1 /* crypto/ec/ecp_smpl.c */
2 /* Includes code written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
3 * for the OpenSSL project.
4 * Includes code written by Bodo Moeller for the OpenSSL project.
6 /* ====================================================================
7 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
9 * Redistribution and use in source and binary forms, with or without
10 * modification, are permitted provided that the following conditions
13 * 1. Redistributions of source code must retain the above copyright
14 * notice, this list of conditions and the following disclaimer.
16 * 2. Redistributions in binary form must reproduce the above copyright
17 * notice, this list of conditions and the following disclaimer in
18 * the documentation and/or other materials provided with the
21 * 3. All advertising materials mentioning features or use of this
22 * software must display the following acknowledgment:
23 * "This product includes software developed by the OpenSSL Project
24 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
26 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
27 * endorse or promote products derived from this software without
28 * prior written permission. For written permission, please contact
29 * openssl-core@openssl.org.
31 * 5. Products derived from this software may not be called "OpenSSL"
32 * nor may "OpenSSL" appear in their names without prior written
33 * permission of the OpenSSL Project.
35 * 6. Redistributions of any form whatsoever must retain the following
37 * "This product includes software developed by the OpenSSL Project
38 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
40 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
41 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
42 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
43 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
44 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
45 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
46 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
47 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
48 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
49 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
50 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
51 * OF THE POSSIBILITY OF SUCH DAMAGE.
52 * ====================================================================
54 * This product includes cryptographic software written by Eric Young
55 * (eay@cryptsoft.com). This product includes software written by Tim
56 * Hudson (tjh@cryptsoft.com).
59 /* ====================================================================
60 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
61 * Portions of this software developed by SUN MICROSYSTEMS, INC.,
62 * and contributed to the OpenSSL project.
65 #include <openssl/err.h>
66 #include <openssl/symhacks.h>
69 #include <openssl/fips.h>
74 const EC_METHOD *EC_GFp_simple_method(void)
76 static const EC_METHOD ret = {
78 NID_X9_62_prime_field,
79 ec_GFp_simple_group_init,
80 ec_GFp_simple_group_finish,
81 ec_GFp_simple_group_clear_finish,
82 ec_GFp_simple_group_copy,
83 ec_GFp_simple_group_set_curve,
84 ec_GFp_simple_group_get_curve,
85 ec_GFp_simple_group_get_degree,
86 ec_GFp_simple_group_check_discriminant,
87 ec_GFp_simple_point_init,
88 ec_GFp_simple_point_finish,
89 ec_GFp_simple_point_clear_finish,
90 ec_GFp_simple_point_copy,
91 ec_GFp_simple_point_set_to_infinity,
92 ec_GFp_simple_set_Jprojective_coordinates_GFp,
93 ec_GFp_simple_get_Jprojective_coordinates_GFp,
94 ec_GFp_simple_point_set_affine_coordinates,
95 ec_GFp_simple_point_get_affine_coordinates,
100 ec_GFp_simple_is_at_infinity,
101 ec_GFp_simple_is_on_curve,
103 ec_GFp_simple_make_affine,
104 ec_GFp_simple_points_make_affine,
106 0 /* precompute_mult */,
107 0 /* have_precompute_mult */,
108 ec_GFp_simple_field_mul,
109 ec_GFp_simple_field_sqr,
111 0 /* field_encode */,
112 0 /* field_decode */,
113 0 /* field_set_to_one */ };
117 return fips_ec_gfp_simple_method();
124 /* Most method functions in this file are designed to work with
125 * non-trivial representations of field elements if necessary
126 * (see ecp_mont.c): while standard modular addition and subtraction
127 * are used, the field_mul and field_sqr methods will be used for
128 * multiplication, and field_encode and field_decode (if defined)
129 * will be used for converting between representations.
131 * Functions ec_GFp_simple_points_make_affine() and
132 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
133 * that if a non-trivial representation is used, it is a Montgomery
134 * representation (i.e. 'encoding' means multiplying by some factor R).
138 int ec_GFp_simple_group_init(EC_GROUP *group)
140 BN_init(&group->field);
143 group->a_is_minus3 = 0;
148 void ec_GFp_simple_group_finish(EC_GROUP *group)
150 BN_free(&group->field);
156 void ec_GFp_simple_group_clear_finish(EC_GROUP *group)
158 BN_clear_free(&group->field);
159 BN_clear_free(&group->a);
160 BN_clear_free(&group->b);
164 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
166 if (!BN_copy(&dest->field, &src->field)) return 0;
167 if (!BN_copy(&dest->a, &src->a)) return 0;
168 if (!BN_copy(&dest->b, &src->b)) return 0;
170 dest->a_is_minus3 = src->a_is_minus3;
176 int ec_GFp_simple_group_set_curve(EC_GROUP *group,
177 const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
180 BN_CTX *new_ctx = NULL;
183 /* p must be a prime > 3 */
184 if (BN_num_bits(p) <= 2 || !BN_is_odd(p))
186 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD);
192 ctx = new_ctx = BN_CTX_new();
198 tmp_a = BN_CTX_get(ctx);
199 if (tmp_a == NULL) goto err;
202 if (!BN_copy(&group->field, p)) goto err;
203 BN_set_negative(&group->field, 0);
206 if (!BN_nnmod(tmp_a, a, p, ctx)) goto err;
207 if (group->meth->field_encode)
208 { if (!group->meth->field_encode(group, &group->a, tmp_a, ctx)) goto err; }
210 if (!BN_copy(&group->a, tmp_a)) goto err;
213 if (!BN_nnmod(&group->b, b, p, ctx)) goto err;
214 if (group->meth->field_encode)
215 if (!group->meth->field_encode(group, &group->b, &group->b, ctx)) goto err;
217 /* group->a_is_minus3 */
218 if (!BN_add_word(tmp_a, 3)) goto err;
219 group->a_is_minus3 = (0 == BN_cmp(tmp_a, &group->field));
226 BN_CTX_free(new_ctx);
231 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
234 BN_CTX *new_ctx = NULL;
238 if (!BN_copy(p, &group->field)) return 0;
241 if (a != NULL || b != NULL)
243 if (group->meth->field_decode)
247 ctx = new_ctx = BN_CTX_new();
253 if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err;
257 if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err;
264 if (!BN_copy(a, &group->a)) goto err;
268 if (!BN_copy(b, &group->b)) goto err;
277 BN_CTX_free(new_ctx);
282 int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
284 return BN_num_bits(&group->field);
288 int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
291 BIGNUM *a,*b,*order,*tmp_1,*tmp_2;
292 const BIGNUM *p = &group->field;
293 BN_CTX *new_ctx = NULL;
297 ctx = new_ctx = BN_CTX_new();
300 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT, ERR_R_MALLOC_FAILURE);
307 tmp_1 = BN_CTX_get(ctx);
308 tmp_2 = BN_CTX_get(ctx);
309 order = BN_CTX_get(ctx);
310 if (order == NULL) goto err;
312 if (group->meth->field_decode)
314 if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err;
315 if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err;
319 if (!BN_copy(a, &group->a)) goto err;
320 if (!BN_copy(b, &group->b)) goto err;
323 /* check the discriminant:
324 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
328 if (BN_is_zero(b)) goto err;
330 else if (!BN_is_zero(b))
332 if (!BN_mod_sqr(tmp_1, a, p, ctx)) goto err;
333 if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx)) goto err;
334 if (!BN_lshift(tmp_1, tmp_2, 2)) goto err;
337 if (!BN_mod_sqr(tmp_2, b, p, ctx)) goto err;
338 if (!BN_mul_word(tmp_2, 27)) goto err;
341 if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx)) goto err;
342 if (BN_is_zero(a)) goto err;
350 BN_CTX_free(new_ctx);
355 int ec_GFp_simple_point_init(EC_POINT *point)
366 void ec_GFp_simple_point_finish(EC_POINT *point)
374 void ec_GFp_simple_point_clear_finish(EC_POINT *point)
376 BN_clear_free(&point->X);
377 BN_clear_free(&point->Y);
378 BN_clear_free(&point->Z);
383 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
385 if (!BN_copy(&dest->X, &src->X)) return 0;
386 if (!BN_copy(&dest->Y, &src->Y)) return 0;
387 if (!BN_copy(&dest->Z, &src->Z)) return 0;
388 dest->Z_is_one = src->Z_is_one;
394 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, EC_POINT *point)
402 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group, EC_POINT *point,
403 const BIGNUM *x, const BIGNUM *y, const BIGNUM *z, BN_CTX *ctx)
405 BN_CTX *new_ctx = NULL;
410 ctx = new_ctx = BN_CTX_new();
417 if (!BN_nnmod(&point->X, x, &group->field, ctx)) goto err;
418 if (group->meth->field_encode)
420 if (!group->meth->field_encode(group, &point->X, &point->X, ctx)) goto err;
426 if (!BN_nnmod(&point->Y, y, &group->field, ctx)) goto err;
427 if (group->meth->field_encode)
429 if (!group->meth->field_encode(group, &point->Y, &point->Y, ctx)) goto err;
437 if (!BN_nnmod(&point->Z, z, &group->field, ctx)) goto err;
438 Z_is_one = BN_is_one(&point->Z);
439 if (group->meth->field_encode)
441 if (Z_is_one && (group->meth->field_set_to_one != 0))
443 if (!group->meth->field_set_to_one(group, &point->Z, ctx)) goto err;
447 if (!group->meth->field_encode(group, &point->Z, &point->Z, ctx)) goto err;
450 point->Z_is_one = Z_is_one;
457 BN_CTX_free(new_ctx);
462 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group, const EC_POINT *point,
463 BIGNUM *x, BIGNUM *y, BIGNUM *z, BN_CTX *ctx)
465 BN_CTX *new_ctx = NULL;
468 if (group->meth->field_decode != 0)
472 ctx = new_ctx = BN_CTX_new();
479 if (!group->meth->field_decode(group, x, &point->X, ctx)) goto err;
483 if (!group->meth->field_decode(group, y, &point->Y, ctx)) goto err;
487 if (!group->meth->field_decode(group, z, &point->Z, ctx)) goto err;
494 if (!BN_copy(x, &point->X)) goto err;
498 if (!BN_copy(y, &point->Y)) goto err;
502 if (!BN_copy(z, &point->Z)) goto err;
510 BN_CTX_free(new_ctx);
515 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group, EC_POINT *point,
516 const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx)
518 if (x == NULL || y == NULL)
520 /* unlike for projective coordinates, we do not tolerate this */
521 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES, ERR_R_PASSED_NULL_PARAMETER);
525 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y, BN_value_one(), ctx);
529 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point,
530 BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
532 BN_CTX *new_ctx = NULL;
533 BIGNUM *Z, *Z_1, *Z_2, *Z_3;
537 if (EC_POINT_is_at_infinity(group, point))
539 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, EC_R_POINT_AT_INFINITY);
545 ctx = new_ctx = BN_CTX_new();
552 Z_1 = BN_CTX_get(ctx);
553 Z_2 = BN_CTX_get(ctx);
554 Z_3 = BN_CTX_get(ctx);
555 if (Z_3 == NULL) goto err;
557 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
559 if (group->meth->field_decode)
561 if (!group->meth->field_decode(group, Z, &point->Z, ctx)) goto err;
571 if (group->meth->field_decode)
575 if (!group->meth->field_decode(group, x, &point->X, ctx)) goto err;
579 if (!group->meth->field_decode(group, y, &point->Y, ctx)) goto err;
586 if (!BN_copy(x, &point->X)) goto err;
590 if (!BN_copy(y, &point->Y)) goto err;
596 if (!BN_mod_inverse(Z_1, Z_, &group->field, ctx))
598 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB);
602 if (group->meth->field_encode == 0)
604 /* field_sqr works on standard representation */
605 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx)) goto err;
609 if (!BN_mod_sqr(Z_2, Z_1, &group->field, ctx)) goto err;
614 /* in the Montgomery case, field_mul will cancel out Montgomery factor in X: */
615 if (!group->meth->field_mul(group, x, &point->X, Z_2, ctx)) goto err;
620 if (group->meth->field_encode == 0)
622 /* field_mul works on standard representation */
623 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx)) goto err;
627 if (!BN_mod_mul(Z_3, Z_2, Z_1, &group->field, ctx)) goto err;
630 /* in the Montgomery case, field_mul will cancel out Montgomery factor in Y: */
631 if (!group->meth->field_mul(group, y, &point->Y, Z_3, ctx)) goto err;
640 BN_CTX_free(new_ctx);
644 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
646 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
647 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
649 BN_CTX *new_ctx = NULL;
650 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
654 return EC_POINT_dbl(group, r, a, ctx);
655 if (EC_POINT_is_at_infinity(group, a))
656 return EC_POINT_copy(r, b);
657 if (EC_POINT_is_at_infinity(group, b))
658 return EC_POINT_copy(r, a);
660 field_mul = group->meth->field_mul;
661 field_sqr = group->meth->field_sqr;
666 ctx = new_ctx = BN_CTX_new();
672 n0 = BN_CTX_get(ctx);
673 n1 = BN_CTX_get(ctx);
674 n2 = BN_CTX_get(ctx);
675 n3 = BN_CTX_get(ctx);
676 n4 = BN_CTX_get(ctx);
677 n5 = BN_CTX_get(ctx);
678 n6 = BN_CTX_get(ctx);
679 if (n6 == NULL) goto end;
681 /* Note that in this function we must not read components of 'a' or 'b'
682 * once we have written the corresponding components of 'r'.
683 * ('r' might be one of 'a' or 'b'.)
689 if (!BN_copy(n1, &a->X)) goto end;
690 if (!BN_copy(n2, &a->Y)) goto end;
696 if (!field_sqr(group, n0, &b->Z, ctx)) goto end;
697 if (!field_mul(group, n1, &a->X, n0, ctx)) goto end;
698 /* n1 = X_a * Z_b^2 */
700 if (!field_mul(group, n0, n0, &b->Z, ctx)) goto end;
701 if (!field_mul(group, n2, &a->Y, n0, ctx)) goto end;
702 /* n2 = Y_a * Z_b^3 */
708 if (!BN_copy(n3, &b->X)) goto end;
709 if (!BN_copy(n4, &b->Y)) goto end;
715 if (!field_sqr(group, n0, &a->Z, ctx)) goto end;
716 if (!field_mul(group, n3, &b->X, n0, ctx)) goto end;
717 /* n3 = X_b * Z_a^2 */
719 if (!field_mul(group, n0, n0, &a->Z, ctx)) goto end;
720 if (!field_mul(group, n4, &b->Y, n0, ctx)) goto end;
721 /* n4 = Y_b * Z_a^3 */
725 if (!BN_mod_sub_quick(n5, n1, n3, p)) goto end;
726 if (!BN_mod_sub_quick(n6, n2, n4, p)) goto end;
734 /* a is the same point as b */
736 ret = EC_POINT_dbl(group, r, a, ctx);
742 /* a is the inverse of b */
751 if (!BN_mod_add_quick(n1, n1, n3, p)) goto end;
752 if (!BN_mod_add_quick(n2, n2, n4, p)) goto end;
757 if (a->Z_is_one && b->Z_is_one)
759 if (!BN_copy(&r->Z, n5)) goto end;
764 { if (!BN_copy(n0, &b->Z)) goto end; }
765 else if (b->Z_is_one)
766 { if (!BN_copy(n0, &a->Z)) goto end; }
768 { if (!field_mul(group, n0, &a->Z, &b->Z, ctx)) goto end; }
769 if (!field_mul(group, &r->Z, n0, n5, ctx)) goto end;
772 /* Z_r = Z_a * Z_b * n5 */
775 if (!field_sqr(group, n0, n6, ctx)) goto end;
776 if (!field_sqr(group, n4, n5, ctx)) goto end;
777 if (!field_mul(group, n3, n1, n4, ctx)) goto end;
778 if (!BN_mod_sub_quick(&r->X, n0, n3, p)) goto end;
779 /* X_r = n6^2 - n5^2 * 'n7' */
782 if (!BN_mod_lshift1_quick(n0, &r->X, p)) goto end;
783 if (!BN_mod_sub_quick(n0, n3, n0, p)) goto end;
784 /* n9 = n5^2 * 'n7' - 2 * X_r */
787 if (!field_mul(group, n0, n0, n6, ctx)) goto end;
788 if (!field_mul(group, n5, n4, n5, ctx)) goto end; /* now n5 is n5^3 */
789 if (!field_mul(group, n1, n2, n5, ctx)) goto end;
790 if (!BN_mod_sub_quick(n0, n0, n1, p)) goto end;
792 if (!BN_add(n0, n0, p)) goto end;
793 /* now 0 <= n0 < 2*p, and n0 is even */
794 if (!BN_rshift1(&r->Y, n0)) goto end;
795 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
800 if (ctx) /* otherwise we already called BN_CTX_end */
803 BN_CTX_free(new_ctx);
808 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, BN_CTX *ctx)
810 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
811 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
813 BN_CTX *new_ctx = NULL;
814 BIGNUM *n0, *n1, *n2, *n3;
817 if (EC_POINT_is_at_infinity(group, a))
824 field_mul = group->meth->field_mul;
825 field_sqr = group->meth->field_sqr;
830 ctx = new_ctx = BN_CTX_new();
836 n0 = BN_CTX_get(ctx);
837 n1 = BN_CTX_get(ctx);
838 n2 = BN_CTX_get(ctx);
839 n3 = BN_CTX_get(ctx);
840 if (n3 == NULL) goto err;
842 /* Note that in this function we must not read components of 'a'
843 * once we have written the corresponding components of 'r'.
844 * ('r' might the same as 'a'.)
850 if (!field_sqr(group, n0, &a->X, ctx)) goto err;
851 if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
852 if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
853 if (!BN_mod_add_quick(n1, n0, &group->a, p)) goto err;
854 /* n1 = 3 * X_a^2 + a_curve */
856 else if (group->a_is_minus3)
858 if (!field_sqr(group, n1, &a->Z, ctx)) goto err;
859 if (!BN_mod_add_quick(n0, &a->X, n1, p)) goto err;
860 if (!BN_mod_sub_quick(n2, &a->X, n1, p)) goto err;
861 if (!field_mul(group, n1, n0, n2, ctx)) goto err;
862 if (!BN_mod_lshift1_quick(n0, n1, p)) goto err;
863 if (!BN_mod_add_quick(n1, n0, n1, p)) goto err;
864 /* n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
865 * = 3 * X_a^2 - 3 * Z_a^4 */
869 if (!field_sqr(group, n0, &a->X, ctx)) goto err;
870 if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
871 if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
872 if (!field_sqr(group, n1, &a->Z, ctx)) goto err;
873 if (!field_sqr(group, n1, n1, ctx)) goto err;
874 if (!field_mul(group, n1, n1, &group->a, ctx)) goto err;
875 if (!BN_mod_add_quick(n1, n1, n0, p)) goto err;
876 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
882 if (!BN_copy(n0, &a->Y)) goto err;
886 if (!field_mul(group, n0, &a->Y, &a->Z, ctx)) goto err;
888 if (!BN_mod_lshift1_quick(&r->Z, n0, p)) goto err;
890 /* Z_r = 2 * Y_a * Z_a */
893 if (!field_sqr(group, n3, &a->Y, ctx)) goto err;
894 if (!field_mul(group, n2, &a->X, n3, ctx)) goto err;
895 if (!BN_mod_lshift_quick(n2, n2, 2, p)) goto err;
896 /* n2 = 4 * X_a * Y_a^2 */
899 if (!BN_mod_lshift1_quick(n0, n2, p)) goto err;
900 if (!field_sqr(group, &r->X, n1, ctx)) goto err;
901 if (!BN_mod_sub_quick(&r->X, &r->X, n0, p)) goto err;
902 /* X_r = n1^2 - 2 * n2 */
905 if (!field_sqr(group, n0, n3, ctx)) goto err;
906 if (!BN_mod_lshift_quick(n3, n0, 3, p)) goto err;
910 if (!BN_mod_sub_quick(n0, n2, &r->X, p)) goto err;
911 if (!field_mul(group, n0, n1, n0, ctx)) goto err;
912 if (!BN_mod_sub_quick(&r->Y, n0, n3, p)) goto err;
913 /* Y_r = n1 * (n2 - X_r) - n3 */
920 BN_CTX_free(new_ctx);
925 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
927 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(&point->Y))
928 /* point is its own inverse */
931 return BN_usub(&point->Y, &group->field, &point->Y);
935 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
937 return BN_is_zero(&point->Z);
941 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point, BN_CTX *ctx)
943 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
944 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
946 BN_CTX *new_ctx = NULL;
947 BIGNUM *rh, *tmp, *Z4, *Z6;
950 if (EC_POINT_is_at_infinity(group, point))
953 field_mul = group->meth->field_mul;
954 field_sqr = group->meth->field_sqr;
959 ctx = new_ctx = BN_CTX_new();
965 rh = BN_CTX_get(ctx);
966 tmp = BN_CTX_get(ctx);
967 Z4 = BN_CTX_get(ctx);
968 Z6 = BN_CTX_get(ctx);
969 if (Z6 == NULL) goto err;
971 /* We have a curve defined by a Weierstrass equation
972 * y^2 = x^3 + a*x + b.
973 * The point to consider is given in Jacobian projective coordinates
974 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
975 * Substituting this and multiplying by Z^6 transforms the above equation into
976 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
977 * To test this, we add up the right-hand side in 'rh'.
981 if (!field_sqr(group, rh, &point->X, ctx)) goto err;
983 if (!point->Z_is_one)
985 if (!field_sqr(group, tmp, &point->Z, ctx)) goto err;
986 if (!field_sqr(group, Z4, tmp, ctx)) goto err;
987 if (!field_mul(group, Z6, Z4, tmp, ctx)) goto err;
989 /* rh := (rh + a*Z^4)*X */
990 if (group->a_is_minus3)
992 if (!BN_mod_lshift1_quick(tmp, Z4, p)) goto err;
993 if (!BN_mod_add_quick(tmp, tmp, Z4, p)) goto err;
994 if (!BN_mod_sub_quick(rh, rh, tmp, p)) goto err;
995 if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
999 if (!field_mul(group, tmp, Z4, &group->a, ctx)) goto err;
1000 if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
1001 if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
1004 /* rh := rh + b*Z^6 */
1005 if (!field_mul(group, tmp, &group->b, Z6, ctx)) goto err;
1006 if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
1010 /* point->Z_is_one */
1012 /* rh := (rh + a)*X */
1013 if (!BN_mod_add_quick(rh, rh, &group->a, p)) goto err;
1014 if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
1016 if (!BN_mod_add_quick(rh, rh, &group->b, p)) goto err;
1020 if (!field_sqr(group, tmp, &point->Y, ctx)) goto err;
1022 ret = (0 == BN_ucmp(tmp, rh));
1026 if (new_ctx != NULL)
1027 BN_CTX_free(new_ctx);
1032 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
1036 * 0 equal (in affine coordinates)
1040 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
1041 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1042 BN_CTX *new_ctx = NULL;
1043 BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1044 const BIGNUM *tmp1_, *tmp2_;
1047 if (EC_POINT_is_at_infinity(group, a))
1049 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1052 if (EC_POINT_is_at_infinity(group, b))
1055 if (a->Z_is_one && b->Z_is_one)
1057 return ((BN_cmp(&a->X, &b->X) == 0) && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
1060 field_mul = group->meth->field_mul;
1061 field_sqr = group->meth->field_sqr;
1065 ctx = new_ctx = BN_CTX_new();
1071 tmp1 = BN_CTX_get(ctx);
1072 tmp2 = BN_CTX_get(ctx);
1073 Za23 = BN_CTX_get(ctx);
1074 Zb23 = BN_CTX_get(ctx);
1075 if (Zb23 == NULL) goto end;
1077 /* We have to decide whether
1078 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1079 * or equivalently, whether
1080 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1085 if (!field_sqr(group, Zb23, &b->Z, ctx)) goto end;
1086 if (!field_mul(group, tmp1, &a->X, Zb23, ctx)) goto end;
1093 if (!field_sqr(group, Za23, &a->Z, ctx)) goto end;
1094 if (!field_mul(group, tmp2, &b->X, Za23, ctx)) goto end;
1100 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1101 if (BN_cmp(tmp1_, tmp2_) != 0)
1103 ret = 1; /* points differ */
1110 if (!field_mul(group, Zb23, Zb23, &b->Z, ctx)) goto end;
1111 if (!field_mul(group, tmp1, &a->Y, Zb23, ctx)) goto end;
1118 if (!field_mul(group, Za23, Za23, &a->Z, ctx)) goto end;
1119 if (!field_mul(group, tmp2, &b->Y, Za23, ctx)) goto end;
1125 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1126 if (BN_cmp(tmp1_, tmp2_) != 0)
1128 ret = 1; /* points differ */
1132 /* points are equal */
1137 if (new_ctx != NULL)
1138 BN_CTX_free(new_ctx);
1143 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
1145 BN_CTX *new_ctx = NULL;
1149 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1154 ctx = new_ctx = BN_CTX_new();
1160 x = BN_CTX_get(ctx);
1161 y = BN_CTX_get(ctx);
1162 if (y == NULL) goto err;
1164 if (!EC_POINT_get_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
1165 if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
1166 if (!point->Z_is_one)
1168 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
1176 if (new_ctx != NULL)
1177 BN_CTX_free(new_ctx);
1182 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num, EC_POINT *points[], BN_CTX *ctx)
1184 BN_CTX *new_ctx = NULL;
1185 BIGNUM *tmp, *tmp_Z;
1186 BIGNUM **prod_Z = NULL;
1195 ctx = new_ctx = BN_CTX_new();
1201 tmp = BN_CTX_get(ctx);
1202 tmp_Z = BN_CTX_get(ctx);
1203 if (tmp == NULL || tmp_Z == NULL) goto err;
1205 prod_Z = OPENSSL_malloc(num * sizeof prod_Z[0]);
1206 if (prod_Z == NULL) goto err;
1207 for (i = 0; i < num; i++)
1209 prod_Z[i] = BN_new();
1210 if (prod_Z[i] == NULL) goto err;
1213 /* Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1214 * skipping any zero-valued inputs (pretend that they're 1). */
1216 if (!BN_is_zero(&points[0]->Z))
1218 if (!BN_copy(prod_Z[0], &points[0]->Z)) goto err;
1222 if (group->meth->field_set_to_one != 0)
1224 if (!group->meth->field_set_to_one(group, prod_Z[0], ctx)) goto err;
1228 if (!BN_one(prod_Z[0])) goto err;
1232 for (i = 1; i < num; i++)
1234 if (!BN_is_zero(&points[i]->Z))
1236 if (!group->meth->field_mul(group, prod_Z[i], prod_Z[i - 1], &points[i]->Z, ctx)) goto err;
1240 if (!BN_copy(prod_Z[i], prod_Z[i - 1])) goto err;
1244 /* Now use a single explicit inversion to replace every
1245 * non-zero points[i]->Z by its inverse. */
1247 if (!BN_mod_inverse(tmp, prod_Z[num - 1], &group->field, ctx))
1249 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
1252 if (group->meth->field_encode != 0)
1254 /* In the Montgomery case, we just turned R*H (representing H)
1255 * into 1/(R*H), but we need R*(1/H) (representing 1/H);
1256 * i.e. we need to multiply by the Montgomery factor twice. */
1257 if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err;
1258 if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err;
1261 for (i = num - 1; i > 0; --i)
1263 /* Loop invariant: tmp is the product of the inverses of
1264 * points[0]->Z .. points[i]->Z (zero-valued inputs skipped). */
1265 if (!BN_is_zero(&points[i]->Z))
1267 /* Set tmp_Z to the inverse of points[i]->Z (as product
1268 * of Z inverses 0 .. i, Z values 0 .. i - 1). */
1269 if (!group->meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx)) goto err;
1270 /* Update tmp to satisfy the loop invariant for i - 1. */
1271 if (!group->meth->field_mul(group, tmp, tmp, &points[i]->Z, ctx)) goto err;
1272 /* Replace points[i]->Z by its inverse. */
1273 if (!BN_copy(&points[i]->Z, tmp_Z)) goto err;
1277 if (!BN_is_zero(&points[0]->Z))
1279 /* Replace points[0]->Z by its inverse. */
1280 if (!BN_copy(&points[0]->Z, tmp)) goto err;
1283 /* Finally, fix up the X and Y coordinates for all points. */
1285 for (i = 0; i < num; i++)
1287 EC_POINT *p = points[i];
1289 if (!BN_is_zero(&p->Z))
1291 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1293 if (!group->meth->field_sqr(group, tmp, &p->Z, ctx)) goto err;
1294 if (!group->meth->field_mul(group, &p->X, &p->X, tmp, ctx)) goto err;
1296 if (!group->meth->field_mul(group, tmp, tmp, &p->Z, ctx)) goto err;
1297 if (!group->meth->field_mul(group, &p->Y, &p->Y, tmp, ctx)) goto err;
1299 if (group->meth->field_set_to_one != 0)
1301 if (!group->meth->field_set_to_one(group, &p->Z, ctx)) goto err;
1305 if (!BN_one(&p->Z)) goto err;
1315 if (new_ctx != NULL)
1316 BN_CTX_free(new_ctx);
1319 for (i = 0; i < num; i++)
1321 if (prod_Z[i] == NULL) break;
1322 BN_clear_free(prod_Z[i]);
1324 OPENSSL_free(prod_Z);
1330 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1332 return BN_mod_mul(r, a, b, &group->field, ctx);
1336 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx)
1338 return BN_mod_sqr(r, a, &group->field, ctx);