1 /* crypto/ec/ec2_mult.c */
2 /* ====================================================================
3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
4 *
5 * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
6 * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
7 * to the OpenSSL project.
8 *
9 * The ECC Code is licensed pursuant to the OpenSSL open source
10 * license provided below.
11 *
12 * The software is originally written by Sheueling Chang Shantz and
13 * Douglas Stebila of Sun Microsystems Laboratories.
14 *
15 */
16 /* ====================================================================
17 * Copyright (c) 1998-2003 The OpenSSL Project. All rights reserved.
18 *
19 * Redistribution and use in source and binary forms, with or without
20 * modification, are permitted provided that the following conditions
21 * are met:
22 *
23 * 1. Redistributions of source code must retain the above copyright
24 * notice, this list of conditions and the following disclaimer.
25 *
26 * 2. Redistributions in binary form must reproduce the above copyright
27 * notice, this list of conditions and the following disclaimer in
28 * the documentation and/or other materials provided with the
29 * distribution.
30 *
31 * 3. All advertising materials mentioning features or use of this
32 * software must display the following acknowledgment:
33 * "This product includes software developed by the OpenSSL Project
34 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
35 *
36 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
37 * endorse or promote products derived from this software without
38 * prior written permission. For written permission, please contact
39 * openssl-core@openssl.org.
40 *
41 * 5. Products derived from this software may not be called "OpenSSL"
42 * nor may "OpenSSL" appear in their names without prior written
43 * permission of the OpenSSL Project.
44 *
45 * 6. Redistributions of any form whatsoever must retain the following
46 * acknowledgment:
47 * "This product includes software developed by the OpenSSL Project
48 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
49 *
50 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
51 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
52 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
53 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
54 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
55 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
56 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
57 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
58 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
59 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
60 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
61 * OF THE POSSIBILITY OF SUCH DAMAGE.
62 * ====================================================================
63 *
64 * This product includes cryptographic software written by Eric Young
65 * (eay@cryptsoft.com). This product includes software written by Tim
66 * Hudson (tjh@cryptsoft.com).
67 *
68 */
69
70 #include <openssl/err.h>
71
72 #include "ec_lcl.h"
73
74 #ifndef OPENSSL_NO_EC2M
75
76
77 /* Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective
78 * coordinates.
79 * Uses algorithm Mdouble in appendix of
80 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
81 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
82 * modified to not require precomputation of c=b^{2^{m-1}}.
83 */
gf2m_Mdouble(const EC_GROUP * group,BIGNUM * x,BIGNUM * z,BN_CTX * ctx)84 static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx)
85 {
86 BIGNUM *t1;
87 int ret = 0;
88
89 /* Since Mdouble is static we can guarantee that ctx != NULL. */
90 BN_CTX_start(ctx);
91 t1 = BN_CTX_get(ctx);
92 if (t1 == NULL) goto err;
93
94 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
95 if (!group->meth->field_sqr(group, t1, z, ctx)) goto err;
96 if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err;
97 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
98 if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err;
99 if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err;
100 if (!BN_GF2m_add(x, x, t1)) goto err;
101
102 ret = 1;
103
104 err:
105 BN_CTX_end(ctx);
106 return ret;
107 }
108
109 /* Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery
110 * projective coordinates.
111 * Uses algorithm Madd in appendix of
112 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
113 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
114 */
gf2m_Madd(const EC_GROUP * group,const BIGNUM * x,BIGNUM * x1,BIGNUM * z1,const BIGNUM * x2,const BIGNUM * z2,BN_CTX * ctx)115 static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1,
116 const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx)
117 {
118 BIGNUM *t1, *t2;
119 int ret = 0;
120
121 /* Since Madd is static we can guarantee that ctx != NULL. */
122 BN_CTX_start(ctx);
123 t1 = BN_CTX_get(ctx);
124 t2 = BN_CTX_get(ctx);
125 if (t2 == NULL) goto err;
126
127 if (!BN_copy(t1, x)) goto err;
128 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err;
129 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err;
130 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err;
131 if (!BN_GF2m_add(z1, z1, x1)) goto err;
132 if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err;
133 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err;
134 if (!BN_GF2m_add(x1, x1, t2)) goto err;
135
136 ret = 1;
137
138 err:
139 BN_CTX_end(ctx);
140 return ret;
141 }
142
143 /* Compute the x, y affine coordinates from the point (x1, z1) (x2, z2)
144 * using Montgomery point multiplication algorithm Mxy() in appendix of
145 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
146 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
147 * Returns:
148 * 0 on error
149 * 1 if return value should be the point at infinity
150 * 2 otherwise
151 */
gf2m_Mxy(const EC_GROUP * group,const BIGNUM * x,const BIGNUM * y,BIGNUM * x1,BIGNUM * z1,BIGNUM * x2,BIGNUM * z2,BN_CTX * ctx)152 static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1,
153 BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx)
154 {
155 BIGNUM *t3, *t4, *t5;
156 int ret = 0;
157
158 if (BN_is_zero(z1))
159 {
160 BN_zero(x2);
161 BN_zero(z2);
162 return 1;
163 }
164
165 if (BN_is_zero(z2))
166 {
167 if (!BN_copy(x2, x)) return 0;
168 if (!BN_GF2m_add(z2, x, y)) return 0;
169 return 2;
170 }
171
172 /* Since Mxy is static we can guarantee that ctx != NULL. */
173 BN_CTX_start(ctx);
174 t3 = BN_CTX_get(ctx);
175 t4 = BN_CTX_get(ctx);
176 t5 = BN_CTX_get(ctx);
177 if (t5 == NULL) goto err;
178
179 if (!BN_one(t5)) goto err;
180
181 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err;
182
183 if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err;
184 if (!BN_GF2m_add(z1, z1, x1)) goto err;
185 if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err;
186 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err;
187 if (!BN_GF2m_add(z2, z2, x2)) goto err;
188
189 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err;
190 if (!group->meth->field_sqr(group, t4, x, ctx)) goto err;
191 if (!BN_GF2m_add(t4, t4, y)) goto err;
192 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err;
193 if (!BN_GF2m_add(t4, t4, z2)) goto err;
194
195 if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err;
196 if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err;
197 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err;
198 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err;
199 if (!BN_GF2m_add(z2, x2, x)) goto err;
200
201 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err;
202 if (!BN_GF2m_add(z2, z2, y)) goto err;
203
204 ret = 2;
205
206 err:
207 BN_CTX_end(ctx);
208 return ret;
209 }
210
211
212 /* Computes scalar*point and stores the result in r.
213 * point can not equal r.
214 * Uses a modified algorithm 2P of
215 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
216 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
217 *
218 * To protect against side-channel attack the function uses constant time swap,
219 * avoiding conditional branches.
220 */
ec_GF2m_montgomery_point_multiply(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,const EC_POINT * point,BN_CTX * ctx)221 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
222 const EC_POINT *point, BN_CTX *ctx)
223 {
224 BIGNUM *x1, *x2, *z1, *z2;
225 int ret = 0, i;
226 BN_ULONG mask,word;
227
228 if (r == point)
229 {
230 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
231 return 0;
232 }
233
234 /* if result should be point at infinity */
235 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
236 EC_POINT_is_at_infinity(group, point))
237 {
238 return EC_POINT_set_to_infinity(group, r);
239 }
240
241 /* only support affine coordinates */
242 if (!point->Z_is_one) return 0;
243
244 /* Since point_multiply is static we can guarantee that ctx != NULL. */
245 BN_CTX_start(ctx);
246 x1 = BN_CTX_get(ctx);
247 z1 = BN_CTX_get(ctx);
248 if (z1 == NULL) goto err;
249
250 x2 = &r->X;
251 z2 = &r->Y;
252
253 bn_wexpand(x1, group->field.top);
254 bn_wexpand(z1, group->field.top);
255 bn_wexpand(x2, group->field.top);
256 bn_wexpand(z2, group->field.top);
257
258 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */
259 if (!BN_one(z1)) goto err; /* z1 = 1 */
260 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
261 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
262 if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */
263
264 /* find top most bit and go one past it */
265 i = scalar->top - 1;
266 mask = BN_TBIT;
267 word = scalar->d[i];
268 while (!(word & mask)) mask >>= 1;
269 mask >>= 1;
270 /* if top most bit was at word break, go to next word */
271 if (!mask)
272 {
273 i--;
274 mask = BN_TBIT;
275 }
276
277 for (; i >= 0; i--)
278 {
279 word = scalar->d[i];
280 while (mask)
281 {
282 BN_consttime_swap(word & mask, x1, x2, group->field.top);
283 BN_consttime_swap(word & mask, z1, z2, group->field.top);
284 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err;
285 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
286 BN_consttime_swap(word & mask, x1, x2, group->field.top);
287 BN_consttime_swap(word & mask, z1, z2, group->field.top);
288 mask >>= 1;
289 }
290 mask = BN_TBIT;
291 }
292
293 /* convert out of "projective" coordinates */
294 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
295 if (i == 0) goto err;
296 else if (i == 1)
297 {
298 if (!EC_POINT_set_to_infinity(group, r)) goto err;
299 }
300 else
301 {
302 if (!BN_one(&r->Z)) goto err;
303 r->Z_is_one = 1;
304 }
305
306 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
307 BN_set_negative(&r->X, 0);
308 BN_set_negative(&r->Y, 0);
309
310 ret = 1;
311
312 err:
313 BN_CTX_end(ctx);
314 return ret;
315 }
316
317
318 /* Computes the sum
319 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
320 * gracefully ignoring NULL scalar values.
321 */
ec_GF2m_simple_mul(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,size_t num,const EC_POINT * points[],const BIGNUM * scalars[],BN_CTX * ctx)322 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
323 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
324 {
325 BN_CTX *new_ctx = NULL;
326 int ret = 0;
327 size_t i;
328 EC_POINT *p=NULL;
329 EC_POINT *acc = NULL;
330
331 if (ctx == NULL)
332 {
333 ctx = new_ctx = BN_CTX_new();
334 if (ctx == NULL)
335 return 0;
336 }
337
338 /* This implementation is more efficient than the wNAF implementation for 2
339 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points,
340 * or if we can perform a fast multiplication based on precomputation.
341 */
342 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
343 {
344 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
345 goto err;
346 }
347
348 if ((p = EC_POINT_new(group)) == NULL) goto err;
349 if ((acc = EC_POINT_new(group)) == NULL) goto err;
350
351 if (!EC_POINT_set_to_infinity(group, acc)) goto err;
352
353 if (scalar)
354 {
355 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
356 if (BN_is_negative(scalar))
357 if (!group->meth->invert(group, p, ctx)) goto err;
358 if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
359 }
360
361 for (i = 0; i < num; i++)
362 {
363 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
364 if (BN_is_negative(scalars[i]))
365 if (!group->meth->invert(group, p, ctx)) goto err;
366 if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
367 }
368
369 if (!EC_POINT_copy(r, acc)) goto err;
370
371 ret = 1;
372
373 err:
374 if (p) EC_POINT_free(p);
375 if (acc) EC_POINT_free(acc);
376 if (new_ctx != NULL)
377 BN_CTX_free(new_ctx);
378 return ret;
379 }
380
381
382 /* Precomputation for point multiplication: fall back to wNAF methods
383 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
384
ec_GF2m_precompute_mult(EC_GROUP * group,BN_CTX * ctx)385 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
386 {
387 return ec_wNAF_precompute_mult(group, ctx);
388 }
389
ec_GF2m_have_precompute_mult(const EC_GROUP * group)390 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
391 {
392 return ec_wNAF_have_precompute_mult(group);
393 }
394
395 #endif
396