1 /* crypto/ec/ec2_mult.c */
2 /* ====================================================================
3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
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.
9 * The ECC Code is licensed pursuant to the OpenSSL open source
10 * license provided below.
12 * The software is originally written by Sheueling Chang Shantz and
13 * Douglas Stebila of Sun Microsystems Laboratories.
16 /* ====================================================================
17 * Copyright (c) 1998-2003 The OpenSSL Project. All rights reserved.
19 * Redistribution and use in source and binary forms, with or without
20 * modification, are permitted provided that the following conditions
23 * 1. Redistributions of source code must retain the above copyright
24 * notice, this list of conditions and the following disclaimer.
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
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/)"
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.
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.
45 * 6. Redistributions of any form whatsoever must retain the following
47 * "This product includes software developed by the OpenSSL Project
48 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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 * ====================================================================
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).
70 #include <openssl/err.h>
74 #ifndef OPENSSL_NO_EC2M
78 * Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective
80 * Uses algorithm Mdouble in appendix of
81 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
82 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
83 * modified to not require precomputation of c=b^{2^{m-1}}.
85 static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx)
90 /* Since Mdouble is static we can guarantee that ctx != NULL. */
93 if (t1 == NULL) goto err;
95 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
96 if (!group->meth->field_sqr(group, t1, z, ctx)) goto err;
97 if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err;
98 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
99 if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err;
100 if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err;
101 if (!BN_GF2m_add(x, x, t1)) goto err;
111 * Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery
112 * projective coordinates.
113 * Uses algorithm Madd in appendix of
114 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
115 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
117 static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1,
118 const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx)
123 /* Since Madd is static we can guarantee that ctx != NULL. */
125 t1 = BN_CTX_get(ctx);
126 t2 = BN_CTX_get(ctx);
127 if (t2 == NULL) goto err;
129 if (!BN_copy(t1, x)) goto err;
130 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err;
131 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err;
132 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err;
133 if (!BN_GF2m_add(z1, z1, x1)) goto err;
134 if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err;
135 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err;
136 if (!BN_GF2m_add(x1, x1, t2)) goto err;
146 * Compute the x, y affine coordinates from the point (x1, z1) (x2, z2)
147 * using Montgomery point multiplication algorithm Mxy() in appendix of
148 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
149 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
152 * 1 if return value should be the point at infinity
155 static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1,
156 BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx)
158 BIGNUM *t3, *t4, *t5;
170 if (!BN_copy(x2, x)) return 0;
171 if (!BN_GF2m_add(z2, x, y)) return 0;
175 /* Since Mxy is static we can guarantee that ctx != NULL. */
177 t3 = BN_CTX_get(ctx);
178 t4 = BN_CTX_get(ctx);
179 t5 = BN_CTX_get(ctx);
180 if (t5 == NULL) goto err;
182 if (!BN_one(t5)) goto err;
184 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err;
186 if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err;
187 if (!BN_GF2m_add(z1, z1, x1)) goto err;
188 if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err;
189 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err;
190 if (!BN_GF2m_add(z2, z2, x2)) goto err;
192 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err;
193 if (!group->meth->field_sqr(group, t4, x, ctx)) goto err;
194 if (!BN_GF2m_add(t4, t4, y)) goto err;
195 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err;
196 if (!BN_GF2m_add(t4, t4, z2)) goto err;
198 if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err;
199 if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err;
200 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err;
201 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err;
202 if (!BN_GF2m_add(z2, x2, x)) goto err;
204 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err;
205 if (!BN_GF2m_add(z2, z2, y)) goto err;
216 * Computes scalar*point and stores the result in r.
217 * point can not equal r.
218 * Uses a modified algorithm 2P of
219 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
220 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
222 * To protect against side-channel attack the function uses constant time swap,
223 * avoiding conditional branches.
225 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
226 const EC_POINT *point, BN_CTX *ctx)
228 BIGNUM *x1, *x2, *z1, *z2;
234 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
238 /* if result should be point at infinity */
239 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
240 EC_POINT_is_at_infinity(group, point))
242 return EC_POINT_set_to_infinity(group, r);
245 /* only support affine coordinates */
246 if (!point->Z_is_one) return 0;
248 /* Since point_multiply is static we can guarantee that ctx != NULL. */
250 x1 = BN_CTX_get(ctx);
251 z1 = BN_CTX_get(ctx);
252 if (z1 == NULL) goto err;
257 bn_wexpand(x1, group->field.top);
258 bn_wexpand(z1, group->field.top);
259 bn_wexpand(x2, group->field.top);
260 bn_wexpand(z2, group->field.top);
262 if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */
263 if (!BN_one(z1)) goto err; /* z1 = 1 */
264 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
265 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
266 if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */
268 /* find top most bit and go one past it */
272 while (!(word & mask)) mask >>= 1;
274 /* if top most bit was at word break, go to next word */
286 BN_consttime_swap(word & mask, x1, x2, group->field.top);
287 BN_consttime_swap(word & mask, z1, z2, group->field.top);
288 if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err;
289 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
290 BN_consttime_swap(word & mask, x1, x2, group->field.top);
291 BN_consttime_swap(word & mask, z1, z2, group->field.top);
297 /* convert out of "projective" coordinates */
298 i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
299 if (i == 0) goto err;
302 if (!EC_POINT_set_to_infinity(group, r)) goto err;
306 if (!BN_one(&r->Z)) goto err;
310 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
311 BN_set_negative(&r->X, 0);
312 BN_set_negative(&r->Y, 0);
324 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
325 * gracefully ignoring NULL scalar values.
327 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
328 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
330 BN_CTX *new_ctx = NULL;
334 EC_POINT *acc = NULL;
338 ctx = new_ctx = BN_CTX_new();
343 /* This implementation is more efficient than the wNAF implementation for 2
344 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points,
345 * or if we can perform a fast multiplication based on precomputation.
347 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
349 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
353 if ((p = EC_POINT_new(group)) == NULL) goto err;
354 if ((acc = EC_POINT_new(group)) == NULL) goto err;
356 if (!EC_POINT_set_to_infinity(group, acc)) goto err;
360 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
361 if (BN_is_negative(scalar))
362 if (!group->meth->invert(group, p, ctx)) goto err;
363 if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
366 for (i = 0; i < num; i++)
368 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
369 if (BN_is_negative(scalars[i]))
370 if (!group->meth->invert(group, p, ctx)) goto err;
371 if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
374 if (!EC_POINT_copy(r, acc)) goto err;
379 if (p) EC_POINT_free(p);
380 if (acc) EC_POINT_free(acc);
382 BN_CTX_free(new_ctx);
387 /* Precomputation for point multiplication: fall back to wNAF methods
388 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
390 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
392 return ec_wNAF_precompute_mult(group, ctx);
395 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
397 return ec_wNAF_have_precompute_mult(group);