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).
72 #include <openssl/err.h>
74 #include "internal/bn_int.h"
77 #ifndef OPENSSL_NO_EC2M
81 * Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective
83 * Uses algorithm Mdouble in appendix of
84 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
85 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
86 * modified to not require precomputation of c=b^{2^{m-1}}.
88 static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx)
93 /* Since Mdouble is static we can guarantee that ctx != NULL. */
96 if (t1 == NULL) goto err;
98 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
99 if (!group->meth->field_sqr(group, t1, z, ctx)) goto err;
100 if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err;
101 if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
102 if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err;
103 if (!group->meth->field_mul(group, t1, group->b, t1, ctx)) goto err;
104 if (!BN_GF2m_add(x, x, t1)) goto err;
114 * Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery
115 * projective coordinates.
116 * Uses algorithm Madd in appendix of
117 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
118 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
120 static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1,
121 const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx)
126 /* Since Madd is static we can guarantee that ctx != NULL. */
128 t1 = BN_CTX_get(ctx);
129 t2 = BN_CTX_get(ctx);
130 if (t2 == NULL) goto err;
132 if (!BN_copy(t1, x)) goto err;
133 if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err;
134 if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err;
135 if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err;
136 if (!BN_GF2m_add(z1, z1, x1)) goto err;
137 if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err;
138 if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err;
139 if (!BN_GF2m_add(x1, x1, t2)) goto err;
149 * Compute the x, y affine coordinates from the point (x1, z1) (x2, z2)
150 * using Montgomery point multiplication algorithm Mxy() in appendix of
151 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
152 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
155 * 1 if return value should be the point at infinity
158 static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1,
159 BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx)
161 BIGNUM *t3, *t4, *t5;
173 if (!BN_copy(x2, x)) return 0;
174 if (!BN_GF2m_add(z2, x, y)) return 0;
178 /* Since Mxy is static we can guarantee that ctx != NULL. */
180 t3 = BN_CTX_get(ctx);
181 t4 = BN_CTX_get(ctx);
182 t5 = BN_CTX_get(ctx);
183 if (t5 == NULL) goto err;
185 if (!BN_one(t5)) goto err;
187 if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err;
189 if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err;
190 if (!BN_GF2m_add(z1, z1, x1)) goto err;
191 if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err;
192 if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err;
193 if (!BN_GF2m_add(z2, z2, x2)) goto err;
195 if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err;
196 if (!group->meth->field_sqr(group, t4, x, ctx)) goto err;
197 if (!BN_GF2m_add(t4, t4, y)) goto err;
198 if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err;
199 if (!BN_GF2m_add(t4, t4, z2)) goto err;
201 if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err;
202 if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err;
203 if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err;
204 if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err;
205 if (!BN_GF2m_add(z2, x2, x)) goto err;
207 if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err;
208 if (!BN_GF2m_add(z2, z2, y)) goto err;
219 * Computes scalar*point and stores the result in r.
220 * point can not equal r.
221 * Uses a modified algorithm 2P of
222 * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
223 * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
225 * To protect against side-channel attack the function uses constant time swap,
226 * avoiding conditional branches.
228 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
229 const EC_POINT *point, BN_CTX *ctx)
231 BIGNUM *x1, *x2, *z1, *z2;
237 ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
241 /* if result should be point at infinity */
242 if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
243 EC_POINT_is_at_infinity(group, point))
245 return EC_POINT_set_to_infinity(group, r);
248 /* only support affine coordinates */
249 if (!point->Z_is_one) return 0;
251 /* Since point_multiply is static we can guarantee that ctx != NULL. */
253 x1 = BN_CTX_get(ctx);
254 z1 = BN_CTX_get(ctx);
255 if (z1 == NULL) goto err;
260 bn_wexpand(x1, bn_get_top(group->field));
261 bn_wexpand(z1, bn_get_top(group->field));
262 bn_wexpand(x2, bn_get_top(group->field));
263 bn_wexpand(z2, bn_get_top(group->field));
265 if (!BN_GF2m_mod_arr(x1, point->X, group->poly)) goto err; /* x1 = x */
266 if (!BN_one(z1)) goto err; /* z1 = 1 */
267 if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
268 if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
269 if (!BN_GF2m_add(x2, x2, group->b)) goto err; /* x2 = x^4 + b */
271 /* find top most bit and go one past it */
272 i = bn_get_top(scalar) - 1;
274 word = bn_get_words(scalar)[i];
275 while (!(word & mask)) mask >>= 1;
277 /* if top most bit was at word break, go to next word */
286 word = bn_get_words(scalar)[i];
289 BN_consttime_swap(word & mask, x1, x2, bn_get_top(group->field));
290 BN_consttime_swap(word & mask, z1, z2, bn_get_top(group->field));
291 if (!gf2m_Madd(group, point->X, x2, z2, x1, z1, ctx)) goto err;
292 if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
293 BN_consttime_swap(word & mask, x1, x2, bn_get_top(group->field));
294 BN_consttime_swap(word & mask, z1, z2, bn_get_top(group->field));
300 /* convert out of "projective" coordinates */
301 i = gf2m_Mxy(group, point->X, point->Y, x1, z1, x2, z2, ctx);
302 if (i == 0) goto err;
305 if (!EC_POINT_set_to_infinity(group, r)) goto err;
309 if (!BN_one(r->Z)) goto err;
313 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
314 BN_set_negative(r->X, 0);
315 BN_set_negative(r->Y, 0);
327 * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
328 * gracefully ignoring NULL scalar values.
330 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
331 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
333 BN_CTX *new_ctx = NULL;
337 EC_POINT *acc = NULL;
341 ctx = new_ctx = BN_CTX_new();
346 /* This implementation is more efficient than the wNAF implementation for 2
347 * or fewer points. Use the ec_wNAF_mul implementation for 3 or more points,
348 * or if we can perform a fast multiplication based on precomputation.
350 if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
352 ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
356 if ((p = EC_POINT_new(group)) == NULL) goto err;
357 if ((acc = EC_POINT_new(group)) == NULL) goto err;
359 if (!EC_POINT_set_to_infinity(group, acc)) goto err;
363 if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
364 if (BN_is_negative(scalar))
365 if (!group->meth->invert(group, p, ctx)) goto err;
366 if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
369 for (i = 0; i < num; i++)
371 if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
372 if (BN_is_negative(scalars[i]))
373 if (!group->meth->invert(group, p, ctx)) goto err;
374 if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
377 if (!EC_POINT_copy(r, acc)) goto err;
382 if (p) EC_POINT_free(p);
383 if (acc) EC_POINT_free(acc);
385 BN_CTX_free(new_ctx);
390 /* Precomputation for point multiplication: fall back to wNAF methods
391 * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
393 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
395 return ec_wNAF_precompute_mult(group, ctx);
398 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
400 return ec_wNAF_have_precompute_mult(group);