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>
70 const EC_METHOD *EC_GFp_simple_method(void)
72 static const EC_METHOD ret = {
74 NID_X9_62_prime_field,
75 ec_GFp_simple_group_init,
76 ec_GFp_simple_group_finish,
77 ec_GFp_simple_group_clear_finish,
78 ec_GFp_simple_group_copy,
79 ec_GFp_simple_group_set_curve,
80 ec_GFp_simple_group_get_curve,
81 ec_GFp_simple_group_get_degree,
82 ec_GFp_simple_group_check_discriminant,
83 ec_GFp_simple_point_init,
84 ec_GFp_simple_point_finish,
85 ec_GFp_simple_point_clear_finish,
86 ec_GFp_simple_point_copy,
87 ec_GFp_simple_point_set_to_infinity,
88 ec_GFp_simple_set_Jprojective_coordinates_GFp,
89 ec_GFp_simple_get_Jprojective_coordinates_GFp,
90 ec_GFp_simple_point_set_affine_coordinates,
91 ec_GFp_simple_point_get_affine_coordinates,
96 ec_GFp_simple_is_at_infinity,
97 ec_GFp_simple_is_on_curve,
99 ec_GFp_simple_make_affine,
100 ec_GFp_simple_points_make_affine,
102 0 /* precompute_mult */,
103 0 /* have_precompute_mult */,
104 ec_GFp_simple_field_mul,
105 ec_GFp_simple_field_sqr,
107 0 /* field_encode */,
108 0 /* field_decode */,
109 0 /* field_set_to_one */ };
115 /* Most method functions in this file are designed to work with
116 * non-trivial representations of field elements if necessary
117 * (see ecp_mont.c): while standard modular addition and subtraction
118 * are used, the field_mul and field_sqr methods will be used for
119 * multiplication, and field_encode and field_decode (if defined)
120 * will be used for converting between representations.
122 * Functions ec_GFp_simple_points_make_affine() and
123 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
124 * that if a non-trivial representation is used, it is a Montgomery
125 * representation (i.e. 'encoding' means multiplying by some factor R).
129 int ec_GFp_simple_group_init(EC_GROUP *group)
131 BN_init(&group->field);
134 group->a_is_minus3 = 0;
139 void ec_GFp_simple_group_finish(EC_GROUP *group)
141 BN_free(&group->field);
147 void ec_GFp_simple_group_clear_finish(EC_GROUP *group)
149 BN_clear_free(&group->field);
150 BN_clear_free(&group->a);
151 BN_clear_free(&group->b);
155 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
157 if (!BN_copy(&dest->field, &src->field)) return 0;
158 if (!BN_copy(&dest->a, &src->a)) return 0;
159 if (!BN_copy(&dest->b, &src->b)) return 0;
161 dest->a_is_minus3 = src->a_is_minus3;
167 int ec_GFp_simple_group_set_curve(EC_GROUP *group,
168 const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
171 BN_CTX *new_ctx = NULL;
174 /* p must be a prime > 3 */
175 if (BN_num_bits(p) <= 2 || !BN_is_odd(p))
177 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD);
183 ctx = new_ctx = BN_CTX_new();
189 tmp_a = BN_CTX_get(ctx);
190 if (tmp_a == NULL) goto err;
193 if (!BN_copy(&group->field, p)) goto err;
194 BN_set_negative(&group->field, 0);
197 if (!BN_nnmod(tmp_a, a, p, ctx)) goto err;
198 if (group->meth->field_encode)
199 { if (!group->meth->field_encode(group, &group->a, tmp_a, ctx)) goto err; }
201 if (!BN_copy(&group->a, tmp_a)) goto err;
204 if (!BN_nnmod(&group->b, b, p, ctx)) goto err;
205 if (group->meth->field_encode)
206 if (!group->meth->field_encode(group, &group->b, &group->b, ctx)) goto err;
208 /* group->a_is_minus3 */
209 if (!BN_add_word(tmp_a, 3)) goto err;
210 group->a_is_minus3 = (0 == BN_cmp(tmp_a, &group->field));
217 BN_CTX_free(new_ctx);
222 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
225 BN_CTX *new_ctx = NULL;
229 if (!BN_copy(p, &group->field)) return 0;
232 if (a != NULL || b != NULL)
234 if (group->meth->field_decode)
238 ctx = new_ctx = BN_CTX_new();
244 if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err;
248 if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err;
255 if (!BN_copy(a, &group->a)) goto err;
259 if (!BN_copy(b, &group->b)) goto err;
268 BN_CTX_free(new_ctx);
273 int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
275 return BN_num_bits(&group->field);
279 int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
282 BIGNUM *a,*b,*order,*tmp_1,*tmp_2;
283 const BIGNUM *p = &group->field;
284 BN_CTX *new_ctx = NULL;
288 ctx = new_ctx = BN_CTX_new();
291 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT, ERR_R_MALLOC_FAILURE);
298 tmp_1 = BN_CTX_get(ctx);
299 tmp_2 = BN_CTX_get(ctx);
300 order = BN_CTX_get(ctx);
301 if (order == NULL) goto err;
303 if (group->meth->field_decode)
305 if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err;
306 if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err;
310 if (!BN_copy(a, &group->a)) goto err;
311 if (!BN_copy(b, &group->b)) goto err;
314 /* check the discriminant:
315 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
319 if (BN_is_zero(b)) goto err;
321 else if (!BN_is_zero(b))
323 if (!BN_mod_sqr(tmp_1, a, p, ctx)) goto err;
324 if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx)) goto err;
325 if (!BN_lshift(tmp_1, tmp_2, 2)) goto err;
328 if (!BN_mod_sqr(tmp_2, b, p, ctx)) goto err;
329 if (!BN_mul_word(tmp_2, 27)) goto err;
332 if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx)) goto err;
333 if (BN_is_zero(a)) goto err;
341 BN_CTX_free(new_ctx);
346 int ec_GFp_simple_point_init(EC_POINT *point)
357 void ec_GFp_simple_point_finish(EC_POINT *point)
365 void ec_GFp_simple_point_clear_finish(EC_POINT *point)
367 BN_clear_free(&point->X);
368 BN_clear_free(&point->Y);
369 BN_clear_free(&point->Z);
374 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
376 if (!BN_copy(&dest->X, &src->X)) return 0;
377 if (!BN_copy(&dest->Y, &src->Y)) return 0;
378 if (!BN_copy(&dest->Z, &src->Z)) return 0;
379 dest->Z_is_one = src->Z_is_one;
385 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, EC_POINT *point)
393 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group, EC_POINT *point,
394 const BIGNUM *x, const BIGNUM *y, const BIGNUM *z, BN_CTX *ctx)
396 BN_CTX *new_ctx = NULL;
401 ctx = new_ctx = BN_CTX_new();
408 if (!BN_nnmod(&point->X, x, &group->field, ctx)) goto err;
409 if (group->meth->field_encode)
411 if (!group->meth->field_encode(group, &point->X, &point->X, ctx)) goto err;
417 if (!BN_nnmod(&point->Y, y, &group->field, ctx)) goto err;
418 if (group->meth->field_encode)
420 if (!group->meth->field_encode(group, &point->Y, &point->Y, ctx)) goto err;
428 if (!BN_nnmod(&point->Z, z, &group->field, ctx)) goto err;
429 Z_is_one = BN_is_one(&point->Z);
430 if (group->meth->field_encode)
432 if (Z_is_one && (group->meth->field_set_to_one != 0))
434 if (!group->meth->field_set_to_one(group, &point->Z, ctx)) goto err;
438 if (!group->meth->field_encode(group, &point->Z, &point->Z, ctx)) goto err;
441 point->Z_is_one = Z_is_one;
448 BN_CTX_free(new_ctx);
453 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group, const EC_POINT *point,
454 BIGNUM *x, BIGNUM *y, BIGNUM *z, BN_CTX *ctx)
456 BN_CTX *new_ctx = NULL;
459 if (group->meth->field_decode != 0)
463 ctx = new_ctx = BN_CTX_new();
470 if (!group->meth->field_decode(group, x, &point->X, ctx)) goto err;
474 if (!group->meth->field_decode(group, y, &point->Y, ctx)) goto err;
478 if (!group->meth->field_decode(group, z, &point->Z, ctx)) goto err;
485 if (!BN_copy(x, &point->X)) goto err;
489 if (!BN_copy(y, &point->Y)) goto err;
493 if (!BN_copy(z, &point->Z)) goto err;
501 BN_CTX_free(new_ctx);
506 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group, EC_POINT *point,
507 const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx)
509 if (x == NULL || y == NULL)
511 /* unlike for projective coordinates, we do not tolerate this */
512 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES, ERR_R_PASSED_NULL_PARAMETER);
516 return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y, BN_value_one(), ctx);
520 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point,
521 BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
523 BN_CTX *new_ctx = NULL;
524 BIGNUM *Z, *Z_1, *Z_2, *Z_3;
528 if (EC_POINT_is_at_infinity(group, point))
530 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, EC_R_POINT_AT_INFINITY);
536 ctx = new_ctx = BN_CTX_new();
543 Z_1 = BN_CTX_get(ctx);
544 Z_2 = BN_CTX_get(ctx);
545 Z_3 = BN_CTX_get(ctx);
546 if (Z_3 == NULL) goto err;
548 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
550 if (group->meth->field_decode)
552 if (!group->meth->field_decode(group, Z, &point->Z, ctx)) goto err;
562 if (group->meth->field_decode)
566 if (!group->meth->field_decode(group, x, &point->X, ctx)) goto err;
570 if (!group->meth->field_decode(group, y, &point->Y, ctx)) goto err;
577 if (!BN_copy(x, &point->X)) goto err;
581 if (!BN_copy(y, &point->Y)) goto err;
587 if (!BN_mod_inverse(Z_1, Z_, &group->field, ctx))
589 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB);
593 if (group->meth->field_encode == 0)
595 /* field_sqr works on standard representation */
596 if (!group->meth->field_sqr(group, Z_2, Z_1, ctx)) goto err;
600 if (!BN_mod_sqr(Z_2, Z_1, &group->field, ctx)) goto err;
605 /* in the Montgomery case, field_mul will cancel out Montgomery factor in X: */
606 if (!group->meth->field_mul(group, x, &point->X, Z_2, ctx)) goto err;
611 if (group->meth->field_encode == 0)
613 /* field_mul works on standard representation */
614 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx)) goto err;
618 if (!BN_mod_mul(Z_3, Z_2, Z_1, &group->field, ctx)) goto err;
621 /* in the Montgomery case, field_mul will cancel out Montgomery factor in Y: */
622 if (!group->meth->field_mul(group, y, &point->Y, Z_3, ctx)) goto err;
631 BN_CTX_free(new_ctx);
635 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
637 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
638 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
640 BN_CTX *new_ctx = NULL;
641 BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
645 return EC_POINT_dbl(group, r, a, ctx);
646 if (EC_POINT_is_at_infinity(group, a))
647 return EC_POINT_copy(r, b);
648 if (EC_POINT_is_at_infinity(group, b))
649 return EC_POINT_copy(r, a);
651 field_mul = group->meth->field_mul;
652 field_sqr = group->meth->field_sqr;
657 ctx = new_ctx = BN_CTX_new();
663 n0 = BN_CTX_get(ctx);
664 n1 = BN_CTX_get(ctx);
665 n2 = BN_CTX_get(ctx);
666 n3 = BN_CTX_get(ctx);
667 n4 = BN_CTX_get(ctx);
668 n5 = BN_CTX_get(ctx);
669 n6 = BN_CTX_get(ctx);
670 if (n6 == NULL) goto end;
672 /* Note that in this function we must not read components of 'a' or 'b'
673 * once we have written the corresponding components of 'r'.
674 * ('r' might be one of 'a' or 'b'.)
680 if (!BN_copy(n1, &a->X)) goto end;
681 if (!BN_copy(n2, &a->Y)) goto end;
687 if (!field_sqr(group, n0, &b->Z, ctx)) goto end;
688 if (!field_mul(group, n1, &a->X, n0, ctx)) goto end;
689 /* n1 = X_a * Z_b^2 */
691 if (!field_mul(group, n0, n0, &b->Z, ctx)) goto end;
692 if (!field_mul(group, n2, &a->Y, n0, ctx)) goto end;
693 /* n2 = Y_a * Z_b^3 */
699 if (!BN_copy(n3, &b->X)) goto end;
700 if (!BN_copy(n4, &b->Y)) goto end;
706 if (!field_sqr(group, n0, &a->Z, ctx)) goto end;
707 if (!field_mul(group, n3, &b->X, n0, ctx)) goto end;
708 /* n3 = X_b * Z_a^2 */
710 if (!field_mul(group, n0, n0, &a->Z, ctx)) goto end;
711 if (!field_mul(group, n4, &b->Y, n0, ctx)) goto end;
712 /* n4 = Y_b * Z_a^3 */
716 if (!BN_mod_sub_quick(n5, n1, n3, p)) goto end;
717 if (!BN_mod_sub_quick(n6, n2, n4, p)) goto end;
725 /* a is the same point as b */
727 ret = EC_POINT_dbl(group, r, a, ctx);
733 /* a is the inverse of b */
742 if (!BN_mod_add_quick(n1, n1, n3, p)) goto end;
743 if (!BN_mod_add_quick(n2, n2, n4, p)) goto end;
748 if (a->Z_is_one && b->Z_is_one)
750 if (!BN_copy(&r->Z, n5)) goto end;
755 { if (!BN_copy(n0, &b->Z)) goto end; }
756 else if (b->Z_is_one)
757 { if (!BN_copy(n0, &a->Z)) goto end; }
759 { if (!field_mul(group, n0, &a->Z, &b->Z, ctx)) goto end; }
760 if (!field_mul(group, &r->Z, n0, n5, ctx)) goto end;
763 /* Z_r = Z_a * Z_b * n5 */
766 if (!field_sqr(group, n0, n6, ctx)) goto end;
767 if (!field_sqr(group, n4, n5, ctx)) goto end;
768 if (!field_mul(group, n3, n1, n4, ctx)) goto end;
769 if (!BN_mod_sub_quick(&r->X, n0, n3, p)) goto end;
770 /* X_r = n6^2 - n5^2 * 'n7' */
773 if (!BN_mod_lshift1_quick(n0, &r->X, p)) goto end;
774 if (!BN_mod_sub_quick(n0, n3, n0, p)) goto end;
775 /* n9 = n5^2 * 'n7' - 2 * X_r */
778 if (!field_mul(group, n0, n0, n6, ctx)) goto end;
779 if (!field_mul(group, n5, n4, n5, ctx)) goto end; /* now n5 is n5^3 */
780 if (!field_mul(group, n1, n2, n5, ctx)) goto end;
781 if (!BN_mod_sub_quick(n0, n0, n1, p)) goto end;
783 if (!BN_add(n0, n0, p)) goto end;
784 /* now 0 <= n0 < 2*p, and n0 is even */
785 if (!BN_rshift1(&r->Y, n0)) goto end;
786 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
791 if (ctx) /* otherwise we already called BN_CTX_end */
794 BN_CTX_free(new_ctx);
799 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, BN_CTX *ctx)
801 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, 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))
815 field_mul = group->meth->field_mul;
816 field_sqr = group->meth->field_sqr;
821 ctx = new_ctx = BN_CTX_new();
827 n0 = BN_CTX_get(ctx);
828 n1 = BN_CTX_get(ctx);
829 n2 = BN_CTX_get(ctx);
830 n3 = BN_CTX_get(ctx);
831 if (n3 == NULL) goto err;
833 /* Note that in this function we must not read components of 'a'
834 * once we have written the corresponding components of 'r'.
835 * ('r' might the same as 'a'.)
841 if (!field_sqr(group, n0, &a->X, ctx)) goto err;
842 if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
843 if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
844 if (!BN_mod_add_quick(n1, n0, &group->a, p)) goto err;
845 /* n1 = 3 * X_a^2 + a_curve */
847 else if (group->a_is_minus3)
849 if (!field_sqr(group, n1, &a->Z, ctx)) goto err;
850 if (!BN_mod_add_quick(n0, &a->X, n1, p)) goto err;
851 if (!BN_mod_sub_quick(n2, &a->X, n1, p)) goto err;
852 if (!field_mul(group, n1, n0, n2, ctx)) goto err;
853 if (!BN_mod_lshift1_quick(n0, n1, p)) goto err;
854 if (!BN_mod_add_quick(n1, n0, n1, p)) goto err;
855 /* n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
856 * = 3 * X_a^2 - 3 * Z_a^4 */
860 if (!field_sqr(group, n0, &a->X, ctx)) goto err;
861 if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
862 if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
863 if (!field_sqr(group, n1, &a->Z, ctx)) goto err;
864 if (!field_sqr(group, n1, n1, ctx)) goto err;
865 if (!field_mul(group, n1, n1, &group->a, ctx)) goto err;
866 if (!BN_mod_add_quick(n1, n1, n0, p)) goto err;
867 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
873 if (!BN_copy(n0, &a->Y)) goto err;
877 if (!field_mul(group, n0, &a->Y, &a->Z, ctx)) goto err;
879 if (!BN_mod_lshift1_quick(&r->Z, n0, p)) goto err;
881 /* Z_r = 2 * Y_a * Z_a */
884 if (!field_sqr(group, n3, &a->Y, ctx)) goto err;
885 if (!field_mul(group, n2, &a->X, n3, ctx)) goto err;
886 if (!BN_mod_lshift_quick(n2, n2, 2, p)) goto err;
887 /* n2 = 4 * X_a * Y_a^2 */
890 if (!BN_mod_lshift1_quick(n0, n2, p)) goto err;
891 if (!field_sqr(group, &r->X, n1, ctx)) goto err;
892 if (!BN_mod_sub_quick(&r->X, &r->X, n0, p)) goto err;
893 /* X_r = n1^2 - 2 * n2 */
896 if (!field_sqr(group, n0, n3, ctx)) goto err;
897 if (!BN_mod_lshift_quick(n3, n0, 3, p)) goto err;
901 if (!BN_mod_sub_quick(n0, n2, &r->X, p)) goto err;
902 if (!field_mul(group, n0, n1, n0, ctx)) goto err;
903 if (!BN_mod_sub_quick(&r->Y, n0, n3, p)) goto err;
904 /* Y_r = n1 * (n2 - X_r) - n3 */
911 BN_CTX_free(new_ctx);
916 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
918 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(&point->Y))
919 /* point is its own inverse */
922 return BN_usub(&point->Y, &group->field, &point->Y);
926 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
928 return BN_is_zero(&point->Z);
932 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point, BN_CTX *ctx)
934 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
935 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
937 BN_CTX *new_ctx = NULL;
938 BIGNUM *rh, *tmp, *Z4, *Z6;
941 if (EC_POINT_is_at_infinity(group, point))
944 field_mul = group->meth->field_mul;
945 field_sqr = group->meth->field_sqr;
950 ctx = new_ctx = BN_CTX_new();
956 rh = BN_CTX_get(ctx);
957 tmp = BN_CTX_get(ctx);
958 Z4 = BN_CTX_get(ctx);
959 Z6 = BN_CTX_get(ctx);
960 if (Z6 == NULL) goto err;
962 /* We have a curve defined by a Weierstrass equation
963 * y^2 = x^3 + a*x + b.
964 * The point to consider is given in Jacobian projective coordinates
965 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
966 * Substituting this and multiplying by Z^6 transforms the above equation into
967 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
968 * To test this, we add up the right-hand side in 'rh'.
972 if (!field_sqr(group, rh, &point->X, ctx)) goto err;
974 if (!point->Z_is_one)
976 if (!field_sqr(group, tmp, &point->Z, ctx)) goto err;
977 if (!field_sqr(group, Z4, tmp, ctx)) goto err;
978 if (!field_mul(group, Z6, Z4, tmp, ctx)) goto err;
980 /* rh := (rh + a*Z^4)*X */
981 if (group->a_is_minus3)
983 if (!BN_mod_lshift1_quick(tmp, Z4, p)) goto err;
984 if (!BN_mod_add_quick(tmp, tmp, Z4, p)) goto err;
985 if (!BN_mod_sub_quick(rh, rh, tmp, p)) goto err;
986 if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
990 if (!field_mul(group, tmp, Z4, &group->a, ctx)) goto err;
991 if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
992 if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
995 /* rh := rh + b*Z^6 */
996 if (!field_mul(group, tmp, &group->b, Z6, ctx)) goto err;
997 if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
1001 /* point->Z_is_one */
1003 /* rh := (rh + a)*X */
1004 if (!BN_mod_add_quick(rh, rh, &group->a, p)) goto err;
1005 if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
1007 if (!BN_mod_add_quick(rh, rh, &group->b, p)) goto err;
1011 if (!field_sqr(group, tmp, &point->Y, ctx)) goto err;
1013 ret = (0 == BN_ucmp(tmp, rh));
1017 if (new_ctx != NULL)
1018 BN_CTX_free(new_ctx);
1023 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
1027 * 0 equal (in affine coordinates)
1031 int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
1032 int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1033 BN_CTX *new_ctx = NULL;
1034 BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1035 const BIGNUM *tmp1_, *tmp2_;
1038 if (EC_POINT_is_at_infinity(group, a))
1040 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1043 if (EC_POINT_is_at_infinity(group, b))
1046 if (a->Z_is_one && b->Z_is_one)
1048 return ((BN_cmp(&a->X, &b->X) == 0) && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
1051 field_mul = group->meth->field_mul;
1052 field_sqr = group->meth->field_sqr;
1056 ctx = new_ctx = BN_CTX_new();
1062 tmp1 = BN_CTX_get(ctx);
1063 tmp2 = BN_CTX_get(ctx);
1064 Za23 = BN_CTX_get(ctx);
1065 Zb23 = BN_CTX_get(ctx);
1066 if (Zb23 == NULL) goto end;
1068 /* We have to decide whether
1069 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1070 * or equivalently, whether
1071 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1076 if (!field_sqr(group, Zb23, &b->Z, ctx)) goto end;
1077 if (!field_mul(group, tmp1, &a->X, Zb23, ctx)) goto end;
1084 if (!field_sqr(group, Za23, &a->Z, ctx)) goto end;
1085 if (!field_mul(group, tmp2, &b->X, Za23, ctx)) goto end;
1091 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1092 if (BN_cmp(tmp1_, tmp2_) != 0)
1094 ret = 1; /* points differ */
1101 if (!field_mul(group, Zb23, Zb23, &b->Z, ctx)) goto end;
1102 if (!field_mul(group, tmp1, &a->Y, Zb23, ctx)) goto end;
1109 if (!field_mul(group, Za23, Za23, &a->Z, ctx)) goto end;
1110 if (!field_mul(group, tmp2, &b->Y, Za23, ctx)) goto end;
1116 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1117 if (BN_cmp(tmp1_, tmp2_) != 0)
1119 ret = 1; /* points differ */
1123 /* points are equal */
1128 if (new_ctx != NULL)
1129 BN_CTX_free(new_ctx);
1134 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
1136 BN_CTX *new_ctx = NULL;
1140 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1145 ctx = new_ctx = BN_CTX_new();
1151 x = BN_CTX_get(ctx);
1152 y = BN_CTX_get(ctx);
1153 if (y == NULL) goto err;
1155 if (!EC_POINT_get_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
1156 if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
1157 if (!point->Z_is_one)
1159 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
1167 if (new_ctx != NULL)
1168 BN_CTX_free(new_ctx);
1173 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num, EC_POINT *points[], BN_CTX *ctx)
1175 BN_CTX *new_ctx = NULL;
1176 BIGNUM *tmp0, *tmp1;
1178 BIGNUM **heap = NULL;
1187 ctx = new_ctx = BN_CTX_new();
1193 tmp0 = BN_CTX_get(ctx);
1194 tmp1 = BN_CTX_get(ctx);
1195 if (tmp0 == NULL || tmp1 == NULL) goto err;
1197 /* Before converting the individual points, compute inverses of all Z values.
1198 * Modular inversion is rather slow, but luckily we can do with a single
1199 * explicit inversion, plus about 3 multiplications per input value.
1205 /* Now pow2 is the smallest power of 2 satifsying pow2 >= num.
1206 * We need twice that. */
1209 heap = OPENSSL_malloc(pow2 * sizeof heap[0]);
1210 if (heap == NULL) goto err;
1212 /* The array is used as a binary tree, exactly as in heapsort:
1216 * heap[4] heap[5] heap[6] heap[7]
1217 * heap[8]heap[9] heap[10]heap[11] heap[12]heap[13] heap[14] heap[15]
1219 * We put the Z's in the last line;
1220 * then we set each other node to the product of its two child-nodes (where
1221 * empty or 0 entries are treated as ones);
1222 * then we invert heap[1];
1223 * then we invert each other node by replacing it by the product of its
1224 * parent (after inversion) and its sibling (before inversion).
1227 for (i = pow2/2 - 1; i > 0; i--)
1229 for (i = 0; i < num; i++)
1230 heap[pow2/2 + i] = &points[i]->Z;
1231 for (i = pow2/2 + num; i < pow2; i++)
1234 /* set each node to the product of its children */
1235 for (i = pow2/2 - 1; i > 0; i--)
1238 if (heap[i] == NULL) goto err;
1240 if (heap[2*i] != NULL)
1242 if ((heap[2*i + 1] == NULL) || BN_is_zero(heap[2*i + 1]))
1244 if (!BN_copy(heap[i], heap[2*i])) goto err;
1248 if (BN_is_zero(heap[2*i]))
1250 if (!BN_copy(heap[i], heap[2*i + 1])) goto err;
1254 if (!group->meth->field_mul(group, heap[i],
1255 heap[2*i], heap[2*i + 1], ctx)) goto err;
1261 /* invert heap[1] */
1262 if (!BN_is_zero(heap[1]))
1264 if (!BN_mod_inverse(heap[1], heap[1], &group->field, ctx))
1266 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
1270 if (group->meth->field_encode != 0)
1272 /* in the Montgomery case, we just turned R*H (representing H)
1273 * into 1/(R*H), but we need R*(1/H) (representing 1/H);
1274 * i.e. we have need to multiply by the Montgomery factor twice */
1275 if (!group->meth->field_encode(group, heap[1], heap[1], ctx)) goto err;
1276 if (!group->meth->field_encode(group, heap[1], heap[1], ctx)) goto err;
1279 /* set other heap[i]'s to their inverses */
1280 for (i = 2; i < pow2/2 + num; i += 2)
1283 if ((heap[i + 1] != NULL) && !BN_is_zero(heap[i + 1]))
1285 if (!group->meth->field_mul(group, tmp0, heap[i/2], heap[i + 1], ctx)) goto err;
1286 if (!group->meth->field_mul(group, tmp1, heap[i/2], heap[i], ctx)) goto err;
1287 if (!BN_copy(heap[i], tmp0)) goto err;
1288 if (!BN_copy(heap[i + 1], tmp1)) goto err;
1292 if (!BN_copy(heap[i], heap[i/2])) goto err;
1296 /* we have replaced all non-zero Z's by their inverses, now fix up all the points */
1297 for (i = 0; i < num; i++)
1299 EC_POINT *p = points[i];
1301 if (!BN_is_zero(&p->Z))
1303 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1305 if (!group->meth->field_sqr(group, tmp1, &p->Z, ctx)) goto err;
1306 if (!group->meth->field_mul(group, &p->X, &p->X, tmp1, ctx)) goto err;
1308 if (!group->meth->field_mul(group, tmp1, tmp1, &p->Z, ctx)) goto err;
1309 if (!group->meth->field_mul(group, &p->Y, &p->Y, tmp1, ctx)) goto err;
1311 if (group->meth->field_set_to_one != 0)
1313 if (!group->meth->field_set_to_one(group, &p->Z, ctx)) goto err;
1317 if (!BN_one(&p->Z)) goto err;
1327 if (new_ctx != NULL)
1328 BN_CTX_free(new_ctx);
1331 /* heap[pow2/2] .. heap[pow2-1] have not been allocated locally! */
1332 for (i = pow2/2 - 1; i > 0; i--)
1334 if (heap[i] != NULL)
1335 BN_clear_free(heap[i]);
1343 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1345 return BN_mod_mul(r, a, b, &group->field, ctx);
1349 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx)
1351 return BN_mod_sqr(r, a, &group->field, ctx);