2 * Copyright 2001-2020 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved
5 * Licensed under the Apache License 2.0 (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
12 * ECDSA low level APIs are deprecated for public use, but still ok for
15 #include "internal/deprecated.h"
17 #include <openssl/err.h>
18 #include <openssl/symhacks.h>
22 const EC_METHOD *EC_GFp_simple_method(void)
24 static const EC_METHOD ret = {
26 NID_X9_62_prime_field,
27 ossl_ec_GFp_simple_group_init,
28 ossl_ec_GFp_simple_group_finish,
29 ossl_ec_GFp_simple_group_clear_finish,
30 ossl_ec_GFp_simple_group_copy,
31 ossl_ec_GFp_simple_group_set_curve,
32 ossl_ec_GFp_simple_group_get_curve,
33 ossl_ec_GFp_simple_group_get_degree,
34 ossl_ec_group_simple_order_bits,
35 ossl_ec_GFp_simple_group_check_discriminant,
36 ossl_ec_GFp_simple_point_init,
37 ossl_ec_GFp_simple_point_finish,
38 ossl_ec_GFp_simple_point_clear_finish,
39 ossl_ec_GFp_simple_point_copy,
40 ossl_ec_GFp_simple_point_set_to_infinity,
41 ossl_ec_GFp_simple_point_set_affine_coordinates,
42 ossl_ec_GFp_simple_point_get_affine_coordinates,
44 ossl_ec_GFp_simple_add,
45 ossl_ec_GFp_simple_dbl,
46 ossl_ec_GFp_simple_invert,
47 ossl_ec_GFp_simple_is_at_infinity,
48 ossl_ec_GFp_simple_is_on_curve,
49 ossl_ec_GFp_simple_cmp,
50 ossl_ec_GFp_simple_make_affine,
51 ossl_ec_GFp_simple_points_make_affine,
53 0 /* precompute_mult */ ,
54 0 /* have_precompute_mult */ ,
55 ossl_ec_GFp_simple_field_mul,
56 ossl_ec_GFp_simple_field_sqr,
58 ossl_ec_GFp_simple_field_inv,
59 0 /* field_encode */ ,
60 0 /* field_decode */ ,
61 0, /* field_set_to_one */
62 ossl_ec_key_simple_priv2oct,
63 ossl_ec_key_simple_oct2priv,
65 ossl_ec_key_simple_generate_key,
66 ossl_ec_key_simple_check_key,
67 ossl_ec_key_simple_generate_public_key,
70 ossl_ecdh_simple_compute_key,
71 ossl_ecdsa_simple_sign_setup,
72 ossl_ecdsa_simple_sign_sig,
73 ossl_ecdsa_simple_verify_sig,
74 0, /* field_inverse_mod_ord */
75 ossl_ec_GFp_simple_blind_coordinates,
76 ossl_ec_GFp_simple_ladder_pre,
77 ossl_ec_GFp_simple_ladder_step,
78 ossl_ec_GFp_simple_ladder_post
85 * Most method functions in this file are designed to work with
86 * non-trivial representations of field elements if necessary
87 * (see ecp_mont.c): while standard modular addition and subtraction
88 * are used, the field_mul and field_sqr methods will be used for
89 * multiplication, and field_encode and field_decode (if defined)
90 * will be used for converting between representations.
92 * Functions ec_GFp_simple_points_make_affine() and
93 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
94 * that if a non-trivial representation is used, it is a Montgomery
95 * representation (i.e. 'encoding' means multiplying by some factor R).
98 int ossl_ec_GFp_simple_group_init(EC_GROUP *group)
100 group->field = BN_new();
103 if (group->field == NULL || group->a == NULL || group->b == NULL) {
104 BN_free(group->field);
109 group->a_is_minus3 = 0;
113 void ossl_ec_GFp_simple_group_finish(EC_GROUP *group)
115 BN_free(group->field);
120 void ossl_ec_GFp_simple_group_clear_finish(EC_GROUP *group)
122 BN_clear_free(group->field);
123 BN_clear_free(group->a);
124 BN_clear_free(group->b);
127 int ossl_ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
129 if (!BN_copy(dest->field, src->field))
131 if (!BN_copy(dest->a, src->a))
133 if (!BN_copy(dest->b, src->b))
136 dest->a_is_minus3 = src->a_is_minus3;
141 int ossl_ec_GFp_simple_group_set_curve(EC_GROUP *group,
142 const BIGNUM *p, const BIGNUM *a,
143 const BIGNUM *b, BN_CTX *ctx)
146 BN_CTX *new_ctx = NULL;
149 /* p must be a prime > 3 */
150 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
151 ERR_raise(ERR_LIB_EC, EC_R_INVALID_FIELD);
156 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
162 tmp_a = BN_CTX_get(ctx);
167 if (!BN_copy(group->field, p))
169 BN_set_negative(group->field, 0);
172 if (!BN_nnmod(tmp_a, a, p, ctx))
174 if (group->meth->field_encode) {
175 if (!group->meth->field_encode(group, group->a, tmp_a, ctx))
177 } else if (!BN_copy(group->a, tmp_a))
181 if (!BN_nnmod(group->b, b, p, ctx))
183 if (group->meth->field_encode)
184 if (!group->meth->field_encode(group, group->b, group->b, ctx))
187 /* group->a_is_minus3 */
188 if (!BN_add_word(tmp_a, 3))
190 group->a_is_minus3 = (0 == BN_cmp(tmp_a, group->field));
196 BN_CTX_free(new_ctx);
200 int ossl_ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p,
201 BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
204 BN_CTX *new_ctx = NULL;
207 if (!BN_copy(p, group->field))
211 if (a != NULL || b != NULL) {
212 if (group->meth->field_decode) {
214 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
219 if (!group->meth->field_decode(group, a, group->a, ctx))
223 if (!group->meth->field_decode(group, b, group->b, ctx))
228 if (!BN_copy(a, group->a))
232 if (!BN_copy(b, group->b))
241 BN_CTX_free(new_ctx);
245 int ossl_ec_GFp_simple_group_get_degree(const EC_GROUP *group)
247 return BN_num_bits(group->field);
250 int ossl_ec_GFp_simple_group_check_discriminant(const EC_GROUP *group,
254 BIGNUM *a, *b, *order, *tmp_1, *tmp_2;
255 const BIGNUM *p = group->field;
256 BN_CTX *new_ctx = NULL;
259 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
261 ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
268 tmp_1 = BN_CTX_get(ctx);
269 tmp_2 = BN_CTX_get(ctx);
270 order = BN_CTX_get(ctx);
274 if (group->meth->field_decode) {
275 if (!group->meth->field_decode(group, a, group->a, ctx))
277 if (!group->meth->field_decode(group, b, group->b, ctx))
280 if (!BN_copy(a, group->a))
282 if (!BN_copy(b, group->b))
287 * check the discriminant:
288 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
294 } else if (!BN_is_zero(b)) {
295 if (!BN_mod_sqr(tmp_1, a, p, ctx))
297 if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx))
299 if (!BN_lshift(tmp_1, tmp_2, 2))
303 if (!BN_mod_sqr(tmp_2, b, p, ctx))
305 if (!BN_mul_word(tmp_2, 27))
309 if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx))
318 BN_CTX_free(new_ctx);
322 int ossl_ec_GFp_simple_point_init(EC_POINT *point)
329 if (point->X == NULL || point->Y == NULL || point->Z == NULL) {
338 void ossl_ec_GFp_simple_point_finish(EC_POINT *point)
345 void ossl_ec_GFp_simple_point_clear_finish(EC_POINT *point)
347 BN_clear_free(point->X);
348 BN_clear_free(point->Y);
349 BN_clear_free(point->Z);
353 int ossl_ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
355 if (!BN_copy(dest->X, src->X))
357 if (!BN_copy(dest->Y, src->Y))
359 if (!BN_copy(dest->Z, src->Z))
361 dest->Z_is_one = src->Z_is_one;
362 dest->curve_name = src->curve_name;
367 int ossl_ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
375 int ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group,
382 BN_CTX *new_ctx = NULL;
386 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
392 if (!BN_nnmod(point->X, x, group->field, ctx))
394 if (group->meth->field_encode) {
395 if (!group->meth->field_encode(group, point->X, point->X, ctx))
401 if (!BN_nnmod(point->Y, y, group->field, ctx))
403 if (group->meth->field_encode) {
404 if (!group->meth->field_encode(group, point->Y, point->Y, ctx))
412 if (!BN_nnmod(point->Z, z, group->field, ctx))
414 Z_is_one = BN_is_one(point->Z);
415 if (group->meth->field_encode) {
416 if (Z_is_one && (group->meth->field_set_to_one != 0)) {
417 if (!group->meth->field_set_to_one(group, point->Z, ctx))
421 meth->field_encode(group, point->Z, point->Z, ctx))
425 point->Z_is_one = Z_is_one;
431 BN_CTX_free(new_ctx);
435 int ossl_ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group,
436 const EC_POINT *point,
437 BIGNUM *x, BIGNUM *y,
438 BIGNUM *z, BN_CTX *ctx)
440 BN_CTX *new_ctx = NULL;
443 if (group->meth->field_decode != 0) {
445 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
451 if (!group->meth->field_decode(group, x, point->X, ctx))
455 if (!group->meth->field_decode(group, y, point->Y, ctx))
459 if (!group->meth->field_decode(group, z, point->Z, ctx))
464 if (!BN_copy(x, point->X))
468 if (!BN_copy(y, point->Y))
472 if (!BN_copy(z, point->Z))
480 BN_CTX_free(new_ctx);
484 int ossl_ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
487 const BIGNUM *y, BN_CTX *ctx)
489 if (x == NULL || y == NULL) {
491 * unlike for projective coordinates, we do not tolerate this
493 ERR_raise(ERR_LIB_EC, ERR_R_PASSED_NULL_PARAMETER);
497 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y,
498 BN_value_one(), ctx);
501 int ossl_ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group,
502 const EC_POINT *point,
503 BIGNUM *x, BIGNUM *y,
506 BN_CTX *new_ctx = NULL;
507 BIGNUM *Z, *Z_1, *Z_2, *Z_3;
511 if (EC_POINT_is_at_infinity(group, point)) {
512 ERR_raise(ERR_LIB_EC, EC_R_POINT_AT_INFINITY);
517 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
524 Z_1 = BN_CTX_get(ctx);
525 Z_2 = BN_CTX_get(ctx);
526 Z_3 = BN_CTX_get(ctx);
530 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
532 if (group->meth->field_decode) {
533 if (!group->meth->field_decode(group, Z, point->Z, ctx))
541 if (group->meth->field_decode) {
543 if (!group->meth->field_decode(group, x, point->X, ctx))
547 if (!group->meth->field_decode(group, y, point->Y, ctx))
552 if (!BN_copy(x, point->X))
556 if (!BN_copy(y, point->Y))
561 if (!group->meth->field_inv(group, Z_1, Z_, ctx)) {
562 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
566 if (group->meth->field_encode == 0) {
567 /* field_sqr works on standard representation */
568 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx))
571 if (!BN_mod_sqr(Z_2, Z_1, group->field, ctx))
577 * in the Montgomery case, field_mul will cancel out Montgomery
580 if (!group->meth->field_mul(group, x, point->X, Z_2, ctx))
585 if (group->meth->field_encode == 0) {
587 * field_mul works on standard representation
589 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx))
592 if (!BN_mod_mul(Z_3, Z_2, Z_1, group->field, ctx))
597 * in the Montgomery case, field_mul will cancel out Montgomery
600 if (!group->meth->field_mul(group, y, point->Y, Z_3, ctx))
609 BN_CTX_free(new_ctx);
613 int ossl_ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
614 const EC_POINT *b, BN_CTX *ctx)
616 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
617 const BIGNUM *, BN_CTX *);
618 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
620 BN_CTX *new_ctx = NULL;
621 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
625 return EC_POINT_dbl(group, r, a, ctx);
626 if (EC_POINT_is_at_infinity(group, a))
627 return EC_POINT_copy(r, b);
628 if (EC_POINT_is_at_infinity(group, b))
629 return EC_POINT_copy(r, a);
631 field_mul = group->meth->field_mul;
632 field_sqr = group->meth->field_sqr;
636 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
642 n0 = BN_CTX_get(ctx);
643 n1 = BN_CTX_get(ctx);
644 n2 = BN_CTX_get(ctx);
645 n3 = BN_CTX_get(ctx);
646 n4 = BN_CTX_get(ctx);
647 n5 = BN_CTX_get(ctx);
648 n6 = BN_CTX_get(ctx);
653 * Note that in this function we must not read components of 'a' or 'b'
654 * once we have written the corresponding components of 'r'. ('r' might
655 * be one of 'a' or 'b'.)
660 if (!BN_copy(n1, a->X))
662 if (!BN_copy(n2, a->Y))
667 if (!field_sqr(group, n0, b->Z, ctx))
669 if (!field_mul(group, n1, a->X, n0, ctx))
671 /* n1 = X_a * Z_b^2 */
673 if (!field_mul(group, n0, n0, b->Z, ctx))
675 if (!field_mul(group, n2, a->Y, n0, ctx))
677 /* n2 = Y_a * Z_b^3 */
682 if (!BN_copy(n3, b->X))
684 if (!BN_copy(n4, b->Y))
689 if (!field_sqr(group, n0, a->Z, ctx))
691 if (!field_mul(group, n3, b->X, n0, ctx))
693 /* n3 = X_b * Z_a^2 */
695 if (!field_mul(group, n0, n0, a->Z, ctx))
697 if (!field_mul(group, n4, b->Y, n0, ctx))
699 /* n4 = Y_b * Z_a^3 */
703 if (!BN_mod_sub_quick(n5, n1, n3, p))
705 if (!BN_mod_sub_quick(n6, n2, n4, p))
710 if (BN_is_zero(n5)) {
711 if (BN_is_zero(n6)) {
712 /* a is the same point as b */
714 ret = EC_POINT_dbl(group, r, a, ctx);
718 /* a is the inverse of b */
727 if (!BN_mod_add_quick(n1, n1, n3, p))
729 if (!BN_mod_add_quick(n2, n2, n4, p))
735 if (a->Z_is_one && b->Z_is_one) {
736 if (!BN_copy(r->Z, n5))
740 if (!BN_copy(n0, b->Z))
742 } else if (b->Z_is_one) {
743 if (!BN_copy(n0, a->Z))
746 if (!field_mul(group, n0, a->Z, b->Z, ctx))
749 if (!field_mul(group, r->Z, n0, n5, ctx))
753 /* Z_r = Z_a * Z_b * n5 */
756 if (!field_sqr(group, n0, n6, ctx))
758 if (!field_sqr(group, n4, n5, ctx))
760 if (!field_mul(group, n3, n1, n4, ctx))
762 if (!BN_mod_sub_quick(r->X, n0, n3, p))
764 /* X_r = n6^2 - n5^2 * 'n7' */
767 if (!BN_mod_lshift1_quick(n0, r->X, p))
769 if (!BN_mod_sub_quick(n0, n3, n0, p))
771 /* n9 = n5^2 * 'n7' - 2 * X_r */
774 if (!field_mul(group, n0, n0, n6, ctx))
776 if (!field_mul(group, n5, n4, n5, ctx))
777 goto end; /* now n5 is n5^3 */
778 if (!field_mul(group, n1, n2, n5, ctx))
780 if (!BN_mod_sub_quick(n0, n0, n1, p))
783 if (!BN_add(n0, n0, p))
785 /* now 0 <= n0 < 2*p, and n0 is even */
786 if (!BN_rshift1(r->Y, n0))
788 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
794 BN_CTX_free(new_ctx);
798 int ossl_ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
801 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
802 const BIGNUM *, BN_CTX *);
803 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
805 BN_CTX *new_ctx = NULL;
806 BIGNUM *n0, *n1, *n2, *n3;
809 if (EC_POINT_is_at_infinity(group, a)) {
815 field_mul = group->meth->field_mul;
816 field_sqr = group->meth->field_sqr;
820 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
826 n0 = BN_CTX_get(ctx);
827 n1 = BN_CTX_get(ctx);
828 n2 = BN_CTX_get(ctx);
829 n3 = BN_CTX_get(ctx);
834 * Note that in this function we must not read components of 'a' once we
835 * have written the corresponding components of 'r'. ('r' might the same
841 if (!field_sqr(group, n0, a->X, ctx))
843 if (!BN_mod_lshift1_quick(n1, n0, p))
845 if (!BN_mod_add_quick(n0, n0, n1, p))
847 if (!BN_mod_add_quick(n1, n0, group->a, p))
849 /* n1 = 3 * X_a^2 + a_curve */
850 } else if (group->a_is_minus3) {
851 if (!field_sqr(group, n1, a->Z, ctx))
853 if (!BN_mod_add_quick(n0, a->X, n1, p))
855 if (!BN_mod_sub_quick(n2, a->X, n1, p))
857 if (!field_mul(group, n1, n0, n2, ctx))
859 if (!BN_mod_lshift1_quick(n0, n1, p))
861 if (!BN_mod_add_quick(n1, n0, n1, p))
864 * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
865 * = 3 * X_a^2 - 3 * Z_a^4
868 if (!field_sqr(group, n0, a->X, ctx))
870 if (!BN_mod_lshift1_quick(n1, n0, p))
872 if (!BN_mod_add_quick(n0, n0, n1, p))
874 if (!field_sqr(group, n1, a->Z, ctx))
876 if (!field_sqr(group, n1, n1, ctx))
878 if (!field_mul(group, n1, n1, group->a, ctx))
880 if (!BN_mod_add_quick(n1, n1, n0, p))
882 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
887 if (!BN_copy(n0, a->Y))
890 if (!field_mul(group, n0, a->Y, a->Z, ctx))
893 if (!BN_mod_lshift1_quick(r->Z, n0, p))
896 /* Z_r = 2 * Y_a * Z_a */
899 if (!field_sqr(group, n3, a->Y, ctx))
901 if (!field_mul(group, n2, a->X, n3, ctx))
903 if (!BN_mod_lshift_quick(n2, n2, 2, p))
905 /* n2 = 4 * X_a * Y_a^2 */
908 if (!BN_mod_lshift1_quick(n0, n2, p))
910 if (!field_sqr(group, r->X, n1, ctx))
912 if (!BN_mod_sub_quick(r->X, r->X, n0, p))
914 /* X_r = n1^2 - 2 * n2 */
917 if (!field_sqr(group, n0, n3, ctx))
919 if (!BN_mod_lshift_quick(n3, n0, 3, p))
924 if (!BN_mod_sub_quick(n0, n2, r->X, p))
926 if (!field_mul(group, n0, n1, n0, ctx))
928 if (!BN_mod_sub_quick(r->Y, n0, n3, p))
930 /* Y_r = n1 * (n2 - X_r) - n3 */
936 BN_CTX_free(new_ctx);
940 int ossl_ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point,
943 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
944 /* point is its own inverse */
947 return BN_usub(point->Y, group->field, point->Y);
950 int ossl_ec_GFp_simple_is_at_infinity(const EC_GROUP *group,
951 const EC_POINT *point)
953 return BN_is_zero(point->Z);
956 int ossl_ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
959 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
960 const BIGNUM *, BN_CTX *);
961 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
963 BN_CTX *new_ctx = NULL;
964 BIGNUM *rh, *tmp, *Z4, *Z6;
967 if (EC_POINT_is_at_infinity(group, point))
970 field_mul = group->meth->field_mul;
971 field_sqr = group->meth->field_sqr;
975 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
981 rh = BN_CTX_get(ctx);
982 tmp = BN_CTX_get(ctx);
983 Z4 = BN_CTX_get(ctx);
984 Z6 = BN_CTX_get(ctx);
989 * We have a curve defined by a Weierstrass equation
990 * y^2 = x^3 + a*x + b.
991 * The point to consider is given in Jacobian projective coordinates
992 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
993 * Substituting this and multiplying by Z^6 transforms the above equation into
994 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
995 * To test this, we add up the right-hand side in 'rh'.
999 if (!field_sqr(group, rh, point->X, ctx))
1002 if (!point->Z_is_one) {
1003 if (!field_sqr(group, tmp, point->Z, ctx))
1005 if (!field_sqr(group, Z4, tmp, ctx))
1007 if (!field_mul(group, Z6, Z4, tmp, ctx))
1010 /* rh := (rh + a*Z^4)*X */
1011 if (group->a_is_minus3) {
1012 if (!BN_mod_lshift1_quick(tmp, Z4, p))
1014 if (!BN_mod_add_quick(tmp, tmp, Z4, p))
1016 if (!BN_mod_sub_quick(rh, rh, tmp, p))
1018 if (!field_mul(group, rh, rh, point->X, ctx))
1021 if (!field_mul(group, tmp, Z4, group->a, ctx))
1023 if (!BN_mod_add_quick(rh, rh, tmp, p))
1025 if (!field_mul(group, rh, rh, point->X, ctx))
1029 /* rh := rh + b*Z^6 */
1030 if (!field_mul(group, tmp, group->b, Z6, ctx))
1032 if (!BN_mod_add_quick(rh, rh, tmp, p))
1035 /* point->Z_is_one */
1037 /* rh := (rh + a)*X */
1038 if (!BN_mod_add_quick(rh, rh, group->a, p))
1040 if (!field_mul(group, rh, rh, point->X, ctx))
1043 if (!BN_mod_add_quick(rh, rh, group->b, p))
1048 if (!field_sqr(group, tmp, point->Y, ctx))
1051 ret = (0 == BN_ucmp(tmp, rh));
1055 BN_CTX_free(new_ctx);
1059 int ossl_ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
1060 const EC_POINT *b, BN_CTX *ctx)
1065 * 0 equal (in affine coordinates)
1069 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
1070 const BIGNUM *, BN_CTX *);
1071 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1072 BN_CTX *new_ctx = NULL;
1073 BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1074 const BIGNUM *tmp1_, *tmp2_;
1077 if (EC_POINT_is_at_infinity(group, a)) {
1078 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1081 if (EC_POINT_is_at_infinity(group, b))
1084 if (a->Z_is_one && b->Z_is_one) {
1085 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
1088 field_mul = group->meth->field_mul;
1089 field_sqr = group->meth->field_sqr;
1092 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
1098 tmp1 = BN_CTX_get(ctx);
1099 tmp2 = BN_CTX_get(ctx);
1100 Za23 = BN_CTX_get(ctx);
1101 Zb23 = BN_CTX_get(ctx);
1106 * We have to decide whether
1107 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1108 * or equivalently, whether
1109 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1113 if (!field_sqr(group, Zb23, b->Z, ctx))
1115 if (!field_mul(group, tmp1, a->X, Zb23, ctx))
1121 if (!field_sqr(group, Za23, a->Z, ctx))
1123 if (!field_mul(group, tmp2, b->X, Za23, ctx))
1129 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1130 if (BN_cmp(tmp1_, tmp2_) != 0) {
1131 ret = 1; /* points differ */
1136 if (!field_mul(group, Zb23, Zb23, b->Z, ctx))
1138 if (!field_mul(group, tmp1, a->Y, Zb23, ctx))
1144 if (!field_mul(group, Za23, Za23, a->Z, ctx))
1146 if (!field_mul(group, tmp2, b->Y, Za23, ctx))
1152 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1153 if (BN_cmp(tmp1_, tmp2_) != 0) {
1154 ret = 1; /* points differ */
1158 /* points are equal */
1163 BN_CTX_free(new_ctx);
1167 int ossl_ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
1170 BN_CTX *new_ctx = NULL;
1174 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1178 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
1184 x = BN_CTX_get(ctx);
1185 y = BN_CTX_get(ctx);
1189 if (!EC_POINT_get_affine_coordinates(group, point, x, y, ctx))
1191 if (!EC_POINT_set_affine_coordinates(group, point, x, y, ctx))
1193 if (!point->Z_is_one) {
1194 ERR_raise(ERR_LIB_EC, ERR_R_INTERNAL_ERROR);
1202 BN_CTX_free(new_ctx);
1206 int ossl_ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num,
1207 EC_POINT *points[], BN_CTX *ctx)
1209 BN_CTX *new_ctx = NULL;
1210 BIGNUM *tmp, *tmp_Z;
1211 BIGNUM **prod_Z = NULL;
1219 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
1225 tmp = BN_CTX_get(ctx);
1226 tmp_Z = BN_CTX_get(ctx);
1230 prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0]));
1233 for (i = 0; i < num; i++) {
1234 prod_Z[i] = BN_new();
1235 if (prod_Z[i] == NULL)
1240 * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1241 * skipping any zero-valued inputs (pretend that they're 1).
1244 if (!BN_is_zero(points[0]->Z)) {
1245 if (!BN_copy(prod_Z[0], points[0]->Z))
1248 if (group->meth->field_set_to_one != 0) {
1249 if (!group->meth->field_set_to_one(group, prod_Z[0], ctx))
1252 if (!BN_one(prod_Z[0]))
1257 for (i = 1; i < num; i++) {
1258 if (!BN_is_zero(points[i]->Z)) {
1260 meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z,
1264 if (!BN_copy(prod_Z[i], prod_Z[i - 1]))
1270 * Now use a single explicit inversion to replace every non-zero
1271 * points[i]->Z by its inverse.
1274 if (!group->meth->field_inv(group, tmp, prod_Z[num - 1], ctx)) {
1275 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1278 if (group->meth->field_encode != 0) {
1280 * In the Montgomery case, we just turned R*H (representing H) into
1281 * 1/(R*H), but we need R*(1/H) (representing 1/H); i.e. we need to
1282 * multiply by the Montgomery factor twice.
1284 if (!group->meth->field_encode(group, tmp, tmp, ctx))
1286 if (!group->meth->field_encode(group, tmp, tmp, ctx))
1290 for (i = num - 1; i > 0; --i) {
1292 * Loop invariant: tmp is the product of the inverses of points[0]->Z
1293 * .. points[i]->Z (zero-valued inputs skipped).
1295 if (!BN_is_zero(points[i]->Z)) {
1297 * Set tmp_Z to the inverse of points[i]->Z (as product of Z
1298 * inverses 0 .. i, Z values 0 .. i - 1).
1301 meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx))
1304 * Update tmp to satisfy the loop invariant for i - 1.
1306 if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx))
1308 /* Replace points[i]->Z by its inverse. */
1309 if (!BN_copy(points[i]->Z, tmp_Z))
1314 if (!BN_is_zero(points[0]->Z)) {
1315 /* Replace points[0]->Z by its inverse. */
1316 if (!BN_copy(points[0]->Z, tmp))
1320 /* Finally, fix up the X and Y coordinates for all points. */
1322 for (i = 0; i < num; i++) {
1323 EC_POINT *p = points[i];
1325 if (!BN_is_zero(p->Z)) {
1326 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1328 if (!group->meth->field_sqr(group, tmp, p->Z, ctx))
1330 if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx))
1333 if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx))
1335 if (!group->meth->field_mul(group, p->Y, p->Y, tmp, ctx))
1338 if (group->meth->field_set_to_one != 0) {
1339 if (!group->meth->field_set_to_one(group, p->Z, ctx))
1353 BN_CTX_free(new_ctx);
1354 if (prod_Z != NULL) {
1355 for (i = 0; i < num; i++) {
1356 if (prod_Z[i] == NULL)
1358 BN_clear_free(prod_Z[i]);
1360 OPENSSL_free(prod_Z);
1365 int ossl_ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1366 const BIGNUM *b, BN_CTX *ctx)
1368 return BN_mod_mul(r, a, b, group->field, ctx);
1371 int ossl_ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1374 return BN_mod_sqr(r, a, group->field, ctx);
1378 * Computes the multiplicative inverse of a in GF(p), storing the result in r.
1379 * If a is zero (or equivalent), you'll get a EC_R_CANNOT_INVERT error.
1380 * Since we don't have a Mont structure here, SCA hardening is with blinding.
1381 * NB: "a" must be in _decoded_ form. (i.e. field_decode must precede.)
1383 int ossl_ec_GFp_simple_field_inv(const EC_GROUP *group, BIGNUM *r,
1384 const BIGNUM *a, BN_CTX *ctx)
1387 BN_CTX *new_ctx = NULL;
1391 && (ctx = new_ctx = BN_CTX_secure_new_ex(group->libctx)) == NULL)
1395 if ((e = BN_CTX_get(ctx)) == NULL)
1399 if (!BN_priv_rand_range_ex(e, group->field, ctx))
1401 } while (BN_is_zero(e));
1404 if (!group->meth->field_mul(group, r, a, e, ctx))
1406 /* r := 1/(a * e) */
1407 if (!BN_mod_inverse(r, r, group->field, ctx)) {
1408 ERR_raise(ERR_LIB_EC, EC_R_CANNOT_INVERT);
1411 /* r := e/(a * e) = 1/a */
1412 if (!group->meth->field_mul(group, r, r, e, ctx))
1419 BN_CTX_free(new_ctx);
1424 * Apply randomization of EC point projective coordinates:
1426 * (X, Y ,Z ) = (lambda^2*X, lambda^3*Y, lambda*Z)
1427 * lambda = [1,group->field)
1430 int ossl_ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p,
1434 BIGNUM *lambda = NULL;
1435 BIGNUM *temp = NULL;
1438 lambda = BN_CTX_get(ctx);
1439 temp = BN_CTX_get(ctx);
1441 ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
1446 * Make sure lambda is not zero.
1447 * If the RNG fails, we cannot blind but nevertheless want
1448 * code to continue smoothly and not clobber the error stack.
1452 ret = BN_priv_rand_range_ex(lambda, group->field, ctx);
1458 } while (BN_is_zero(lambda));
1460 /* if field_encode defined convert between representations */
1461 if ((group->meth->field_encode != NULL
1462 && !group->meth->field_encode(group, lambda, lambda, ctx))
1463 || !group->meth->field_mul(group, p->Z, p->Z, lambda, ctx)
1464 || !group->meth->field_sqr(group, temp, lambda, ctx)
1465 || !group->meth->field_mul(group, p->X, p->X, temp, ctx)
1466 || !group->meth->field_mul(group, temp, temp, lambda, ctx)
1467 || !group->meth->field_mul(group, p->Y, p->Y, temp, ctx))
1480 * - p: affine coordinates
1483 * - s := p, r := 2p: blinded projective (homogeneous) coordinates
1485 * For doubling we use Formula 3 from Izu-Takagi "A fast parallel elliptic curve
1486 * multiplication resistant against side channel attacks" appendix, described at
1487 * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#doubling-dbl-2002-it-2
1488 * simplified for Z1=1.
1490 * Blinding uses the equivalence relation (\lambda X, \lambda Y, \lambda Z)
1491 * for any non-zero \lambda that holds for projective (homogeneous) coords.
1493 int ossl_ec_GFp_simple_ladder_pre(const EC_GROUP *group,
1494 EC_POINT *r, EC_POINT *s,
1495 EC_POINT *p, BN_CTX *ctx)
1497 BIGNUM *t1, *t2, *t3, *t4, *t5 = NULL;
1505 if (!p->Z_is_one /* r := 2p */
1506 || !group->meth->field_sqr(group, t3, p->X, ctx)
1507 || !BN_mod_sub_quick(t4, t3, group->a, group->field)
1508 || !group->meth->field_sqr(group, t4, t4, ctx)
1509 || !group->meth->field_mul(group, t5, p->X, group->b, ctx)
1510 || !BN_mod_lshift_quick(t5, t5, 3, group->field)
1511 /* r->X coord output */
1512 || !BN_mod_sub_quick(r->X, t4, t5, group->field)
1513 || !BN_mod_add_quick(t1, t3, group->a, group->field)
1514 || !group->meth->field_mul(group, t2, p->X, t1, ctx)
1515 || !BN_mod_add_quick(t2, group->b, t2, group->field)
1516 /* r->Z coord output */
1517 || !BN_mod_lshift_quick(r->Z, t2, 2, group->field))
1520 /* make sure lambda (r->Y here for storage) is not zero */
1522 if (!BN_priv_rand_range_ex(r->Y, group->field, ctx))
1524 } while (BN_is_zero(r->Y));
1526 /* make sure lambda (s->Z here for storage) is not zero */
1528 if (!BN_priv_rand_range_ex(s->Z, group->field, ctx))
1530 } while (BN_is_zero(s->Z));
1532 /* if field_encode defined convert between representations */
1533 if (group->meth->field_encode != NULL
1534 && (!group->meth->field_encode(group, r->Y, r->Y, ctx)
1535 || !group->meth->field_encode(group, s->Z, s->Z, ctx)))
1538 /* blind r and s independently */
1539 if (!group->meth->field_mul(group, r->Z, r->Z, r->Y, ctx)
1540 || !group->meth->field_mul(group, r->X, r->X, r->Y, ctx)
1541 || !group->meth->field_mul(group, s->X, p->X, s->Z, ctx)) /* s := p */
1552 * - s, r: projective (homogeneous) coordinates
1553 * - p: affine coordinates
1556 * - s := r + s, r := 2r: projective (homogeneous) coordinates
1558 * Differential addition-and-doubling using Eq. (9) and (10) from Izu-Takagi
1559 * "A fast parallel elliptic curve multiplication resistant against side channel
1560 * attacks", as described at
1561 * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-mladd-2002-it-4
1563 int ossl_ec_GFp_simple_ladder_step(const EC_GROUP *group,
1564 EC_POINT *r, EC_POINT *s,
1565 EC_POINT *p, BN_CTX *ctx)
1568 BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
1571 t0 = BN_CTX_get(ctx);
1572 t1 = BN_CTX_get(ctx);
1573 t2 = BN_CTX_get(ctx);
1574 t3 = BN_CTX_get(ctx);
1575 t4 = BN_CTX_get(ctx);
1576 t5 = BN_CTX_get(ctx);
1577 t6 = BN_CTX_get(ctx);
1580 || !group->meth->field_mul(group, t6, r->X, s->X, ctx)
1581 || !group->meth->field_mul(group, t0, r->Z, s->Z, ctx)
1582 || !group->meth->field_mul(group, t4, r->X, s->Z, ctx)
1583 || !group->meth->field_mul(group, t3, r->Z, s->X, ctx)
1584 || !group->meth->field_mul(group, t5, group->a, t0, ctx)
1585 || !BN_mod_add_quick(t5, t6, t5, group->field)
1586 || !BN_mod_add_quick(t6, t3, t4, group->field)
1587 || !group->meth->field_mul(group, t5, t6, t5, ctx)
1588 || !group->meth->field_sqr(group, t0, t0, ctx)
1589 || !BN_mod_lshift_quick(t2, group->b, 2, group->field)
1590 || !group->meth->field_mul(group, t0, t2, t0, ctx)
1591 || !BN_mod_lshift1_quick(t5, t5, group->field)
1592 || !BN_mod_sub_quick(t3, t4, t3, group->field)
1593 /* s->Z coord output */
1594 || !group->meth->field_sqr(group, s->Z, t3, ctx)
1595 || !group->meth->field_mul(group, t4, s->Z, p->X, ctx)
1596 || !BN_mod_add_quick(t0, t0, t5, group->field)
1597 /* s->X coord output */
1598 || !BN_mod_sub_quick(s->X, t0, t4, group->field)
1599 || !group->meth->field_sqr(group, t4, r->X, ctx)
1600 || !group->meth->field_sqr(group, t5, r->Z, ctx)
1601 || !group->meth->field_mul(group, t6, t5, group->a, ctx)
1602 || !BN_mod_add_quick(t1, r->X, r->Z, group->field)
1603 || !group->meth->field_sqr(group, t1, t1, ctx)
1604 || !BN_mod_sub_quick(t1, t1, t4, group->field)
1605 || !BN_mod_sub_quick(t1, t1, t5, group->field)
1606 || !BN_mod_sub_quick(t3, t4, t6, group->field)
1607 || !group->meth->field_sqr(group, t3, t3, ctx)
1608 || !group->meth->field_mul(group, t0, t5, t1, ctx)
1609 || !group->meth->field_mul(group, t0, t2, t0, ctx)
1610 /* r->X coord output */
1611 || !BN_mod_sub_quick(r->X, t3, t0, group->field)
1612 || !BN_mod_add_quick(t3, t4, t6, group->field)
1613 || !group->meth->field_sqr(group, t4, t5, ctx)
1614 || !group->meth->field_mul(group, t4, t4, t2, ctx)
1615 || !group->meth->field_mul(group, t1, t1, t3, ctx)
1616 || !BN_mod_lshift1_quick(t1, t1, group->field)
1617 /* r->Z coord output */
1618 || !BN_mod_add_quick(r->Z, t4, t1, group->field))
1630 * - s, r: projective (homogeneous) coordinates
1631 * - p: affine coordinates
1634 * - r := (x,y): affine coordinates
1636 * Recovers the y-coordinate of r using Eq. (8) from Brier-Joye, "Weierstrass
1637 * Elliptic Curves and Side-Channel Attacks", modified to work in mixed
1638 * projective coords, i.e. p is affine and (r,s) in projective (homogeneous)
1639 * coords, and return r in affine coordinates.
1641 * X4 = two*Y1*X2*Z3*Z2;
1642 * Y4 = two*b*Z3*SQR(Z2) + Z3*(a*Z2+X1*X2)*(X1*Z2+X2) - X3*SQR(X1*Z2-X2);
1643 * Z4 = two*Y1*Z3*SQR(Z2);
1646 * - Z2==0 implies r is at infinity (handled by the BN_is_zero(r->Z) branch);
1647 * - Z3==0 implies s is at infinity (handled by the BN_is_zero(s->Z) branch);
1648 * - Y1==0 implies p has order 2, so either r or s are infinity and handled by
1649 * one of the BN_is_zero(...) branches.
1651 int ossl_ec_GFp_simple_ladder_post(const EC_GROUP *group,
1652 EC_POINT *r, EC_POINT *s,
1653 EC_POINT *p, BN_CTX *ctx)
1656 BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
1658 if (BN_is_zero(r->Z))
1659 return EC_POINT_set_to_infinity(group, r);
1661 if (BN_is_zero(s->Z)) {
1662 if (!EC_POINT_copy(r, p)
1663 || !EC_POINT_invert(group, r, ctx))
1669 t0 = BN_CTX_get(ctx);
1670 t1 = BN_CTX_get(ctx);
1671 t2 = BN_CTX_get(ctx);
1672 t3 = BN_CTX_get(ctx);
1673 t4 = BN_CTX_get(ctx);
1674 t5 = BN_CTX_get(ctx);
1675 t6 = BN_CTX_get(ctx);
1678 || !BN_mod_lshift1_quick(t4, p->Y, group->field)
1679 || !group->meth->field_mul(group, t6, r->X, t4, ctx)
1680 || !group->meth->field_mul(group, t6, s->Z, t6, ctx)
1681 || !group->meth->field_mul(group, t5, r->Z, t6, ctx)
1682 || !BN_mod_lshift1_quick(t1, group->b, group->field)
1683 || !group->meth->field_mul(group, t1, s->Z, t1, ctx)
1684 || !group->meth->field_sqr(group, t3, r->Z, ctx)
1685 || !group->meth->field_mul(group, t2, t3, t1, ctx)
1686 || !group->meth->field_mul(group, t6, r->Z, group->a, ctx)
1687 || !group->meth->field_mul(group, t1, p->X, r->X, ctx)
1688 || !BN_mod_add_quick(t1, t1, t6, group->field)
1689 || !group->meth->field_mul(group, t1, s->Z, t1, ctx)
1690 || !group->meth->field_mul(group, t0, p->X, r->Z, ctx)
1691 || !BN_mod_add_quick(t6, r->X, t0, group->field)
1692 || !group->meth->field_mul(group, t6, t6, t1, ctx)
1693 || !BN_mod_add_quick(t6, t6, t2, group->field)
1694 || !BN_mod_sub_quick(t0, t0, r->X, group->field)
1695 || !group->meth->field_sqr(group, t0, t0, ctx)
1696 || !group->meth->field_mul(group, t0, t0, s->X, ctx)
1697 || !BN_mod_sub_quick(t0, t6, t0, group->field)
1698 || !group->meth->field_mul(group, t1, s->Z, t4, ctx)
1699 || !group->meth->field_mul(group, t1, t3, t1, ctx)
1700 || (group->meth->field_decode != NULL
1701 && !group->meth->field_decode(group, t1, t1, ctx))
1702 || !group->meth->field_inv(group, t1, t1, ctx)
1703 || (group->meth->field_encode != NULL
1704 && !group->meth->field_encode(group, t1, t1, ctx))
1705 || !group->meth->field_mul(group, r->X, t5, t1, ctx)
1706 || !group->meth->field_mul(group, r->Y, t0, t1, ctx))
1709 if (group->meth->field_set_to_one != NULL) {
1710 if (!group->meth->field_set_to_one(group, r->Z, ctx))