| /* 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/ec.h> |
| |
| #include <string.h> |
| |
| #include <openssl/bn.h> |
| #include <openssl/err.h> |
| #include <openssl/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_JACOBIAN *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_JACOBIAN *dest, const EC_JACOBIAN *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_JACOBIAN *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_JACOBIAN *point) { |
| ec_felem_neg(group, &point->Y, &point->Y); |
| } |
| |
| int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, |
| const EC_JACOBIAN *point) { |
| return ec_felem_non_zero_mask(group, &point->Z) == 0; |
| } |
| |
| int ec_GFp_simple_is_on_curve(const EC_GROUP *group, |
| const EC_JACOBIAN *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_JACOBIAN *a, |
| const EC_JACOBIAN *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_JACOBIAN *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_JACOBIAN *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); |
| bn_words_to_big_endian(out, len, in->words, group->field.width); |
| *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; |
| } |
| |
| bn_big_endian_to_words(out->words, group->field.width, in, len); |
| |
| 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; |
| } |