2 * Copyright 2002-2018 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved
5 * Licensed under the OpenSSL license (the "License"). You may not use
6 * this file except in compliance with the License. You can obtain a copy
7 * in the file LICENSE in the source distribution or at
8 * https://www.openssl.org/source/license.html
11 #include <openssl/err.h>
13 #include "internal/bn_int.h"
16 #ifndef OPENSSL_NO_EC2M
18 const EC_METHOD *EC_GF2m_simple_method(void)
20 static const EC_METHOD ret = {
22 NID_X9_62_characteristic_two_field,
23 ec_GF2m_simple_group_init,
24 ec_GF2m_simple_group_finish,
25 ec_GF2m_simple_group_clear_finish,
26 ec_GF2m_simple_group_copy,
27 ec_GF2m_simple_group_set_curve,
28 ec_GF2m_simple_group_get_curve,
29 ec_GF2m_simple_group_get_degree,
30 ec_group_simple_order_bits,
31 ec_GF2m_simple_group_check_discriminant,
32 ec_GF2m_simple_point_init,
33 ec_GF2m_simple_point_finish,
34 ec_GF2m_simple_point_clear_finish,
35 ec_GF2m_simple_point_copy,
36 ec_GF2m_simple_point_set_to_infinity,
37 0 /* set_Jprojective_coordinates_GFp */ ,
38 0 /* get_Jprojective_coordinates_GFp */ ,
39 ec_GF2m_simple_point_set_affine_coordinates,
40 ec_GF2m_simple_point_get_affine_coordinates,
44 ec_GF2m_simple_invert,
45 ec_GF2m_simple_is_at_infinity,
46 ec_GF2m_simple_is_on_curve,
48 ec_GF2m_simple_make_affine,
49 ec_GF2m_simple_points_make_affine,
51 0 /* precompute_mul */,
52 0 /* have_precompute_mul */,
53 ec_GF2m_simple_field_mul,
54 ec_GF2m_simple_field_sqr,
55 ec_GF2m_simple_field_div,
56 0 /* field_encode */ ,
57 0 /* field_decode */ ,
58 0, /* field_set_to_one */
59 ec_key_simple_priv2oct,
60 ec_key_simple_oct2priv,
62 ec_key_simple_generate_key,
63 ec_key_simple_check_key,
64 ec_key_simple_generate_public_key,
67 ecdh_simple_compute_key,
68 0, /* field_inverse_mod_ord */
69 0 /* blind_coordinates */
76 * Initialize a GF(2^m)-based EC_GROUP structure. Note that all other members
77 * are handled by EC_GROUP_new.
79 int ec_GF2m_simple_group_init(EC_GROUP *group)
81 group->field = BN_new();
85 if (group->field == NULL || group->a == NULL || group->b == NULL) {
86 BN_free(group->field);
95 * Free a GF(2^m)-based EC_GROUP structure. Note that all other members are
96 * handled by EC_GROUP_free.
98 void ec_GF2m_simple_group_finish(EC_GROUP *group)
100 BN_free(group->field);
106 * Clear and free a GF(2^m)-based EC_GROUP structure. Note that all other
107 * members are handled by EC_GROUP_clear_free.
109 void ec_GF2m_simple_group_clear_finish(EC_GROUP *group)
111 BN_clear_free(group->field);
112 BN_clear_free(group->a);
113 BN_clear_free(group->b);
123 * Copy a GF(2^m)-based EC_GROUP structure. Note that all other members are
124 * handled by EC_GROUP_copy.
126 int ec_GF2m_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
128 if (!BN_copy(dest->field, src->field))
130 if (!BN_copy(dest->a, src->a))
132 if (!BN_copy(dest->b, src->b))
134 dest->poly[0] = src->poly[0];
135 dest->poly[1] = src->poly[1];
136 dest->poly[2] = src->poly[2];
137 dest->poly[3] = src->poly[3];
138 dest->poly[4] = src->poly[4];
139 dest->poly[5] = src->poly[5];
140 if (bn_wexpand(dest->a, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2) ==
143 if (bn_wexpand(dest->b, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2) ==
146 bn_set_all_zero(dest->a);
147 bn_set_all_zero(dest->b);
151 /* Set the curve parameters of an EC_GROUP structure. */
152 int ec_GF2m_simple_group_set_curve(EC_GROUP *group,
153 const BIGNUM *p, const BIGNUM *a,
154 const BIGNUM *b, BN_CTX *ctx)
159 if (!BN_copy(group->field, p))
161 i = BN_GF2m_poly2arr(group->field, group->poly, 6) - 1;
162 if ((i != 5) && (i != 3)) {
163 ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_SET_CURVE, EC_R_UNSUPPORTED_FIELD);
168 if (!BN_GF2m_mod_arr(group->a, a, group->poly))
170 if (bn_wexpand(group->a, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2)
173 bn_set_all_zero(group->a);
176 if (!BN_GF2m_mod_arr(group->b, b, group->poly))
178 if (bn_wexpand(group->b, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2)
181 bn_set_all_zero(group->b);
189 * Get the curve parameters of an EC_GROUP structure. If p, a, or b are NULL
190 * then there values will not be set but the method will return with success.
192 int ec_GF2m_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p,
193 BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
198 if (!BN_copy(p, group->field))
203 if (!BN_copy(a, group->a))
208 if (!BN_copy(b, group->b))
219 * Gets the degree of the field. For a curve over GF(2^m) this is the value
222 int ec_GF2m_simple_group_get_degree(const EC_GROUP *group)
224 return BN_num_bits(group->field) - 1;
228 * Checks the discriminant of the curve. y^2 + x*y = x^3 + a*x^2 + b is an
229 * elliptic curve <=> b != 0 (mod p)
231 int ec_GF2m_simple_group_check_discriminant(const EC_GROUP *group,
236 BN_CTX *new_ctx = NULL;
239 ctx = new_ctx = BN_CTX_new();
241 ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_CHECK_DISCRIMINANT,
242 ERR_R_MALLOC_FAILURE);
251 if (!BN_GF2m_mod_arr(b, group->b, group->poly))
255 * check the discriminant: y^2 + x*y = x^3 + a*x^2 + b is an elliptic
256 * curve <=> b != 0 (mod p)
266 BN_CTX_free(new_ctx);
270 /* Initializes an EC_POINT. */
271 int ec_GF2m_simple_point_init(EC_POINT *point)
277 if (point->X == NULL || point->Y == NULL || point->Z == NULL) {
286 /* Frees an EC_POINT. */
287 void ec_GF2m_simple_point_finish(EC_POINT *point)
294 /* Clears and frees an EC_POINT. */
295 void ec_GF2m_simple_point_clear_finish(EC_POINT *point)
297 BN_clear_free(point->X);
298 BN_clear_free(point->Y);
299 BN_clear_free(point->Z);
304 * Copy the contents of one EC_POINT into another. Assumes dest is
307 int ec_GF2m_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
309 if (!BN_copy(dest->X, src->X))
311 if (!BN_copy(dest->Y, src->Y))
313 if (!BN_copy(dest->Z, src->Z))
315 dest->Z_is_one = src->Z_is_one;
316 dest->curve_name = src->curve_name;
322 * Set an EC_POINT to the point at infinity. A point at infinity is
323 * represented by having Z=0.
325 int ec_GF2m_simple_point_set_to_infinity(const EC_GROUP *group,
334 * Set the coordinates of an EC_POINT using affine coordinates. Note that
335 * the simple implementation only uses affine coordinates.
337 int ec_GF2m_simple_point_set_affine_coordinates(const EC_GROUP *group,
340 const BIGNUM *y, BN_CTX *ctx)
343 if (x == NULL || y == NULL) {
344 ECerr(EC_F_EC_GF2M_SIMPLE_POINT_SET_AFFINE_COORDINATES,
345 ERR_R_PASSED_NULL_PARAMETER);
349 if (!BN_copy(point->X, x))
351 BN_set_negative(point->X, 0);
352 if (!BN_copy(point->Y, y))
354 BN_set_negative(point->Y, 0);
355 if (!BN_copy(point->Z, BN_value_one()))
357 BN_set_negative(point->Z, 0);
366 * Gets the affine coordinates of an EC_POINT. Note that the simple
367 * implementation only uses affine coordinates.
369 int ec_GF2m_simple_point_get_affine_coordinates(const EC_GROUP *group,
370 const EC_POINT *point,
371 BIGNUM *x, BIGNUM *y,
376 if (EC_POINT_is_at_infinity(group, point)) {
377 ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES,
378 EC_R_POINT_AT_INFINITY);
382 if (BN_cmp(point->Z, BN_value_one())) {
383 ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES,
384 ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
388 if (!BN_copy(x, point->X))
390 BN_set_negative(x, 0);
393 if (!BN_copy(y, point->Y))
395 BN_set_negative(y, 0);
404 * Computes a + b and stores the result in r. r could be a or b, a could be
405 * b. Uses algorithm A.10.2 of IEEE P1363.
407 int ec_GF2m_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
408 const EC_POINT *b, BN_CTX *ctx)
410 BN_CTX *new_ctx = NULL;
411 BIGNUM *x0, *y0, *x1, *y1, *x2, *y2, *s, *t;
414 if (EC_POINT_is_at_infinity(group, a)) {
415 if (!EC_POINT_copy(r, b))
420 if (EC_POINT_is_at_infinity(group, b)) {
421 if (!EC_POINT_copy(r, a))
427 ctx = new_ctx = BN_CTX_new();
433 x0 = BN_CTX_get(ctx);
434 y0 = BN_CTX_get(ctx);
435 x1 = BN_CTX_get(ctx);
436 y1 = BN_CTX_get(ctx);
437 x2 = BN_CTX_get(ctx);
438 y2 = BN_CTX_get(ctx);
445 if (!BN_copy(x0, a->X))
447 if (!BN_copy(y0, a->Y))
450 if (!EC_POINT_get_affine_coordinates_GF2m(group, a, x0, y0, ctx))
454 if (!BN_copy(x1, b->X))
456 if (!BN_copy(y1, b->Y))
459 if (!EC_POINT_get_affine_coordinates_GF2m(group, b, x1, y1, ctx))
463 if (BN_GF2m_cmp(x0, x1)) {
464 if (!BN_GF2m_add(t, x0, x1))
466 if (!BN_GF2m_add(s, y0, y1))
468 if (!group->meth->field_div(group, s, s, t, ctx))
470 if (!group->meth->field_sqr(group, x2, s, ctx))
472 if (!BN_GF2m_add(x2, x2, group->a))
474 if (!BN_GF2m_add(x2, x2, s))
476 if (!BN_GF2m_add(x2, x2, t))
479 if (BN_GF2m_cmp(y0, y1) || BN_is_zero(x1)) {
480 if (!EC_POINT_set_to_infinity(group, r))
485 if (!group->meth->field_div(group, s, y1, x1, ctx))
487 if (!BN_GF2m_add(s, s, x1))
490 if (!group->meth->field_sqr(group, x2, s, ctx))
492 if (!BN_GF2m_add(x2, x2, s))
494 if (!BN_GF2m_add(x2, x2, group->a))
498 if (!BN_GF2m_add(y2, x1, x2))
500 if (!group->meth->field_mul(group, y2, y2, s, ctx))
502 if (!BN_GF2m_add(y2, y2, x2))
504 if (!BN_GF2m_add(y2, y2, y1))
507 if (!EC_POINT_set_affine_coordinates_GF2m(group, r, x2, y2, ctx))
514 BN_CTX_free(new_ctx);
519 * Computes 2 * a and stores the result in r. r could be a. Uses algorithm
520 * A.10.2 of IEEE P1363.
522 int ec_GF2m_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
525 return ec_GF2m_simple_add(group, r, a, a, ctx);
528 int ec_GF2m_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
530 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
531 /* point is its own inverse */
534 if (!EC_POINT_make_affine(group, point, ctx))
536 return BN_GF2m_add(point->Y, point->X, point->Y);
539 /* Indicates whether the given point is the point at infinity. */
540 int ec_GF2m_simple_is_at_infinity(const EC_GROUP *group,
541 const EC_POINT *point)
543 return BN_is_zero(point->Z);
547 * Determines whether the given EC_POINT is an actual point on the curve defined
548 * in the EC_GROUP. A point is valid if it satisfies the Weierstrass equation:
549 * y^2 + x*y = x^3 + a*x^2 + b.
551 int ec_GF2m_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
555 BN_CTX *new_ctx = NULL;
557 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
558 const BIGNUM *, BN_CTX *);
559 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
561 if (EC_POINT_is_at_infinity(group, point))
564 field_mul = group->meth->field_mul;
565 field_sqr = group->meth->field_sqr;
567 /* only support affine coordinates */
568 if (!point->Z_is_one)
572 ctx = new_ctx = BN_CTX_new();
578 y2 = BN_CTX_get(ctx);
579 lh = BN_CTX_get(ctx);
584 * We have a curve defined by a Weierstrass equation
585 * y^2 + x*y = x^3 + a*x^2 + b.
586 * <=> x^3 + a*x^2 + x*y + b + y^2 = 0
587 * <=> ((x + a) * x + y ) * x + b + y^2 = 0
589 if (!BN_GF2m_add(lh, point->X, group->a))
591 if (!field_mul(group, lh, lh, point->X, ctx))
593 if (!BN_GF2m_add(lh, lh, point->Y))
595 if (!field_mul(group, lh, lh, point->X, ctx))
597 if (!BN_GF2m_add(lh, lh, group->b))
599 if (!field_sqr(group, y2, point->Y, ctx))
601 if (!BN_GF2m_add(lh, lh, y2))
603 ret = BN_is_zero(lh);
607 BN_CTX_free(new_ctx);
612 * Indicates whether two points are equal.
615 * 0 equal (in affine coordinates)
618 int ec_GF2m_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
619 const EC_POINT *b, BN_CTX *ctx)
621 BIGNUM *aX, *aY, *bX, *bY;
622 BN_CTX *new_ctx = NULL;
625 if (EC_POINT_is_at_infinity(group, a)) {
626 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
629 if (EC_POINT_is_at_infinity(group, b))
632 if (a->Z_is_one && b->Z_is_one) {
633 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
637 ctx = new_ctx = BN_CTX_new();
643 aX = BN_CTX_get(ctx);
644 aY = BN_CTX_get(ctx);
645 bX = BN_CTX_get(ctx);
646 bY = BN_CTX_get(ctx);
650 if (!EC_POINT_get_affine_coordinates_GF2m(group, a, aX, aY, ctx))
652 if (!EC_POINT_get_affine_coordinates_GF2m(group, b, bX, bY, ctx))
654 ret = ((BN_cmp(aX, bX) == 0) && BN_cmp(aY, bY) == 0) ? 0 : 1;
658 BN_CTX_free(new_ctx);
662 /* Forces the given EC_POINT to internally use affine coordinates. */
663 int ec_GF2m_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
666 BN_CTX *new_ctx = NULL;
670 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
674 ctx = new_ctx = BN_CTX_new();
685 if (!EC_POINT_get_affine_coordinates_GF2m(group, point, x, y, ctx))
687 if (!BN_copy(point->X, x))
689 if (!BN_copy(point->Y, y))
691 if (!BN_one(point->Z))
699 BN_CTX_free(new_ctx);
704 * Forces each of the EC_POINTs in the given array to use affine coordinates.
706 int ec_GF2m_simple_points_make_affine(const EC_GROUP *group, size_t num,
707 EC_POINT *points[], BN_CTX *ctx)
711 for (i = 0; i < num; i++) {
712 if (!group->meth->make_affine(group, points[i], ctx))
719 /* Wrapper to simple binary polynomial field multiplication implementation. */
720 int ec_GF2m_simple_field_mul(const EC_GROUP *group, BIGNUM *r,
721 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
723 return BN_GF2m_mod_mul_arr(r, a, b, group->poly, ctx);
726 /* Wrapper to simple binary polynomial field squaring implementation. */
727 int ec_GF2m_simple_field_sqr(const EC_GROUP *group, BIGNUM *r,
728 const BIGNUM *a, BN_CTX *ctx)
730 return BN_GF2m_mod_sqr_arr(r, a, group->poly, ctx);
733 /* Wrapper to simple binary polynomial field division implementation. */
734 int ec_GF2m_simple_field_div(const EC_GROUP *group, BIGNUM *r,
735 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
737 return BN_GF2m_mod_div(r, a, b, group->field, ctx);