2 * Copyright 2001-2019 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* ====================================================================
11 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
12 * Portions of this software developed by SUN MICROSYSTEMS, INC.,
13 * and contributed to the OpenSSL project.
16 #include <openssl/err.h>
17 #include <openssl/symhacks.h>
21 const EC_METHOD *EC_GFp_simple_method(void)
23 static const EC_METHOD ret = {
25 NID_X9_62_prime_field,
26 ec_GFp_simple_group_init,
27 ec_GFp_simple_group_finish,
28 ec_GFp_simple_group_clear_finish,
29 ec_GFp_simple_group_copy,
30 ec_GFp_simple_group_set_curve,
31 ec_GFp_simple_group_get_curve,
32 ec_GFp_simple_group_get_degree,
33 ec_group_simple_order_bits,
34 ec_GFp_simple_group_check_discriminant,
35 ec_GFp_simple_point_init,
36 ec_GFp_simple_point_finish,
37 ec_GFp_simple_point_clear_finish,
38 ec_GFp_simple_point_copy,
39 ec_GFp_simple_point_set_to_infinity,
40 ec_GFp_simple_set_Jprojective_coordinates_GFp,
41 ec_GFp_simple_get_Jprojective_coordinates_GFp,
42 ec_GFp_simple_point_set_affine_coordinates,
43 ec_GFp_simple_point_get_affine_coordinates,
48 ec_GFp_simple_is_at_infinity,
49 ec_GFp_simple_is_on_curve,
51 ec_GFp_simple_make_affine,
52 ec_GFp_simple_points_make_affine,
54 0 /* precompute_mult */ ,
55 0 /* have_precompute_mult */ ,
56 ec_GFp_simple_field_mul,
57 ec_GFp_simple_field_sqr,
59 ec_GFp_simple_field_inv,
60 0 /* field_encode */ ,
61 0 /* field_decode */ ,
62 0, /* field_set_to_one */
63 ec_key_simple_priv2oct,
64 ec_key_simple_oct2priv,
66 ec_key_simple_generate_key,
67 ec_key_simple_check_key,
68 ec_key_simple_generate_public_key,
71 ecdh_simple_compute_key,
72 ec_GFp_simple_blind_coordinates
79 * Most method functions in this file are designed to work with
80 * non-trivial representations of field elements if necessary
81 * (see ecp_mont.c): while standard modular addition and subtraction
82 * are used, the field_mul and field_sqr methods will be used for
83 * multiplication, and field_encode and field_decode (if defined)
84 * will be used for converting between representations.
86 * Functions ec_GFp_simple_points_make_affine() and
87 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
88 * that if a non-trivial representation is used, it is a Montgomery
89 * representation (i.e. 'encoding' means multiplying by some factor R).
92 int ec_GFp_simple_group_init(EC_GROUP *group)
94 group->field = BN_new();
97 if (group->field == NULL || group->a == NULL || group->b == NULL) {
98 BN_free(group->field);
103 group->a_is_minus3 = 0;
107 void ec_GFp_simple_group_finish(EC_GROUP *group)
109 BN_free(group->field);
114 void ec_GFp_simple_group_clear_finish(EC_GROUP *group)
116 BN_clear_free(group->field);
117 BN_clear_free(group->a);
118 BN_clear_free(group->b);
121 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
123 if (!BN_copy(dest->field, src->field))
125 if (!BN_copy(dest->a, src->a))
127 if (!BN_copy(dest->b, src->b))
130 dest->a_is_minus3 = src->a_is_minus3;
135 int ec_GFp_simple_group_set_curve(EC_GROUP *group,
136 const BIGNUM *p, const BIGNUM *a,
137 const BIGNUM *b, BN_CTX *ctx)
140 BN_CTX *new_ctx = NULL;
143 /* p must be a prime > 3 */
144 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
145 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD);
150 ctx = new_ctx = BN_CTX_new();
156 tmp_a = BN_CTX_get(ctx);
161 if (!BN_copy(group->field, p))
163 BN_set_negative(group->field, 0);
166 if (!BN_nnmod(tmp_a, a, p, ctx))
168 if (group->meth->field_encode) {
169 if (!group->meth->field_encode(group, group->a, tmp_a, ctx))
171 } else if (!BN_copy(group->a, tmp_a))
175 if (!BN_nnmod(group->b, b, p, ctx))
177 if (group->meth->field_encode)
178 if (!group->meth->field_encode(group, group->b, group->b, ctx))
181 /* group->a_is_minus3 */
182 if (!BN_add_word(tmp_a, 3))
184 group->a_is_minus3 = (0 == BN_cmp(tmp_a, group->field));
190 BN_CTX_free(new_ctx);
194 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
195 BIGNUM *b, BN_CTX *ctx)
198 BN_CTX *new_ctx = NULL;
201 if (!BN_copy(p, group->field))
205 if (a != NULL || b != NULL) {
206 if (group->meth->field_decode) {
208 ctx = new_ctx = BN_CTX_new();
213 if (!group->meth->field_decode(group, a, group->a, ctx))
217 if (!group->meth->field_decode(group, b, group->b, ctx))
222 if (!BN_copy(a, group->a))
226 if (!BN_copy(b, group->b))
235 BN_CTX_free(new_ctx);
239 int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
241 return BN_num_bits(group->field);
244 int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
247 BIGNUM *a, *b, *order, *tmp_1, *tmp_2;
248 const BIGNUM *p = group->field;
249 BN_CTX *new_ctx = NULL;
252 ctx = new_ctx = BN_CTX_new();
254 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT,
255 ERR_R_MALLOC_FAILURE);
262 tmp_1 = BN_CTX_get(ctx);
263 tmp_2 = BN_CTX_get(ctx);
264 order = BN_CTX_get(ctx);
268 if (group->meth->field_decode) {
269 if (!group->meth->field_decode(group, a, group->a, ctx))
271 if (!group->meth->field_decode(group, b, group->b, ctx))
274 if (!BN_copy(a, group->a))
276 if (!BN_copy(b, group->b))
281 * check the discriminant:
282 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
288 } else if (!BN_is_zero(b)) {
289 if (!BN_mod_sqr(tmp_1, a, p, ctx))
291 if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx))
293 if (!BN_lshift(tmp_1, tmp_2, 2))
297 if (!BN_mod_sqr(tmp_2, b, p, ctx))
299 if (!BN_mul_word(tmp_2, 27))
303 if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx))
313 BN_CTX_free(new_ctx);
317 int ec_GFp_simple_point_init(EC_POINT *point)
324 if (point->X == NULL || point->Y == NULL || point->Z == NULL) {
333 void ec_GFp_simple_point_finish(EC_POINT *point)
340 void ec_GFp_simple_point_clear_finish(EC_POINT *point)
342 BN_clear_free(point->X);
343 BN_clear_free(point->Y);
344 BN_clear_free(point->Z);
348 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
350 if (!BN_copy(dest->X, src->X))
352 if (!BN_copy(dest->Y, src->Y))
354 if (!BN_copy(dest->Z, src->Z))
356 dest->Z_is_one = src->Z_is_one;
357 dest->curve_name = src->curve_name;
362 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
370 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group,
377 BN_CTX *new_ctx = NULL;
381 ctx = new_ctx = BN_CTX_new();
387 if (!BN_nnmod(point->X, x, group->field, ctx))
389 if (group->meth->field_encode) {
390 if (!group->meth->field_encode(group, point->X, point->X, ctx))
396 if (!BN_nnmod(point->Y, y, group->field, ctx))
398 if (group->meth->field_encode) {
399 if (!group->meth->field_encode(group, point->Y, point->Y, ctx))
407 if (!BN_nnmod(point->Z, z, group->field, ctx))
409 Z_is_one = BN_is_one(point->Z);
410 if (group->meth->field_encode) {
411 if (Z_is_one && (group->meth->field_set_to_one != 0)) {
412 if (!group->meth->field_set_to_one(group, point->Z, ctx))
416 meth->field_encode(group, point->Z, point->Z, ctx))
420 point->Z_is_one = Z_is_one;
426 BN_CTX_free(new_ctx);
430 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group,
431 const EC_POINT *point,
432 BIGNUM *x, BIGNUM *y,
433 BIGNUM *z, BN_CTX *ctx)
435 BN_CTX *new_ctx = NULL;
438 if (group->meth->field_decode != 0) {
440 ctx = new_ctx = BN_CTX_new();
446 if (!group->meth->field_decode(group, x, point->X, ctx))
450 if (!group->meth->field_decode(group, y, point->Y, ctx))
454 if (!group->meth->field_decode(group, z, point->Z, ctx))
459 if (!BN_copy(x, point->X))
463 if (!BN_copy(y, point->Y))
467 if (!BN_copy(z, point->Z))
475 BN_CTX_free(new_ctx);
479 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
482 const BIGNUM *y, BN_CTX *ctx)
484 if (x == NULL || y == NULL) {
486 * unlike for projective coordinates, we do not tolerate this
488 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES,
489 ERR_R_PASSED_NULL_PARAMETER);
493 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y,
494 BN_value_one(), ctx);
497 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group,
498 const EC_POINT *point,
499 BIGNUM *x, BIGNUM *y,
502 BN_CTX *new_ctx = NULL;
503 BIGNUM *Z, *Z_1, *Z_2, *Z_3;
507 if (EC_POINT_is_at_infinity(group, point)) {
508 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES,
509 EC_R_POINT_AT_INFINITY);
514 ctx = new_ctx = BN_CTX_new();
521 Z_1 = BN_CTX_get(ctx);
522 Z_2 = BN_CTX_get(ctx);
523 Z_3 = BN_CTX_get(ctx);
527 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
529 if (group->meth->field_decode) {
530 if (!group->meth->field_decode(group, Z, point->Z, ctx))
538 if (group->meth->field_decode) {
540 if (!group->meth->field_decode(group, x, point->X, ctx))
544 if (!group->meth->field_decode(group, y, point->Y, ctx))
549 if (!BN_copy(x, point->X))
553 if (!BN_copy(y, point->Y))
558 if (!group->meth->field_inv(group, Z_1, Z_, ctx)) {
559 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES,
564 if (group->meth->field_encode == 0) {
565 /* field_sqr works on standard representation */
566 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx))
569 if (!BN_mod_sqr(Z_2, Z_1, group->field, ctx))
575 * in the Montgomery case, field_mul will cancel out Montgomery
578 if (!group->meth->field_mul(group, x, point->X, Z_2, ctx))
583 if (group->meth->field_encode == 0) {
585 * field_mul works on standard representation
587 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx))
590 if (!BN_mod_mul(Z_3, Z_2, Z_1, group->field, ctx))
595 * in the Montgomery case, field_mul will cancel out Montgomery
598 if (!group->meth->field_mul(group, y, point->Y, Z_3, ctx))
607 BN_CTX_free(new_ctx);
611 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
612 const EC_POINT *b, BN_CTX *ctx)
614 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
615 const BIGNUM *, BN_CTX *);
616 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
618 BN_CTX *new_ctx = NULL;
619 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
623 return EC_POINT_dbl(group, r, a, ctx);
624 if (EC_POINT_is_at_infinity(group, a))
625 return EC_POINT_copy(r, b);
626 if (EC_POINT_is_at_infinity(group, b))
627 return EC_POINT_copy(r, a);
629 field_mul = group->meth->field_mul;
630 field_sqr = group->meth->field_sqr;
634 ctx = new_ctx = BN_CTX_new();
640 n0 = BN_CTX_get(ctx);
641 n1 = BN_CTX_get(ctx);
642 n2 = BN_CTX_get(ctx);
643 n3 = BN_CTX_get(ctx);
644 n4 = BN_CTX_get(ctx);
645 n5 = BN_CTX_get(ctx);
646 n6 = BN_CTX_get(ctx);
651 * Note that in this function we must not read components of 'a' or 'b'
652 * once we have written the corresponding components of 'r'. ('r' might
653 * be one of 'a' or 'b'.)
658 if (!BN_copy(n1, a->X))
660 if (!BN_copy(n2, a->Y))
665 if (!field_sqr(group, n0, b->Z, ctx))
667 if (!field_mul(group, n1, a->X, n0, ctx))
669 /* n1 = X_a * Z_b^2 */
671 if (!field_mul(group, n0, n0, b->Z, ctx))
673 if (!field_mul(group, n2, a->Y, n0, ctx))
675 /* n2 = Y_a * Z_b^3 */
680 if (!BN_copy(n3, b->X))
682 if (!BN_copy(n4, b->Y))
687 if (!field_sqr(group, n0, a->Z, ctx))
689 if (!field_mul(group, n3, b->X, n0, ctx))
691 /* n3 = X_b * Z_a^2 */
693 if (!field_mul(group, n0, n0, a->Z, ctx))
695 if (!field_mul(group, n4, b->Y, n0, ctx))
697 /* n4 = Y_b * Z_a^3 */
701 if (!BN_mod_sub_quick(n5, n1, n3, p))
703 if (!BN_mod_sub_quick(n6, n2, n4, p))
708 if (BN_is_zero(n5)) {
709 if (BN_is_zero(n6)) {
710 /* a is the same point as b */
712 ret = EC_POINT_dbl(group, r, a, ctx);
716 /* a is the inverse of b */
725 if (!BN_mod_add_quick(n1, n1, n3, p))
727 if (!BN_mod_add_quick(n2, n2, n4, p))
733 if (a->Z_is_one && b->Z_is_one) {
734 if (!BN_copy(r->Z, n5))
738 if (!BN_copy(n0, b->Z))
740 } else if (b->Z_is_one) {
741 if (!BN_copy(n0, a->Z))
744 if (!field_mul(group, n0, a->Z, b->Z, ctx))
747 if (!field_mul(group, r->Z, n0, n5, ctx))
751 /* Z_r = Z_a * Z_b * n5 */
754 if (!field_sqr(group, n0, n6, ctx))
756 if (!field_sqr(group, n4, n5, ctx))
758 if (!field_mul(group, n3, n1, n4, ctx))
760 if (!BN_mod_sub_quick(r->X, n0, n3, p))
762 /* X_r = n6^2 - n5^2 * 'n7' */
765 if (!BN_mod_lshift1_quick(n0, r->X, p))
767 if (!BN_mod_sub_quick(n0, n3, n0, p))
769 /* n9 = n5^2 * 'n7' - 2 * X_r */
772 if (!field_mul(group, n0, n0, n6, ctx))
774 if (!field_mul(group, n5, n4, n5, ctx))
775 goto end; /* now n5 is n5^3 */
776 if (!field_mul(group, n1, n2, n5, ctx))
778 if (!BN_mod_sub_quick(n0, n0, n1, p))
781 if (!BN_add(n0, n0, p))
783 /* now 0 <= n0 < 2*p, and n0 is even */
784 if (!BN_rshift1(r->Y, n0))
786 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
791 if (ctx) /* otherwise we already called BN_CTX_end */
793 BN_CTX_free(new_ctx);
797 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
800 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
801 const BIGNUM *, BN_CTX *);
802 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
804 BN_CTX *new_ctx = NULL;
805 BIGNUM *n0, *n1, *n2, *n3;
808 if (EC_POINT_is_at_infinity(group, a)) {
814 field_mul = group->meth->field_mul;
815 field_sqr = group->meth->field_sqr;
819 ctx = new_ctx = BN_CTX_new();
825 n0 = BN_CTX_get(ctx);
826 n1 = BN_CTX_get(ctx);
827 n2 = BN_CTX_get(ctx);
828 n3 = BN_CTX_get(ctx);
833 * Note that in this function we must not read components of 'a' once we
834 * have written the corresponding components of 'r'. ('r' might the same
840 if (!field_sqr(group, n0, a->X, ctx))
842 if (!BN_mod_lshift1_quick(n1, n0, p))
844 if (!BN_mod_add_quick(n0, n0, n1, p))
846 if (!BN_mod_add_quick(n1, n0, group->a, p))
848 /* n1 = 3 * X_a^2 + a_curve */
849 } else if (group->a_is_minus3) {
850 if (!field_sqr(group, n1, a->Z, ctx))
852 if (!BN_mod_add_quick(n0, a->X, n1, p))
854 if (!BN_mod_sub_quick(n2, a->X, n1, p))
856 if (!field_mul(group, n1, n0, n2, ctx))
858 if (!BN_mod_lshift1_quick(n0, n1, p))
860 if (!BN_mod_add_quick(n1, n0, n1, p))
863 * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
864 * = 3 * X_a^2 - 3 * Z_a^4
867 if (!field_sqr(group, n0, a->X, ctx))
869 if (!BN_mod_lshift1_quick(n1, n0, p))
871 if (!BN_mod_add_quick(n0, n0, n1, p))
873 if (!field_sqr(group, n1, a->Z, ctx))
875 if (!field_sqr(group, n1, n1, ctx))
877 if (!field_mul(group, n1, n1, group->a, ctx))
879 if (!BN_mod_add_quick(n1, n1, n0, p))
881 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
886 if (!BN_copy(n0, a->Y))
889 if (!field_mul(group, n0, a->Y, a->Z, ctx))
892 if (!BN_mod_lshift1_quick(r->Z, n0, p))
895 /* Z_r = 2 * Y_a * Z_a */
898 if (!field_sqr(group, n3, a->Y, ctx))
900 if (!field_mul(group, n2, a->X, n3, ctx))
902 if (!BN_mod_lshift_quick(n2, n2, 2, p))
904 /* n2 = 4 * X_a * Y_a^2 */
907 if (!BN_mod_lshift1_quick(n0, n2, p))
909 if (!field_sqr(group, r->X, n1, ctx))
911 if (!BN_mod_sub_quick(r->X, r->X, n0, p))
913 /* X_r = n1^2 - 2 * n2 */
916 if (!field_sqr(group, n0, n3, ctx))
918 if (!BN_mod_lshift_quick(n3, n0, 3, p))
923 if (!BN_mod_sub_quick(n0, n2, r->X, p))
925 if (!field_mul(group, n0, n1, n0, ctx))
927 if (!BN_mod_sub_quick(r->Y, n0, n3, p))
929 /* Y_r = n1 * (n2 - X_r) - n3 */
935 BN_CTX_free(new_ctx);
939 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
941 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
942 /* point is its own inverse */
945 return BN_usub(point->Y, group->field, point->Y);
948 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
950 return BN_is_zero(point->Z);
953 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
956 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
957 const BIGNUM *, BN_CTX *);
958 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
960 BN_CTX *new_ctx = NULL;
961 BIGNUM *rh, *tmp, *Z4, *Z6;
964 if (EC_POINT_is_at_infinity(group, point))
967 field_mul = group->meth->field_mul;
968 field_sqr = group->meth->field_sqr;
972 ctx = new_ctx = BN_CTX_new();
978 rh = BN_CTX_get(ctx);
979 tmp = BN_CTX_get(ctx);
980 Z4 = BN_CTX_get(ctx);
981 Z6 = BN_CTX_get(ctx);
986 * We have a curve defined by a Weierstrass equation
987 * y^2 = x^3 + a*x + b.
988 * The point to consider is given in Jacobian projective coordinates
989 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
990 * Substituting this and multiplying by Z^6 transforms the above equation into
991 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
992 * To test this, we add up the right-hand side in 'rh'.
996 if (!field_sqr(group, rh, point->X, ctx))
999 if (!point->Z_is_one) {
1000 if (!field_sqr(group, tmp, point->Z, ctx))
1002 if (!field_sqr(group, Z4, tmp, ctx))
1004 if (!field_mul(group, Z6, Z4, tmp, ctx))
1007 /* rh := (rh + a*Z^4)*X */
1008 if (group->a_is_minus3) {
1009 if (!BN_mod_lshift1_quick(tmp, Z4, p))
1011 if (!BN_mod_add_quick(tmp, tmp, Z4, p))
1013 if (!BN_mod_sub_quick(rh, rh, tmp, p))
1015 if (!field_mul(group, rh, rh, point->X, ctx))
1018 if (!field_mul(group, tmp, Z4, group->a, ctx))
1020 if (!BN_mod_add_quick(rh, rh, tmp, p))
1022 if (!field_mul(group, rh, rh, point->X, ctx))
1026 /* rh := rh + b*Z^6 */
1027 if (!field_mul(group, tmp, group->b, Z6, ctx))
1029 if (!BN_mod_add_quick(rh, rh, tmp, p))
1032 /* point->Z_is_one */
1034 /* rh := (rh + a)*X */
1035 if (!BN_mod_add_quick(rh, rh, group->a, p))
1037 if (!field_mul(group, rh, rh, point->X, ctx))
1040 if (!BN_mod_add_quick(rh, rh, group->b, p))
1045 if (!field_sqr(group, tmp, point->Y, ctx))
1048 ret = (0 == BN_ucmp(tmp, rh));
1052 BN_CTX_free(new_ctx);
1056 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
1057 const EC_POINT *b, BN_CTX *ctx)
1062 * 0 equal (in affine coordinates)
1066 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
1067 const BIGNUM *, BN_CTX *);
1068 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1069 BN_CTX *new_ctx = NULL;
1070 BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1071 const BIGNUM *tmp1_, *tmp2_;
1074 if (EC_POINT_is_at_infinity(group, a)) {
1075 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1078 if (EC_POINT_is_at_infinity(group, b))
1081 if (a->Z_is_one && b->Z_is_one) {
1082 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
1085 field_mul = group->meth->field_mul;
1086 field_sqr = group->meth->field_sqr;
1089 ctx = new_ctx = BN_CTX_new();
1095 tmp1 = BN_CTX_get(ctx);
1096 tmp2 = BN_CTX_get(ctx);
1097 Za23 = BN_CTX_get(ctx);
1098 Zb23 = BN_CTX_get(ctx);
1103 * We have to decide whether
1104 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1105 * or equivalently, whether
1106 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1110 if (!field_sqr(group, Zb23, b->Z, ctx))
1112 if (!field_mul(group, tmp1, a->X, Zb23, ctx))
1118 if (!field_sqr(group, Za23, a->Z, ctx))
1120 if (!field_mul(group, tmp2, b->X, Za23, ctx))
1126 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1127 if (BN_cmp(tmp1_, tmp2_) != 0) {
1128 ret = 1; /* points differ */
1133 if (!field_mul(group, Zb23, Zb23, b->Z, ctx))
1135 if (!field_mul(group, tmp1, a->Y, Zb23, ctx))
1141 if (!field_mul(group, Za23, Za23, a->Z, ctx))
1143 if (!field_mul(group, tmp2, b->Y, Za23, ctx))
1149 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1150 if (BN_cmp(tmp1_, tmp2_) != 0) {
1151 ret = 1; /* points differ */
1155 /* points are equal */
1160 BN_CTX_free(new_ctx);
1164 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
1167 BN_CTX *new_ctx = NULL;
1171 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1175 ctx = new_ctx = BN_CTX_new();
1181 x = BN_CTX_get(ctx);
1182 y = BN_CTX_get(ctx);
1186 if (!EC_POINT_get_affine_coordinates_GFp(group, point, x, y, ctx))
1188 if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx))
1190 if (!point->Z_is_one) {
1191 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
1199 BN_CTX_free(new_ctx);
1203 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num,
1204 EC_POINT *points[], BN_CTX *ctx)
1206 BN_CTX *new_ctx = NULL;
1207 BIGNUM *tmp, *tmp_Z;
1208 BIGNUM **prod_Z = NULL;
1216 ctx = new_ctx = BN_CTX_new();
1222 tmp = BN_CTX_get(ctx);
1223 tmp_Z = BN_CTX_get(ctx);
1224 if (tmp == NULL || tmp_Z == NULL)
1227 prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0]));
1230 for (i = 0; i < num; i++) {
1231 prod_Z[i] = BN_new();
1232 if (prod_Z[i] == NULL)
1237 * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1238 * skipping any zero-valued inputs (pretend that they're 1).
1241 if (!BN_is_zero(points[0]->Z)) {
1242 if (!BN_copy(prod_Z[0], points[0]->Z))
1245 if (group->meth->field_set_to_one != 0) {
1246 if (!group->meth->field_set_to_one(group, prod_Z[0], ctx))
1249 if (!BN_one(prod_Z[0]))
1254 for (i = 1; i < num; i++) {
1255 if (!BN_is_zero(points[i]->Z)) {
1257 meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z,
1261 if (!BN_copy(prod_Z[i], prod_Z[i - 1]))
1267 * Now use a single explicit inversion to replace every non-zero
1268 * points[i]->Z by its inverse.
1271 if (!group->meth->field_inv(group, tmp, prod_Z[num - 1], ctx)) {
1272 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
1275 if (group->meth->field_encode != 0) {
1277 * In the Montgomery case, we just turned R*H (representing H) into
1278 * 1/(R*H), but we need R*(1/H) (representing 1/H); i.e. we need to
1279 * multiply by the Montgomery factor twice.
1281 if (!group->meth->field_encode(group, tmp, tmp, ctx))
1283 if (!group->meth->field_encode(group, tmp, tmp, ctx))
1287 for (i = num - 1; i > 0; --i) {
1289 * Loop invariant: tmp is the product of the inverses of points[0]->Z
1290 * .. points[i]->Z (zero-valued inputs skipped).
1292 if (!BN_is_zero(points[i]->Z)) {
1294 * Set tmp_Z to the inverse of points[i]->Z (as product of Z
1295 * inverses 0 .. i, Z values 0 .. i - 1).
1298 meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx))
1301 * Update tmp to satisfy the loop invariant for i - 1.
1303 if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx))
1305 /* Replace points[i]->Z by its inverse. */
1306 if (!BN_copy(points[i]->Z, tmp_Z))
1311 if (!BN_is_zero(points[0]->Z)) {
1312 /* Replace points[0]->Z by its inverse. */
1313 if (!BN_copy(points[0]->Z, tmp))
1317 /* Finally, fix up the X and Y coordinates for all points. */
1319 for (i = 0; i < num; i++) {
1320 EC_POINT *p = points[i];
1322 if (!BN_is_zero(p->Z)) {
1323 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1325 if (!group->meth->field_sqr(group, tmp, p->Z, ctx))
1327 if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx))
1330 if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx))
1332 if (!group->meth->field_mul(group, p->Y, p->Y, tmp, ctx))
1335 if (group->meth->field_set_to_one != 0) {
1336 if (!group->meth->field_set_to_one(group, p->Z, ctx))
1350 BN_CTX_free(new_ctx);
1351 if (prod_Z != NULL) {
1352 for (i = 0; i < num; i++) {
1353 if (prod_Z[i] == NULL)
1355 BN_clear_free(prod_Z[i]);
1357 OPENSSL_free(prod_Z);
1362 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1363 const BIGNUM *b, BN_CTX *ctx)
1365 return BN_mod_mul(r, a, b, group->field, ctx);
1368 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1371 return BN_mod_sqr(r, a, group->field, ctx);
1375 * Computes the multiplicative inverse of a in GF(p), storing the result in r.
1376 * If a is zero (or equivalent), you'll get a EC_R_CANNOT_INVERT error.
1377 * Since we don't have a Mont structure here, SCA hardening is with blinding.
1379 int ec_GFp_simple_field_inv(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1383 BN_CTX *new_ctx = NULL;
1386 if (ctx == NULL && (ctx = new_ctx = BN_CTX_secure_new()) == NULL)
1390 if ((e = BN_CTX_get(ctx)) == NULL)
1394 if (!BN_rand_range(e, group->field))
1396 } while (BN_is_zero(e));
1399 if (!group->meth->field_mul(group, r, a, e, ctx))
1401 /* r := 1/(a * e) */
1402 if (!BN_mod_inverse(r, r, group->field, ctx)) {
1403 ECerr(EC_F_EC_GFP_SIMPLE_FIELD_INV, EC_R_CANNOT_INVERT);
1406 /* r := e/(a * e) = 1/a */
1407 if (!group->meth->field_mul(group, r, r, e, ctx))
1414 BN_CTX_free(new_ctx);
1419 * Apply randomization of EC point projective coordinates:
1421 * (X, Y ,Z ) = (lambda^2*X, lambda^3*Y, lambda*Z)
1422 * lambda = [1,group->field)
1425 int ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p,
1429 BIGNUM *lambda = NULL;
1430 BIGNUM *temp = NULL;
1433 lambda = BN_CTX_get(ctx);
1434 temp = BN_CTX_get(ctx);
1436 ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_MALLOC_FAILURE);
1440 /* make sure lambda is not zero */
1442 if (!BN_rand_range(lambda, group->field)) {
1443 ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_BN_LIB);
1446 } while (BN_is_zero(lambda));
1448 /* if field_encode defined convert between representations */
1449 if (group->meth->field_encode != NULL
1450 && !group->meth->field_encode(group, lambda, lambda, ctx))
1452 if (!group->meth->field_mul(group, p->Z, p->Z, lambda, ctx))
1454 if (!group->meth->field_sqr(group, temp, lambda, ctx))
1456 if (!group->meth->field_mul(group, p->X, p->X, temp, ctx))
1458 if (!group->meth->field_mul(group, temp, temp, lambda, ctx))
1460 if (!group->meth->field_mul(group, p->Y, p->Y, temp, ctx))