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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 /* Computes scalar*point and stores the result in r.
212  * point can not equal r.
213  * Uses algorithm 2P of
214  *     Lopez, J. and Dahab, R.  "Fast multiplication on elliptic curves over
215  *     GF(2^m) without precomputation" (CHES '99, LNCS 1717).
216  */
ec_GF2m_montgomery_point_multiply(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,const EC_POINT * point,BN_CTX * ctx)217 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
218 	const EC_POINT *point, BN_CTX *ctx)
219 	{
220 	BIGNUM *x1, *x2, *z1, *z2;
221 	int ret = 0, i;
222 	BN_ULONG mask,word;
223 
224 	if (r == point)
225 		{
226 		ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
227 		return 0;
228 		}
229 
230 	/* if result should be point at infinity */
231 	if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
232 		EC_POINT_is_at_infinity(group, point))
233 		{
234 		return EC_POINT_set_to_infinity(group, r);
235 		}
236 
237 	/* only support affine coordinates */
238 	if (!point->Z_is_one) return 0;
239 
240 	/* Since point_multiply is static we can guarantee that ctx != NULL. */
241 	BN_CTX_start(ctx);
242 	x1 = BN_CTX_get(ctx);
243 	z1 = BN_CTX_get(ctx);
244 	if (z1 == NULL) goto err;
245 
246 	x2 = &r->X;
247 	z2 = &r->Y;
248 
249 	if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */
250 	if (!BN_one(z1)) goto err; /* z1 = 1 */
251 	if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
252 	if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
253 	if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */
254 
255 	/* find top most bit and go one past it */
256 	i = scalar->top - 1;
257 	mask = BN_TBIT;
258 	word = scalar->d[i];
259 	while (!(word & mask)) mask >>= 1;
260 	mask >>= 1;
261 	/* if top most bit was at word break, go to next word */
262 	if (!mask)
263 		{
264 		i--;
265 		mask = BN_TBIT;
266 		}
267 
268 	for (; i >= 0; i--)
269 		{
270 		word = scalar->d[i];
271 		while (mask)
272 			{
273 			if (word & mask)
274 				{
275 				if (!gf2m_Madd(group, &point->X, x1, z1, x2, z2, ctx)) goto err;
276 				if (!gf2m_Mdouble(group, x2, z2, ctx)) goto err;
277 				}
278 			else
279 				{
280 				if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err;
281 				if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
282 				}
283 			mask >>= 1;
284 			}
285 		mask = BN_TBIT;
286 		}
287 
288 	/* convert out of "projective" coordinates */
289 	i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
290 	if (i == 0) goto err;
291 	else if (i == 1)
292 		{
293 		if (!EC_POINT_set_to_infinity(group, r)) goto err;
294 		}
295 	else
296 		{
297 		if (!BN_one(&r->Z)) goto err;
298 		r->Z_is_one = 1;
299 		}
300 
301 	/* GF(2^m) field elements should always have BIGNUM::neg = 0 */
302 	BN_set_negative(&r->X, 0);
303 	BN_set_negative(&r->Y, 0);
304 
305 	ret = 1;
306 
307  err:
308 	BN_CTX_end(ctx);
309 	return ret;
310 	}
311 
312 
313 /* Computes the sum
314  *     scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
315  * gracefully ignoring NULL scalar values.
316  */
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)317 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
318 	size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
319 	{
320 	BN_CTX *new_ctx = NULL;
321 	int ret = 0;
322 	size_t i;
323 	EC_POINT *p=NULL;
324 	EC_POINT *acc = NULL;
325 
326 	if (ctx == NULL)
327 		{
328 		ctx = new_ctx = BN_CTX_new();
329 		if (ctx == NULL)
330 			return 0;
331 		}
332 
333 	/* This implementation is more efficient than the wNAF implementation for 2
334 	 * or fewer points.  Use the ec_wNAF_mul implementation for 3 or more points,
335 	 * or if we can perform a fast multiplication based on precomputation.
336 	 */
337 	if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
338 		{
339 		ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
340 		goto err;
341 		}
342 
343 	if ((p = EC_POINT_new(group)) == NULL) goto err;
344 	if ((acc = EC_POINT_new(group)) == NULL) goto err;
345 
346 	if (!EC_POINT_set_to_infinity(group, acc)) goto err;
347 
348 	if (scalar)
349 		{
350 		if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
351 		if (BN_is_negative(scalar))
352 			if (!group->meth->invert(group, p, ctx)) goto err;
353 		if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
354 		}
355 
356 	for (i = 0; i < num; i++)
357 		{
358 		if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
359 		if (BN_is_negative(scalars[i]))
360 			if (!group->meth->invert(group, p, ctx)) goto err;
361 		if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
362 		}
363 
364 	if (!EC_POINT_copy(r, acc)) goto err;
365 
366 	ret = 1;
367 
368   err:
369 	if (p) EC_POINT_free(p);
370 	if (acc) EC_POINT_free(acc);
371 	if (new_ctx != NULL)
372 		BN_CTX_free(new_ctx);
373 	return ret;
374 	}
375 
376 
377 /* Precomputation for point multiplication: fall back to wNAF methods
378  * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
379 
ec_GF2m_precompute_mult(EC_GROUP * group,BN_CTX * ctx)380 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
381 	{
382 	return ec_wNAF_precompute_mult(group, ctx);
383  	}
384 
ec_GF2m_have_precompute_mult(const EC_GROUP * group)385 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
386 	{
387 	return ec_wNAF_have_precompute_mult(group);
388  	}
389 
390 #endif
391