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.
67 #include <openssl/err.h>
68 #include <openssl/symhacks.h>
72 const EC_METHOD *EC_GFp_simple_method(void)
74 static const EC_METHOD ret = {
76 NID_X9_62_prime_field,
77 ec_GFp_simple_group_init,
78 ec_GFp_simple_group_finish,
79 ec_GFp_simple_group_clear_finish,
80 ec_GFp_simple_group_copy,
81 ec_GFp_simple_group_set_curve,
82 ec_GFp_simple_group_get_curve,
83 ec_GFp_simple_group_get_degree,
84 ec_GFp_simple_group_check_discriminant,
85 ec_GFp_simple_point_init,
86 ec_GFp_simple_point_finish,
87 ec_GFp_simple_point_clear_finish,
88 ec_GFp_simple_point_copy,
89 ec_GFp_simple_point_set_to_infinity,
90 ec_GFp_simple_set_Jprojective_coordinates_GFp,
91 ec_GFp_simple_get_Jprojective_coordinates_GFp,
92 ec_GFp_simple_point_set_affine_coordinates,
93 ec_GFp_simple_point_get_affine_coordinates,
98 ec_GFp_simple_is_at_infinity,
99 ec_GFp_simple_is_on_curve,
101 ec_GFp_simple_make_affine,
102 ec_GFp_simple_points_make_affine,
104 0 /* precompute_mult */,
105 0 /* have_precompute_mult */,
106 ec_GFp_simple_field_mul,
107 ec_GFp_simple_field_sqr,
109 0 /* field_encode */,
110 0 /* field_decode */,
111 0 /* field_set_to_one */ };
117 /* Most method functions in this file are designed to work with
118 * non-trivial representations of field elements if necessary
119 * (see ecp_mont.c): while standard modular addition and subtraction
120 * are used, the field_mul and field_sqr methods will be used for
121 * multiplication, and field_encode and field_decode (if defined)
122 * will be used for converting between representations.
124 * Functions ec_GFp_simple_points_make_affine() and
125 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
126 * that if a non-trivial representation is used, it is a Montgomery
127 * representation (i.e. 'encoding' means multiplying by some factor R).
131 int ec_GFp_simple_group_init(EC_GROUP *group)
133 group->field = BN_new();
136 if(!group->field || !group->a || !group->b)
138 if(!group->field) BN_free(group->field);
139 if(!group->a) BN_free(group->a);
140 if(!group->b) BN_free(group->b);
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)
362 if(!point->X || !point->Y || !point->Z)
364 if(point->X) BN_free(point->X);
365 if(point->Y) BN_free(point->Y);
366 if(point->Z) BN_free(point->Z);
373 void ec_GFp_simple_point_finish(EC_POINT *point)
381 void ec_GFp_simple_point_clear_finish(EC_POINT *point)
383 BN_clear_free(point->X);
384 BN_clear_free(point->Y);
385 BN_clear_free(point->Z);
390 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
392 if (!BN_copy(dest->X, src->X)) return 0;
393 if (!BN_copy(dest->Y, src->Y)) return 0;
394 if (!BN_copy(dest->Z, src->Z)) return 0;
395 dest->Z_is_one = src->Z_is_one;
401 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, EC_POINT *point)
409 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group, EC_POINT *point,
410 const BIGNUM *x, const BIGNUM *y, const BIGNUM *z, BN_CTX *ctx)
412 BN_CTX *new_ctx = NULL;
417 ctx = new_ctx = BN_CTX_new();
424 if (!BN_nnmod(point->X, x, group->field, ctx)) goto err;
425 if (group->meth->field_encode)
427 if (!group->meth->field_encode(group, point->X, point->X, ctx)) goto err;
433 if (!BN_nnmod(point->Y, y, group->field, ctx)) goto err;
434 if (group->meth->field_encode)
436 if (!group->meth->field_encode(group, point->Y, point->Y, ctx)) goto err;
444 if (!BN_nnmod(point->Z, z, group->field, ctx)) goto err;
445 Z_is_one = BN_is_one(point->Z);
446 if (group->meth->field_encode)
448 if (Z_is_one && (group->meth->field_set_to_one != 0))
450 if (!group->meth->field_set_to_one(group, point->Z, ctx)) goto err;
454 if (!group->meth->field_encode(group, point->Z, point->Z, ctx)) goto err;
457 point->Z_is_one = Z_is_one;
464 BN_CTX_free(new_ctx);
469 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group, const EC_POINT *point,
470 BIGNUM *x, BIGNUM *y, BIGNUM *z, BN_CTX *ctx)
472 BN_CTX *new_ctx = NULL;
475 if (group->meth->field_decode != 0)
479 ctx = new_ctx = BN_CTX_new();
486 if (!group->meth->field_decode(group, x, point->X, ctx)) goto err;
490 if (!group->meth->field_decode(group, y, point->Y, ctx)) goto err;
494 if (!group->meth->field_decode(group, z, point->Z, ctx)) goto err;
501 if (!BN_copy(x, point->X)) goto err;
505 if (!BN_copy(y, point->Y)) goto err;
509 if (!BN_copy(z, point->Z)) goto err;
517 BN_CTX_free(new_ctx);
522 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group, EC_POINT *point,
523 const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx)
525 if (x == NULL || y == NULL)
527 /* unlike for projective coordinates, we do not tolerate this */
528 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES, ERR_R_PASSED_NULL_PARAMETER);
532 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y, BN_value_one(), ctx);
536 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point,
537 BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
539 BN_CTX *new_ctx = NULL;
540 BIGNUM *Z, *Z_1, *Z_2, *Z_3;
544 if (EC_POINT_is_at_infinity(group, point))
546 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, EC_R_POINT_AT_INFINITY);
552 ctx = new_ctx = BN_CTX_new();
559 Z_1 = BN_CTX_get(ctx);
560 Z_2 = BN_CTX_get(ctx);
561 Z_3 = BN_CTX_get(ctx);
562 if (Z_3 == NULL) goto err;
564 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
566 if (group->meth->field_decode)
568 if (!group->meth->field_decode(group, Z, point->Z, ctx)) goto err;
578 if (group->meth->field_decode)
582 if (!group->meth->field_decode(group, x, point->X, ctx)) goto err;
586 if (!group->meth->field_decode(group, y, point->Y, ctx)) goto err;
593 if (!BN_copy(x, point->X)) goto err;
597 if (!BN_copy(y, point->Y)) goto err;
603 if (!BN_mod_inverse(Z_1, Z_, group->field, ctx))
605 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB);
609 if (group->meth->field_encode == 0)
611 /* field_sqr works on standard representation */
612 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx)) goto err;
616 if (!BN_mod_sqr(Z_2, Z_1, group->field, ctx)) goto err;
621 /* in the Montgomery case, field_mul will cancel out Montgomery factor in X: */
622 if (!group->meth->field_mul(group, x, point->X, Z_2, ctx)) goto err;
627 if (group->meth->field_encode == 0)
629 /* field_mul works on standard representation */
630 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx)) goto err;
634 if (!BN_mod_mul(Z_3, Z_2, Z_1, group->field, ctx)) goto err;
637 /* in the Montgomery case, field_mul will cancel out Montgomery factor in Y: */
638 if (!group->meth->field_mul(group, y, point->Y, Z_3, ctx)) goto err;
647 BN_CTX_free(new_ctx);
651 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
653 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
654 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
656 BN_CTX *new_ctx = NULL;
657 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
661 return EC_POINT_dbl(group, r, a, ctx);
662 if (EC_POINT_is_at_infinity(group, a))
663 return EC_POINT_copy(r, b);
664 if (EC_POINT_is_at_infinity(group, b))
665 return EC_POINT_copy(r, a);
667 field_mul = group->meth->field_mul;
668 field_sqr = group->meth->field_sqr;
673 ctx = new_ctx = BN_CTX_new();
679 n0 = BN_CTX_get(ctx);
680 n1 = BN_CTX_get(ctx);
681 n2 = BN_CTX_get(ctx);
682 n3 = BN_CTX_get(ctx);
683 n4 = BN_CTX_get(ctx);
684 n5 = BN_CTX_get(ctx);
685 n6 = BN_CTX_get(ctx);
686 if (n6 == NULL) goto end;
688 /* Note that in this function we must not read components of 'a' or 'b'
689 * once we have written the corresponding components of 'r'.
690 * ('r' might be one of 'a' or 'b'.)
696 if (!BN_copy(n1, a->X)) goto end;
697 if (!BN_copy(n2, a->Y)) goto end;
703 if (!field_sqr(group, n0, b->Z, ctx)) goto end;
704 if (!field_mul(group, n1, a->X, n0, ctx)) goto end;
705 /* n1 = X_a * Z_b^2 */
707 if (!field_mul(group, n0, n0, b->Z, ctx)) goto end;
708 if (!field_mul(group, n2, a->Y, n0, ctx)) goto end;
709 /* n2 = Y_a * Z_b^3 */
715 if (!BN_copy(n3, b->X)) goto end;
716 if (!BN_copy(n4, b->Y)) goto end;
722 if (!field_sqr(group, n0, a->Z, ctx)) goto end;
723 if (!field_mul(group, n3, b->X, n0, ctx)) goto end;
724 /* n3 = X_b * Z_a^2 */
726 if (!field_mul(group, n0, n0, a->Z, ctx)) goto end;
727 if (!field_mul(group, n4, b->Y, n0, ctx)) goto end;
728 /* n4 = Y_b * Z_a^3 */
732 if (!BN_mod_sub_quick(n5, n1, n3, p)) goto end;
733 if (!BN_mod_sub_quick(n6, n2, n4, p)) goto end;
741 /* a is the same point as b */
743 ret = EC_POINT_dbl(group, r, a, ctx);
749 /* a is the inverse of b */
758 if (!BN_mod_add_quick(n1, n1, n3, p)) goto end;
759 if (!BN_mod_add_quick(n2, n2, n4, p)) goto end;
764 if (a->Z_is_one && b->Z_is_one)
766 if (!BN_copy(r->Z, n5)) goto end;
771 { if (!BN_copy(n0, b->Z)) goto end; }
772 else if (b->Z_is_one)
773 { if (!BN_copy(n0, a->Z)) goto end; }
775 { if (!field_mul(group, n0, a->Z, b->Z, ctx)) goto end; }
776 if (!field_mul(group, r->Z, n0, n5, ctx)) goto end;
779 /* Z_r = Z_a * Z_b * n5 */
782 if (!field_sqr(group, n0, n6, ctx)) goto end;
783 if (!field_sqr(group, n4, n5, ctx)) goto end;
784 if (!field_mul(group, n3, n1, n4, ctx)) goto end;
785 if (!BN_mod_sub_quick(r->X, n0, n3, p)) goto end;
786 /* X_r = n6^2 - n5^2 * 'n7' */
789 if (!BN_mod_lshift1_quick(n0, r->X, p)) goto end;
790 if (!BN_mod_sub_quick(n0, n3, n0, p)) goto end;
791 /* n9 = n5^2 * 'n7' - 2 * X_r */
794 if (!field_mul(group, n0, n0, n6, ctx)) goto end;
795 if (!field_mul(group, n5, n4, n5, ctx)) goto end; /* now n5 is n5^3 */
796 if (!field_mul(group, n1, n2, n5, ctx)) goto end;
797 if (!BN_mod_sub_quick(n0, n0, n1, p)) goto end;
799 if (!BN_add(n0, n0, p)) goto end;
800 /* now 0 <= n0 < 2*p, and n0 is even */
801 if (!BN_rshift1(r->Y, n0)) goto end;
802 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
807 if (ctx) /* otherwise we already called BN_CTX_end */
810 BN_CTX_free(new_ctx);
815 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, BN_CTX *ctx)
817 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
818 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
820 BN_CTX *new_ctx = NULL;
821 BIGNUM *n0, *n1, *n2, *n3;
824 if (EC_POINT_is_at_infinity(group, a))
831 field_mul = group->meth->field_mul;
832 field_sqr = group->meth->field_sqr;
837 ctx = new_ctx = BN_CTX_new();
843 n0 = BN_CTX_get(ctx);
844 n1 = BN_CTX_get(ctx);
845 n2 = BN_CTX_get(ctx);
846 n3 = BN_CTX_get(ctx);
847 if (n3 == NULL) goto err;
849 /* Note that in this function we must not read components of 'a'
850 * once we have written the corresponding components of 'r'.
851 * ('r' might the same as 'a'.)
857 if (!field_sqr(group, n0, a->X, ctx)) goto err;
858 if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
859 if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
860 if (!BN_mod_add_quick(n1, n0, group->a, p)) goto err;
861 /* n1 = 3 * X_a^2 + a_curve */
863 else if (group->a_is_minus3)
865 if (!field_sqr(group, n1, a->Z, ctx)) goto err;
866 if (!BN_mod_add_quick(n0, a->X, n1, p)) goto err;
867 if (!BN_mod_sub_quick(n2, a->X, n1, p)) goto err;
868 if (!field_mul(group, n1, n0, n2, ctx)) goto err;
869 if (!BN_mod_lshift1_quick(n0, n1, p)) goto err;
870 if (!BN_mod_add_quick(n1, n0, n1, p)) goto err;
871 /* n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
872 * = 3 * X_a^2 - 3 * Z_a^4 */
876 if (!field_sqr(group, n0, a->X, ctx)) goto err;
877 if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
878 if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
879 if (!field_sqr(group, n1, a->Z, ctx)) goto err;
880 if (!field_sqr(group, n1, n1, ctx)) goto err;
881 if (!field_mul(group, n1, n1, group->a, ctx)) goto err;
882 if (!BN_mod_add_quick(n1, n1, n0, p)) goto err;
883 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
889 if (!BN_copy(n0, a->Y)) goto err;
893 if (!field_mul(group, n0, a->Y, a->Z, ctx)) goto err;
895 if (!BN_mod_lshift1_quick(r->Z, n0, p)) goto err;
897 /* Z_r = 2 * Y_a * Z_a */
900 if (!field_sqr(group, n3, a->Y, ctx)) goto err;
901 if (!field_mul(group, n2, a->X, n3, ctx)) goto err;
902 if (!BN_mod_lshift_quick(n2, n2, 2, p)) goto err;
903 /* n2 = 4 * X_a * Y_a^2 */
906 if (!BN_mod_lshift1_quick(n0, n2, p)) goto err;
907 if (!field_sqr(group, r->X, n1, ctx)) goto err;
908 if (!BN_mod_sub_quick(r->X, r->X, n0, p)) goto err;
909 /* X_r = n1^2 - 2 * n2 */
912 if (!field_sqr(group, n0, n3, ctx)) goto err;
913 if (!BN_mod_lshift_quick(n3, n0, 3, p)) goto err;
917 if (!BN_mod_sub_quick(n0, n2, r->X, p)) goto err;
918 if (!field_mul(group, n0, n1, n0, ctx)) goto err;
919 if (!BN_mod_sub_quick(r->Y, n0, n3, p)) goto err;
920 /* Y_r = n1 * (n2 - X_r) - n3 */
927 BN_CTX_free(new_ctx);
932 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
934 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
935 /* point is its own inverse */
938 return BN_usub(point->Y, group->field, point->Y);
942 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
944 return BN_is_zero(point->Z);
948 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point, BN_CTX *ctx)
950 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
951 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
953 BN_CTX *new_ctx = NULL;
954 BIGNUM *rh, *tmp, *Z4, *Z6;
957 if (EC_POINT_is_at_infinity(group, point))
960 field_mul = group->meth->field_mul;
961 field_sqr = group->meth->field_sqr;
966 ctx = new_ctx = BN_CTX_new();
972 rh = BN_CTX_get(ctx);
973 tmp = BN_CTX_get(ctx);
974 Z4 = BN_CTX_get(ctx);
975 Z6 = BN_CTX_get(ctx);
976 if (Z6 == NULL) goto err;
978 /* We have a curve defined by a Weierstrass equation
979 * y^2 = x^3 + a*x + b.
980 * The point to consider is given in Jacobian projective coordinates
981 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
982 * Substituting this and multiplying by Z^6 transforms the above equation into
983 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
984 * To test this, we add up the right-hand side in 'rh'.
988 if (!field_sqr(group, rh, point->X, ctx)) goto err;
990 if (!point->Z_is_one)
992 if (!field_sqr(group, tmp, point->Z, ctx)) goto err;
993 if (!field_sqr(group, Z4, tmp, ctx)) goto err;
994 if (!field_mul(group, Z6, Z4, tmp, ctx)) goto err;
996 /* rh := (rh + a*Z^4)*X */
997 if (group->a_is_minus3)
999 if (!BN_mod_lshift1_quick(tmp, Z4, p)) goto err;
1000 if (!BN_mod_add_quick(tmp, tmp, Z4, p)) goto err;
1001 if (!BN_mod_sub_quick(rh, rh, tmp, p)) goto err;
1002 if (!field_mul(group, rh, rh, point->X, ctx)) goto err;
1006 if (!field_mul(group, tmp, Z4, group->a, ctx)) goto err;
1007 if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
1008 if (!field_mul(group, rh, rh, point->X, ctx)) goto err;
1011 /* rh := rh + b*Z^6 */
1012 if (!field_mul(group, tmp, group->b, Z6, ctx)) goto err;
1013 if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
1017 /* point->Z_is_one */
1019 /* rh := (rh + a)*X */
1020 if (!BN_mod_add_quick(rh, rh, group->a, p)) goto err;
1021 if (!field_mul(group, rh, rh, point->X, ctx)) goto err;
1023 if (!BN_mod_add_quick(rh, rh, group->b, p)) goto err;
1027 if (!field_sqr(group, tmp, point->Y, ctx)) goto err;
1029 ret = (0 == BN_ucmp(tmp, rh));
1033 if (new_ctx != NULL)
1034 BN_CTX_free(new_ctx);
1039 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
1043 * 0 equal (in affine coordinates)
1047 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
1048 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1049 BN_CTX *new_ctx = NULL;
1050 BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1051 const BIGNUM *tmp1_, *tmp2_;
1054 if (EC_POINT_is_at_infinity(group, a))
1056 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1059 if (EC_POINT_is_at_infinity(group, b))
1062 if (a->Z_is_one && b->Z_is_one)
1064 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
1067 field_mul = group->meth->field_mul;
1068 field_sqr = group->meth->field_sqr;
1072 ctx = new_ctx = BN_CTX_new();
1078 tmp1 = BN_CTX_get(ctx);
1079 tmp2 = BN_CTX_get(ctx);
1080 Za23 = BN_CTX_get(ctx);
1081 Zb23 = BN_CTX_get(ctx);
1082 if (Zb23 == NULL) goto end;
1084 /* We have to decide whether
1085 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1086 * or equivalently, whether
1087 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1092 if (!field_sqr(group, Zb23, b->Z, ctx)) goto end;
1093 if (!field_mul(group, tmp1, a->X, Zb23, ctx)) goto end;
1100 if (!field_sqr(group, Za23, a->Z, ctx)) goto end;
1101 if (!field_mul(group, tmp2, b->X, Za23, ctx)) goto end;
1107 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1108 if (BN_cmp(tmp1_, tmp2_) != 0)
1110 ret = 1; /* points differ */
1117 if (!field_mul(group, Zb23, Zb23, b->Z, ctx)) goto end;
1118 if (!field_mul(group, tmp1, a->Y, Zb23, ctx)) goto end;
1125 if (!field_mul(group, Za23, Za23, a->Z, ctx)) goto end;
1126 if (!field_mul(group, tmp2, b->Y, Za23, ctx)) goto end;
1132 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1133 if (BN_cmp(tmp1_, tmp2_) != 0)
1135 ret = 1; /* points differ */
1139 /* points are equal */
1144 if (new_ctx != NULL)
1145 BN_CTX_free(new_ctx);
1150 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
1152 BN_CTX *new_ctx = NULL;
1156 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1161 ctx = new_ctx = BN_CTX_new();
1167 x = BN_CTX_get(ctx);
1168 y = BN_CTX_get(ctx);
1169 if (y == NULL) goto err;
1171 if (!EC_POINT_get_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
1172 if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
1173 if (!point->Z_is_one)
1175 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
1183 if (new_ctx != NULL)
1184 BN_CTX_free(new_ctx);
1189 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num, EC_POINT *points[], BN_CTX *ctx)
1191 BN_CTX *new_ctx = NULL;
1192 BIGNUM *tmp, *tmp_Z;
1193 BIGNUM **prod_Z = NULL;
1202 ctx = new_ctx = BN_CTX_new();
1208 tmp = BN_CTX_get(ctx);
1209 tmp_Z = BN_CTX_get(ctx);
1210 if (tmp == NULL || tmp_Z == NULL) goto err;
1212 prod_Z = OPENSSL_malloc(num * sizeof prod_Z[0]);
1213 if (prod_Z == NULL) goto err;
1214 for (i = 0; i < num; i++)
1216 prod_Z[i] = BN_new();
1217 if (prod_Z[i] == NULL) goto err;
1220 /* Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1221 * skipping any zero-valued inputs (pretend that they're 1). */
1223 if (!BN_is_zero(points[0]->Z))
1225 if (!BN_copy(prod_Z[0], points[0]->Z)) goto err;
1229 if (group->meth->field_set_to_one != 0)
1231 if (!group->meth->field_set_to_one(group, prod_Z[0], ctx)) goto err;
1235 if (!BN_one(prod_Z[0])) goto err;
1239 for (i = 1; i < num; i++)
1241 if (!BN_is_zero(points[i]->Z))
1243 if (!group->meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z, ctx)) goto err;
1247 if (!BN_copy(prod_Z[i], prod_Z[i - 1])) goto err;
1251 /* Now use a single explicit inversion to replace every
1252 * non-zero points[i]->Z by its inverse. */
1254 if (!BN_mod_inverse(tmp, prod_Z[num - 1], group->field, ctx))
1256 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
1259 if (group->meth->field_encode != 0)
1261 /* In the Montgomery case, we just turned R*H (representing H)
1262 * into 1/(R*H), but we need R*(1/H) (representing 1/H);
1263 * i.e. we need to multiply by the Montgomery factor twice. */
1264 if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err;
1265 if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err;
1268 for (i = num - 1; i > 0; --i)
1270 /* Loop invariant: tmp is the product of the inverses of
1271 * points[0]->Z .. points[i]->Z (zero-valued inputs skipped). */
1272 if (!BN_is_zero(points[i]->Z))
1274 /* Set tmp_Z to the inverse of points[i]->Z (as product
1275 * of Z inverses 0 .. i, Z values 0 .. i - 1). */
1276 if (!group->meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx)) goto err;
1277 /* Update tmp to satisfy the loop invariant for i - 1. */
1278 if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx)) goto err;
1279 /* Replace points[i]->Z by its inverse. */
1280 if (!BN_copy(points[i]->Z, tmp_Z)) goto err;
1284 if (!BN_is_zero(points[0]->Z))
1286 /* Replace points[0]->Z by its inverse. */
1287 if (!BN_copy(points[0]->Z, tmp)) goto err;
1290 /* Finally, fix up the X and Y coordinates for all points. */
1292 for (i = 0; i < num; i++)
1294 EC_POINT *p = points[i];
1296 if (!BN_is_zero(p->Z))
1298 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1300 if (!group->meth->field_sqr(group, tmp, p->Z, ctx)) goto err;
1301 if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx)) goto err;
1303 if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx)) goto err;
1304 if (!group->meth->field_mul(group, p->Y, p->Y, tmp, ctx)) goto err;
1306 if (group->meth->field_set_to_one != 0)
1308 if (!group->meth->field_set_to_one(group, p->Z, ctx)) goto err;
1312 if (!BN_one(p->Z)) goto err;
1322 if (new_ctx != NULL)
1323 BN_CTX_free(new_ctx);
1326 for (i = 0; i < num; i++)
1328 if (prod_Z[i] == NULL) break;
1329 BN_clear_free(prod_Z[i]);
1331 OPENSSL_free(prod_Z);
1337 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1339 return BN_mod_mul(r, a, b, group->field, ctx);
1343 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx)
1345 return BN_mod_sqr(r, a, group->field, ctx);