/* * Copyright 2001-2018 The OpenSSL Project Authors. All Rights Reserved. * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved * * Licensed under the OpenSSL license (the "License"). You may not use * this file except in compliance with the License. You can obtain a copy * in the file LICENSE in the source distribution or at * https://www.openssl.org/source/license.html */ #include #include #include "ec_lcl.h" const EC_METHOD *EC_GFp_simple_method(void) { static const EC_METHOD ret = { EC_FLAGS_DEFAULT_OCT, NID_X9_62_prime_field, 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_group_simple_order_bits, 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 */ ec_key_simple_priv2oct, ec_key_simple_oct2priv, 0, /* set private */ ec_key_simple_generate_key, ec_key_simple_check_key, ec_key_simple_generate_public_key, 0, /* keycopy */ 0, /* keyfinish */ ecdh_simple_compute_key, 0, /* field_inverse_mod_ord */ ec_GFp_simple_blind_coordinates, ec_GFp_simple_ladder_pre, ec_GFp_simple_ladder_step, ec_GFp_simple_ladder_post }; 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) { group->field = BN_new(); group->a = BN_new(); group->b = BN_new(); if (group->field == NULL || group->a == NULL || group->b == NULL) { BN_free(group->field); BN_free(group->a); BN_free(group->b); return 0; } 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)) return 0; if (!BN_copy(dest->a, src->a)) return 0; if (!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)) { ECerr(EC_F_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); 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: 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) { ECerr(EC_F_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); BN_CTX_free(new_ctx); return ret; } int ec_GFp_simple_point_init(EC_POINT *point) { point->X = BN_new(); point->Y = BN_new(); point->Z = BN_new(); point->Z_is_one = 0; if (point->X == NULL || point->Y == NULL || point->Z == NULL) { BN_free(point->X); BN_free(point->Y); BN_free(point->Z); return 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; dest->curve_name = src->curve_name; 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 */ ECerr(EC_F_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)) { ECerr(EC_F_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)) { ECerr(EC_F_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); 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); 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); 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 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); 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); 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) { ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, 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 *tmp, *tmp_Z; BIGNUM **prod_Z = 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); tmp = BN_CTX_get(ctx); tmp_Z = BN_CTX_get(ctx); if (tmp_Z == NULL) goto err; prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0])); if (prod_Z == NULL) goto err; for (i = 0; i < num; i++) { prod_Z[i] = BN_new(); if (prod_Z[i] == NULL) goto err; } /* * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z, * skipping any zero-valued inputs (pretend that they're 1). */ if (!BN_is_zero(points[0]->Z)) { if (!BN_copy(prod_Z[0], points[0]->Z)) goto err; } else { if (group->meth->field_set_to_one != 0) { if (!group->meth->field_set_to_one(group, prod_Z[0], ctx)) goto err; } else { if (!BN_one(prod_Z[0])) goto err; } } for (i = 1; i < num; i++) { if (!BN_is_zero(points[i]->Z)) { if (!group-> meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z, ctx)) goto err; } else { if (!BN_copy(prod_Z[i], prod_Z[i - 1])) goto err; } } /* * Now use a single explicit inversion to replace every non-zero * points[i]->Z by its inverse. */ if (!BN_mod_inverse(tmp, prod_Z[num - 1], group->field, ctx)) { ECerr(EC_F_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 need to * multiply by the Montgomery factor twice. */ if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err; if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err; } for (i = num - 1; i > 0; --i) { /* * Loop invariant: tmp is the product of the inverses of points[0]->Z * .. points[i]->Z (zero-valued inputs skipped). */ if (!BN_is_zero(points[i]->Z)) { /* * Set tmp_Z to the inverse of points[i]->Z (as product of Z * inverses 0 .. i, Z values 0 .. i - 1). */ if (!group-> meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx)) goto err; /* * Update tmp to satisfy the loop invariant for i - 1. */ if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx)) goto err; /* Replace points[i]->Z by its inverse. */ if (!BN_copy(points[i]->Z, tmp_Z)) goto err; } } if (!BN_is_zero(points[0]->Z)) { /* Replace points[0]->Z by its inverse. */ if (!BN_copy(points[0]->Z, tmp)) goto err; } /* Finally, fix up the X and Y coordinates for all 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, tmp, p->Z, ctx)) goto err; if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx)) goto err; if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx)) goto err; if (!group->meth->field_mul(group, p->Y, p->Y, tmp, 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 (prod_Z != NULL) { for (i = 0; i < num; i++) { if (prod_Z[i] == NULL) break; BN_clear_free(prod_Z[i]); } OPENSSL_free(prod_Z); } 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) * lambda = [1,group->field) * */ int ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p, BN_CTX *ctx) { int ret = 0; BIGNUM *lambda = NULL; BIGNUM *temp = NULL; BN_CTX_start(ctx); lambda = BN_CTX_get(ctx); temp = BN_CTX_get(ctx); if (temp == NULL) { ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_MALLOC_FAILURE); goto err; } /* make sure lambda is not zero */ do { if (!BN_priv_rand_range(lambda, group->field)) { ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_BN_LIB); goto err; } } while (BN_is_zero(lambda)); /* if field_encode defined convert between representations */ if (group->meth->field_encode != NULL && !group->meth->field_encode(group, lambda, lambda, ctx)) goto err; if (!group->meth->field_mul(group, p->Z, p->Z, lambda, ctx)) goto err; if (!group->meth->field_sqr(group, temp, lambda, ctx)) goto err; if (!group->meth->field_mul(group, p->X, p->X, temp, ctx)) goto err; if (!group->meth->field_mul(group, temp, temp, lambda, ctx)) goto err; if (!group->meth->field_mul(group, p->Y, p->Y, temp, ctx)) goto err; p->Z_is_one = 0; ret = 1; err: BN_CTX_end(ctx); return ret; } /*- * Set s := p, r := 2p. * * For doubling we use Formula 3 from Izu-Takagi "A fast parallel elliptic curve * multiplication resistant against side channel attacks" appendix, as described * at * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#doubling-dbl-2002-it-2 * * The input point p will be in randomized Jacobian projective coords: * x = X/Z**2, y=Y/Z**3 * * The output points p, s, and r are converted to standard (homogeneous) * projective coords: * x = X/Z, y=Y/Z */ int ec_GFp_simple_ladder_pre(const EC_GROUP *group, EC_POINT *r, EC_POINT *s, EC_POINT *p, BN_CTX *ctx) { BIGNUM *t1, *t2, *t3, *t4, *t5, *t6 = NULL; t1 = r->Z; t2 = r->Y; t3 = s->X; t4 = r->X; t5 = s->Y; t6 = s->Z; /* convert p: (X,Y,Z) -> (XZ,Y,Z**3) */ if (!group->meth->field_mul(group, p->X, p->X, p->Z, ctx) || !group->meth->field_sqr(group, t1, p->Z, ctx) || !group->meth->field_mul(group, p->Z, p->Z, t1, ctx) /* r := 2p */ || !group->meth->field_sqr(group, t2, p->X, ctx) || !group->meth->field_sqr(group, t3, p->Z, ctx) || !group->meth->field_mul(group, t4, t3, group->a, ctx) || !BN_mod_sub_quick(t5, t2, t4, group->field) || !BN_mod_add_quick(t2, t2, t4, group->field) || !group->meth->field_sqr(group, t5, t5, ctx) || !group->meth->field_mul(group, t6, t3, group->b, ctx) || !group->meth->field_mul(group, t1, p->X, p->Z, ctx) || !group->meth->field_mul(group, t4, t1, t6, ctx) || !BN_mod_lshift_quick(t4, t4, 3, group->field) /* r->X coord output */ || !BN_mod_sub_quick(r->X, t5, t4, group->field) || !group->meth->field_mul(group, t1, t1, t2, ctx) || !group->meth->field_mul(group, t2, t3, t6, ctx) || !BN_mod_add_quick(t1, t1, t2, group->field) /* r->Z coord output */ || !BN_mod_lshift_quick(r->Z, t1, 2, group->field) || !EC_POINT_copy(s, p)) return 0; r->Z_is_one = 0; s->Z_is_one = 0; p->Z_is_one = 0; return 1; } /*- * Differential addition-and-doubling using Eq. (8) and (10) from Izu-Takagi * "A fast parallel elliptic curve multiplication resistant against side channel * attacks", as described at * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-ladd-2002-it-3 */ int ec_GFp_simple_ladder_step(const EC_GROUP *group, EC_POINT *r, EC_POINT *s, EC_POINT *p, BN_CTX *ctx) { int ret = 0; BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6, *t7 = NULL; BN_CTX_start(ctx); t0 = BN_CTX_get(ctx); t1 = BN_CTX_get(ctx); t2 = BN_CTX_get(ctx); t3 = BN_CTX_get(ctx); t4 = BN_CTX_get(ctx); t5 = BN_CTX_get(ctx); t6 = BN_CTX_get(ctx); t7 = BN_CTX_get(ctx); if (t7 == NULL || !group->meth->field_mul(group, t0, r->X, s->X, ctx) || !group->meth->field_mul(group, t1, r->Z, s->Z, ctx) || !group->meth->field_mul(group, t2, r->X, s->Z, ctx) || !group->meth->field_mul(group, t3, r->Z, s->X, ctx) || !group->meth->field_mul(group, t4, group->a, t1, ctx) || !BN_mod_sub_quick(t4, t0, t4, group->field) || !BN_mod_add_quick(t5, t3, t2, group->field) || !group->meth->field_sqr(group, t4, t4, ctx) || !group->meth->field_mul(group, t5, t1, t5, ctx) || !BN_mod_lshift_quick(t0, group->b, 2, group->field) || !group->meth->field_mul(group, t5, t0, t5, ctx) || !BN_mod_sub_quick(t5, t4, t5, group->field) /* s->X coord output */ || !group->meth->field_mul(group, s->X, t5, p->Z, ctx) || !BN_mod_sub_quick(t3, t2, t3, group->field) || !group->meth->field_sqr(group, t3, t3, ctx) /* s->Z coord output */ || !group->meth->field_mul(group, s->Z, t3, p->X, ctx) || !group->meth->field_sqr(group, t2, r->X, ctx) || !group->meth->field_sqr(group, t4, r->Z, ctx) || !group->meth->field_mul(group, t1, t4, group->a, ctx) || !BN_mod_add_quick(t6, r->X, r->Z, group->field) || !group->meth->field_sqr(group, t6, t6, ctx) || !BN_mod_sub_quick(t6, t6, t2, group->field) || !BN_mod_sub_quick(t6, t6, t4, group->field) || !BN_mod_sub_quick(t7, t2, t1, group->field) || !group->meth->field_sqr(group, t7, t7, ctx) || !group->meth->field_mul(group, t5, t4, t6, ctx) || !group->meth->field_mul(group, t5, t0, t5, ctx) /* r->X coord output */ || !BN_mod_sub_quick(r->X, t7, t5, group->field) || !BN_mod_add_quick(t2, t2, t1, group->field) || !group->meth->field_sqr(group, t5, t4, ctx) || !group->meth->field_mul(group, t5, t5, t0, ctx) || !group->meth->field_mul(group, t6, t6, t2, ctx) || !BN_mod_lshift1_quick(t6, t6, group->field) /* r->Z coord output */ || !BN_mod_add_quick(r->Z, t5, t6, group->field)) goto err; ret = 1; err: BN_CTX_end(ctx); return ret; } /*- * Recovers the y-coordinate of r using Eq. (8) from Brier-Joye, "Weierstrass * Elliptic Curves and Side-Channel Attacks", modified to work in projective * coordinates and return r in Jacobian projective coordinates. * * X4 = two*Y1*X2*Z3*Z2*Z1; * Y4 = two*b*Z3*SQR(Z2*Z1) + Z3*(a*Z2*Z1+X1*X2)*(X1*Z2+X2*Z1) - X3*SQR(X1*Z2-X2*Z1); * Z4 = two*Y1*Z3*SQR(Z2)*Z1; * * Z4 != 0 because: * - Z1==0 implies p is at infinity, which would have caused an early exit in * the caller; * - Z2==0 implies r is at infinity (handled by the BN_is_zero(r->Z) branch); * - Z3==0 implies s is at infinity (handled by the BN_is_zero(s->Z) branch); * - Y1==0 implies p has order 2, so either r or s are infinity and handled by * one of the BN_is_zero(...) branches. */ int ec_GFp_simple_ladder_post(const EC_GROUP *group, EC_POINT *r, EC_POINT *s, EC_POINT *p, BN_CTX *ctx) { int ret = 0; BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL; if (BN_is_zero(r->Z)) return EC_POINT_set_to_infinity(group, r); if (BN_is_zero(s->Z)) { /* (X,Y,Z) -> (XZ,YZ**2,Z) */ if (!group->meth->field_mul(group, r->X, p->X, p->Z, ctx) || !group->meth->field_sqr(group, r->Z, p->Z, ctx) || !group->meth->field_mul(group, r->Y, p->Y, r->Z, ctx) || !BN_copy(r->Z, p->Z) || !EC_POINT_invert(group, r, ctx)) return 0; return 1; } BN_CTX_start(ctx); t0 = BN_CTX_get(ctx); t1 = BN_CTX_get(ctx); t2 = BN_CTX_get(ctx); t3 = BN_CTX_get(ctx); t4 = BN_CTX_get(ctx); t5 = BN_CTX_get(ctx); t6 = BN_CTX_get(ctx); if (t6 == NULL || !BN_mod_lshift1_quick(t0, p->Y, group->field) || !group->meth->field_mul(group, t1, r->X, p->Z, ctx) || !group->meth->field_mul(group, t2, r->Z, s->Z, ctx) || !group->meth->field_mul(group, t2, t1, t2, ctx) || !group->meth->field_mul(group, t3, t2, t0, ctx) || !group->meth->field_mul(group, t2, r->Z, p->Z, ctx) || !group->meth->field_sqr(group, t4, t2, ctx) || !BN_mod_lshift1_quick(t5, group->b, group->field) || !group->meth->field_mul(group, t4, t4, t5, ctx) || !group->meth->field_mul(group, t6, t2, group->a, ctx) || !group->meth->field_mul(group, t5, r->X, p->X, ctx) || !BN_mod_add_quick(t5, t6, t5, group->field) || !group->meth->field_mul(group, t6, r->Z, p->X, ctx) || !BN_mod_add_quick(t2, t6, t1, group->field) || !group->meth->field_mul(group, t5, t5, t2, ctx) || !BN_mod_sub_quick(t6, t6, t1, group->field) || !group->meth->field_sqr(group, t6, t6, ctx) || !group->meth->field_mul(group, t6, t6, s->X, ctx) || !BN_mod_add_quick(t4, t5, t4, group->field) || !group->meth->field_mul(group, t4, t4, s->Z, ctx) || !BN_mod_sub_quick(t4, t4, t6, group->field) || !group->meth->field_sqr(group, t5, r->Z, ctx) || !group->meth->field_mul(group, r->Z, p->Z, s->Z, ctx) || !group->meth->field_mul(group, r->Z, t5, r->Z, ctx) || !group->meth->field_mul(group, r->Z, r->Z, t0, ctx) /* t3 := X, t4 := Y */ /* (X,Y,Z) -> (XZ,YZ**2,Z) */ || !group->meth->field_mul(group, r->X, t3, r->Z, ctx) || !group->meth->field_sqr(group, t3, r->Z, ctx) || !group->meth->field_mul(group, r->Y, t4, t3, ctx)) goto err; ret = 1; err: BN_CTX_end(ctx); return ret; }