ec2_mult.c revision 296465 
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".
80 * modified to not require precomputation of c=b^{2^{m-1}}.
81 */
82static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z,
83                        BN_CTX *ctx)
84{
85    BIGNUM *t1;
86    int ret = 0;
87
88    /* Since Mdouble is static we can guarantee that ctx != NULL. */
89    BN_CTX_start(ctx);
90    t1 = BN_CTX_get(ctx);
91    if (t1 == NULL)
92        goto err;
93
94    if (!group->meth->field_sqr(group, x, x, ctx))
95        goto err;
96    if (!group->meth->field_sqr(group, t1, z, ctx))
97        goto err;
98    if (!group->meth->field_mul(group, z, x, t1, ctx))
99        goto err;
100    if (!group->meth->field_sqr(group, x, x, ctx))
101        goto err;
102    if (!group->meth->field_sqr(group, t1, t1, ctx))
103        goto err;
104    if (!group->meth->field_mul(group, t1, &group->b, t1, ctx))
105        goto err;
106    if (!BN_GF2m_add(x, x, t1))
107        goto err;
108
109    ret = 1;
110
111 err:
112    BN_CTX_end(ctx);
113    return ret;
114}
115
116/*-
117 * Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery
118 * projective coordinates.
119 * Uses algorithm Madd in appendix of
120 *     Lopex, J. and Dahab, R.  "Fast multiplication on elliptic curves over
121 *     GF(2^m) without precomputation".
122 */
123static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1,
124                     BIGNUM *z1, const BIGNUM *x2, const BIGNUM *z2,
125                     BN_CTX *ctx)
126{
127    BIGNUM *t1, *t2;
128    int ret = 0;
129
130    /* Since Madd is static we can guarantee that ctx != NULL. */
131    BN_CTX_start(ctx);
132    t1 = BN_CTX_get(ctx);
133    t2 = BN_CTX_get(ctx);
134    if (t2 == NULL)
135        goto err;
136
137    if (!BN_copy(t1, x))
138        goto err;
139    if (!group->meth->field_mul(group, x1, x1, z2, ctx))
140        goto err;
141    if (!group->meth->field_mul(group, z1, z1, x2, ctx))
142        goto err;
143    if (!group->meth->field_mul(group, t2, x1, z1, ctx))
144        goto err;
145    if (!BN_GF2m_add(z1, z1, x1))
146        goto err;
147    if (!group->meth->field_sqr(group, z1, z1, ctx))
148        goto err;
149    if (!group->meth->field_mul(group, x1, z1, t1, ctx))
150        goto err;
151    if (!BN_GF2m_add(x1, x1, t2))
152        goto err;
153
154    ret = 1;
155
156 err:
157    BN_CTX_end(ctx);
158    return ret;
159}
160
161/*-
162 * Compute the x, y affine coordinates from the point (x1, z1) (x2, z2)
163 * using Montgomery point multiplication algorithm Mxy() in appendix of
164 *     Lopex, J. and Dahab, R.  "Fast multiplication on elliptic curves over
165 *     GF(2^m) without precomputation".
166 * Returns:
167 *     0 on error
168 *     1 if return value should be the point at infinity
169 *     2 otherwise
170 */
171static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y,
172                    BIGNUM *x1, BIGNUM *z1, BIGNUM *x2, BIGNUM *z2,
173                    BN_CTX *ctx)
174{
175    BIGNUM *t3, *t4, *t5;
176    int ret = 0;
177
178    if (BN_is_zero(z1)) {
179        BN_zero(x2);
180        BN_zero(z2);
181        return 1;
182    }
183
184    if (BN_is_zero(z2)) {
185        if (!BN_copy(x2, x))
186            return 0;
187        if (!BN_GF2m_add(z2, x, y))
188            return 0;
189        return 2;
190    }
191
192    /* Since Mxy is static we can guarantee that ctx != NULL. */
193    BN_CTX_start(ctx);
194    t3 = BN_CTX_get(ctx);
195    t4 = BN_CTX_get(ctx);
196    t5 = BN_CTX_get(ctx);
197    if (t5 == NULL)
198        goto err;
199
200    if (!BN_one(t5))
201        goto err;
202
203    if (!group->meth->field_mul(group, t3, z1, z2, ctx))
204        goto err;
205
206    if (!group->meth->field_mul(group, z1, z1, x, ctx))
207        goto err;
208    if (!BN_GF2m_add(z1, z1, x1))
209        goto err;
210    if (!group->meth->field_mul(group, z2, z2, x, ctx))
211        goto err;
212    if (!group->meth->field_mul(group, x1, z2, x1, ctx))
213        goto err;
214    if (!BN_GF2m_add(z2, z2, x2))
215        goto err;
216
217    if (!group->meth->field_mul(group, z2, z2, z1, ctx))
218        goto err;
219    if (!group->meth->field_sqr(group, t4, x, ctx))
220        goto err;
221    if (!BN_GF2m_add(t4, t4, y))
222        goto err;
223    if (!group->meth->field_mul(group, t4, t4, t3, ctx))
224        goto err;
225    if (!BN_GF2m_add(t4, t4, z2))
226        goto err;
227
228    if (!group->meth->field_mul(group, t3, t3, x, ctx))
229        goto err;
230    if (!group->meth->field_div(group, t3, t5, t3, ctx))
231        goto err;
232    if (!group->meth->field_mul(group, t4, t3, t4, ctx))
233        goto err;
234    if (!group->meth->field_mul(group, x2, x1, t3, ctx))
235        goto err;
236    if (!BN_GF2m_add(z2, x2, x))
237        goto err;
238
239    if (!group->meth->field_mul(group, z2, z2, t4, ctx))
240        goto err;
241    if (!BN_GF2m_add(z2, z2, y))
242        goto err;
243
244    ret = 2;
245
246 err:
247    BN_CTX_end(ctx);
248    return ret;
249}
250
251/*-
252 * Computes scalar*point and stores the result in r.
253 * point can not equal r.
254 * Uses a modified algorithm 2P of
255 *     Lopex, J. and Dahab, R.  "Fast multiplication on elliptic curves over
256 *     GF(2^m) without precomputation".
257 *
258 * To protect against side-channel attack the function uses constant time
259 * swap avoiding conditional branches.
260 */
261static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group,
262                                             EC_POINT *r,
263                                             const BIGNUM *scalar,
264                                             const EC_POINT *point,
265                                             BN_CTX *ctx)
266{
267    BIGNUM *x1, *x2, *z1, *z2;
268    int ret = 0, i, j;
269    BN_ULONG mask;
270
271    if (r == point) {
272        ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
273        return 0;
274    }
275
276    /* if result should be point at infinity */
277    if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
278        EC_POINT_is_at_infinity(group, point)) {
279        return EC_POINT_set_to_infinity(group, r);
280    }
281
282    /* only support affine coordinates */
283    if (!point->Z_is_one)
284        return 0;
285
286    /*
287     * Since point_multiply is static we can guarantee that ctx != NULL.
288     */
289    BN_CTX_start(ctx);
290    x1 = BN_CTX_get(ctx);
291    z1 = BN_CTX_get(ctx);
292    if (z1 == NULL)
293        goto err;
294
295    x2 = &r->X;
296    z2 = &r->Y;
297
298    bn_wexpand(x1, group->field.top);
299    bn_wexpand(z1, group->field.top);
300    bn_wexpand(x2, group->field.top);
301    bn_wexpand(z2, group->field.top);
302
303    if (!BN_GF2m_mod_arr(x1, &point->X, group->poly))
304        goto err;               /* x1 = x */
305    if (!BN_one(z1))
306        goto err;               /* z1 = 1 */
307    if (!group->meth->field_sqr(group, z2, x1, ctx))
308        goto err;               /* z2 = x1^2 = x^2 */
309    if (!group->meth->field_sqr(group, x2, z2, ctx))
310        goto err;
311    if (!BN_GF2m_add(x2, x2, &group->b))
312        goto err;               /* x2 = x^4 + b */
313
314    /* find top most bit and go one past it */
315    i = scalar->top - 1;
316    j = BN_BITS2 - 1;
317    mask = BN_TBIT;
318    while (!(scalar->d[i] & mask)) {
319        mask >>= 1;
320        j--;
321    }
322    mask >>= 1;
323    j--;
324    /* if top most bit was at word break, go to next word */
325    if (!mask) {
326        i--;
327        j = BN_BITS2 - 1;
328        mask = BN_TBIT;
329    }
330
331    for (; i >= 0; i--) {
332        for (; j >= 0; j--) {
333            BN_consttime_swap(scalar->d[i] & mask, x1, x2, group->field.top);
334            BN_consttime_swap(scalar->d[i] & mask, z1, z2, group->field.top);
335            if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx))
336                goto err;
337            if (!gf2m_Mdouble(group, x1, z1, ctx))
338                goto err;
339            BN_consttime_swap(scalar->d[i] & mask, x1, x2, group->field.top);
340            BN_consttime_swap(scalar->d[i] & mask, z1, z2, group->field.top);
341            mask >>= 1;
342        }
343        j = BN_BITS2 - 1;
344        mask = BN_TBIT;
345    }
346
347    /* convert out of "projective" coordinates */
348    i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
349    if (i == 0)
350        goto err;
351    else if (i == 1) {
352        if (!EC_POINT_set_to_infinity(group, r))
353            goto err;
354    } else {
355        if (!BN_one(&r->Z))
356            goto err;
357        r->Z_is_one = 1;
358    }
359
360    /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
361    BN_set_negative(&r->X, 0);
362    BN_set_negative(&r->Y, 0);
363
364    ret = 1;
365
366 err:
367    BN_CTX_end(ctx);
368    return ret;
369}
370
371/*-
372 * Computes the sum
373 *     scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
374 * gracefully ignoring NULL scalar values.
375 */
376int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r,
377                       const BIGNUM *scalar, size_t num,
378                       const EC_POINT *points[], const BIGNUM *scalars[],
379                       BN_CTX *ctx)
380{
381    BN_CTX *new_ctx = NULL;
382    int ret = 0;
383    size_t i;
384    EC_POINT *p = NULL;
385    EC_POINT *acc = NULL;
386
387    if (ctx == NULL) {
388        ctx = new_ctx = BN_CTX_new();
389        if (ctx == NULL)
390            return 0;
391    }
392
393    /*
394     * This implementation is more efficient than the wNAF implementation for
395     * 2 or fewer points.  Use the ec_wNAF_mul implementation for 3 or more
396     * points, or if we can perform a fast multiplication based on
397     * precomputation.
398     */
399    if ((scalar && (num > 1)) || (num > 2)
400        || (num == 0 && EC_GROUP_have_precompute_mult(group))) {
401        ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
402        goto err;
403    }
404
405    if ((p = EC_POINT_new(group)) == NULL)
406        goto err;
407    if ((acc = EC_POINT_new(group)) == NULL)
408        goto err;
409
410    if (!EC_POINT_set_to_infinity(group, acc))
411        goto err;
412
413    if (scalar) {
414        if (!ec_GF2m_montgomery_point_multiply
415            (group, p, scalar, group->generator, ctx))
416            goto err;
417        if (BN_is_negative(scalar))
418            if (!group->meth->invert(group, p, ctx))
419                goto err;
420        if (!group->meth->add(group, acc, acc, p, ctx))
421            goto err;
422    }
423
424    for (i = 0; i < num; i++) {
425        if (!ec_GF2m_montgomery_point_multiply
426            (group, p, scalars[i], points[i], ctx))
427            goto err;
428        if (BN_is_negative(scalars[i]))
429            if (!group->meth->invert(group, p, ctx))
430                goto err;
431        if (!group->meth->add(group, acc, acc, p, ctx))
432            goto err;
433    }
434
435    if (!EC_POINT_copy(r, acc))
436        goto err;
437
438    ret = 1;
439
440 err:
441    if (p)
442        EC_POINT_free(p);
443    if (acc)
444        EC_POINT_free(acc);
445    if (new_ctx != NULL)
446        BN_CTX_free(new_ctx);
447    return ret;
448}
449
450/*
451 * Precomputation for point multiplication: fall back to wNAF methods because
452 * ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate
453 */
454
455int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
456{
457    return ec_wNAF_precompute_mult(group, ctx);
458}
459
460int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
461{
462    return ec_wNAF_have_precompute_mult(group);
463}
464