2 * Copyright 2001-2018 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved
5 * Licensed under the OpenSSL license (the "License"). You may not use
6 * this file except in compliance with the License. You can obtain a copy
7 * in the file LICENSE in the source distribution or at
8 * https://www.openssl.org/source/license.html
11 #include <openssl/err.h>
12 #include <openssl/symhacks.h>
16 const EC_METHOD *EC_GFp_simple_method(void)
18 static const EC_METHOD ret = {
20 NID_X9_62_prime_field,
21 ec_GFp_simple_group_init,
22 ec_GFp_simple_group_finish,
23 ec_GFp_simple_group_clear_finish,
24 ec_GFp_simple_group_copy,
25 ec_GFp_simple_group_set_curve,
26 ec_GFp_simple_group_get_curve,
27 ec_GFp_simple_group_get_degree,
28 ec_group_simple_order_bits,
29 ec_GFp_simple_group_check_discriminant,
30 ec_GFp_simple_point_init,
31 ec_GFp_simple_point_finish,
32 ec_GFp_simple_point_clear_finish,
33 ec_GFp_simple_point_copy,
34 ec_GFp_simple_point_set_to_infinity,
35 ec_GFp_simple_set_Jprojective_coordinates_GFp,
36 ec_GFp_simple_get_Jprojective_coordinates_GFp,
37 ec_GFp_simple_point_set_affine_coordinates,
38 ec_GFp_simple_point_get_affine_coordinates,
43 ec_GFp_simple_is_at_infinity,
44 ec_GFp_simple_is_on_curve,
46 ec_GFp_simple_make_affine,
47 ec_GFp_simple_points_make_affine,
49 0 /* precompute_mult */ ,
50 0 /* have_precompute_mult */ ,
51 ec_GFp_simple_field_mul,
52 ec_GFp_simple_field_sqr,
54 0 /* field_encode */ ,
55 0 /* field_decode */ ,
56 0, /* field_set_to_one */
57 ec_key_simple_priv2oct,
58 ec_key_simple_oct2priv,
60 ec_key_simple_generate_key,
61 ec_key_simple_check_key,
62 ec_key_simple_generate_public_key,
65 ecdh_simple_compute_key,
66 0, /* field_inverse_mod_ord */
67 ec_GFp_simple_blind_coordinates,
68 ec_GFp_simple_ladder_pre,
69 ec_GFp_simple_ladder_step,
70 ec_GFp_simple_ladder_post
77 * Most method functions in this file are designed to work with
78 * non-trivial representations of field elements if necessary
79 * (see ecp_mont.c): while standard modular addition and subtraction
80 * are used, the field_mul and field_sqr methods will be used for
81 * multiplication, and field_encode and field_decode (if defined)
82 * will be used for converting between representations.
84 * Functions ec_GFp_simple_points_make_affine() and
85 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
86 * that if a non-trivial representation is used, it is a Montgomery
87 * representation (i.e. 'encoding' means multiplying by some factor R).
90 int ec_GFp_simple_group_init(EC_GROUP *group)
92 group->field = BN_new();
95 if (group->field == NULL || group->a == NULL || group->b == NULL) {
96 BN_free(group->field);
101 group->a_is_minus3 = 0;
105 void ec_GFp_simple_group_finish(EC_GROUP *group)
107 BN_free(group->field);
112 void ec_GFp_simple_group_clear_finish(EC_GROUP *group)
114 BN_clear_free(group->field);
115 BN_clear_free(group->a);
116 BN_clear_free(group->b);
119 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
121 if (!BN_copy(dest->field, src->field))
123 if (!BN_copy(dest->a, src->a))
125 if (!BN_copy(dest->b, src->b))
128 dest->a_is_minus3 = src->a_is_minus3;
133 int ec_GFp_simple_group_set_curve(EC_GROUP *group,
134 const BIGNUM *p, const BIGNUM *a,
135 const BIGNUM *b, BN_CTX *ctx)
138 BN_CTX *new_ctx = NULL;
141 /* p must be a prime > 3 */
142 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
143 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD);
148 ctx = new_ctx = BN_CTX_new();
154 tmp_a = BN_CTX_get(ctx);
159 if (!BN_copy(group->field, p))
161 BN_set_negative(group->field, 0);
164 if (!BN_nnmod(tmp_a, a, p, ctx))
166 if (group->meth->field_encode) {
167 if (!group->meth->field_encode(group, group->a, tmp_a, ctx))
169 } else if (!BN_copy(group->a, tmp_a))
173 if (!BN_nnmod(group->b, b, p, ctx))
175 if (group->meth->field_encode)
176 if (!group->meth->field_encode(group, group->b, group->b, ctx))
179 /* group->a_is_minus3 */
180 if (!BN_add_word(tmp_a, 3))
182 group->a_is_minus3 = (0 == BN_cmp(tmp_a, group->field));
188 BN_CTX_free(new_ctx);
192 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
193 BIGNUM *b, BN_CTX *ctx)
196 BN_CTX *new_ctx = NULL;
199 if (!BN_copy(p, group->field))
203 if (a != NULL || b != NULL) {
204 if (group->meth->field_decode) {
206 ctx = new_ctx = BN_CTX_new();
211 if (!group->meth->field_decode(group, a, group->a, ctx))
215 if (!group->meth->field_decode(group, b, group->b, ctx))
220 if (!BN_copy(a, group->a))
224 if (!BN_copy(b, group->b))
233 BN_CTX_free(new_ctx);
237 int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
239 return BN_num_bits(group->field);
242 int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
245 BIGNUM *a, *b, *order, *tmp_1, *tmp_2;
246 const BIGNUM *p = group->field;
247 BN_CTX *new_ctx = NULL;
250 ctx = new_ctx = BN_CTX_new();
252 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT,
253 ERR_R_MALLOC_FAILURE);
260 tmp_1 = BN_CTX_get(ctx);
261 tmp_2 = BN_CTX_get(ctx);
262 order = BN_CTX_get(ctx);
266 if (group->meth->field_decode) {
267 if (!group->meth->field_decode(group, a, group->a, ctx))
269 if (!group->meth->field_decode(group, b, group->b, ctx))
272 if (!BN_copy(a, group->a))
274 if (!BN_copy(b, group->b))
279 * check the discriminant:
280 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
286 } else if (!BN_is_zero(b)) {
287 if (!BN_mod_sqr(tmp_1, a, p, ctx))
289 if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx))
291 if (!BN_lshift(tmp_1, tmp_2, 2))
295 if (!BN_mod_sqr(tmp_2, b, p, ctx))
297 if (!BN_mul_word(tmp_2, 27))
301 if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx))
311 BN_CTX_free(new_ctx);
315 int ec_GFp_simple_point_init(EC_POINT *point)
322 if (point->X == NULL || point->Y == NULL || point->Z == NULL) {
331 void ec_GFp_simple_point_finish(EC_POINT *point)
338 void ec_GFp_simple_point_clear_finish(EC_POINT *point)
340 BN_clear_free(point->X);
341 BN_clear_free(point->Y);
342 BN_clear_free(point->Z);
346 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
348 if (!BN_copy(dest->X, src->X))
350 if (!BN_copy(dest->Y, src->Y))
352 if (!BN_copy(dest->Z, src->Z))
354 dest->Z_is_one = src->Z_is_one;
355 dest->curve_name = src->curve_name;
360 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
368 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group,
375 BN_CTX *new_ctx = NULL;
379 ctx = new_ctx = BN_CTX_new();
385 if (!BN_nnmod(point->X, x, group->field, ctx))
387 if (group->meth->field_encode) {
388 if (!group->meth->field_encode(group, point->X, point->X, ctx))
394 if (!BN_nnmod(point->Y, y, group->field, ctx))
396 if (group->meth->field_encode) {
397 if (!group->meth->field_encode(group, point->Y, point->Y, ctx))
405 if (!BN_nnmod(point->Z, z, group->field, ctx))
407 Z_is_one = BN_is_one(point->Z);
408 if (group->meth->field_encode) {
409 if (Z_is_one && (group->meth->field_set_to_one != 0)) {
410 if (!group->meth->field_set_to_one(group, point->Z, ctx))
414 meth->field_encode(group, point->Z, point->Z, ctx))
418 point->Z_is_one = Z_is_one;
424 BN_CTX_free(new_ctx);
428 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group,
429 const EC_POINT *point,
430 BIGNUM *x, BIGNUM *y,
431 BIGNUM *z, BN_CTX *ctx)
433 BN_CTX *new_ctx = NULL;
436 if (group->meth->field_decode != 0) {
438 ctx = new_ctx = BN_CTX_new();
444 if (!group->meth->field_decode(group, x, point->X, ctx))
448 if (!group->meth->field_decode(group, y, point->Y, ctx))
452 if (!group->meth->field_decode(group, z, point->Z, ctx))
457 if (!BN_copy(x, point->X))
461 if (!BN_copy(y, point->Y))
465 if (!BN_copy(z, point->Z))
473 BN_CTX_free(new_ctx);
477 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
480 const BIGNUM *y, BN_CTX *ctx)
482 if (x == NULL || y == NULL) {
484 * unlike for projective coordinates, we do not tolerate this
486 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES,
487 ERR_R_PASSED_NULL_PARAMETER);
491 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y,
492 BN_value_one(), ctx);
495 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group,
496 const EC_POINT *point,
497 BIGNUM *x, BIGNUM *y,
500 BN_CTX *new_ctx = NULL;
501 BIGNUM *Z, *Z_1, *Z_2, *Z_3;
505 if (EC_POINT_is_at_infinity(group, point)) {
506 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES,
507 EC_R_POINT_AT_INFINITY);
512 ctx = new_ctx = BN_CTX_new();
519 Z_1 = BN_CTX_get(ctx);
520 Z_2 = BN_CTX_get(ctx);
521 Z_3 = BN_CTX_get(ctx);
525 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
527 if (group->meth->field_decode) {
528 if (!group->meth->field_decode(group, Z, point->Z, ctx))
536 if (group->meth->field_decode) {
538 if (!group->meth->field_decode(group, x, point->X, ctx))
542 if (!group->meth->field_decode(group, y, point->Y, ctx))
547 if (!BN_copy(x, point->X))
551 if (!BN_copy(y, point->Y))
556 if (!BN_mod_inverse(Z_1, Z_, group->field, ctx)) {
557 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES,
562 if (group->meth->field_encode == 0) {
563 /* field_sqr works on standard representation */
564 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx))
567 if (!BN_mod_sqr(Z_2, Z_1, group->field, ctx))
573 * in the Montgomery case, field_mul will cancel out Montgomery
576 if (!group->meth->field_mul(group, x, point->X, Z_2, ctx))
581 if (group->meth->field_encode == 0) {
583 * field_mul works on standard representation
585 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx))
588 if (!BN_mod_mul(Z_3, Z_2, Z_1, group->field, ctx))
593 * in the Montgomery case, field_mul will cancel out Montgomery
596 if (!group->meth->field_mul(group, y, point->Y, Z_3, ctx))
605 BN_CTX_free(new_ctx);
609 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
610 const EC_POINT *b, BN_CTX *ctx)
612 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
613 const BIGNUM *, BN_CTX *);
614 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
616 BN_CTX *new_ctx = NULL;
617 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
621 return EC_POINT_dbl(group, r, a, ctx);
622 if (EC_POINT_is_at_infinity(group, a))
623 return EC_POINT_copy(r, b);
624 if (EC_POINT_is_at_infinity(group, b))
625 return EC_POINT_copy(r, a);
627 field_mul = group->meth->field_mul;
628 field_sqr = group->meth->field_sqr;
632 ctx = new_ctx = BN_CTX_new();
638 n0 = BN_CTX_get(ctx);
639 n1 = BN_CTX_get(ctx);
640 n2 = BN_CTX_get(ctx);
641 n3 = BN_CTX_get(ctx);
642 n4 = BN_CTX_get(ctx);
643 n5 = BN_CTX_get(ctx);
644 n6 = BN_CTX_get(ctx);
649 * Note that in this function we must not read components of 'a' or 'b'
650 * once we have written the corresponding components of 'r'. ('r' might
651 * be one of 'a' or 'b'.)
656 if (!BN_copy(n1, a->X))
658 if (!BN_copy(n2, a->Y))
663 if (!field_sqr(group, n0, b->Z, ctx))
665 if (!field_mul(group, n1, a->X, n0, ctx))
667 /* n1 = X_a * Z_b^2 */
669 if (!field_mul(group, n0, n0, b->Z, ctx))
671 if (!field_mul(group, n2, a->Y, n0, ctx))
673 /* n2 = Y_a * Z_b^3 */
678 if (!BN_copy(n3, b->X))
680 if (!BN_copy(n4, b->Y))
685 if (!field_sqr(group, n0, a->Z, ctx))
687 if (!field_mul(group, n3, b->X, n0, ctx))
689 /* n3 = X_b * Z_a^2 */
691 if (!field_mul(group, n0, n0, a->Z, ctx))
693 if (!field_mul(group, n4, b->Y, n0, ctx))
695 /* n4 = Y_b * Z_a^3 */
699 if (!BN_mod_sub_quick(n5, n1, n3, p))
701 if (!BN_mod_sub_quick(n6, n2, n4, p))
706 if (BN_is_zero(n5)) {
707 if (BN_is_zero(n6)) {
708 /* a is the same point as b */
710 ret = EC_POINT_dbl(group, r, a, ctx);
714 /* a is the inverse of b */
723 if (!BN_mod_add_quick(n1, n1, n3, p))
725 if (!BN_mod_add_quick(n2, n2, n4, p))
731 if (a->Z_is_one && b->Z_is_one) {
732 if (!BN_copy(r->Z, n5))
736 if (!BN_copy(n0, b->Z))
738 } else if (b->Z_is_one) {
739 if (!BN_copy(n0, a->Z))
742 if (!field_mul(group, n0, a->Z, b->Z, ctx))
745 if (!field_mul(group, r->Z, n0, n5, ctx))
749 /* Z_r = Z_a * Z_b * n5 */
752 if (!field_sqr(group, n0, n6, ctx))
754 if (!field_sqr(group, n4, n5, ctx))
756 if (!field_mul(group, n3, n1, n4, ctx))
758 if (!BN_mod_sub_quick(r->X, n0, n3, p))
760 /* X_r = n6^2 - n5^2 * 'n7' */
763 if (!BN_mod_lshift1_quick(n0, r->X, p))
765 if (!BN_mod_sub_quick(n0, n3, n0, p))
767 /* n9 = n5^2 * 'n7' - 2 * X_r */
770 if (!field_mul(group, n0, n0, n6, ctx))
772 if (!field_mul(group, n5, n4, n5, ctx))
773 goto end; /* now n5 is n5^3 */
774 if (!field_mul(group, n1, n2, n5, ctx))
776 if (!BN_mod_sub_quick(n0, n0, n1, p))
779 if (!BN_add(n0, n0, p))
781 /* now 0 <= n0 < 2*p, and n0 is even */
782 if (!BN_rshift1(r->Y, n0))
784 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
789 if (ctx) /* otherwise we already called BN_CTX_end */
791 BN_CTX_free(new_ctx);
795 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
798 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
799 const BIGNUM *, BN_CTX *);
800 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
802 BN_CTX *new_ctx = NULL;
803 BIGNUM *n0, *n1, *n2, *n3;
806 if (EC_POINT_is_at_infinity(group, a)) {
812 field_mul = group->meth->field_mul;
813 field_sqr = group->meth->field_sqr;
817 ctx = new_ctx = BN_CTX_new();
823 n0 = BN_CTX_get(ctx);
824 n1 = BN_CTX_get(ctx);
825 n2 = BN_CTX_get(ctx);
826 n3 = BN_CTX_get(ctx);
831 * Note that in this function we must not read components of 'a' once we
832 * have written the corresponding components of 'r'. ('r' might the same
838 if (!field_sqr(group, n0, a->X, ctx))
840 if (!BN_mod_lshift1_quick(n1, n0, p))
842 if (!BN_mod_add_quick(n0, n0, n1, p))
844 if (!BN_mod_add_quick(n1, n0, group->a, p))
846 /* n1 = 3 * X_a^2 + a_curve */
847 } else if (group->a_is_minus3) {
848 if (!field_sqr(group, n1, a->Z, ctx))
850 if (!BN_mod_add_quick(n0, a->X, n1, p))
852 if (!BN_mod_sub_quick(n2, a->X, n1, p))
854 if (!field_mul(group, n1, n0, n2, ctx))
856 if (!BN_mod_lshift1_quick(n0, n1, p))
858 if (!BN_mod_add_quick(n1, n0, n1, p))
861 * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
862 * = 3 * X_a^2 - 3 * Z_a^4
865 if (!field_sqr(group, n0, a->X, ctx))
867 if (!BN_mod_lshift1_quick(n1, n0, p))
869 if (!BN_mod_add_quick(n0, n0, n1, p))
871 if (!field_sqr(group, n1, a->Z, ctx))
873 if (!field_sqr(group, n1, n1, ctx))
875 if (!field_mul(group, n1, n1, group->a, ctx))
877 if (!BN_mod_add_quick(n1, n1, n0, p))
879 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
884 if (!BN_copy(n0, a->Y))
887 if (!field_mul(group, n0, a->Y, a->Z, ctx))
890 if (!BN_mod_lshift1_quick(r->Z, n0, p))
893 /* Z_r = 2 * Y_a * Z_a */
896 if (!field_sqr(group, n3, a->Y, ctx))
898 if (!field_mul(group, n2, a->X, n3, ctx))
900 if (!BN_mod_lshift_quick(n2, n2, 2, p))
902 /* n2 = 4 * X_a * Y_a^2 */
905 if (!BN_mod_lshift1_quick(n0, n2, p))
907 if (!field_sqr(group, r->X, n1, ctx))
909 if (!BN_mod_sub_quick(r->X, r->X, n0, p))
911 /* X_r = n1^2 - 2 * n2 */
914 if (!field_sqr(group, n0, n3, ctx))
916 if (!BN_mod_lshift_quick(n3, n0, 3, p))
921 if (!BN_mod_sub_quick(n0, n2, r->X, p))
923 if (!field_mul(group, n0, n1, n0, ctx))
925 if (!BN_mod_sub_quick(r->Y, n0, n3, p))
927 /* Y_r = n1 * (n2 - X_r) - n3 */
933 BN_CTX_free(new_ctx);
937 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
939 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
940 /* point is its own inverse */
943 return BN_usub(point->Y, group->field, point->Y);
946 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
948 return BN_is_zero(point->Z);
951 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
954 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
955 const BIGNUM *, BN_CTX *);
956 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
958 BN_CTX *new_ctx = NULL;
959 BIGNUM *rh, *tmp, *Z4, *Z6;
962 if (EC_POINT_is_at_infinity(group, point))
965 field_mul = group->meth->field_mul;
966 field_sqr = group->meth->field_sqr;
970 ctx = new_ctx = BN_CTX_new();
976 rh = BN_CTX_get(ctx);
977 tmp = BN_CTX_get(ctx);
978 Z4 = BN_CTX_get(ctx);
979 Z6 = BN_CTX_get(ctx);
984 * We have a curve defined by a Weierstrass equation
985 * y^2 = x^3 + a*x + b.
986 * The point to consider is given in Jacobian projective coordinates
987 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
988 * Substituting this and multiplying by Z^6 transforms the above equation into
989 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
990 * To test this, we add up the right-hand side in 'rh'.
994 if (!field_sqr(group, rh, point->X, ctx))
997 if (!point->Z_is_one) {
998 if (!field_sqr(group, tmp, point->Z, ctx))
1000 if (!field_sqr(group, Z4, tmp, ctx))
1002 if (!field_mul(group, Z6, Z4, tmp, ctx))
1005 /* rh := (rh + a*Z^4)*X */
1006 if (group->a_is_minus3) {
1007 if (!BN_mod_lshift1_quick(tmp, Z4, p))
1009 if (!BN_mod_add_quick(tmp, tmp, Z4, p))
1011 if (!BN_mod_sub_quick(rh, rh, tmp, p))
1013 if (!field_mul(group, rh, rh, point->X, ctx))
1016 if (!field_mul(group, tmp, Z4, group->a, ctx))
1018 if (!BN_mod_add_quick(rh, rh, tmp, p))
1020 if (!field_mul(group, rh, rh, point->X, ctx))
1024 /* rh := rh + b*Z^6 */
1025 if (!field_mul(group, tmp, group->b, Z6, ctx))
1027 if (!BN_mod_add_quick(rh, rh, tmp, p))
1030 /* point->Z_is_one */
1032 /* rh := (rh + a)*X */
1033 if (!BN_mod_add_quick(rh, rh, group->a, p))
1035 if (!field_mul(group, rh, rh, point->X, ctx))
1038 if (!BN_mod_add_quick(rh, rh, group->b, p))
1043 if (!field_sqr(group, tmp, point->Y, ctx))
1046 ret = (0 == BN_ucmp(tmp, rh));
1050 BN_CTX_free(new_ctx);
1054 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
1055 const EC_POINT *b, BN_CTX *ctx)
1060 * 0 equal (in affine coordinates)
1064 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
1065 const BIGNUM *, BN_CTX *);
1066 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1067 BN_CTX *new_ctx = NULL;
1068 BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1069 const BIGNUM *tmp1_, *tmp2_;
1072 if (EC_POINT_is_at_infinity(group, a)) {
1073 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1076 if (EC_POINT_is_at_infinity(group, b))
1079 if (a->Z_is_one && b->Z_is_one) {
1080 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
1083 field_mul = group->meth->field_mul;
1084 field_sqr = group->meth->field_sqr;
1087 ctx = new_ctx = BN_CTX_new();
1093 tmp1 = BN_CTX_get(ctx);
1094 tmp2 = BN_CTX_get(ctx);
1095 Za23 = BN_CTX_get(ctx);
1096 Zb23 = BN_CTX_get(ctx);
1101 * We have to decide whether
1102 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1103 * or equivalently, whether
1104 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1108 if (!field_sqr(group, Zb23, b->Z, ctx))
1110 if (!field_mul(group, tmp1, a->X, Zb23, ctx))
1116 if (!field_sqr(group, Za23, a->Z, ctx))
1118 if (!field_mul(group, tmp2, b->X, Za23, ctx))
1124 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1125 if (BN_cmp(tmp1_, tmp2_) != 0) {
1126 ret = 1; /* points differ */
1131 if (!field_mul(group, Zb23, Zb23, b->Z, ctx))
1133 if (!field_mul(group, tmp1, a->Y, Zb23, ctx))
1139 if (!field_mul(group, Za23, Za23, a->Z, ctx))
1141 if (!field_mul(group, tmp2, b->Y, Za23, ctx))
1147 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1148 if (BN_cmp(tmp1_, tmp2_) != 0) {
1149 ret = 1; /* points differ */
1153 /* points are equal */
1158 BN_CTX_free(new_ctx);
1162 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
1165 BN_CTX *new_ctx = NULL;
1169 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1173 ctx = new_ctx = BN_CTX_new();
1179 x = BN_CTX_get(ctx);
1180 y = BN_CTX_get(ctx);
1184 if (!EC_POINT_get_affine_coordinates(group, point, x, y, ctx))
1186 if (!EC_POINT_set_affine_coordinates(group, point, x, y, ctx))
1188 if (!point->Z_is_one) {
1189 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
1197 BN_CTX_free(new_ctx);
1201 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num,
1202 EC_POINT *points[], BN_CTX *ctx)
1204 BN_CTX *new_ctx = NULL;
1205 BIGNUM *tmp, *tmp_Z;
1206 BIGNUM **prod_Z = NULL;
1214 ctx = new_ctx = BN_CTX_new();
1220 tmp = BN_CTX_get(ctx);
1221 tmp_Z = BN_CTX_get(ctx);
1225 prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0]));
1228 for (i = 0; i < num; i++) {
1229 prod_Z[i] = BN_new();
1230 if (prod_Z[i] == NULL)
1235 * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1236 * skipping any zero-valued inputs (pretend that they're 1).
1239 if (!BN_is_zero(points[0]->Z)) {
1240 if (!BN_copy(prod_Z[0], points[0]->Z))
1243 if (group->meth->field_set_to_one != 0) {
1244 if (!group->meth->field_set_to_one(group, prod_Z[0], ctx))
1247 if (!BN_one(prod_Z[0]))
1252 for (i = 1; i < num; i++) {
1253 if (!BN_is_zero(points[i]->Z)) {
1255 meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z,
1259 if (!BN_copy(prod_Z[i], prod_Z[i - 1]))
1265 * Now use a single explicit inversion to replace every non-zero
1266 * points[i]->Z by its inverse.
1269 if (!BN_mod_inverse(tmp, prod_Z[num - 1], group->field, ctx)) {
1270 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
1273 if (group->meth->field_encode != 0) {
1275 * In the Montgomery case, we just turned R*H (representing H) into
1276 * 1/(R*H), but we need R*(1/H) (representing 1/H); i.e. we need to
1277 * multiply by the Montgomery factor twice.
1279 if (!group->meth->field_encode(group, tmp, tmp, ctx))
1281 if (!group->meth->field_encode(group, tmp, tmp, ctx))
1285 for (i = num - 1; i > 0; --i) {
1287 * Loop invariant: tmp is the product of the inverses of points[0]->Z
1288 * .. points[i]->Z (zero-valued inputs skipped).
1290 if (!BN_is_zero(points[i]->Z)) {
1292 * Set tmp_Z to the inverse of points[i]->Z (as product of Z
1293 * inverses 0 .. i, Z values 0 .. i - 1).
1296 meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx))
1299 * Update tmp to satisfy the loop invariant for i - 1.
1301 if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx))
1303 /* Replace points[i]->Z by its inverse. */
1304 if (!BN_copy(points[i]->Z, tmp_Z))
1309 if (!BN_is_zero(points[0]->Z)) {
1310 /* Replace points[0]->Z by its inverse. */
1311 if (!BN_copy(points[0]->Z, tmp))
1315 /* Finally, fix up the X and Y coordinates for all points. */
1317 for (i = 0; i < num; i++) {
1318 EC_POINT *p = points[i];
1320 if (!BN_is_zero(p->Z)) {
1321 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1323 if (!group->meth->field_sqr(group, tmp, p->Z, ctx))
1325 if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx))
1328 if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx))
1330 if (!group->meth->field_mul(group, p->Y, p->Y, tmp, ctx))
1333 if (group->meth->field_set_to_one != 0) {
1334 if (!group->meth->field_set_to_one(group, p->Z, ctx))
1348 BN_CTX_free(new_ctx);
1349 if (prod_Z != NULL) {
1350 for (i = 0; i < num; i++) {
1351 if (prod_Z[i] == NULL)
1353 BN_clear_free(prod_Z[i]);
1355 OPENSSL_free(prod_Z);
1360 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1361 const BIGNUM *b, BN_CTX *ctx)
1363 return BN_mod_mul(r, a, b, group->field, ctx);
1366 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1369 return BN_mod_sqr(r, a, group->field, ctx);
1373 * Apply randomization of EC point projective coordinates:
1375 * (X, Y ,Z ) = (lambda^2*X, lambda^3*Y, lambda*Z)
1376 * lambda = [1,group->field)
1379 int ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p,
1383 BIGNUM *lambda = NULL;
1384 BIGNUM *temp = NULL;
1387 lambda = BN_CTX_get(ctx);
1388 temp = BN_CTX_get(ctx);
1390 ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_MALLOC_FAILURE);
1394 /* make sure lambda is not zero */
1396 if (!BN_priv_rand_range(lambda, group->field)) {
1397 ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_BN_LIB);
1400 } while (BN_is_zero(lambda));
1402 /* if field_encode defined convert between representations */
1403 if (group->meth->field_encode != NULL
1404 && !group->meth->field_encode(group, lambda, lambda, ctx))
1406 if (!group->meth->field_mul(group, p->Z, p->Z, lambda, ctx))
1408 if (!group->meth->field_sqr(group, temp, lambda, ctx))
1410 if (!group->meth->field_mul(group, p->X, p->X, temp, ctx))
1412 if (!group->meth->field_mul(group, temp, temp, lambda, ctx))
1414 if (!group->meth->field_mul(group, p->Y, p->Y, temp, ctx))
1426 * Set s := p, r := 2p.
1428 * For doubling we use Formula 3 from Izu-Takagi "A fast parallel elliptic curve
1429 * multiplication resistant against side channel attacks" appendix, as described
1431 * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#doubling-dbl-2002-it-2
1433 * The input point p will be in randomized Jacobian projective coords:
1434 * x = X/Z**2, y=Y/Z**3
1436 * The output points p, s, and r are converted to standard (homogeneous)
1437 * projective coords:
1440 int ec_GFp_simple_ladder_pre(const EC_GROUP *group,
1441 EC_POINT *r, EC_POINT *s,
1442 EC_POINT *p, BN_CTX *ctx)
1444 BIGNUM *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
1453 /* convert p: (X,Y,Z) -> (XZ,Y,Z**3) */
1454 if (!group->meth->field_mul(group, p->X, p->X, p->Z, ctx)
1455 || !group->meth->field_sqr(group, t1, p->Z, ctx)
1456 || !group->meth->field_mul(group, p->Z, p->Z, t1, ctx)
1458 || !group->meth->field_sqr(group, t2, p->X, ctx)
1459 || !group->meth->field_sqr(group, t3, p->Z, ctx)
1460 || !group->meth->field_mul(group, t4, t3, group->a, ctx)
1461 || !BN_mod_sub_quick(t5, t2, t4, group->field)
1462 || !BN_mod_add_quick(t2, t2, t4, group->field)
1463 || !group->meth->field_sqr(group, t5, t5, ctx)
1464 || !group->meth->field_mul(group, t6, t3, group->b, ctx)
1465 || !group->meth->field_mul(group, t1, p->X, p->Z, ctx)
1466 || !group->meth->field_mul(group, t4, t1, t6, ctx)
1467 || !BN_mod_lshift_quick(t4, t4, 3, group->field)
1468 /* r->X coord output */
1469 || !BN_mod_sub_quick(r->X, t5, t4, group->field)
1470 || !group->meth->field_mul(group, t1, t1, t2, ctx)
1471 || !group->meth->field_mul(group, t2, t3, t6, ctx)
1472 || !BN_mod_add_quick(t1, t1, t2, group->field)
1473 /* r->Z coord output */
1474 || !BN_mod_lshift_quick(r->Z, t1, 2, group->field)
1475 || !EC_POINT_copy(s, p))
1486 * Differential addition-and-doubling using Eq. (9) and (10) from Izu-Takagi
1487 * "A fast parallel elliptic curve multiplication resistant against side channel
1488 * attacks", as described at
1489 * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-ladd-2002-it-4
1491 int ec_GFp_simple_ladder_step(const EC_GROUP *group,
1492 EC_POINT *r, EC_POINT *s,
1493 EC_POINT *p, BN_CTX *ctx)
1496 BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6, *t7 = NULL;
1499 t0 = BN_CTX_get(ctx);
1500 t1 = BN_CTX_get(ctx);
1501 t2 = BN_CTX_get(ctx);
1502 t3 = BN_CTX_get(ctx);
1503 t4 = BN_CTX_get(ctx);
1504 t5 = BN_CTX_get(ctx);
1505 t6 = BN_CTX_get(ctx);
1506 t7 = BN_CTX_get(ctx);
1509 || !group->meth->field_mul(group, t0, r->X, s->X, ctx)
1510 || !group->meth->field_mul(group, t1, r->Z, s->Z, ctx)
1511 || !group->meth->field_mul(group, t2, r->X, s->Z, ctx)
1512 || !group->meth->field_mul(group, t3, r->Z, s->X, ctx)
1513 || !group->meth->field_mul(group, t4, group->a, t1, ctx)
1514 || !BN_mod_add_quick(t0, t0, t4, group->field)
1515 || !BN_mod_add_quick(t4, t3, t2, group->field)
1516 || !group->meth->field_mul(group, t0, t4, t0, ctx)
1517 || !group->meth->field_sqr(group, t1, t1, ctx)
1518 || !BN_mod_lshift_quick(t7, group->b, 2, group->field)
1519 || !group->meth->field_mul(group, t1, t7, t1, ctx)
1520 || !BN_mod_lshift1_quick(t0, t0, group->field)
1521 || !BN_mod_add_quick(t0, t1, t0, group->field)
1522 || !BN_mod_sub_quick(t1, t2, t3, group->field)
1523 || !group->meth->field_sqr(group, t1, t1, ctx)
1524 || !group->meth->field_mul(group, t3, t1, p->X, ctx)
1525 || !group->meth->field_mul(group, t0, p->Z, t0, ctx)
1526 /* s->X coord output */
1527 || !BN_mod_sub_quick(s->X, t0, t3, group->field)
1528 /* s->Z coord output */
1529 || !group->meth->field_mul(group, s->Z, p->Z, t1, ctx)
1530 || !group->meth->field_sqr(group, t3, r->X, ctx)
1531 || !group->meth->field_sqr(group, t2, r->Z, ctx)
1532 || !group->meth->field_mul(group, t4, t2, group->a, ctx)
1533 || !BN_mod_add_quick(t5, r->X, r->Z, group->field)
1534 || !group->meth->field_sqr(group, t5, t5, ctx)
1535 || !BN_mod_sub_quick(t5, t5, t3, group->field)
1536 || !BN_mod_sub_quick(t5, t5, t2, group->field)
1537 || !BN_mod_sub_quick(t6, t3, t4, group->field)
1538 || !group->meth->field_sqr(group, t6, t6, ctx)
1539 || !group->meth->field_mul(group, t0, t2, t5, ctx)
1540 || !group->meth->field_mul(group, t0, t7, t0, ctx)
1541 /* r->X coord output */
1542 || !BN_mod_sub_quick(r->X, t6, t0, group->field)
1543 || !BN_mod_add_quick(t6, t3, t4, group->field)
1544 || !group->meth->field_sqr(group, t3, t2, ctx)
1545 || !group->meth->field_mul(group, t7, t3, t7, ctx)
1546 || !group->meth->field_mul(group, t5, t5, t6, ctx)
1547 || !BN_mod_lshift1_quick(t5, t5, group->field)
1548 /* r->Z coord output */
1549 || !BN_mod_add_quick(r->Z, t7, t5, group->field))
1560 * Recovers the y-coordinate of r using Eq. (8) from Brier-Joye, "Weierstrass
1561 * Elliptic Curves and Side-Channel Attacks", modified to work in projective
1562 * coordinates and return r in Jacobian projective coordinates.
1564 * X4 = two*Y1*X2*Z3*Z2*Z1;
1565 * Y4 = two*b*Z3*SQR(Z2*Z1) + Z3*(a*Z2*Z1+X1*X2)*(X1*Z2+X2*Z1) - X3*SQR(X1*Z2-X2*Z1);
1566 * Z4 = two*Y1*Z3*SQR(Z2)*Z1;
1569 * - Z1==0 implies p is at infinity, which would have caused an early exit in
1571 * - Z2==0 implies r is at infinity (handled by the BN_is_zero(r->Z) branch);
1572 * - Z3==0 implies s is at infinity (handled by the BN_is_zero(s->Z) branch);
1573 * - Y1==0 implies p has order 2, so either r or s are infinity and handled by
1574 * one of the BN_is_zero(...) branches.
1576 int ec_GFp_simple_ladder_post(const EC_GROUP *group,
1577 EC_POINT *r, EC_POINT *s,
1578 EC_POINT *p, BN_CTX *ctx)
1581 BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
1583 if (BN_is_zero(r->Z))
1584 return EC_POINT_set_to_infinity(group, r);
1586 if (BN_is_zero(s->Z)) {
1587 /* (X,Y,Z) -> (XZ,YZ**2,Z) */
1588 if (!group->meth->field_mul(group, r->X, p->X, p->Z, ctx)
1589 || !group->meth->field_sqr(group, r->Z, p->Z, ctx)
1590 || !group->meth->field_mul(group, r->Y, p->Y, r->Z, ctx)
1591 || !BN_copy(r->Z, p->Z)
1592 || !EC_POINT_invert(group, r, ctx))
1598 t0 = BN_CTX_get(ctx);
1599 t1 = BN_CTX_get(ctx);
1600 t2 = BN_CTX_get(ctx);
1601 t3 = BN_CTX_get(ctx);
1602 t4 = BN_CTX_get(ctx);
1603 t5 = BN_CTX_get(ctx);
1604 t6 = BN_CTX_get(ctx);
1607 || !BN_mod_lshift1_quick(t0, p->Y, group->field)
1608 || !group->meth->field_mul(group, t1, r->X, p->Z, ctx)
1609 || !group->meth->field_mul(group, t2, r->Z, s->Z, ctx)
1610 || !group->meth->field_mul(group, t2, t1, t2, ctx)
1611 || !group->meth->field_mul(group, t3, t2, t0, ctx)
1612 || !group->meth->field_mul(group, t2, r->Z, p->Z, ctx)
1613 || !group->meth->field_sqr(group, t4, t2, ctx)
1614 || !BN_mod_lshift1_quick(t5, group->b, group->field)
1615 || !group->meth->field_mul(group, t4, t4, t5, ctx)
1616 || !group->meth->field_mul(group, t6, t2, group->a, ctx)
1617 || !group->meth->field_mul(group, t5, r->X, p->X, ctx)
1618 || !BN_mod_add_quick(t5, t6, t5, group->field)
1619 || !group->meth->field_mul(group, t6, r->Z, p->X, ctx)
1620 || !BN_mod_add_quick(t2, t6, t1, group->field)
1621 || !group->meth->field_mul(group, t5, t5, t2, ctx)
1622 || !BN_mod_sub_quick(t6, t6, t1, group->field)
1623 || !group->meth->field_sqr(group, t6, t6, ctx)
1624 || !group->meth->field_mul(group, t6, t6, s->X, ctx)
1625 || !BN_mod_add_quick(t4, t5, t4, group->field)
1626 || !group->meth->field_mul(group, t4, t4, s->Z, ctx)
1627 || !BN_mod_sub_quick(t4, t4, t6, group->field)
1628 || !group->meth->field_sqr(group, t5, r->Z, ctx)
1629 || !group->meth->field_mul(group, r->Z, p->Z, s->Z, ctx)
1630 || !group->meth->field_mul(group, r->Z, t5, r->Z, ctx)
1631 || !group->meth->field_mul(group, r->Z, r->Z, t0, ctx)
1632 /* t3 := X, t4 := Y */
1633 /* (X,Y,Z) -> (XZ,YZ**2,Z) */
1634 || !group->meth->field_mul(group, r->X, t3, r->Z, ctx)
1635 || !group->meth->field_sqr(group, t3, r->Z, ctx)
1636 || !group->meth->field_mul(group, r->Y, t4, t3, ctx))