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
75 /* Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective
76 * coordinates.
77 * Uses algorithm Mdouble in appendix of
78 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
79 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
80 * modified to not require precomputation of c=b^{2^{m-1}}.
81 */
gf2m_Mdouble(const EC_GROUP * group,BIGNUM * x,BIGNUM * z,BN_CTX * ctx)82 static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx)
83 {
84 BIGNUM *t1;
85 int ret = 0;
86
87 /* Since Mdouble is static we can guarantee that ctx != NULL. */
88 BN_CTX_start(ctx);
89 t1 = BN_CTX_get(ctx);
90 if (t1 == NULL) goto err;
91
92 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
93 if (!group->meth->field_sqr(group, t1, z, ctx)) goto err;
94 if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err;
95 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
96 if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err;
97 if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err;
98 if (!BN_GF2m_add(x, x, t1)) goto err;
99
100 ret = 1;
101
102 err:
103 BN_CTX_end(ctx);
104 return ret;
105 }
106
107 /* Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery
108 * projective coordinates.
109 * Uses algorithm Madd in appendix of
110 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
111 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
112 */
gf2m_Madd(const EC_GROUP * group,const BIGNUM * x,BIGNUM * x1,BIGNUM * z1,const BIGNUM * x2,const BIGNUM * z2,BN_CTX * ctx)113 static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1,
114 const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx)
115 {
116 BIGNUM *t1, *t2;
117 int ret = 0;
118
119 /* Since Madd is static we can guarantee that ctx != NULL. */
120 BN_CTX_start(ctx);
121 t1 = BN_CTX_get(ctx);
122 t2 = BN_CTX_get(ctx);
123 if (t2 == NULL) goto err;
124
125 if (!BN_copy(t1, x)) goto err;
126 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err;
127 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err;
128 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err;
129 if (!BN_GF2m_add(z1, z1, x1)) goto err;
130 if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err;
131 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err;
132 if (!BN_GF2m_add(x1, x1, t2)) goto err;
133
134 ret = 1;
135
136 err:
137 BN_CTX_end(ctx);
138 return ret;
139 }
140
141 /* Compute the x, y affine coordinates from the point (x1, z1) (x2, z2)
142 * using Montgomery point multiplication algorithm Mxy() in appendix of
143 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
144 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
145 * Returns:
146 * 0 on error
147 * 1 if return value should be the point at infinity
148 * 2 otherwise
149 */
gf2m_Mxy(const EC_GROUP * group,const BIGNUM * x,const BIGNUM * y,BIGNUM * x1,BIGNUM * z1,BIGNUM * x2,BIGNUM * z2,BN_CTX * ctx)150 static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1,
151 BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx)
152 {
153 BIGNUM *t3, *t4, *t5;
154 int ret = 0;
155
156 if (BN_is_zero(z1))
157 {
158 BN_zero(x2);
159 BN_zero(z2);
160 return 1;
161 }
162
163 if (BN_is_zero(z2))
164 {
165 if (!BN_copy(x2, x)) return 0;
166 if (!BN_GF2m_add(z2, x, y)) return 0;
167 return 2;
168 }
169
170 /* Since Mxy is static we can guarantee that ctx != NULL. */
171 BN_CTX_start(ctx);
172 t3 = BN_CTX_get(ctx);
173 t4 = BN_CTX_get(ctx);
174 t5 = BN_CTX_get(ctx);
175 if (t5 == NULL) goto err;
176
177 if (!BN_one(t5)) goto err;
178
179 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err;
180
181 if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err;
182 if (!BN_GF2m_add(z1, z1, x1)) goto err;
183 if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err;
184 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err;
185 if (!BN_GF2m_add(z2, z2, x2)) goto err;
186
187 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err;
188 if (!group->meth->field_sqr(group, t4, x, ctx)) goto err;
189 if (!BN_GF2m_add(t4, t4, y)) goto err;
190 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err;
191 if (!BN_GF2m_add(t4, t4, z2)) goto err;
192
193 if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err;
194 if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err;
195 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err;
196 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err;
197 if (!BN_GF2m_add(z2, x2, x)) goto err;
198
199 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err;
200 if (!BN_GF2m_add(z2, z2, y)) goto err;
201
202 ret = 2;
203
204 err:
205 BN_CTX_end(ctx);
206 return ret;
207 }
208
209 /* Computes scalar*point and stores the result in r.
210 * point can not equal r.
211 * Uses algorithm 2P of
212 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
213 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
214 */
ec_GF2m_montgomery_point_multiply(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,const EC_POINT * point,BN_CTX * ctx)215 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
216 const EC_POINT *point, BN_CTX *ctx)
217 {
218 BIGNUM *x1, *x2, *z1, *z2;
219 int ret = 0, i;
220 BN_ULONG mask,word;
221
222 if (r == point)
223 {
224 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
225 return 0;
226 }
227
228 /* if result should be point at infinity */
229 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
230 EC_POINT_is_at_infinity(group, point))
231 {
232 return EC_POINT_set_to_infinity(group, r);
233 }
234
235 /* only support affine coordinates */
236 if (!point->Z_is_one) return 0;
237
238 /* Since point_multiply is static we can guarantee that ctx != NULL. */
239 BN_CTX_start(ctx);
240 x1 = BN_CTX_get(ctx);
241 z1 = BN_CTX_get(ctx);
242 if (z1 == NULL) goto err;
243
244 x2 = &r->X;
245 z2 = &r->Y;
246
247 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */
248 if (!BN_one(z1)) goto err; /* z1 = 1 */
249 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
250 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
251 if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */
252
253 /* find top most bit and go one past it */
254 i = scalar->top - 1;
255 mask = BN_TBIT;
256 word = scalar->d[i];
257 while (!(word & mask)) mask >>= 1;
258 mask >>= 1;
259 /* if top most bit was at word break, go to next word */
260 if (!mask)
261 {
262 i--;
263 mask = BN_TBIT;
264 }
265
266 for (; i >= 0; i--)
267 {
268 word = scalar->d[i];
269 while (mask)
270 {
271 if (word & mask)
272 {
273 if (!gf2m_Madd(group, &point->X, x1, z1, x2, z2, ctx)) goto err;
274 if (!gf2m_Mdouble(group, x2, z2, ctx)) goto err;
275 }
276 else
277 {
278 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err;
279 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
280 }
281 mask >>= 1;
282 }
283 mask = BN_TBIT;
284 }
285
286 /* convert out of "projective" coordinates */
287 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
288 if (i == 0) goto err;
289 else if (i == 1)
290 {
291 if (!EC_POINT_set_to_infinity(group, r)) goto err;
292 }
293 else
294 {
295 if (!BN_one(&r->Z)) goto err;
296 r->Z_is_one = 1;
297 }
298
299 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
300 BN_set_negative(&r->X, 0);
301 BN_set_negative(&r->Y, 0);
302
303 ret = 1;
304
305 err:
306 BN_CTX_end(ctx);
307 return ret;
308 }
309
310
311 /* Computes the sum
312 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
313 * gracefully ignoring NULL scalar values.
314 */
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)315 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
316 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
317 {
318 BN_CTX *new_ctx = NULL;
319 int ret = 0;
320 size_t i;
321 EC_POINT *p=NULL;
322
323 if (ctx == NULL)
324 {
325 ctx = new_ctx = BN_CTX_new();
326 if (ctx == NULL)
327 return 0;
328 }
329
330 /* This implementation is more efficient than the wNAF implementation for 2
331 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points,
332 * or if we can perform a fast multiplication based on precomputation.
333 */
334 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
335 {
336 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
337 goto err;
338 }
339
340 if ((p = EC_POINT_new(group)) == NULL) goto err;
341
342 if (!EC_POINT_set_to_infinity(group, r)) goto err;
343
344 if (scalar)
345 {
346 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
347 if (BN_is_negative(scalar))
348 if (!group->meth->invert(group, p, ctx)) goto err;
349 if (!group->meth->add(group, r, r, p, ctx)) goto err;
350 }
351
352 for (i = 0; i < num; i++)
353 {
354 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
355 if (BN_is_negative(scalars[i]))
356 if (!group->meth->invert(group, p, ctx)) goto err;
357 if (!group->meth->add(group, r, r, p, ctx)) goto err;
358 }
359
360 ret = 1;
361
362 err:
363 if (p) EC_POINT_free(p);
364 if (new_ctx != NULL)
365 BN_CTX_free(new_ctx);
366 return ret;
367 }
368
369
370 /* Precomputation for point multiplication: fall back to wNAF methods
371 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
372
ec_GF2m_precompute_mult(EC_GROUP * group,BN_CTX * ctx)373 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
374 {
375 return ec_wNAF_precompute_mult(group, ctx);
376 }
377
ec_GF2m_have_precompute_mult(const EC_GROUP * group)378 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
379 {
380 return ec_wNAF_have_precompute_mult(group);
381 }
382