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1726 lines
40 KiB
C
1726 lines
40 KiB
C
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/* $OpenBSD: ecp_smpl.c,v 1.29 2018/11/15 05:53:31 tb Exp $ */
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/* Includes code written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
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* for the OpenSSL project.
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* Includes code written by Bodo Moeller for the OpenSSL project.
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*/
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/* ====================================================================
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* Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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*
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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*
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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*
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* 3. All advertising materials mentioning features or use of this
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* software must display the following acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
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*
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* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
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* endorse or promote products derived from this software without
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* prior written permission. For written permission, please contact
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* openssl-core@openssl.org.
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*
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* 5. Products derived from this software may not be called "OpenSSL"
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* nor may "OpenSSL" appear in their names without prior written
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* permission of the OpenSSL Project.
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*
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* 6. Redistributions of any form whatsoever must retain the following
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* acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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*
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* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
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* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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|
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
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* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
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* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
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* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
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* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
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* OF THE POSSIBILITY OF SUCH DAMAGE.
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* ====================================================================
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*
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* This product includes cryptographic software written by Eric Young
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* (eay@cryptsoft.com). This product includes software written by Tim
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* Hudson (tjh@cryptsoft.com).
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*
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*/
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/* ====================================================================
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* Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
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* Portions of this software developed by SUN MICROSYSTEMS, INC.,
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* and contributed to the OpenSSL project.
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*/
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#include <openssl/err.h>
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#include "bn_lcl.h"
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#include "ec_lcl.h"
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const EC_METHOD *
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EC_GFp_simple_method(void)
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{
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static const EC_METHOD ret = {
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.flags = EC_FLAGS_DEFAULT_OCT,
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.field_type = NID_X9_62_prime_field,
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.group_init = ec_GFp_simple_group_init,
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.group_finish = ec_GFp_simple_group_finish,
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.group_clear_finish = ec_GFp_simple_group_clear_finish,
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.group_copy = ec_GFp_simple_group_copy,
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.group_set_curve = ec_GFp_simple_group_set_curve,
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.group_get_curve = ec_GFp_simple_group_get_curve,
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.group_get_degree = ec_GFp_simple_group_get_degree,
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.group_check_discriminant =
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ec_GFp_simple_group_check_discriminant,
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.point_init = ec_GFp_simple_point_init,
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.point_finish = ec_GFp_simple_point_finish,
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.point_clear_finish = ec_GFp_simple_point_clear_finish,
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.point_copy = ec_GFp_simple_point_copy,
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.point_set_to_infinity = ec_GFp_simple_point_set_to_infinity,
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.point_set_Jprojective_coordinates_GFp =
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ec_GFp_simple_set_Jprojective_coordinates_GFp,
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.point_get_Jprojective_coordinates_GFp =
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ec_GFp_simple_get_Jprojective_coordinates_GFp,
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.point_set_affine_coordinates =
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ec_GFp_simple_point_set_affine_coordinates,
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.point_get_affine_coordinates =
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ec_GFp_simple_point_get_affine_coordinates,
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.add = ec_GFp_simple_add,
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.dbl = ec_GFp_simple_dbl,
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.invert = ec_GFp_simple_invert,
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.is_at_infinity = ec_GFp_simple_is_at_infinity,
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.is_on_curve = ec_GFp_simple_is_on_curve,
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.point_cmp = ec_GFp_simple_cmp,
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.make_affine = ec_GFp_simple_make_affine,
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.points_make_affine = ec_GFp_simple_points_make_affine,
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.mul_generator_ct = ec_GFp_simple_mul_generator_ct,
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.mul_single_ct = ec_GFp_simple_mul_single_ct,
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.mul_double_nonct = ec_GFp_simple_mul_double_nonct,
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.field_mul = ec_GFp_simple_field_mul,
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.field_sqr = ec_GFp_simple_field_sqr,
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.blind_coordinates = ec_GFp_simple_blind_coordinates,
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};
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return &ret;
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}
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/* Most method functions in this file are designed to work with
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* non-trivial representations of field elements if necessary
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* (see ecp_mont.c): while standard modular addition and subtraction
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* are used, the field_mul and field_sqr methods will be used for
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* multiplication, and field_encode and field_decode (if defined)
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* will be used for converting between representations.
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* Functions ec_GFp_simple_points_make_affine() and
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* ec_GFp_simple_point_get_affine_coordinates() specifically assume
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* that if a non-trivial representation is used, it is a Montgomery
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* representation (i.e. 'encoding' means multiplying by some factor R).
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*/
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int
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ec_GFp_simple_group_init(EC_GROUP * group)
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{
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BN_init(&group->field);
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BN_init(&group->a);
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BN_init(&group->b);
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group->a_is_minus3 = 0;
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return 1;
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}
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void
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ec_GFp_simple_group_finish(EC_GROUP * group)
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{
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BN_free(&group->field);
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BN_free(&group->a);
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BN_free(&group->b);
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}
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void
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ec_GFp_simple_group_clear_finish(EC_GROUP * group)
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{
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BN_clear_free(&group->field);
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BN_clear_free(&group->a);
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BN_clear_free(&group->b);
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}
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int
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ec_GFp_simple_group_copy(EC_GROUP * dest, const EC_GROUP * src)
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{
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if (!BN_copy(&dest->field, &src->field))
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return 0;
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if (!BN_copy(&dest->a, &src->a))
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return 0;
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if (!BN_copy(&dest->b, &src->b))
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return 0;
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dest->a_is_minus3 = src->a_is_minus3;
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return 1;
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}
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int
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ec_GFp_simple_group_set_curve(EC_GROUP * group,
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const BIGNUM * p, const BIGNUM * a, const BIGNUM * b, BN_CTX * ctx)
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{
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int ret = 0;
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BN_CTX *new_ctx = NULL;
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BIGNUM *tmp_a;
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/* p must be a prime > 3 */
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if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
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ECerror(EC_R_INVALID_FIELD);
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return 0;
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}
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if (ctx == NULL) {
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ctx = new_ctx = BN_CTX_new();
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if (ctx == NULL)
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return 0;
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}
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BN_CTX_start(ctx);
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if ((tmp_a = BN_CTX_get(ctx)) == NULL)
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goto err;
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/* group->field */
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if (!BN_copy(&group->field, p))
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goto err;
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BN_set_negative(&group->field, 0);
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/* group->a */
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if (!BN_nnmod(tmp_a, a, p, ctx))
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goto err;
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if (group->meth->field_encode) {
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if (!group->meth->field_encode(group, &group->a, tmp_a, ctx))
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goto err;
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} else if (!BN_copy(&group->a, tmp_a))
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goto err;
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/* group->b */
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if (!BN_nnmod(&group->b, b, p, ctx))
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goto err;
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if (group->meth->field_encode)
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if (!group->meth->field_encode(group, &group->b, &group->b, ctx))
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goto err;
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/* group->a_is_minus3 */
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if (!BN_add_word(tmp_a, 3))
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goto err;
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group->a_is_minus3 = (0 == BN_cmp(tmp_a, &group->field));
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ret = 1;
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err:
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BN_CTX_end(ctx);
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BN_CTX_free(new_ctx);
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return ret;
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}
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int
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ec_GFp_simple_group_get_curve(const EC_GROUP * group, BIGNUM * p, BIGNUM * a, BIGNUM * b, BN_CTX * ctx)
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{
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int ret = 0;
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BN_CTX *new_ctx = NULL;
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if (p != NULL) {
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if (!BN_copy(p, &group->field))
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return 0;
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}
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if (a != NULL || b != NULL) {
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if (group->meth->field_decode) {
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if (ctx == NULL) {
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ctx = new_ctx = BN_CTX_new();
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if (ctx == NULL)
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return 0;
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||
|
}
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if (a != NULL) {
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if (!group->meth->field_decode(group, a, &group->a, ctx))
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goto err;
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||
|
}
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||
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if (b != NULL) {
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if (!group->meth->field_decode(group, b, &group->b, ctx))
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goto err;
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||
|
}
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} else {
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if (a != NULL) {
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if (!BN_copy(a, &group->a))
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goto err;
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}
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if (b != NULL) {
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if (!BN_copy(b, &group->b))
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goto err;
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}
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}
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}
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ret = 1;
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||
|
|
||
|
err:
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||
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BN_CTX_free(new_ctx);
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||
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return ret;
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|
}
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|
|
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|
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int
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ec_GFp_simple_group_get_degree(const EC_GROUP * group)
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{
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return BN_num_bits(&group->field);
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||
|
}
|
||
|
|
||
|
|
||
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int
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||
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ec_GFp_simple_group_check_discriminant(const EC_GROUP * group, BN_CTX * ctx)
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||
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{
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||
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int ret = 0;
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||
|
BIGNUM *a, *b, *order, *tmp_1, *tmp_2;
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const BIGNUM *p = &group->field;
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||
|
BN_CTX *new_ctx = NULL;
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||
|
|
||
|
if (ctx == NULL) {
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||
|
ctx = new_ctx = BN_CTX_new();
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||
|
if (ctx == NULL) {
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||
|
ECerror(ERR_R_MALLOC_FAILURE);
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
BN_CTX_start(ctx);
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||
|
if ((a = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((b = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((tmp_1 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((tmp_2 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((order = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
if (group->meth->field_decode) {
|
||
|
if (!group->meth->field_decode(group, a, &group->a, ctx))
|
||
|
goto err;
|
||
|
if (!group->meth->field_decode(group, b, &group->b, ctx))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!BN_copy(a, &group->a))
|
||
|
goto err;
|
||
|
if (!BN_copy(b, &group->b))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
/*
|
||
|
* check the discriminant: y^2 = x^3 + a*x + b is an elliptic curve
|
||
|
* <=> 4*a^3 + 27*b^2 != 0 (mod p) 0 =< a, b < p
|
||
|
*/
|
||
|
if (BN_is_zero(a)) {
|
||
|
if (BN_is_zero(b))
|
||
|
goto err;
|
||
|
} else if (!BN_is_zero(b)) {
|
||
|
if (!BN_mod_sqr(tmp_1, a, p, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx))
|
||
|
goto err;
|
||
|
if (!BN_lshift(tmp_1, tmp_2, 2))
|
||
|
goto err;
|
||
|
/* tmp_1 = 4*a^3 */
|
||
|
|
||
|
if (!BN_mod_sqr(tmp_2, b, p, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mul_word(tmp_2, 27))
|
||
|
goto err;
|
||
|
/* tmp_2 = 27*b^2 */
|
||
|
|
||
|
if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx))
|
||
|
goto err;
|
||
|
if (BN_is_zero(a))
|
||
|
goto err;
|
||
|
}
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
if (ctx != NULL)
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_point_init(EC_POINT * point)
|
||
|
{
|
||
|
BN_init(&point->X);
|
||
|
BN_init(&point->Y);
|
||
|
BN_init(&point->Z);
|
||
|
point->Z_is_one = 0;
|
||
|
|
||
|
return 1;
|
||
|
}
|
||
|
|
||
|
|
||
|
void
|
||
|
ec_GFp_simple_point_finish(EC_POINT * point)
|
||
|
{
|
||
|
BN_free(&point->X);
|
||
|
BN_free(&point->Y);
|
||
|
BN_free(&point->Z);
|
||
|
}
|
||
|
|
||
|
|
||
|
void
|
||
|
ec_GFp_simple_point_clear_finish(EC_POINT * point)
|
||
|
{
|
||
|
BN_clear_free(&point->X);
|
||
|
BN_clear_free(&point->Y);
|
||
|
BN_clear_free(&point->Z);
|
||
|
point->Z_is_one = 0;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_point_copy(EC_POINT * dest, const EC_POINT * src)
|
||
|
{
|
||
|
if (!BN_copy(&dest->X, &src->X))
|
||
|
return 0;
|
||
|
if (!BN_copy(&dest->Y, &src->Y))
|
||
|
return 0;
|
||
|
if (!BN_copy(&dest->Z, &src->Z))
|
||
|
return 0;
|
||
|
dest->Z_is_one = src->Z_is_one;
|
||
|
|
||
|
return 1;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_point_set_to_infinity(const EC_GROUP * group, EC_POINT * point)
|
||
|
{
|
||
|
point->Z_is_one = 0;
|
||
|
BN_zero(&point->Z);
|
||
|
return 1;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP * group, EC_POINT * point,
|
||
|
const BIGNUM * x, const BIGNUM * y, const BIGNUM * z, BN_CTX * ctx)
|
||
|
{
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
if (x != NULL) {
|
||
|
if (!BN_nnmod(&point->X, x, &group->field, ctx))
|
||
|
goto err;
|
||
|
if (group->meth->field_encode) {
|
||
|
if (!group->meth->field_encode(group, &point->X, &point->X, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
if (y != NULL) {
|
||
|
if (!BN_nnmod(&point->Y, y, &group->field, ctx))
|
||
|
goto err;
|
||
|
if (group->meth->field_encode) {
|
||
|
if (!group->meth->field_encode(group, &point->Y, &point->Y, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
if (z != NULL) {
|
||
|
int Z_is_one;
|
||
|
|
||
|
if (!BN_nnmod(&point->Z, z, &group->field, ctx))
|
||
|
goto err;
|
||
|
Z_is_one = BN_is_one(&point->Z);
|
||
|
if (group->meth->field_encode) {
|
||
|
if (Z_is_one && (group->meth->field_set_to_one != 0)) {
|
||
|
if (!group->meth->field_set_to_one(group, &point->Z, ctx))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!group->meth->field_encode(group, &point->Z, &point->Z, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
point->Z_is_one = Z_is_one;
|
||
|
}
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP * group, const EC_POINT * point,
|
||
|
BIGNUM * x, BIGNUM * y, BIGNUM * z, BN_CTX * ctx)
|
||
|
{
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (group->meth->field_decode != 0) {
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
if (x != NULL) {
|
||
|
if (!group->meth->field_decode(group, x, &point->X, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
if (y != NULL) {
|
||
|
if (!group->meth->field_decode(group, y, &point->Y, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
if (z != NULL) {
|
||
|
if (!group->meth->field_decode(group, z, &point->Z, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
} else {
|
||
|
if (x != NULL) {
|
||
|
if (!BN_copy(x, &point->X))
|
||
|
goto err;
|
||
|
}
|
||
|
if (y != NULL) {
|
||
|
if (!BN_copy(y, &point->Y))
|
||
|
goto err;
|
||
|
}
|
||
|
if (z != NULL) {
|
||
|
if (!BN_copy(z, &point->Z))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP * group, EC_POINT * point,
|
||
|
const BIGNUM * x, const BIGNUM * y, BN_CTX * ctx)
|
||
|
{
|
||
|
if (x == NULL || y == NULL) {
|
||
|
/* unlike for projective coordinates, we do not tolerate this */
|
||
|
ECerror(ERR_R_PASSED_NULL_PARAMETER);
|
||
|
return 0;
|
||
|
}
|
||
|
return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y, BN_value_one(), ctx);
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP * group, const EC_POINT * point,
|
||
|
BIGNUM * x, BIGNUM * y, BN_CTX * ctx)
|
||
|
{
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *Z, *Z_1, *Z_2, *Z_3;
|
||
|
const BIGNUM *Z_;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (EC_POINT_is_at_infinity(group, point) > 0) {
|
||
|
ECerror(EC_R_POINT_AT_INFINITY);
|
||
|
return 0;
|
||
|
}
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((Z = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((Z_1 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((Z_2 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((Z_3 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
/* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
|
||
|
|
||
|
if (group->meth->field_decode) {
|
||
|
if (!group->meth->field_decode(group, Z, &point->Z, ctx))
|
||
|
goto err;
|
||
|
Z_ = Z;
|
||
|
} else {
|
||
|
Z_ = &point->Z;
|
||
|
}
|
||
|
|
||
|
if (BN_is_one(Z_)) {
|
||
|
if (group->meth->field_decode) {
|
||
|
if (x != NULL) {
|
||
|
if (!group->meth->field_decode(group, x, &point->X, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
if (y != NULL) {
|
||
|
if (!group->meth->field_decode(group, y, &point->Y, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
} else {
|
||
|
if (x != NULL) {
|
||
|
if (!BN_copy(x, &point->X))
|
||
|
goto err;
|
||
|
}
|
||
|
if (y != NULL) {
|
||
|
if (!BN_copy(y, &point->Y))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
} else {
|
||
|
if (!BN_mod_inverse_ct(Z_1, Z_, &group->field, ctx)) {
|
||
|
ECerror(ERR_R_BN_LIB);
|
||
|
goto err;
|
||
|
}
|
||
|
if (group->meth->field_encode == 0) {
|
||
|
/* field_sqr works on standard representation */
|
||
|
if (!group->meth->field_sqr(group, Z_2, Z_1, ctx))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!BN_mod_sqr(Z_2, Z_1, &group->field, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
if (x != NULL) {
|
||
|
/*
|
||
|
* in the Montgomery case, field_mul will cancel out
|
||
|
* Montgomery factor in X:
|
||
|
*/
|
||
|
if (!group->meth->field_mul(group, x, &point->X, Z_2, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
if (y != NULL) {
|
||
|
if (group->meth->field_encode == 0) {
|
||
|
/* field_mul works on standard representation */
|
||
|
if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!BN_mod_mul(Z_3, Z_2, Z_1, &group->field, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
/*
|
||
|
* in the Montgomery case, field_mul will cancel out
|
||
|
* Montgomery factor in Y:
|
||
|
*/
|
||
|
if (!group->meth->field_mul(group, y, &point->Y, Z_3, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_add(const EC_GROUP * group, EC_POINT * r, const EC_POINT * a, const EC_POINT * b, BN_CTX * ctx)
|
||
|
{
|
||
|
int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
const BIGNUM *p;
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (a == b)
|
||
|
return EC_POINT_dbl(group, r, a, ctx);
|
||
|
if (EC_POINT_is_at_infinity(group, a) > 0)
|
||
|
return EC_POINT_copy(r, b);
|
||
|
if (EC_POINT_is_at_infinity(group, b) > 0)
|
||
|
return EC_POINT_copy(r, a);
|
||
|
|
||
|
field_mul = group->meth->field_mul;
|
||
|
field_sqr = group->meth->field_sqr;
|
||
|
p = &group->field;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((n0 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((n1 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((n2 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((n3 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((n4 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((n5 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((n6 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
|
||
|
/*
|
||
|
* Note that in this function we must not read components of 'a' or
|
||
|
* 'b' once we have written the corresponding components of 'r'. ('r'
|
||
|
* might be one of 'a' or 'b'.)
|
||
|
*/
|
||
|
|
||
|
/* n1, n2 */
|
||
|
if (b->Z_is_one) {
|
||
|
if (!BN_copy(n1, &a->X))
|
||
|
goto end;
|
||
|
if (!BN_copy(n2, &a->Y))
|
||
|
goto end;
|
||
|
/* n1 = X_a */
|
||
|
/* n2 = Y_a */
|
||
|
} else {
|
||
|
if (!field_sqr(group, n0, &b->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, n1, &a->X, n0, ctx))
|
||
|
goto end;
|
||
|
/* n1 = X_a * Z_b^2 */
|
||
|
|
||
|
if (!field_mul(group, n0, n0, &b->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, n2, &a->Y, n0, ctx))
|
||
|
goto end;
|
||
|
/* n2 = Y_a * Z_b^3 */
|
||
|
}
|
||
|
|
||
|
/* n3, n4 */
|
||
|
if (a->Z_is_one) {
|
||
|
if (!BN_copy(n3, &b->X))
|
||
|
goto end;
|
||
|
if (!BN_copy(n4, &b->Y))
|
||
|
goto end;
|
||
|
/* n3 = X_b */
|
||
|
/* n4 = Y_b */
|
||
|
} else {
|
||
|
if (!field_sqr(group, n0, &a->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, n3, &b->X, n0, ctx))
|
||
|
goto end;
|
||
|
/* n3 = X_b * Z_a^2 */
|
||
|
|
||
|
if (!field_mul(group, n0, n0, &a->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, n4, &b->Y, n0, ctx))
|
||
|
goto end;
|
||
|
/* n4 = Y_b * Z_a^3 */
|
||
|
}
|
||
|
|
||
|
/* n5, n6 */
|
||
|
if (!BN_mod_sub_quick(n5, n1, n3, p))
|
||
|
goto end;
|
||
|
if (!BN_mod_sub_quick(n6, n2, n4, p))
|
||
|
goto end;
|
||
|
/* n5 = n1 - n3 */
|
||
|
/* n6 = n2 - n4 */
|
||
|
|
||
|
if (BN_is_zero(n5)) {
|
||
|
if (BN_is_zero(n6)) {
|
||
|
/* a is the same point as b */
|
||
|
BN_CTX_end(ctx);
|
||
|
ret = EC_POINT_dbl(group, r, a, ctx);
|
||
|
ctx = NULL;
|
||
|
goto end;
|
||
|
} else {
|
||
|
/* a is the inverse of b */
|
||
|
BN_zero(&r->Z);
|
||
|
r->Z_is_one = 0;
|
||
|
ret = 1;
|
||
|
goto end;
|
||
|
}
|
||
|
}
|
||
|
/* 'n7', 'n8' */
|
||
|
if (!BN_mod_add_quick(n1, n1, n3, p))
|
||
|
goto end;
|
||
|
if (!BN_mod_add_quick(n2, n2, n4, p))
|
||
|
goto end;
|
||
|
/* 'n7' = n1 + n3 */
|
||
|
/* 'n8' = n2 + n4 */
|
||
|
|
||
|
/* Z_r */
|
||
|
if (a->Z_is_one && b->Z_is_one) {
|
||
|
if (!BN_copy(&r->Z, n5))
|
||
|
goto end;
|
||
|
} else {
|
||
|
if (a->Z_is_one) {
|
||
|
if (!BN_copy(n0, &b->Z))
|
||
|
goto end;
|
||
|
} else if (b->Z_is_one) {
|
||
|
if (!BN_copy(n0, &a->Z))
|
||
|
goto end;
|
||
|
} else {
|
||
|
if (!field_mul(group, n0, &a->Z, &b->Z, ctx))
|
||
|
goto end;
|
||
|
}
|
||
|
if (!field_mul(group, &r->Z, n0, n5, ctx))
|
||
|
goto end;
|
||
|
}
|
||
|
r->Z_is_one = 0;
|
||
|
/* Z_r = Z_a * Z_b * n5 */
|
||
|
|
||
|
/* X_r */
|
||
|
if (!field_sqr(group, n0, n6, ctx))
|
||
|
goto end;
|
||
|
if (!field_sqr(group, n4, n5, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, n3, n1, n4, ctx))
|
||
|
goto end;
|
||
|
if (!BN_mod_sub_quick(&r->X, n0, n3, p))
|
||
|
goto end;
|
||
|
/* X_r = n6^2 - n5^2 * 'n7' */
|
||
|
|
||
|
/* 'n9' */
|
||
|
if (!BN_mod_lshift1_quick(n0, &r->X, p))
|
||
|
goto end;
|
||
|
if (!BN_mod_sub_quick(n0, n3, n0, p))
|
||
|
goto end;
|
||
|
/* n9 = n5^2 * 'n7' - 2 * X_r */
|
||
|
|
||
|
/* Y_r */
|
||
|
if (!field_mul(group, n0, n0, n6, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, n5, n4, n5, ctx))
|
||
|
goto end; /* now n5 is n5^3 */
|
||
|
if (!field_mul(group, n1, n2, n5, ctx))
|
||
|
goto end;
|
||
|
if (!BN_mod_sub_quick(n0, n0, n1, p))
|
||
|
goto end;
|
||
|
if (BN_is_odd(n0))
|
||
|
if (!BN_add(n0, n0, p))
|
||
|
goto end;
|
||
|
/* now 0 <= n0 < 2*p, and n0 is even */
|
||
|
if (!BN_rshift1(&r->Y, n0))
|
||
|
goto end;
|
||
|
/* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
end:
|
||
|
if (ctx) /* otherwise we already called BN_CTX_end */
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_dbl(const EC_GROUP * group, EC_POINT * r, const EC_POINT * a, BN_CTX * ctx)
|
||
|
{
|
||
|
int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
const BIGNUM *p;
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *n0, *n1, *n2, *n3;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (EC_POINT_is_at_infinity(group, a) > 0) {
|
||
|
BN_zero(&r->Z);
|
||
|
r->Z_is_one = 0;
|
||
|
return 1;
|
||
|
}
|
||
|
field_mul = group->meth->field_mul;
|
||
|
field_sqr = group->meth->field_sqr;
|
||
|
p = &group->field;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((n0 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((n1 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((n2 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((n3 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
/*
|
||
|
* Note that in this function we must not read components of 'a' once
|
||
|
* we have written the corresponding components of 'r'. ('r' might
|
||
|
* the same as 'a'.)
|
||
|
*/
|
||
|
|
||
|
/* n1 */
|
||
|
if (a->Z_is_one) {
|
||
|
if (!field_sqr(group, n0, &a->X, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_lshift1_quick(n1, n0, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(n0, n0, n1, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(n1, n0, &group->a, p))
|
||
|
goto err;
|
||
|
/* n1 = 3 * X_a^2 + a_curve */
|
||
|
} else if (group->a_is_minus3) {
|
||
|
if (!field_sqr(group, n1, &a->Z, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(n0, &a->X, n1, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_sub_quick(n2, &a->X, n1, p))
|
||
|
goto err;
|
||
|
if (!field_mul(group, n1, n0, n2, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_lshift1_quick(n0, n1, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(n1, n0, n1, p))
|
||
|
goto err;
|
||
|
/*
|
||
|
* n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2) = 3 * X_a^2 - 3 *
|
||
|
* Z_a^4
|
||
|
*/
|
||
|
} else {
|
||
|
if (!field_sqr(group, n0, &a->X, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_lshift1_quick(n1, n0, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(n0, n0, n1, p))
|
||
|
goto err;
|
||
|
if (!field_sqr(group, n1, &a->Z, ctx))
|
||
|
goto err;
|
||
|
if (!field_sqr(group, n1, n1, ctx))
|
||
|
goto err;
|
||
|
if (!field_mul(group, n1, n1, &group->a, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(n1, n1, n0, p))
|
||
|
goto err;
|
||
|
/* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
|
||
|
}
|
||
|
|
||
|
/* Z_r */
|
||
|
if (a->Z_is_one) {
|
||
|
if (!BN_copy(n0, &a->Y))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!field_mul(group, n0, &a->Y, &a->Z, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
if (!BN_mod_lshift1_quick(&r->Z, n0, p))
|
||
|
goto err;
|
||
|
r->Z_is_one = 0;
|
||
|
/* Z_r = 2 * Y_a * Z_a */
|
||
|
|
||
|
/* n2 */
|
||
|
if (!field_sqr(group, n3, &a->Y, ctx))
|
||
|
goto err;
|
||
|
if (!field_mul(group, n2, &a->X, n3, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_lshift_quick(n2, n2, 2, p))
|
||
|
goto err;
|
||
|
/* n2 = 4 * X_a * Y_a^2 */
|
||
|
|
||
|
/* X_r */
|
||
|
if (!BN_mod_lshift1_quick(n0, n2, p))
|
||
|
goto err;
|
||
|
if (!field_sqr(group, &r->X, n1, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_sub_quick(&r->X, &r->X, n0, p))
|
||
|
goto err;
|
||
|
/* X_r = n1^2 - 2 * n2 */
|
||
|
|
||
|
/* n3 */
|
||
|
if (!field_sqr(group, n0, n3, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_lshift_quick(n3, n0, 3, p))
|
||
|
goto err;
|
||
|
/* n3 = 8 * Y_a^4 */
|
||
|
|
||
|
/* Y_r */
|
||
|
if (!BN_mod_sub_quick(n0, n2, &r->X, p))
|
||
|
goto err;
|
||
|
if (!field_mul(group, n0, n1, n0, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_sub_quick(&r->Y, n0, n3, p))
|
||
|
goto err;
|
||
|
/* Y_r = n1 * (n2 - X_r) - n3 */
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_invert(const EC_GROUP * group, EC_POINT * point, BN_CTX * ctx)
|
||
|
{
|
||
|
if (EC_POINT_is_at_infinity(group, point) > 0 || BN_is_zero(&point->Y))
|
||
|
/* point is its own inverse */
|
||
|
return 1;
|
||
|
|
||
|
return BN_usub(&point->Y, &group->field, &point->Y);
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_is_at_infinity(const EC_GROUP * group, const EC_POINT * point)
|
||
|
{
|
||
|
return BN_is_zero(&point->Z);
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_is_on_curve(const EC_GROUP * group, const EC_POINT * point, BN_CTX * ctx)
|
||
|
{
|
||
|
int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
const BIGNUM *p;
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *rh, *tmp, *Z4, *Z6;
|
||
|
int ret = -1;
|
||
|
|
||
|
if (EC_POINT_is_at_infinity(group, point) > 0)
|
||
|
return 1;
|
||
|
|
||
|
field_mul = group->meth->field_mul;
|
||
|
field_sqr = group->meth->field_sqr;
|
||
|
p = &group->field;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return -1;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((rh = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((tmp = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((Z4 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((Z6 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
/*
|
||
|
* 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'.
|
||
|
*/
|
||
|
|
||
|
/* rh := X^2 */
|
||
|
if (!field_sqr(group, rh, &point->X, ctx))
|
||
|
goto err;
|
||
|
|
||
|
if (!point->Z_is_one) {
|
||
|
if (!field_sqr(group, tmp, &point->Z, ctx))
|
||
|
goto err;
|
||
|
if (!field_sqr(group, Z4, tmp, ctx))
|
||
|
goto err;
|
||
|
if (!field_mul(group, Z6, Z4, tmp, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* rh := (rh + a*Z^4)*X */
|
||
|
if (group->a_is_minus3) {
|
||
|
if (!BN_mod_lshift1_quick(tmp, Z4, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(tmp, tmp, Z4, p))
|
||
|
goto err;
|
||
|
if (!BN_mod_sub_quick(rh, rh, tmp, p))
|
||
|
goto err;
|
||
|
if (!field_mul(group, rh, rh, &point->X, ctx))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!field_mul(group, tmp, Z4, &group->a, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(rh, rh, tmp, p))
|
||
|
goto err;
|
||
|
if (!field_mul(group, rh, rh, &point->X, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
/* rh := rh + b*Z^6 */
|
||
|
if (!field_mul(group, tmp, &group->b, Z6, ctx))
|
||
|
goto err;
|
||
|
if (!BN_mod_add_quick(rh, rh, tmp, p))
|
||
|
goto err;
|
||
|
} else {
|
||
|
/* point->Z_is_one */
|
||
|
|
||
|
/* rh := (rh + a)*X */
|
||
|
if (!BN_mod_add_quick(rh, rh, &group->a, p))
|
||
|
goto err;
|
||
|
if (!field_mul(group, rh, rh, &point->X, ctx))
|
||
|
goto err;
|
||
|
/* rh := rh + b */
|
||
|
if (!BN_mod_add_quick(rh, rh, &group->b, p))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
/* 'lh' := Y^2 */
|
||
|
if (!field_sqr(group, tmp, &point->Y, ctx))
|
||
|
goto err;
|
||
|
|
||
|
ret = (0 == BN_ucmp(tmp, rh));
|
||
|
|
||
|
err:
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_cmp(const EC_GROUP * group, const EC_POINT * a, const EC_POINT * b, BN_CTX * ctx)
|
||
|
{
|
||
|
/*
|
||
|
* return values: -1 error 0 equal (in affine coordinates) 1
|
||
|
* not equal
|
||
|
*/
|
||
|
|
||
|
int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
|
||
|
const BIGNUM *tmp1_, *tmp2_;
|
||
|
int ret = -1;
|
||
|
|
||
|
if (EC_POINT_is_at_infinity(group, a) > 0) {
|
||
|
return EC_POINT_is_at_infinity(group, b) > 0 ? 0 : 1;
|
||
|
}
|
||
|
if (EC_POINT_is_at_infinity(group, b) > 0)
|
||
|
return 1;
|
||
|
|
||
|
if (a->Z_is_one && b->Z_is_one) {
|
||
|
return ((BN_cmp(&a->X, &b->X) == 0) && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
|
||
|
}
|
||
|
field_mul = group->meth->field_mul;
|
||
|
field_sqr = group->meth->field_sqr;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return -1;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((tmp1 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((tmp2 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((Za23 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
if ((Zb23 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto end;
|
||
|
|
||
|
/*
|
||
|
* 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).
|
||
|
*/
|
||
|
|
||
|
if (!b->Z_is_one) {
|
||
|
if (!field_sqr(group, Zb23, &b->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, tmp1, &a->X, Zb23, ctx))
|
||
|
goto end;
|
||
|
tmp1_ = tmp1;
|
||
|
} else
|
||
|
tmp1_ = &a->X;
|
||
|
if (!a->Z_is_one) {
|
||
|
if (!field_sqr(group, Za23, &a->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, tmp2, &b->X, Za23, ctx))
|
||
|
goto end;
|
||
|
tmp2_ = tmp2;
|
||
|
} else
|
||
|
tmp2_ = &b->X;
|
||
|
|
||
|
/* compare X_a*Z_b^2 with X_b*Z_a^2 */
|
||
|
if (BN_cmp(tmp1_, tmp2_) != 0) {
|
||
|
ret = 1; /* points differ */
|
||
|
goto end;
|
||
|
}
|
||
|
if (!b->Z_is_one) {
|
||
|
if (!field_mul(group, Zb23, Zb23, &b->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, tmp1, &a->Y, Zb23, ctx))
|
||
|
goto end;
|
||
|
/* tmp1_ = tmp1 */
|
||
|
} else
|
||
|
tmp1_ = &a->Y;
|
||
|
if (!a->Z_is_one) {
|
||
|
if (!field_mul(group, Za23, Za23, &a->Z, ctx))
|
||
|
goto end;
|
||
|
if (!field_mul(group, tmp2, &b->Y, Za23, ctx))
|
||
|
goto end;
|
||
|
/* tmp2_ = tmp2 */
|
||
|
} else
|
||
|
tmp2_ = &b->Y;
|
||
|
|
||
|
/* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
|
||
|
if (BN_cmp(tmp1_, tmp2_) != 0) {
|
||
|
ret = 1; /* points differ */
|
||
|
goto end;
|
||
|
}
|
||
|
/* points are equal */
|
||
|
ret = 0;
|
||
|
|
||
|
end:
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_make_affine(const EC_GROUP * group, EC_POINT * point, BN_CTX * ctx)
|
||
|
{
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *x, *y;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (point->Z_is_one || EC_POINT_is_at_infinity(group, point) > 0)
|
||
|
return 1;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((x = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((y = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
if (!EC_POINT_get_affine_coordinates_GFp(group, point, x, y, ctx))
|
||
|
goto err;
|
||
|
if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx))
|
||
|
goto err;
|
||
|
if (!point->Z_is_one) {
|
||
|
ECerror(ERR_R_INTERNAL_ERROR);
|
||
|
goto err;
|
||
|
}
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_points_make_affine(const EC_GROUP * group, size_t num, EC_POINT * points[], BN_CTX * ctx)
|
||
|
{
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
BIGNUM *tmp0, *tmp1;
|
||
|
size_t pow2 = 0;
|
||
|
BIGNUM **heap = NULL;
|
||
|
size_t i;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (num == 0)
|
||
|
return 1;
|
||
|
|
||
|
if (ctx == NULL) {
|
||
|
ctx = new_ctx = BN_CTX_new();
|
||
|
if (ctx == NULL)
|
||
|
return 0;
|
||
|
}
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((tmp0 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((tmp1 = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
/*
|
||
|
* Before converting the individual points, compute inverses of all Z
|
||
|
* values. Modular inversion is rather slow, but luckily we can do
|
||
|
* with a single explicit inversion, plus about 3 multiplications per
|
||
|
* input value.
|
||
|
*/
|
||
|
|
||
|
pow2 = 1;
|
||
|
while (num > pow2)
|
||
|
pow2 <<= 1;
|
||
|
/*
|
||
|
* Now pow2 is the smallest power of 2 satifsying pow2 >= num. We
|
||
|
* need twice that.
|
||
|
*/
|
||
|
pow2 <<= 1;
|
||
|
|
||
|
heap = reallocarray(NULL, pow2, sizeof heap[0]);
|
||
|
if (heap == NULL)
|
||
|
goto err;
|
||
|
|
||
|
/*
|
||
|
* The array is used as a binary tree, exactly as in heapsort:
|
||
|
*
|
||
|
* heap[1] heap[2] heap[3] heap[4] heap[5]
|
||
|
* heap[6] heap[7] heap[8]heap[9] heap[10]heap[11]
|
||
|
* heap[12]heap[13] heap[14] heap[15]
|
||
|
*
|
||
|
* We put the Z's in the last line; then we set each other node to the
|
||
|
* product of its two child-nodes (where empty or 0 entries are
|
||
|
* treated as ones); then we invert heap[1]; then we invert each
|
||
|
* other node by replacing it by the product of its parent (after
|
||
|
* inversion) and its sibling (before inversion).
|
||
|
*/
|
||
|
heap[0] = NULL;
|
||
|
for (i = pow2 / 2 - 1; i > 0; i--)
|
||
|
heap[i] = NULL;
|
||
|
for (i = 0; i < num; i++)
|
||
|
heap[pow2 / 2 + i] = &points[i]->Z;
|
||
|
for (i = pow2 / 2 + num; i < pow2; i++)
|
||
|
heap[i] = NULL;
|
||
|
|
||
|
/* set each node to the product of its children */
|
||
|
for (i = pow2 / 2 - 1; i > 0; i--) {
|
||
|
heap[i] = BN_new();
|
||
|
if (heap[i] == NULL)
|
||
|
goto err;
|
||
|
|
||
|
if (heap[2 * i] != NULL) {
|
||
|
if ((heap[2 * i + 1] == NULL) || BN_is_zero(heap[2 * i + 1])) {
|
||
|
if (!BN_copy(heap[i], heap[2 * i]))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (BN_is_zero(heap[2 * i])) {
|
||
|
if (!BN_copy(heap[i], heap[2 * i + 1]))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!group->meth->field_mul(group, heap[i],
|
||
|
heap[2 * i], heap[2 * i + 1], ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
|
||
|
/* invert heap[1] */
|
||
|
if (!BN_is_zero(heap[1])) {
|
||
|
if (!BN_mod_inverse_ct(heap[1], heap[1], &group->field, ctx)) {
|
||
|
ECerror(ERR_R_BN_LIB);
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
if (group->meth->field_encode != 0) {
|
||
|
/*
|
||
|
* in the Montgomery case, we just turned R*H (representing
|
||
|
* H) into 1/(R*H), but we need R*(1/H) (representing
|
||
|
* 1/H); i.e. we have need to multiply by the Montgomery
|
||
|
* factor twice
|
||
|
*/
|
||
|
if (!group->meth->field_encode(group, heap[1], heap[1], ctx))
|
||
|
goto err;
|
||
|
if (!group->meth->field_encode(group, heap[1], heap[1], ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
/* set other heap[i]'s to their inverses */
|
||
|
for (i = 2; i < pow2 / 2 + num; i += 2) {
|
||
|
/* i is even */
|
||
|
if ((heap[i + 1] != NULL) && !BN_is_zero(heap[i + 1])) {
|
||
|
if (!group->meth->field_mul(group, tmp0, heap[i / 2], heap[i + 1], ctx))
|
||
|
goto err;
|
||
|
if (!group->meth->field_mul(group, tmp1, heap[i / 2], heap[i], ctx))
|
||
|
goto err;
|
||
|
if (!BN_copy(heap[i], tmp0))
|
||
|
goto err;
|
||
|
if (!BN_copy(heap[i + 1], tmp1))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!BN_copy(heap[i], heap[i / 2]))
|
||
|
goto err;
|
||
|
}
|
||
|
}
|
||
|
|
||
|
/*
|
||
|
* we have replaced all non-zero Z's by their inverses, now fix up
|
||
|
* all the points
|
||
|
*/
|
||
|
for (i = 0; i < num; i++) {
|
||
|
EC_POINT *p = points[i];
|
||
|
|
||
|
if (!BN_is_zero(&p->Z)) {
|
||
|
/* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
|
||
|
|
||
|
if (!group->meth->field_sqr(group, tmp1, &p->Z, ctx))
|
||
|
goto err;
|
||
|
if (!group->meth->field_mul(group, &p->X, &p->X, tmp1, ctx))
|
||
|
goto err;
|
||
|
|
||
|
if (!group->meth->field_mul(group, tmp1, tmp1, &p->Z, ctx))
|
||
|
goto err;
|
||
|
if (!group->meth->field_mul(group, &p->Y, &p->Y, tmp1, ctx))
|
||
|
goto err;
|
||
|
|
||
|
if (group->meth->field_set_to_one != 0) {
|
||
|
if (!group->meth->field_set_to_one(group, &p->Z, ctx))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!BN_one(&p->Z))
|
||
|
goto err;
|
||
|
}
|
||
|
p->Z_is_one = 1;
|
||
|
}
|
||
|
}
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
if (heap != NULL) {
|
||
|
/*
|
||
|
* heap[pow2/2] .. heap[pow2-1] have not been allocated
|
||
|
* locally!
|
||
|
*/
|
||
|
for (i = pow2 / 2 - 1; i > 0; i--) {
|
||
|
BN_clear_free(heap[i]);
|
||
|
}
|
||
|
free(heap);
|
||
|
}
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_field_mul(const EC_GROUP * group, BIGNUM * r, const BIGNUM * a, const BIGNUM * b, BN_CTX * ctx)
|
||
|
{
|
||
|
return BN_mod_mul(r, a, b, &group->field, ctx);
|
||
|
}
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_field_sqr(const EC_GROUP * group, BIGNUM * r, const BIGNUM * a, BN_CTX * ctx)
|
||
|
{
|
||
|
return BN_mod_sqr(r, a, &group->field, ctx);
|
||
|
}
|
||
|
|
||
|
/*
|
||
|
* Apply randomization of EC point projective coordinates:
|
||
|
*
|
||
|
* (X, Y, Z) = (lambda^2 * X, lambda^3 * Y, lambda * Z)
|
||
|
*
|
||
|
* where lambda is in the interval [1, group->field).
|
||
|
*/
|
||
|
int
|
||
|
ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p, BN_CTX *ctx)
|
||
|
{
|
||
|
BIGNUM *lambda = NULL;
|
||
|
BIGNUM *tmp = NULL;
|
||
|
int ret = 0;
|
||
|
|
||
|
BN_CTX_start(ctx);
|
||
|
if ((lambda = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((tmp = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
/* Generate lambda in [1, group->field - 1] */
|
||
|
if (!bn_rand_interval(lambda, BN_value_one(), &group->field))
|
||
|
goto err;
|
||
|
|
||
|
if (group->meth->field_encode != NULL &&
|
||
|
!group->meth->field_encode(group, lambda, lambda, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* Z = lambda * Z */
|
||
|
if (!group->meth->field_mul(group, &p->Z, lambda, &p->Z, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* tmp = lambda^2 */
|
||
|
if (!group->meth->field_sqr(group, tmp, lambda, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* X = lambda^2 * X */
|
||
|
if (!group->meth->field_mul(group, &p->X, tmp, &p->X, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* tmp = lambda^3 */
|
||
|
if (!group->meth->field_mul(group, tmp, tmp, lambda, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* Y = lambda^3 * Y */
|
||
|
if (!group->meth->field_mul(group, &p->Y, tmp, &p->Y, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* Disable optimized arithmetics after replacing Z by lambda * Z. */
|
||
|
p->Z_is_one = 0;
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
BN_CTX_end(ctx);
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
|
||
|
#define EC_POINT_BN_set_flags(P, flags) do { \
|
||
|
BN_set_flags(&(P)->X, (flags)); \
|
||
|
BN_set_flags(&(P)->Y, (flags)); \
|
||
|
BN_set_flags(&(P)->Z, (flags)); \
|
||
|
} while(0)
|
||
|
|
||
|
#define EC_POINT_CSWAP(c, a, b, w, t) do { \
|
||
|
if (!BN_swap_ct(c, &(a)->X, &(b)->X, w) || \
|
||
|
!BN_swap_ct(c, &(a)->Y, &(b)->Y, w) || \
|
||
|
!BN_swap_ct(c, &(a)->Z, &(b)->Z, w)) \
|
||
|
goto err; \
|
||
|
t = ((a)->Z_is_one ^ (b)->Z_is_one) & (c); \
|
||
|
(a)->Z_is_one ^= (t); \
|
||
|
(b)->Z_is_one ^= (t); \
|
||
|
} while(0)
|
||
|
|
||
|
/*
|
||
|
* This function computes (in constant time) a point multiplication over the
|
||
|
* EC group.
|
||
|
*
|
||
|
* At a high level, it is Montgomery ladder with conditional swaps.
|
||
|
*
|
||
|
* It performs either a fixed point multiplication
|
||
|
* (scalar * generator)
|
||
|
* when point is NULL, or a variable point multiplication
|
||
|
* (scalar * point)
|
||
|
* when point is not NULL.
|
||
|
*
|
||
|
* scalar should be in the range [0,n) otherwise all constant time bets are off.
|
||
|
*
|
||
|
* NB: This says nothing about EC_POINT_add and EC_POINT_dbl,
|
||
|
* which of course are not constant time themselves.
|
||
|
*
|
||
|
* The product is stored in r.
|
||
|
*
|
||
|
* Returns 1 on success, 0 otherwise.
|
||
|
*/
|
||
|
static int
|
||
|
ec_GFp_simple_mul_ct(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
|
||
|
const EC_POINT *point, BN_CTX *ctx)
|
||
|
{
|
||
|
int i, cardinality_bits, group_top, kbit, pbit, Z_is_one;
|
||
|
EC_POINT *s = NULL;
|
||
|
BIGNUM *k = NULL;
|
||
|
BIGNUM *lambda = NULL;
|
||
|
BIGNUM *cardinality = NULL;
|
||
|
BN_CTX *new_ctx = NULL;
|
||
|
int ret = 0;
|
||
|
|
||
|
if (ctx == NULL && (ctx = new_ctx = BN_CTX_new()) == NULL)
|
||
|
return 0;
|
||
|
|
||
|
BN_CTX_start(ctx);
|
||
|
|
||
|
if ((s = EC_POINT_new(group)) == NULL)
|
||
|
goto err;
|
||
|
|
||
|
if (point == NULL) {
|
||
|
if (!EC_POINT_copy(s, group->generator))
|
||
|
goto err;
|
||
|
} else {
|
||
|
if (!EC_POINT_copy(s, point))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
EC_POINT_BN_set_flags(s, BN_FLG_CONSTTIME);
|
||
|
|
||
|
if ((cardinality = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((lambda = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if ((k = BN_CTX_get(ctx)) == NULL)
|
||
|
goto err;
|
||
|
if (!BN_mul(cardinality, &group->order, &group->cofactor, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/*
|
||
|
* Group cardinalities are often on a word boundary.
|
||
|
* So when we pad the scalar, some timing diff might
|
||
|
* pop if it needs to be expanded due to carries.
|
||
|
* So expand ahead of time.
|
||
|
*/
|
||
|
cardinality_bits = BN_num_bits(cardinality);
|
||
|
group_top = cardinality->top;
|
||
|
if ((bn_wexpand(k, group_top + 2) == NULL) ||
|
||
|
(bn_wexpand(lambda, group_top + 2) == NULL))
|
||
|
goto err;
|
||
|
|
||
|
if (!BN_copy(k, scalar))
|
||
|
goto err;
|
||
|
|
||
|
BN_set_flags(k, BN_FLG_CONSTTIME);
|
||
|
|
||
|
if (BN_num_bits(k) > cardinality_bits || BN_is_negative(k)) {
|
||
|
/*
|
||
|
* This is an unusual input, and we don't guarantee
|
||
|
* constant-timeness
|
||
|
*/
|
||
|
if (!BN_nnmod(k, k, cardinality, ctx))
|
||
|
goto err;
|
||
|
}
|
||
|
|
||
|
if (!BN_add(lambda, k, cardinality))
|
||
|
goto err;
|
||
|
BN_set_flags(lambda, BN_FLG_CONSTTIME);
|
||
|
if (!BN_add(k, lambda, cardinality))
|
||
|
goto err;
|
||
|
/*
|
||
|
* lambda := scalar + cardinality
|
||
|
* k := scalar + 2*cardinality
|
||
|
*/
|
||
|
kbit = BN_is_bit_set(lambda, cardinality_bits);
|
||
|
if (!BN_swap_ct(kbit, k, lambda, group_top + 2))
|
||
|
goto err;
|
||
|
|
||
|
group_top = group->field.top;
|
||
|
if ((bn_wexpand(&s->X, group_top) == NULL) ||
|
||
|
(bn_wexpand(&s->Y, group_top) == NULL) ||
|
||
|
(bn_wexpand(&s->Z, group_top) == NULL) ||
|
||
|
(bn_wexpand(&r->X, group_top) == NULL) ||
|
||
|
(bn_wexpand(&r->Y, group_top) == NULL) ||
|
||
|
(bn_wexpand(&r->Z, group_top) == NULL))
|
||
|
goto err;
|
||
|
|
||
|
/*
|
||
|
* Apply coordinate blinding for EC_POINT if the underlying EC_METHOD
|
||
|
* implements it.
|
||
|
*/
|
||
|
if (!ec_point_blind_coordinates(group, s, ctx))
|
||
|
goto err;
|
||
|
|
||
|
/* top bit is a 1, in a fixed pos */
|
||
|
if (!EC_POINT_copy(r, s))
|
||
|
goto err;
|
||
|
|
||
|
EC_POINT_BN_set_flags(r, BN_FLG_CONSTTIME);
|
||
|
|
||
|
if (!EC_POINT_dbl(group, s, s, ctx))
|
||
|
goto err;
|
||
|
|
||
|
pbit = 0;
|
||
|
|
||
|
/*
|
||
|
* The ladder step, with branches, is
|
||
|
*
|
||
|
* k[i] == 0: S = add(R, S), R = dbl(R)
|
||
|
* k[i] == 1: R = add(S, R), S = dbl(S)
|
||
|
*
|
||
|
* Swapping R, S conditionally on k[i] leaves you with state
|
||
|
*
|
||
|
* k[i] == 0: T, U = R, S
|
||
|
* k[i] == 1: T, U = S, R
|
||
|
*
|
||
|
* Then perform the ECC ops.
|
||
|
*
|
||
|
* U = add(T, U)
|
||
|
* T = dbl(T)
|
||
|
*
|
||
|
* Which leaves you with state
|
||
|
*
|
||
|
* k[i] == 0: U = add(R, S), T = dbl(R)
|
||
|
* k[i] == 1: U = add(S, R), T = dbl(S)
|
||
|
*
|
||
|
* Swapping T, U conditionally on k[i] leaves you with state
|
||
|
*
|
||
|
* k[i] == 0: R, S = T, U
|
||
|
* k[i] == 1: R, S = U, T
|
||
|
*
|
||
|
* Which leaves you with state
|
||
|
*
|
||
|
* k[i] == 0: S = add(R, S), R = dbl(R)
|
||
|
* k[i] == 1: R = add(S, R), S = dbl(S)
|
||
|
*
|
||
|
* So we get the same logic, but instead of a branch it's a
|
||
|
* conditional swap, followed by ECC ops, then another conditional swap.
|
||
|
*
|
||
|
* Optimization: The end of iteration i and start of i-1 looks like
|
||
|
*
|
||
|
* ...
|
||
|
* CSWAP(k[i], R, S)
|
||
|
* ECC
|
||
|
* CSWAP(k[i], R, S)
|
||
|
* (next iteration)
|
||
|
* CSWAP(k[i-1], R, S)
|
||
|
* ECC
|
||
|
* CSWAP(k[i-1], R, S)
|
||
|
* ...
|
||
|
*
|
||
|
* So instead of two contiguous swaps, you can merge the condition
|
||
|
* bits and do a single swap.
|
||
|
*
|
||
|
* k[i] k[i-1] Outcome
|
||
|
* 0 0 No Swap
|
||
|
* 0 1 Swap
|
||
|
* 1 0 Swap
|
||
|
* 1 1 No Swap
|
||
|
*
|
||
|
* This is XOR. pbit tracks the previous bit of k.
|
||
|
*/
|
||
|
|
||
|
for (i = cardinality_bits - 1; i >= 0; i--) {
|
||
|
kbit = BN_is_bit_set(k, i) ^ pbit;
|
||
|
EC_POINT_CSWAP(kbit, r, s, group_top, Z_is_one);
|
||
|
if (!EC_POINT_add(group, s, r, s, ctx))
|
||
|
goto err;
|
||
|
if (!EC_POINT_dbl(group, r, r, ctx))
|
||
|
goto err;
|
||
|
/*
|
||
|
* pbit logic merges this cswap with that of the
|
||
|
* next iteration
|
||
|
*/
|
||
|
pbit ^= kbit;
|
||
|
}
|
||
|
/* one final cswap to move the right value into r */
|
||
|
EC_POINT_CSWAP(pbit, r, s, group_top, Z_is_one);
|
||
|
|
||
|
ret = 1;
|
||
|
|
||
|
err:
|
||
|
EC_POINT_free(s);
|
||
|
if (ctx != NULL)
|
||
|
BN_CTX_end(ctx);
|
||
|
BN_CTX_free(new_ctx);
|
||
|
|
||
|
return ret;
|
||
|
}
|
||
|
|
||
|
#undef EC_POINT_BN_set_flags
|
||
|
#undef EC_POINT_CSWAP
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_mul_generator_ct(const EC_GROUP *group, EC_POINT *r,
|
||
|
const BIGNUM *scalar, BN_CTX *ctx)
|
||
|
{
|
||
|
return ec_GFp_simple_mul_ct(group, r, scalar, NULL, ctx);
|
||
|
}
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_mul_single_ct(const EC_GROUP *group, EC_POINT *r,
|
||
|
const BIGNUM *scalar, const EC_POINT *point, BN_CTX *ctx)
|
||
|
{
|
||
|
return ec_GFp_simple_mul_ct(group, r, scalar, point, ctx);
|
||
|
}
|
||
|
|
||
|
int
|
||
|
ec_GFp_simple_mul_double_nonct(const EC_GROUP *group, EC_POINT *r,
|
||
|
const BIGNUM *g_scalar, const BIGNUM *p_scalar, const EC_POINT *point,
|
||
|
BN_CTX *ctx)
|
||
|
{
|
||
|
return ec_wNAF_mul(group, r, g_scalar, 1, &point, &p_scalar, ctx);
|
||
|
}
|