point->X, 0, sizeof(EC_FELEM)); OPENSSL_memset(&point->Y, 0, sizeof(EC_FELEM)); OPENSSL_memset(&point->Z, 0, sizeof(EC_FELEM)); } void ec_GFp_simple_point_copy(EC_RAW_POINT *dest, const EC_RAW_POINT *src) { OPENSSL_memcpy(&dest->X, &src->X, sizeof(EC_FELEM)); OPENSSL_memcpy(&dest->Y, &src->Y, sizeof(EC_FELEM)); OPENSSL_memcpy(&dest->Z, &src->Z, sizeof(EC_FELEM)); } void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, EC_RAW_POINT *point) { // Although it is strictly only necessary to zero Z, we zero the entire point // in case |point| was stack-allocated and yet to be initialized. ec_GFp_simple_point_init(point); } void ec_GFp_simple_invert(const EC_GROUP *group, EC_RAW_POINT *point) { ec_felem_neg(group, &point->Y, &point->Y); } int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_RAW_POINT *point) { return ec_felem_non_zero_mask(group, &point->Z) == 0; } int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_RAW_POINT *point) { // We have a curve defined by a Weierstrass equation // y^2 = x^3 + a*x + b. // The point to consider is given in Jacobian projective coordinates // where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3). // Substituting this and multiplying by Z^6 transforms the above equation // into // Y^2 = X^3 + a*X*Z^4 + b*Z^6. // To test this, we add up the right-hand side in 'rh'. // // This function may be used when double-checking the secret result of a point // multiplication, so we proceed in constant-time. void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, const EC_FELEM *b) = group->meth->felem_mul; void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) = group->meth->felem_sqr; // rh := X^2 EC_FELEM rh; felem_sqr(group, &rh, &point->X); EC_FELEM tmp, Z4, Z6; felem_sqr(group, &tmp, &point->Z); felem_sqr(group, &Z4, &tmp); felem_mul(group, &Z6, &Z4, &tmp); // rh := rh + a*Z^4 if (group->a_is_minus3) { ec_felem_add(group, &tmp, &Z4, &Z4); ec_felem_add(group, &tmp, &tmp, &Z4); ec_felem_sub(group, &rh, &rh, &tmp); } else { felem_mul(group, &tmp, &Z4, &group->a); ec_felem_add(group, &rh, &rh, &tmp); } // rh := (rh + a*Z^4)*X felem_mul(group, &rh, &rh, &point->X); // rh := rh + b*Z^6 felem_mul(group, &tmp, &group->b, &Z6); ec_felem_add(group, &rh, &rh, &tmp); // 'lh' := Y^2 felem_sqr(group, &tmp, &point->Y); ec_felem_sub(group, &tmp, &tmp, &rh); BN_ULONG not_equal = ec_felem_non_zero_mask(group, &tmp); // If Z = 0, the point is infinity, which is always on the curve. BN_ULONG not_infinity = ec_felem_non_zero_mask(group, &point->Z); return 1 & ~(not_infinity & not_equal); } int ec_GFp_simple_points_equal(const EC_GROUP *group, const EC_RAW_POINT *a, const EC_RAW_POINT *b) { // This function is implemented in constant-time for two reasons. First, // although EC points are usually public, their Jacobian Z coordinates may be // secret, or at least are not obviously public. Second, more complex // protocols will sometimes manipulate secret points. // // This does mean that we pay a 6M+2S Jacobian comparison when comparing two // publicly affine points costs no field operations at all. If needed, we can // restore this optimization by keeping better track of affine vs. Jacobian // forms. See https://crbug.com/boringssl/326. // If neither |a| or |b| is infinity, we have to decide whether // (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3), // or equivalently, whether // (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3). void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, const EC_FELEM *b) = group->meth->felem_mul; void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) = group->meth->felem_sqr; EC_FELEM tmp1, tmp2, Za23, Zb23; felem_sqr(group, &Zb23, &b->Z); // Zb23 = Z_b^2 felem_mul(group, &tmp1, &a->X, &Zb23); // tmp1 = X_a * Z_b^2 felem_sqr(group, &Za23, &a->Z); // Za23 = Z_a^2 felem_mul(group, &tmp2, &b->X, &Za23); // tmp2 = X_b * Z_a^2 ec_felem_sub(group, &tmp1, &tmp1, &tmp2); const BN_ULONG x_not_equal = ec_felem_non_zero_mask(group, &tmp1); felem_mul(group, &Zb23, &Zb23, &b->Z); // Zb23 = Z_b^3 felem_mul(group, &tmp1, &a->Y, &Zb23); // tmp1 = Y_a * Z_b^3 felem_mul(group, &Za23, &Za23, &a->Z); // Za23 = Z_a^3 felem_mul(group, &tmp2, &b->Y, &Za23); // tmp2 = Y_b * Z_a^3 ec_felem_sub(group, &tmp1, &tmp1, &tmp2); const BN_ULONG y_not_equal = ec_felem_non_zero_mask(group, &tmp1); const BN_ULONG x_and_y_equal = ~(x_not_equal | y_not_equal); const BN_ULONG a_not_infinity = ec_felem_non_zero_mask(group, &a->Z); const BN_ULONG b_not_infinity = ec_felem_non_zero_mask(group, &b->Z); const BN_ULONG a_and_b_infinity = ~(a_not_infinity | b_not_infinity); const BN_ULONG equal = a_and_b_infinity | (a_not_infinity & b_not_infinity & x_and_y_equal); return equal & 1; } int ec_affine_jacobian_equal(const EC_GROUP *group, const EC_AFFINE *a, const EC_RAW_POINT *b) { // If |b| is not infinity, we have to decide whether // (X_a, Y_a) = (X_b/Z_b^2, Y_b/Z_b^3), // or equivalently, whether // (X_a*Z_b^2, Y_a*Z_b^3) = (X_b, Y_b). void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a, const EC_FELEM *b) = group->meth->felem_mul; void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) = group->meth->felem_sqr; EC_FELEM tmp, Zb2; felem_sqr(group, &Zb2, &b->Z); // Zb2 = Z_b^2 felem_mul(group, &tmp, &a->X, &Zb2); // tmp = X_a * Z_b^2 ec_felem_sub(group, &tmp, &tmp, &b->X); const BN_ULONG x_not_equal = ec_felem_non_zero_mask(group, &tmp); felem_mul(group, &tmp, &a->Y, &Zb2); // tmp = Y_a * Z_b^2 felem_mul(group, &tmp, &tmp, &b->Z); // tmp = Y_a * Z_b^3 ec_felem_sub(group, &tmp, &tmp, &b->Y); const BN_ULONG y_not_equal = ec_felem_non_zero_mask(group, &tmp); const BN_ULONG x_and_y_equal = ~(x_not_equal | y_not_equal); const BN_ULONG b_not_infinity = ec_felem_non_zero_mask(group, &b->Z); const BN_ULONG equal = b_not_infinity & x_and_y_equal; return equal & 1; } int ec_GFp_simple_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)) { // |ec_get_x_coordinate_as_scalar| will check this internally, but this way // we do not push to the error queue. return 0; } EC_SCALAR x; return ec_get_x_coordinate_as_scalar(group, &x, p) && ec_scalar_equal_vartime(group, &x, r); } void ec_GFp_simple_felem_to_bytes(const EC_GROUP *group, uint8_t *out, size_t *out_len, const EC_FELEM *in) { size_t len = BN_num_bytes(&group->field); for (size_t i = 0; i < len; i++) { out[i] = in->bytes[len - 1 - i]; } *out_len = len; } int ec_GFp_simple_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out, const uint8_t *in, size_t len) { if (len != BN_num_bytes(&group->field)) { OPENSSL_PUT_ERROR(EC, EC_R_DECODE_ERROR); return 0; } OPENSSL_memset(out, 0, sizeof(EC_FELEM)); for (size_t i = 0; i < len; i++) { out->bytes[i] = in[len - 1 - i]; } if (!bn_less_than_words(out->words, group->field.d, group->field.width)) { OPENSSL_PUT_ERROR(EC, EC_R_DECODE_ERROR); return 0; } return 1; }