1 /* crypto/ec/ecp_smpl.c */
3 * Includes code written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
4 * for the OpenSSL project. Includes code written by Bodo Moeller for the
7 /* ====================================================================
8 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
10 * Redistribution and use in source and binary forms, with or without
11 * modification, are permitted provided that the following conditions
14 * 1. Redistributions of source code must retain the above copyright
15 * notice, this list of conditions and the following disclaimer.
17 * 2. Redistributions in binary form must reproduce the above copyright
18 * notice, this list of conditions and the following disclaimer in
19 * the documentation and/or other materials provided with the
22 * 3. All advertising materials mentioning features or use of this
23 * software must display the following acknowledgment:
24 * "This product includes software developed by the OpenSSL Project
25 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
27 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
28 * endorse or promote products derived from this software without
29 * prior written permission. For written permission, please contact
30 * openssl-core@openssl.org.
32 * 5. Products derived from this software may not be called "OpenSSL"
33 * nor may "OpenSSL" appear in their names without prior written
34 * permission of the OpenSSL Project.
36 * 6. Redistributions of any form whatsoever must retain the following
38 * "This product includes software developed by the OpenSSL Project
39 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
41 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
42 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
43 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
44 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
45 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
46 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
47 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
48 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
49 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
50 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
51 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
52 * OF THE POSSIBILITY OF SUCH DAMAGE.
53 * ====================================================================
55 * This product includes cryptographic software written by Eric Young
56 * (eay@cryptsoft.com). This product includes software written by Tim
57 * Hudson (tjh@cryptsoft.com).
60 /* ====================================================================
61 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
62 * Portions of this software developed by SUN MICROSYSTEMS, INC.,
63 * and contributed to the OpenSSL project.
66 #include <openssl/err.h>
67 #include <openssl/symhacks.h>
71 const EC_METHOD
*EC_GFp_simple_method(void)
73 static const EC_METHOD ret
= {
75 NID_X9_62_prime_field
,
76 ec_GFp_simple_group_init
,
77 ec_GFp_simple_group_finish
,
78 ec_GFp_simple_group_clear_finish
,
79 ec_GFp_simple_group_copy
,
80 ec_GFp_simple_group_set_curve
,
81 ec_GFp_simple_group_get_curve
,
82 ec_GFp_simple_group_get_degree
,
83 ec_GFp_simple_group_check_discriminant
,
84 ec_GFp_simple_point_init
,
85 ec_GFp_simple_point_finish
,
86 ec_GFp_simple_point_clear_finish
,
87 ec_GFp_simple_point_copy
,
88 ec_GFp_simple_point_set_to_infinity
,
89 ec_GFp_simple_set_Jprojective_coordinates_GFp
,
90 ec_GFp_simple_get_Jprojective_coordinates_GFp
,
91 ec_GFp_simple_point_set_affine_coordinates
,
92 ec_GFp_simple_point_get_affine_coordinates
,
97 ec_GFp_simple_is_at_infinity
,
98 ec_GFp_simple_is_on_curve
,
100 ec_GFp_simple_make_affine
,
101 ec_GFp_simple_points_make_affine
,
103 0 /* precompute_mult */ ,
104 0 /* have_precompute_mult */ ,
105 ec_GFp_simple_field_mul
,
106 ec_GFp_simple_field_sqr
,
108 0 /* field_encode */ ,
109 0 /* field_decode */ ,
110 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).
130 int ec_GFp_simple_group_init(EC_GROUP
*group
)
132 group
->field
= BN_new();
135 if (!group
->field
|| !group
->a
|| !group
->b
) {
137 BN_free(group
->field
);
144 group
->a_is_minus3
= 0;
148 void ec_GFp_simple_group_finish(EC_GROUP
*group
)
150 BN_free(group
->field
);
155 void ec_GFp_simple_group_clear_finish(EC_GROUP
*group
)
157 BN_clear_free(group
->field
);
158 BN_clear_free(group
->a
);
159 BN_clear_free(group
->b
);
162 int ec_GFp_simple_group_copy(EC_GROUP
*dest
, const EC_GROUP
*src
)
164 if (!BN_copy(dest
->field
, src
->field
))
166 if (!BN_copy(dest
->a
, src
->a
))
168 if (!BN_copy(dest
->b
, src
->b
))
171 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
,
178 const BIGNUM
*b
, BN_CTX
*ctx
)
181 BN_CTX
*new_ctx
= NULL
;
184 /* p must be a prime > 3 */
185 if (BN_num_bits(p
) <= 2 || !BN_is_odd(p
)) {
186 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE
, EC_R_INVALID_FIELD
);
191 ctx
= new_ctx
= BN_CTX_new();
197 tmp_a
= BN_CTX_get(ctx
);
202 if (!BN_copy(group
->field
, p
))
204 BN_set_negative(group
->field
, 0);
207 if (!BN_nnmod(tmp_a
, a
, p
, ctx
))
209 if (group
->meth
->field_encode
) {
210 if (!group
->meth
->field_encode(group
, group
->a
, tmp_a
, ctx
))
212 } else if (!BN_copy(group
->a
, tmp_a
))
216 if (!BN_nnmod(group
->b
, b
, p
, ctx
))
218 if (group
->meth
->field_encode
)
219 if (!group
->meth
->field_encode(group
, group
->b
, group
->b
, ctx
))
222 /* group->a_is_minus3 */
223 if (!BN_add_word(tmp_a
, 3))
225 group
->a_is_minus3
= (0 == BN_cmp(tmp_a
, group
->field
));
232 BN_CTX_free(new_ctx
);
236 int ec_GFp_simple_group_get_curve(const EC_GROUP
*group
, BIGNUM
*p
, BIGNUM
*a
,
237 BIGNUM
*b
, BN_CTX
*ctx
)
240 BN_CTX
*new_ctx
= NULL
;
243 if (!BN_copy(p
, group
->field
))
247 if (a
!= NULL
|| b
!= NULL
) {
248 if (group
->meth
->field_decode
) {
250 ctx
= new_ctx
= BN_CTX_new();
255 if (!group
->meth
->field_decode(group
, a
, group
->a
, ctx
))
259 if (!group
->meth
->field_decode(group
, b
, group
->b
, ctx
))
264 if (!BN_copy(a
, group
->a
))
268 if (!BN_copy(b
, group
->b
))
278 BN_CTX_free(new_ctx
);
282 int ec_GFp_simple_group_get_degree(const EC_GROUP
*group
)
284 return BN_num_bits(group
->field
);
287 int ec_GFp_simple_group_check_discriminant(const EC_GROUP
*group
, BN_CTX
*ctx
)
290 BIGNUM
*a
, *b
, *order
, *tmp_1
, *tmp_2
;
291 const BIGNUM
*p
= group
->field
;
292 BN_CTX
*new_ctx
= NULL
;
295 ctx
= new_ctx
= BN_CTX_new();
297 ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT
,
298 ERR_R_MALLOC_FAILURE
);
305 tmp_1
= BN_CTX_get(ctx
);
306 tmp_2
= BN_CTX_get(ctx
);
307 order
= BN_CTX_get(ctx
);
311 if (group
->meth
->field_decode
) {
312 if (!group
->meth
->field_decode(group
, a
, group
->a
, ctx
))
314 if (!group
->meth
->field_decode(group
, b
, group
->b
, ctx
))
317 if (!BN_copy(a
, group
->a
))
319 if (!BN_copy(b
, group
->b
))
324 * check the discriminant:
325 * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
331 } else if (!BN_is_zero(b
)) {
332 if (!BN_mod_sqr(tmp_1
, a
, p
, ctx
))
334 if (!BN_mod_mul(tmp_2
, tmp_1
, a
, p
, ctx
))
336 if (!BN_lshift(tmp_1
, tmp_2
, 2))
340 if (!BN_mod_sqr(tmp_2
, b
, p
, ctx
))
342 if (!BN_mul_word(tmp_2
, 27))
346 if (!BN_mod_add(a
, tmp_1
, tmp_2
, p
, ctx
))
357 BN_CTX_free(new_ctx
);
361 int ec_GFp_simple_point_init(EC_POINT
*point
)
368 if (!point
->X
|| !point
->Y
|| !point
->Z
) {
380 void ec_GFp_simple_point_finish(EC_POINT
*point
)
387 void ec_GFp_simple_point_clear_finish(EC_POINT
*point
)
389 BN_clear_free(point
->X
);
390 BN_clear_free(point
->Y
);
391 BN_clear_free(point
->Z
);
395 int ec_GFp_simple_point_copy(EC_POINT
*dest
, const EC_POINT
*src
)
397 if (!BN_copy(dest
->X
, src
->X
))
399 if (!BN_copy(dest
->Y
, src
->Y
))
401 if (!BN_copy(dest
->Z
, src
->Z
))
403 dest
->Z_is_one
= src
->Z_is_one
;
408 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP
*group
,
416 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP
*group
,
423 BN_CTX
*new_ctx
= NULL
;
427 ctx
= new_ctx
= BN_CTX_new();
433 if (!BN_nnmod(point
->X
, x
, group
->field
, ctx
))
435 if (group
->meth
->field_encode
) {
436 if (!group
->meth
->field_encode(group
, point
->X
, point
->X
, ctx
))
442 if (!BN_nnmod(point
->Y
, y
, group
->field
, ctx
))
444 if (group
->meth
->field_encode
) {
445 if (!group
->meth
->field_encode(group
, point
->Y
, point
->Y
, ctx
))
453 if (!BN_nnmod(point
->Z
, z
, group
->field
, ctx
))
455 Z_is_one
= BN_is_one(point
->Z
);
456 if (group
->meth
->field_encode
) {
457 if (Z_is_one
&& (group
->meth
->field_set_to_one
!= 0)) {
458 if (!group
->meth
->field_set_to_one(group
, point
->Z
, ctx
))
462 meth
->field_encode(group
, point
->Z
, point
->Z
, ctx
))
466 point
->Z_is_one
= Z_is_one
;
473 BN_CTX_free(new_ctx
);
477 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP
*group
,
478 const EC_POINT
*point
,
479 BIGNUM
*x
, BIGNUM
*y
,
480 BIGNUM
*z
, BN_CTX
*ctx
)
482 BN_CTX
*new_ctx
= NULL
;
485 if (group
->meth
->field_decode
!= 0) {
487 ctx
= new_ctx
= BN_CTX_new();
493 if (!group
->meth
->field_decode(group
, x
, point
->X
, ctx
))
497 if (!group
->meth
->field_decode(group
, y
, point
->Y
, ctx
))
501 if (!group
->meth
->field_decode(group
, z
, point
->Z
, ctx
))
506 if (!BN_copy(x
, point
->X
))
510 if (!BN_copy(y
, point
->Y
))
514 if (!BN_copy(z
, point
->Z
))
523 BN_CTX_free(new_ctx
);
527 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP
*group
,
530 const BIGNUM
*y
, BN_CTX
*ctx
)
532 if (x
== NULL
|| y
== NULL
) {
534 * unlike for projective coordinates, we do not tolerate this
536 ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES
,
537 ERR_R_PASSED_NULL_PARAMETER
);
541 return EC_POINT_set_Jprojective_coordinates_GFp(group
, point
, x
, y
,
542 BN_value_one(), ctx
);
545 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP
*group
,
546 const EC_POINT
*point
,
547 BIGNUM
*x
, BIGNUM
*y
,
550 BN_CTX
*new_ctx
= NULL
;
551 BIGNUM
*Z
, *Z_1
, *Z_2
, *Z_3
;
555 if (EC_POINT_is_at_infinity(group
, point
)) {
556 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES
,
557 EC_R_POINT_AT_INFINITY
);
562 ctx
= new_ctx
= BN_CTX_new();
569 Z_1
= BN_CTX_get(ctx
);
570 Z_2
= BN_CTX_get(ctx
);
571 Z_3
= BN_CTX_get(ctx
);
575 /* transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3) */
577 if (group
->meth
->field_decode
) {
578 if (!group
->meth
->field_decode(group
, Z
, point
->Z
, ctx
))
586 if (group
->meth
->field_decode
) {
588 if (!group
->meth
->field_decode(group
, x
, point
->X
, ctx
))
592 if (!group
->meth
->field_decode(group
, y
, point
->Y
, ctx
))
597 if (!BN_copy(x
, point
->X
))
601 if (!BN_copy(y
, point
->Y
))
606 if (!BN_mod_inverse(Z_1
, Z_
, group
->field
, ctx
)) {
607 ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES
,
612 if (group
->meth
->field_encode
== 0) {
613 /* field_sqr works on standard representation */
614 if (!group
->meth
->field_sqr(group
, Z_2
, Z_1
, ctx
))
617 if (!BN_mod_sqr(Z_2
, Z_1
, group
->field
, ctx
))
623 * in the Montgomery case, field_mul will cancel out Montgomery
626 if (!group
->meth
->field_mul(group
, x
, point
->X
, Z_2
, ctx
))
631 if (group
->meth
->field_encode
== 0) {
633 * field_mul works on standard representation
635 if (!group
->meth
->field_mul(group
, Z_3
, Z_2
, Z_1
, ctx
))
638 if (!BN_mod_mul(Z_3
, Z_2
, Z_1
, group
->field
, ctx
))
643 * in the Montgomery case, field_mul will cancel out Montgomery
646 if (!group
->meth
->field_mul(group
, y
, point
->Y
, Z_3
, ctx
))
656 BN_CTX_free(new_ctx
);
660 int ec_GFp_simple_add(const EC_GROUP
*group
, EC_POINT
*r
, const EC_POINT
*a
,
661 const EC_POINT
*b
, BN_CTX
*ctx
)
663 int (*field_mul
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*,
664 const BIGNUM
*, BN_CTX
*);
665 int (*field_sqr
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*, BN_CTX
*);
667 BN_CTX
*new_ctx
= NULL
;
668 BIGNUM
*n0
, *n1
, *n2
, *n3
, *n4
, *n5
, *n6
;
672 return EC_POINT_dbl(group
, r
, a
, ctx
);
673 if (EC_POINT_is_at_infinity(group
, a
))
674 return EC_POINT_copy(r
, b
);
675 if (EC_POINT_is_at_infinity(group
, b
))
676 return EC_POINT_copy(r
, a
);
678 field_mul
= group
->meth
->field_mul
;
679 field_sqr
= group
->meth
->field_sqr
;
683 ctx
= new_ctx
= BN_CTX_new();
689 n0
= BN_CTX_get(ctx
);
690 n1
= BN_CTX_get(ctx
);
691 n2
= BN_CTX_get(ctx
);
692 n3
= BN_CTX_get(ctx
);
693 n4
= BN_CTX_get(ctx
);
694 n5
= BN_CTX_get(ctx
);
695 n6
= BN_CTX_get(ctx
);
700 * Note that in this function we must not read components of 'a' or 'b'
701 * once we have written the corresponding components of 'r'. ('r' might
702 * be one of 'a' or 'b'.)
707 if (!BN_copy(n1
, a
->X
))
709 if (!BN_copy(n2
, a
->Y
))
714 if (!field_sqr(group
, n0
, b
->Z
, ctx
))
716 if (!field_mul(group
, n1
, a
->X
, n0
, ctx
))
718 /* n1 = X_a * Z_b^2 */
720 if (!field_mul(group
, n0
, n0
, b
->Z
, ctx
))
722 if (!field_mul(group
, n2
, a
->Y
, n0
, ctx
))
724 /* n2 = Y_a * Z_b^3 */
729 if (!BN_copy(n3
, b
->X
))
731 if (!BN_copy(n4
, b
->Y
))
736 if (!field_sqr(group
, n0
, a
->Z
, ctx
))
738 if (!field_mul(group
, n3
, b
->X
, n0
, ctx
))
740 /* n3 = X_b * Z_a^2 */
742 if (!field_mul(group
, n0
, n0
, a
->Z
, ctx
))
744 if (!field_mul(group
, n4
, b
->Y
, n0
, ctx
))
746 /* n4 = Y_b * Z_a^3 */
750 if (!BN_mod_sub_quick(n5
, n1
, n3
, p
))
752 if (!BN_mod_sub_quick(n6
, n2
, n4
, p
))
757 if (BN_is_zero(n5
)) {
758 if (BN_is_zero(n6
)) {
759 /* a is the same point as b */
761 ret
= EC_POINT_dbl(group
, r
, a
, ctx
);
765 /* a is the inverse of b */
774 if (!BN_mod_add_quick(n1
, n1
, n3
, p
))
776 if (!BN_mod_add_quick(n2
, n2
, n4
, p
))
782 if (a
->Z_is_one
&& b
->Z_is_one
) {
783 if (!BN_copy(r
->Z
, n5
))
787 if (!BN_copy(n0
, b
->Z
))
789 } else if (b
->Z_is_one
) {
790 if (!BN_copy(n0
, a
->Z
))
793 if (!field_mul(group
, n0
, a
->Z
, b
->Z
, ctx
))
796 if (!field_mul(group
, r
->Z
, n0
, n5
, ctx
))
800 /* Z_r = Z_a * Z_b * n5 */
803 if (!field_sqr(group
, n0
, n6
, ctx
))
805 if (!field_sqr(group
, n4
, n5
, ctx
))
807 if (!field_mul(group
, n3
, n1
, n4
, ctx
))
809 if (!BN_mod_sub_quick(r
->X
, n0
, n3
, p
))
811 /* X_r = n6^2 - n5^2 * 'n7' */
814 if (!BN_mod_lshift1_quick(n0
, r
->X
, p
))
816 if (!BN_mod_sub_quick(n0
, n3
, n0
, p
))
818 /* n9 = n5^2 * 'n7' - 2 * X_r */
821 if (!field_mul(group
, n0
, n0
, n6
, ctx
))
823 if (!field_mul(group
, n5
, n4
, n5
, ctx
))
824 goto end
; /* now n5 is n5^3 */
825 if (!field_mul(group
, n1
, n2
, n5
, ctx
))
827 if (!BN_mod_sub_quick(n0
, n0
, n1
, p
))
830 if (!BN_add(n0
, n0
, p
))
832 /* now 0 <= n0 < 2*p, and n0 is even */
833 if (!BN_rshift1(r
->Y
, n0
))
835 /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
840 if (ctx
) /* otherwise we already called BN_CTX_end */
843 BN_CTX_free(new_ctx
);
847 int ec_GFp_simple_dbl(const EC_GROUP
*group
, EC_POINT
*r
, const EC_POINT
*a
,
850 int (*field_mul
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*,
851 const BIGNUM
*, BN_CTX
*);
852 int (*field_sqr
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*, BN_CTX
*);
854 BN_CTX
*new_ctx
= NULL
;
855 BIGNUM
*n0
, *n1
, *n2
, *n3
;
858 if (EC_POINT_is_at_infinity(group
, a
)) {
864 field_mul
= group
->meth
->field_mul
;
865 field_sqr
= group
->meth
->field_sqr
;
869 ctx
= new_ctx
= BN_CTX_new();
875 n0
= BN_CTX_get(ctx
);
876 n1
= BN_CTX_get(ctx
);
877 n2
= BN_CTX_get(ctx
);
878 n3
= BN_CTX_get(ctx
);
883 * Note that in this function we must not read components of 'a' once we
884 * have written the corresponding components of 'r'. ('r' might the same
890 if (!field_sqr(group
, n0
, a
->X
, ctx
))
892 if (!BN_mod_lshift1_quick(n1
, n0
, p
))
894 if (!BN_mod_add_quick(n0
, n0
, n1
, p
))
896 if (!BN_mod_add_quick(n1
, n0
, group
->a
, p
))
898 /* n1 = 3 * X_a^2 + a_curve */
899 } else if (group
->a_is_minus3
) {
900 if (!field_sqr(group
, n1
, a
->Z
, ctx
))
902 if (!BN_mod_add_quick(n0
, a
->X
, n1
, p
))
904 if (!BN_mod_sub_quick(n2
, a
->X
, n1
, p
))
906 if (!field_mul(group
, n1
, n0
, n2
, ctx
))
908 if (!BN_mod_lshift1_quick(n0
, n1
, p
))
910 if (!BN_mod_add_quick(n1
, n0
, n1
, p
))
913 * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
914 * = 3 * X_a^2 - 3 * Z_a^4
917 if (!field_sqr(group
, n0
, a
->X
, ctx
))
919 if (!BN_mod_lshift1_quick(n1
, n0
, p
))
921 if (!BN_mod_add_quick(n0
, n0
, n1
, p
))
923 if (!field_sqr(group
, n1
, a
->Z
, ctx
))
925 if (!field_sqr(group
, n1
, n1
, ctx
))
927 if (!field_mul(group
, n1
, n1
, group
->a
, ctx
))
929 if (!BN_mod_add_quick(n1
, n1
, n0
, p
))
931 /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
936 if (!BN_copy(n0
, a
->Y
))
939 if (!field_mul(group
, n0
, a
->Y
, a
->Z
, ctx
))
942 if (!BN_mod_lshift1_quick(r
->Z
, n0
, p
))
945 /* Z_r = 2 * Y_a * Z_a */
948 if (!field_sqr(group
, n3
, a
->Y
, ctx
))
950 if (!field_mul(group
, n2
, a
->X
, n3
, ctx
))
952 if (!BN_mod_lshift_quick(n2
, n2
, 2, p
))
954 /* n2 = 4 * X_a * Y_a^2 */
957 if (!BN_mod_lshift1_quick(n0
, n2
, p
))
959 if (!field_sqr(group
, r
->X
, n1
, ctx
))
961 if (!BN_mod_sub_quick(r
->X
, r
->X
, n0
, p
))
963 /* X_r = n1^2 - 2 * n2 */
966 if (!field_sqr(group
, n0
, n3
, ctx
))
968 if (!BN_mod_lshift_quick(n3
, n0
, 3, p
))
973 if (!BN_mod_sub_quick(n0
, n2
, r
->X
, p
))
975 if (!field_mul(group
, n0
, n1
, n0
, ctx
))
977 if (!BN_mod_sub_quick(r
->Y
, n0
, n3
, p
))
979 /* Y_r = n1 * (n2 - X_r) - n3 */
986 BN_CTX_free(new_ctx
);
990 int ec_GFp_simple_invert(const EC_GROUP
*group
, EC_POINT
*point
, BN_CTX
*ctx
)
992 if (EC_POINT_is_at_infinity(group
, point
) || BN_is_zero(point
->Y
))
993 /* point is its own inverse */
996 return BN_usub(point
->Y
, group
->field
, point
->Y
);
999 int ec_GFp_simple_is_at_infinity(const EC_GROUP
*group
, const EC_POINT
*point
)
1001 return BN_is_zero(point
->Z
);
1004 int ec_GFp_simple_is_on_curve(const EC_GROUP
*group
, const EC_POINT
*point
,
1007 int (*field_mul
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*,
1008 const BIGNUM
*, BN_CTX
*);
1009 int (*field_sqr
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*, BN_CTX
*);
1011 BN_CTX
*new_ctx
= NULL
;
1012 BIGNUM
*rh
, *tmp
, *Z4
, *Z6
;
1015 if (EC_POINT_is_at_infinity(group
, point
))
1018 field_mul
= group
->meth
->field_mul
;
1019 field_sqr
= group
->meth
->field_sqr
;
1023 ctx
= new_ctx
= BN_CTX_new();
1029 rh
= BN_CTX_get(ctx
);
1030 tmp
= BN_CTX_get(ctx
);
1031 Z4
= BN_CTX_get(ctx
);
1032 Z6
= BN_CTX_get(ctx
);
1037 * We have a curve defined by a Weierstrass equation
1038 * y^2 = x^3 + a*x + b.
1039 * The point to consider is given in Jacobian projective coordinates
1040 * where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
1041 * Substituting this and multiplying by Z^6 transforms the above equation into
1042 * Y^2 = X^3 + a*X*Z^4 + b*Z^6.
1043 * To test this, we add up the right-hand side in 'rh'.
1047 if (!field_sqr(group
, rh
, point
->X
, ctx
))
1050 if (!point
->Z_is_one
) {
1051 if (!field_sqr(group
, tmp
, point
->Z
, ctx
))
1053 if (!field_sqr(group
, Z4
, tmp
, ctx
))
1055 if (!field_mul(group
, Z6
, Z4
, tmp
, ctx
))
1058 /* rh := (rh + a*Z^4)*X */
1059 if (group
->a_is_minus3
) {
1060 if (!BN_mod_lshift1_quick(tmp
, Z4
, p
))
1062 if (!BN_mod_add_quick(tmp
, tmp
, Z4
, p
))
1064 if (!BN_mod_sub_quick(rh
, rh
, tmp
, p
))
1066 if (!field_mul(group
, rh
, rh
, point
->X
, ctx
))
1069 if (!field_mul(group
, tmp
, Z4
, group
->a
, ctx
))
1071 if (!BN_mod_add_quick(rh
, rh
, tmp
, p
))
1073 if (!field_mul(group
, rh
, rh
, point
->X
, ctx
))
1077 /* rh := rh + b*Z^6 */
1078 if (!field_mul(group
, tmp
, group
->b
, Z6
, ctx
))
1080 if (!BN_mod_add_quick(rh
, rh
, tmp
, p
))
1083 /* point->Z_is_one */
1085 /* rh := (rh + a)*X */
1086 if (!BN_mod_add_quick(rh
, rh
, group
->a
, p
))
1088 if (!field_mul(group
, rh
, rh
, point
->X
, ctx
))
1091 if (!BN_mod_add_quick(rh
, rh
, group
->b
, p
))
1096 if (!field_sqr(group
, tmp
, point
->Y
, ctx
))
1099 ret
= (0 == BN_ucmp(tmp
, rh
));
1103 if (new_ctx
!= NULL
)
1104 BN_CTX_free(new_ctx
);
1108 int ec_GFp_simple_cmp(const EC_GROUP
*group
, const EC_POINT
*a
,
1109 const EC_POINT
*b
, BN_CTX
*ctx
)
1114 * 0 equal (in affine coordinates)
1118 int (*field_mul
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*,
1119 const BIGNUM
*, BN_CTX
*);
1120 int (*field_sqr
) (const EC_GROUP
*, BIGNUM
*, const BIGNUM
*, BN_CTX
*);
1121 BN_CTX
*new_ctx
= NULL
;
1122 BIGNUM
*tmp1
, *tmp2
, *Za23
, *Zb23
;
1123 const BIGNUM
*tmp1_
, *tmp2_
;
1126 if (EC_POINT_is_at_infinity(group
, a
)) {
1127 return EC_POINT_is_at_infinity(group
, b
) ? 0 : 1;
1130 if (EC_POINT_is_at_infinity(group
, b
))
1133 if (a
->Z_is_one
&& b
->Z_is_one
) {
1134 return ((BN_cmp(a
->X
, b
->X
) == 0) && BN_cmp(a
->Y
, b
->Y
) == 0) ? 0 : 1;
1137 field_mul
= group
->meth
->field_mul
;
1138 field_sqr
= group
->meth
->field_sqr
;
1141 ctx
= new_ctx
= BN_CTX_new();
1147 tmp1
= BN_CTX_get(ctx
);
1148 tmp2
= BN_CTX_get(ctx
);
1149 Za23
= BN_CTX_get(ctx
);
1150 Zb23
= BN_CTX_get(ctx
);
1155 * We have to decide whether
1156 * (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1157 * or equivalently, whether
1158 * (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1162 if (!field_sqr(group
, Zb23
, b
->Z
, ctx
))
1164 if (!field_mul(group
, tmp1
, a
->X
, Zb23
, ctx
))
1170 if (!field_sqr(group
, Za23
, a
->Z
, ctx
))
1172 if (!field_mul(group
, tmp2
, b
->X
, Za23
, ctx
))
1178 /* compare X_a*Z_b^2 with X_b*Z_a^2 */
1179 if (BN_cmp(tmp1_
, tmp2_
) != 0) {
1180 ret
= 1; /* points differ */
1185 if (!field_mul(group
, Zb23
, Zb23
, b
->Z
, ctx
))
1187 if (!field_mul(group
, tmp1
, a
->Y
, Zb23
, ctx
))
1193 if (!field_mul(group
, Za23
, Za23
, a
->Z
, ctx
))
1195 if (!field_mul(group
, tmp2
, b
->Y
, Za23
, ctx
))
1201 /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */
1202 if (BN_cmp(tmp1_
, tmp2_
) != 0) {
1203 ret
= 1; /* points differ */
1207 /* points are equal */
1212 if (new_ctx
!= NULL
)
1213 BN_CTX_free(new_ctx
);
1217 int ec_GFp_simple_make_affine(const EC_GROUP
*group
, EC_POINT
*point
,
1220 BN_CTX
*new_ctx
= NULL
;
1224 if (point
->Z_is_one
|| EC_POINT_is_at_infinity(group
, point
))
1228 ctx
= new_ctx
= BN_CTX_new();
1234 x
= BN_CTX_get(ctx
);
1235 y
= BN_CTX_get(ctx
);
1239 if (!EC_POINT_get_affine_coordinates_GFp(group
, point
, x
, y
, ctx
))
1241 if (!EC_POINT_set_affine_coordinates_GFp(group
, point
, x
, y
, ctx
))
1243 if (!point
->Z_is_one
) {
1244 ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE
, ERR_R_INTERNAL_ERROR
);
1252 if (new_ctx
!= NULL
)
1253 BN_CTX_free(new_ctx
);
1257 int ec_GFp_simple_points_make_affine(const EC_GROUP
*group
, size_t num
,
1258 EC_POINT
*points
[], BN_CTX
*ctx
)
1260 BN_CTX
*new_ctx
= NULL
;
1261 BIGNUM
*tmp
, *tmp_Z
;
1262 BIGNUM
**prod_Z
= NULL
;
1270 ctx
= new_ctx
= BN_CTX_new();
1276 tmp
= BN_CTX_get(ctx
);
1277 tmp_Z
= BN_CTX_get(ctx
);
1278 if (tmp
== NULL
|| tmp_Z
== NULL
)
1281 prod_Z
= OPENSSL_malloc(num
* sizeof prod_Z
[0]);
1284 for (i
= 0; i
< num
; i
++) {
1285 prod_Z
[i
] = BN_new();
1286 if (prod_Z
[i
] == NULL
)
1291 * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1292 * skipping any zero-valued inputs (pretend that they're 1).
1295 if (!BN_is_zero(points
[0]->Z
)) {
1296 if (!BN_copy(prod_Z
[0], points
[0]->Z
))
1299 if (group
->meth
->field_set_to_one
!= 0) {
1300 if (!group
->meth
->field_set_to_one(group
, prod_Z
[0], ctx
))
1303 if (!BN_one(prod_Z
[0]))
1308 for (i
= 1; i
< num
; i
++) {
1309 if (!BN_is_zero(points
[i
]->Z
)) {
1311 meth
->field_mul(group
, prod_Z
[i
], prod_Z
[i
- 1], points
[i
]->Z
,
1315 if (!BN_copy(prod_Z
[i
], prod_Z
[i
- 1]))
1321 * Now use a single explicit inversion to replace every non-zero
1322 * points[i]->Z by its inverse.
1325 if (!BN_mod_inverse(tmp
, prod_Z
[num
- 1], group
->field
, ctx
)) {
1326 ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE
, ERR_R_BN_LIB
);
1329 if (group
->meth
->field_encode
!= 0) {
1331 * In the Montgomery case, we just turned R*H (representing H) into
1332 * 1/(R*H), but we need R*(1/H) (representing 1/H); i.e. we need to
1333 * multiply by the Montgomery factor twice.
1335 if (!group
->meth
->field_encode(group
, tmp
, tmp
, ctx
))
1337 if (!group
->meth
->field_encode(group
, tmp
, tmp
, ctx
))
1341 for (i
= num
- 1; i
> 0; --i
) {
1343 * Loop invariant: tmp is the product of the inverses of points[0]->Z
1344 * .. points[i]->Z (zero-valued inputs skipped).
1346 if (!BN_is_zero(points
[i
]->Z
)) {
1348 * Set tmp_Z to the inverse of points[i]->Z (as product of Z
1349 * inverses 0 .. i, Z values 0 .. i - 1).
1352 meth
->field_mul(group
, tmp_Z
, prod_Z
[i
- 1], tmp
, ctx
))
1355 * Update tmp to satisfy the loop invariant for i - 1.
1357 if (!group
->meth
->field_mul(group
, tmp
, tmp
, points
[i
]->Z
, ctx
))
1359 /* Replace points[i]->Z by its inverse. */
1360 if (!BN_copy(points
[i
]->Z
, tmp_Z
))
1365 if (!BN_is_zero(points
[0]->Z
)) {
1366 /* Replace points[0]->Z by its inverse. */
1367 if (!BN_copy(points
[0]->Z
, tmp
))
1371 /* Finally, fix up the X and Y coordinates for all points. */
1373 for (i
= 0; i
< num
; i
++) {
1374 EC_POINT
*p
= points
[i
];
1376 if (!BN_is_zero(p
->Z
)) {
1377 /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */
1379 if (!group
->meth
->field_sqr(group
, tmp
, p
->Z
, ctx
))
1381 if (!group
->meth
->field_mul(group
, p
->X
, p
->X
, tmp
, ctx
))
1384 if (!group
->meth
->field_mul(group
, tmp
, tmp
, p
->Z
, ctx
))
1386 if (!group
->meth
->field_mul(group
, p
->Y
, p
->Y
, tmp
, ctx
))
1389 if (group
->meth
->field_set_to_one
!= 0) {
1390 if (!group
->meth
->field_set_to_one(group
, p
->Z
, ctx
))
1404 if (new_ctx
!= NULL
)
1405 BN_CTX_free(new_ctx
);
1406 if (prod_Z
!= NULL
) {
1407 for (i
= 0; i
< num
; i
++) {
1408 if (prod_Z
[i
] == NULL
)
1410 BN_clear_free(prod_Z
[i
]);
1412 OPENSSL_free(prod_Z
);
1417 int ec_GFp_simple_field_mul(const EC_GROUP
*group
, BIGNUM
*r
, const BIGNUM
*a
,
1418 const BIGNUM
*b
, BN_CTX
*ctx
)
1420 return BN_mod_mul(r
, a
, b
, group
->field
, ctx
);
1423 int ec_GFp_simple_field_sqr(const EC_GROUP
*group
, BIGNUM
*r
, const BIGNUM
*a
,
1426 return BN_mod_sqr(r
, a
, group
->field
, ctx
);