1/* Originally written by Bodo Moeller for the OpenSSL project.
2 * ====================================================================
3 * Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 *
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 *
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in
14 * the documentation and/or other materials provided with the
15 * distribution.
16 *
17 * 3. All advertising materials mentioning features or use of this
18 * software must display the following acknowledgment:
19 * "This product includes software developed by the OpenSSL Project
20 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
21 *
22 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
23 * endorse or promote products derived from this software without
24 * prior written permission. For written permission, please contact
25 * openssl-core@openssl.org.
26 *
27 * 5. Products derived from this software may not be called "OpenSSL"
28 * nor may "OpenSSL" appear in their names without prior written
29 * permission of the OpenSSL Project.
30 *
31 * 6. Redistributions of any form whatsoever must retain the following
32 * acknowledgment:
33 * "This product includes software developed by the OpenSSL Project
34 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
35 *
36 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
37 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
38 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
39 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
40 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
41 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
42 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
43 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
44 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
45 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
46 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
47 * OF THE POSSIBILITY OF SUCH DAMAGE.
48 * ====================================================================
49 *
50 * This product includes cryptographic software written by Eric Young
51 * (eay@cryptsoft.com). This product includes software written by Tim
52 * Hudson (tjh@cryptsoft.com).
53 *
54 */
55/* ====================================================================
56 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
57 *
58 * Portions of the attached software ("Contribution") are developed by
59 * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
60 *
61 * The Contribution is licensed pursuant to the OpenSSL open source
62 * license provided above.
63 *
64 * The elliptic curve binary polynomial software is originally written by
65 * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
66 * Laboratories. */
67
68#include <openssl/ec.h>
69
70#include <string.h>
71
72#include <openssl/bn.h>
73#include <openssl/err.h>
74#include <openssl/mem.h>
75
76#include "internal.h"
77#include "../../internal.h"
78
79
80// Most method functions in this file are designed to work with non-trivial
81// representations of field elements if necessary (see ecp_mont.c): while
82// standard modular addition and subtraction are used, the field_mul and
83// field_sqr methods will be used for multiplication, and field_encode and
84// field_decode (if defined) will be used for converting between
85// representations.
86//
87// Functions here specifically assume that if a non-trivial representation is
88// used, it is a Montgomery representation (i.e. 'encoding' means multiplying
89// by some factor R).
90
91int ec_GFp_simple_group_init(EC_GROUP *group) {
92 BN_init(&group->field);
93 group->a_is_minus3 = 0;
94 return 1;
95}
96
97void ec_GFp_simple_group_finish(EC_GROUP *group) {
98 BN_free(&group->field);
99}
100
101int ec_GFp_simple_group_set_curve(EC_GROUP *group, const BIGNUM *p,
102 const BIGNUM *a, const BIGNUM *b,
103 BN_CTX *ctx) {
104 int ret = 0;
105 BN_CTX *new_ctx = NULL;
106
107 // p must be a prime > 3
108 if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
109 OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD);
110 return 0;
111 }
112
113 if (ctx == NULL) {
114 ctx = new_ctx = BN_CTX_new();
115 if (ctx == NULL) {
116 return 0;
117 }
118 }
119
120 BN_CTX_start(ctx);
121 BIGNUM *tmp = BN_CTX_get(ctx);
122 if (tmp == NULL) {
123 goto err;
124 }
125
126 // group->field
127 if (!BN_copy(&group->field, p)) {
128 goto err;
129 }
130 BN_set_negative(&group->field, 0);
131 // Store the field in minimal form, so it can be used with |BN_ULONG| arrays.
132 bn_set_minimal_width(&group->field);
133
134 // group->a
135 if (!BN_nnmod(tmp, a, &group->field, ctx) ||
136 !ec_bignum_to_felem(group, &group->a, tmp)) {
137 goto err;
138 }
139
140 // group->a_is_minus3
141 if (!BN_add_word(tmp, 3)) {
142 goto err;
143 }
144 group->a_is_minus3 = (0 == BN_cmp(tmp, &group->field));
145
146 // group->b
147 if (!BN_nnmod(tmp, b, &group->field, ctx) ||
148 !ec_bignum_to_felem(group, &group->b, tmp)) {
149 goto err;
150 }
151
152 if (!ec_bignum_to_felem(group, &group->one, BN_value_one())) {
153 goto err;
154 }
155
156 ret = 1;
157
158err:
159 BN_CTX_end(ctx);
160 BN_CTX_free(new_ctx);
161 return ret;
162}
163
164int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
165 BIGNUM *b) {
166 if ((p != NULL && !BN_copy(p, &group->field)) ||
167 (a != NULL && !ec_felem_to_bignum(group, a, &group->a)) ||
168 (b != NULL && !ec_felem_to_bignum(group, b, &group->b))) {
169 return 0;
170 }
171 return 1;
172}
173
174void ec_GFp_simple_point_init(EC_RAW_POINT *point) {
175 OPENSSL_memset(&point->X, 0, sizeof(EC_FELEM));
176 OPENSSL_memset(&point->Y, 0, sizeof(EC_FELEM));
177 OPENSSL_memset(&point->Z, 0, sizeof(EC_FELEM));
178}
179
180void ec_GFp_simple_point_copy(EC_RAW_POINT *dest, const EC_RAW_POINT *src) {
181 OPENSSL_memcpy(&dest->X, &src->X, sizeof(EC_FELEM));
182 OPENSSL_memcpy(&dest->Y, &src->Y, sizeof(EC_FELEM));
183 OPENSSL_memcpy(&dest->Z, &src->Z, sizeof(EC_FELEM));
184}
185
186void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
187 EC_RAW_POINT *point) {
188 // Although it is strictly only necessary to zero Z, we zero the entire point
189 // in case |point| was stack-allocated and yet to be initialized.
190 ec_GFp_simple_point_init(point);
191}
192
193int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
194 EC_RAW_POINT *point,
195 const BIGNUM *x,
196 const BIGNUM *y) {
197 if (x == NULL || y == NULL) {
198 OPENSSL_PUT_ERROR(EC, ERR_R_PASSED_NULL_PARAMETER);
199 return 0;
200 }
201
202 if (!ec_bignum_to_felem(group, &point->X, x) ||
203 !ec_bignum_to_felem(group, &point->Y, y)) {
204 return 0;
205 }
206 OPENSSL_memcpy(&point->Z, &group->one, sizeof(EC_FELEM));
207
208 return 1;
209}
210
211void ec_GFp_simple_invert(const EC_GROUP *group, EC_RAW_POINT *point) {
212 ec_felem_neg(group, &point->Y, &point->Y);
213}
214
215int ec_GFp_simple_is_at_infinity(const EC_GROUP *group,
216 const EC_RAW_POINT *point) {
217 return ec_felem_non_zero_mask(group, &point->Z) == 0;
218}
219
220int ec_GFp_simple_is_on_curve(const EC_GROUP *group,
221 const EC_RAW_POINT *point) {
222 if (ec_GFp_simple_is_at_infinity(group, point)) {
223 return 1;
224 }
225
226 // We have a curve defined by a Weierstrass equation
227 // y^2 = x^3 + a*x + b.
228 // The point to consider is given in Jacobian projective coordinates
229 // where (X, Y, Z) represents (x, y) = (X/Z^2, Y/Z^3).
230 // Substituting this and multiplying by Z^6 transforms the above equation
231 // into
232 // Y^2 = X^3 + a*X*Z^4 + b*Z^6.
233 // To test this, we add up the right-hand side in 'rh'.
234
235 void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
236 const EC_FELEM *b) = group->meth->felem_mul;
237 void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
238 group->meth->felem_sqr;
239
240 // rh := X^2
241 EC_FELEM rh;
242 felem_sqr(group, &rh, &point->X);
243
244 EC_FELEM tmp, Z4, Z6;
245 if (!ec_felem_equal(group, &point->Z, &group->one)) {
246 felem_sqr(group, &tmp, &point->Z);
247 felem_sqr(group, &Z4, &tmp);
248 felem_mul(group, &Z6, &Z4, &tmp);
249
250 // rh := (rh + a*Z^4)*X
251 if (group->a_is_minus3) {
252 ec_felem_add(group, &tmp, &Z4, &Z4);
253 ec_felem_add(group, &tmp, &tmp, &Z4);
254 ec_felem_sub(group, &rh, &rh, &tmp);
255 felem_mul(group, &rh, &rh, &point->X);
256 } else {
257 felem_mul(group, &tmp, &Z4, &group->a);
258 ec_felem_add(group, &rh, &rh, &tmp);
259 felem_mul(group, &rh, &rh, &point->X);
260 }
261
262 // rh := rh + b*Z^6
263 felem_mul(group, &tmp, &group->b, &Z6);
264 ec_felem_add(group, &rh, &rh, &tmp);
265 } else {
266 // rh := (rh + a)*X
267 ec_felem_add(group, &rh, &rh, &group->a);
268 felem_mul(group, &rh, &rh, &point->X);
269 // rh := rh + b
270 ec_felem_add(group, &rh, &rh, &group->b);
271 }
272
273 // 'lh' := Y^2
274 felem_sqr(group, &tmp, &point->Y);
275 return ec_felem_equal(group, &tmp, &rh);
276}
277
278int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_RAW_POINT *a,
279 const EC_RAW_POINT *b) {
280 // Note this function returns zero if |a| and |b| are equal and 1 if they are
281 // not equal.
282 if (ec_GFp_simple_is_at_infinity(group, a)) {
283 return ec_GFp_simple_is_at_infinity(group, b) ? 0 : 1;
284 }
285
286 if (ec_GFp_simple_is_at_infinity(group, b)) {
287 return 1;
288 }
289
290 int a_Z_is_one = ec_felem_equal(group, &a->Z, &group->one);
291 int b_Z_is_one = ec_felem_equal(group, &b->Z, &group->one);
292
293 if (a_Z_is_one && b_Z_is_one) {
294 return !ec_felem_equal(group, &a->X, &b->X) ||
295 !ec_felem_equal(group, &a->Y, &b->Y);
296 }
297
298 void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
299 const EC_FELEM *b) = group->meth->felem_mul;
300 void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
301 group->meth->felem_sqr;
302
303 // We have to decide whether
304 // (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
305 // or equivalently, whether
306 // (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
307
308 EC_FELEM tmp1, tmp2, Za23, Zb23;
309 const EC_FELEM *tmp1_, *tmp2_;
310 if (!b_Z_is_one) {
311 felem_sqr(group, &Zb23, &b->Z);
312 felem_mul(group, &tmp1, &a->X, &Zb23);
313 tmp1_ = &tmp1;
314 } else {
315 tmp1_ = &a->X;
316 }
317 if (!a_Z_is_one) {
318 felem_sqr(group, &Za23, &a->Z);
319 felem_mul(group, &tmp2, &b->X, &Za23);
320 tmp2_ = &tmp2;
321 } else {
322 tmp2_ = &b->X;
323 }
324
325 // Compare X_a*Z_b^2 with X_b*Z_a^2.
326 if (!ec_felem_equal(group, tmp1_, tmp2_)) {
327 return 1; // The points differ.
328 }
329
330 if (!b_Z_is_one) {
331 felem_mul(group, &Zb23, &Zb23, &b->Z);
332 felem_mul(group, &tmp1, &a->Y, &Zb23);
333 // tmp1_ = &tmp1
334 } else {
335 tmp1_ = &a->Y;
336 }
337 if (!a_Z_is_one) {
338 felem_mul(group, &Za23, &Za23, &a->Z);
339 felem_mul(group, &tmp2, &b->Y, &Za23);
340 // tmp2_ = &tmp2
341 } else {
342 tmp2_ = &b->Y;
343 }
344
345 // Compare Y_a*Z_b^3 with Y_b*Z_a^3.
346 if (!ec_felem_equal(group, tmp1_, tmp2_)) {
347 return 1; // The points differ.
348 }
349
350 // The points are equal.
351 return 0;
352}
353
354int ec_GFp_simple_mont_inv_mod_ord_vartime(const EC_GROUP *group,
355 EC_SCALAR *out,
356 const EC_SCALAR *in) {
357 // This implementation (in fact) runs in constant time,
358 // even though for this interface it is not mandatory.
359
360 // out = in^-1 in the Montgomery domain. This is
361 // |ec_scalar_to_montgomery| followed by |ec_scalar_inv_montgomery|, but
362 // |ec_scalar_inv_montgomery| followed by |ec_scalar_from_montgomery| is
363 // equivalent and slightly more efficient.
364 ec_scalar_inv_montgomery(group, out, in);
365 ec_scalar_from_montgomery(group, out, out);
366 return 1;
367}
368
369int ec_GFp_simple_cmp_x_coordinate(const EC_GROUP *group, const EC_RAW_POINT *p,
370 const EC_SCALAR *r) {
371 if (ec_GFp_simple_is_at_infinity(group, p)) {
372 // |ec_get_x_coordinate_as_scalar| will check this internally, but this way
373 // we do not push to the error queue.
374 return 0;
375 }
376
377 EC_SCALAR x;
378 return ec_get_x_coordinate_as_scalar(group, &x, p) &&
379 ec_scalar_equal_vartime(group, &x, r);
380}
381