/* 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 #include #include #include #include "internal.h" const EC_METHOD *EC_GFp_simple_method(void) { static const EC_METHOD ret = {EC_FLAGS_DEFAULT_OCT, ec_GFp_simple_group_init, ec_GFp_simple_group_finish, ec_GFp_simple_group_clear_finish, ec_GFp_simple_group_copy, ec_GFp_simple_group_set_curve, ec_GFp_simple_group_get_curve, ec_GFp_simple_group_get_degree, ec_GFp_simple_group_check_discriminant, ec_GFp_simple_point_init, ec_GFp_simple_point_finish, ec_GFp_simple_point_clear_finish, ec_GFp_simple_point_copy, ec_GFp_simple_point_set_to_infinity, ec_GFp_simple_set_Jprojective_coordinates_GFp, ec_GFp_simple_get_Jprojective_coordinates_GFp, ec_GFp_simple_point_set_affine_coordinates, ec_GFp_simple_point_get_affine_coordinates, 0, 0, 0, ec_GFp_simple_add, ec_GFp_simple_dbl, ec_GFp_simple_invert, ec_GFp_simple_is_at_infinity, ec_GFp_simple_is_on_curve, ec_GFp_simple_cmp, ec_GFp_simple_make_affine, ec_GFp_simple_points_make_affine, 0 /* mul */, 0 /* precompute_mult */, 0 /* have_precompute_mult */, ec_GFp_simple_field_mul, ec_GFp_simple_field_sqr, 0 /* field_div */, 0 /* field_encode */, 0 /* field_decode */, 0 /* field_set_to_one */}; return &ret; } /* 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 ec_GFp_simple_points_make_affine() and * ec_GFp_simple_point_get_affine_coordinates() 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); BN_init(&group->a); BN_init(&group->b); group->a_is_minus3 = 0; return 1; } void ec_GFp_simple_group_finish(EC_GROUP *group) { BN_free(&group->field); BN_free(&group->a); BN_free(&group->b); } void ec_GFp_simple_group_clear_finish(EC_GROUP *group) { BN_clear_free(&group->field); BN_clear_free(&group->a); BN_clear_free(&group->b); } int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src) { if (!BN_copy(&dest->field, &src->field) || !BN_copy(&dest->a, &src->a) || !BN_copy(&dest->b, &src->b)) { return 0; } dest->a_is_minus3 = src->a_is_minus3; return 1; } int ec_GFp_simple_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { int ret = 0; BN_CTX *new_ctx = NULL; BIGNUM *tmp_a; /* p must be a prime > 3 */ if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_group_set_curve, EC_R_INVALID_FIELD); return 0; } if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) return 0; } BN_CTX_start(ctx); tmp_a = BN_CTX_get(ctx); if (tmp_a == NULL) goto err; /* group->field */ if (!BN_copy(&group->field, p)) goto err; BN_set_negative(&group->field, 0); /* group->a */ if (!BN_nnmod(tmp_a, a, p, ctx)) goto err; if (group->meth->field_encode) { if (!group->meth->field_encode(group, &group->a, tmp_a, ctx)) goto err; } else if (!BN_copy(&group->a, tmp_a)) goto err; /* group->b */ if (!BN_nnmod(&group->b, b, p, ctx)) goto err; if (group->meth->field_encode) if (!group->meth->field_encode(group, &group->b, &group->b, ctx)) goto err; /* group->a_is_minus3 */ if (!BN_add_word(tmp_a, 3)) goto err; group->a_is_minus3 = (0 == BN_cmp(tmp_a, &group->field)); ret = 1; err: BN_CTX_end(ctx); if (new_ctx != NULL) BN_CTX_free(new_ctx); return ret; } int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b, BN_CTX *ctx) { int ret = 0; BN_CTX *new_ctx = NULL; if (p != NULL) { if (!BN_copy(p, &group->field)) return 0; } if (a != NULL || b != NULL) { if (group->meth->field_decode) { if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) return 0; } if (a != NULL) { if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err; } if (b != NULL) { if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err; } } else { if (a != NULL) { if (!BN_copy(a, &group->a)) goto err; } if (b != NULL) { if (!BN_copy(b, &group->b)) goto err; } } } ret = 1; err: if (new_ctx) BN_CTX_free(new_ctx); return ret; } int ec_GFp_simple_group_get_degree(const EC_GROUP *group) { return BN_num_bits(&group->field); } int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx) { int ret = 0; BIGNUM *a, *b, *order, *tmp_1, *tmp_2; const BIGNUM *p = &group->field; BN_CTX *new_ctx = NULL; if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_group_check_discriminant, ERR_R_MALLOC_FAILURE); goto err; } } BN_CTX_start(ctx); a = BN_CTX_get(ctx); b = BN_CTX_get(ctx); tmp_1 = BN_CTX_get(ctx); tmp_2 = BN_CTX_get(ctx); order = BN_CTX_get(ctx); if (order == 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); if (new_ctx != NULL) 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: if (new_ctx != NULL) 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: if (new_ctx != NULL) 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 */ OPENSSL_PUT_ERROR(EC, ec_GFp_simple_point_set_affine_coordinates, 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)) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_point_get_affine_coordinates, 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); Z = BN_CTX_get(ctx); Z_1 = BN_CTX_get(ctx); Z_2 = BN_CTX_get(ctx); Z_3 = BN_CTX_get(ctx); if (Z_3 == 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(Z_1, Z_, &group->field, ctx)) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_point_get_affine_coordinates, 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); if (new_ctx != NULL) 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)) return EC_POINT_copy(r, b); if (EC_POINT_is_at_infinity(group, b)) 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); n0 = BN_CTX_get(ctx); n1 = BN_CTX_get(ctx); n2 = BN_CTX_get(ctx); n3 = BN_CTX_get(ctx); n4 = BN_CTX_get(ctx); n5 = BN_CTX_get(ctx); n6 = BN_CTX_get(ctx); if (n6 == 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); if (new_ctx != NULL) 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)) { 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); n0 = BN_CTX_get(ctx); n1 = BN_CTX_get(ctx); n2 = BN_CTX_get(ctx); n3 = BN_CTX_get(ctx); if (n3 == 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); if (new_ctx != NULL) 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) || 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 !point->Z_is_one && 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)) 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); rh = BN_CTX_get(ctx); tmp = BN_CTX_get(ctx); Z4 = BN_CTX_get(ctx); Z6 = BN_CTX_get(ctx); if (Z6 == 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); if (new_ctx != NULL) 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)) { return EC_POINT_is_at_infinity(group, b) ? 0 : 1; } if (EC_POINT_is_at_infinity(group, b)) 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); tmp1 = BN_CTX_get(ctx); tmp2 = BN_CTX_get(ctx); Za23 = BN_CTX_get(ctx); Zb23 = BN_CTX_get(ctx); if (Zb23 == 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); if (new_ctx != NULL) 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)) return 1; if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) return 0; } BN_CTX_start(ctx); x = BN_CTX_get(ctx); y = BN_CTX_get(ctx); if (y == 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) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_make_affine, ERR_R_INTERNAL_ERROR); goto err; } ret = 1; err: BN_CTX_end(ctx); if (new_ctx != NULL) 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); tmp0 = BN_CTX_get(ctx); tmp1 = BN_CTX_get(ctx); if (tmp0 == NULL || tmp1 == 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 = OPENSSL_malloc(pow2 * sizeof heap[0]); if (heap == NULL) goto err; /* TODO(bmoeller): There is no reason to use this tree structure. * We should instead proceed sequentially, exactly as in * ec_GFp_nistp_points_make_affine_internal, which makes everything * much simpler. */ /* 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(heap[1], heap[1], &group->field, ctx)) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_points_make_affine, 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 (!BN_is_zero(heap[i])) { 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 + 1], heap[i / 2])) 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); if (new_ctx != NULL) 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--) { if (heap[i] != NULL) BN_clear_free(heap[i]); } OPENSSL_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); }