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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 (at) 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 (at) cryptsoft.com).  This product includes software written by Tim
     66  * Hudson (tjh (at) 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  */
     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  */
    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  */
    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 
    212 /* Computes scalar*point and stores the result in r.
    213  * point can not equal r.
    214  * Uses a modified algorithm 2P of
    215  *     Lopez, J. and Dahab, R.  "Fast multiplication on elliptic curves over
    216  *     GF(2^m) without precomputation" (CHES '99, LNCS 1717).
    217  *
    218  * To protect against side-channel attack the function uses constant time swap,
    219  * avoiding conditional branches.
    220  */
    221 static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
    222 	const EC_POINT *point, BN_CTX *ctx)
    223 	{
    224 	BIGNUM *x1, *x2, *z1, *z2;
    225 	int ret = 0, i;
    226 	BN_ULONG mask,word;
    227 
    228 	if (r == point)
    229 		{
    230 		ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
    231 		return 0;
    232 		}
    233 
    234 	/* if result should be point at infinity */
    235 	if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
    236 		EC_POINT_is_at_infinity(group, point))
    237 		{
    238 		return EC_POINT_set_to_infinity(group, r);
    239 		}
    240 
    241 	/* only support affine coordinates */
    242 	if (!point->Z_is_one) return 0;
    243 
    244 	/* Since point_multiply is static we can guarantee that ctx != NULL. */
    245 	BN_CTX_start(ctx);
    246 	x1 = BN_CTX_get(ctx);
    247 	z1 = BN_CTX_get(ctx);
    248 	if (z1 == NULL) goto err;
    249 
    250 	x2 = &r->X;
    251 	z2 = &r->Y;
    252 
    253 	bn_wexpand(x1, group->field.top);
    254 	bn_wexpand(z1, group->field.top);
    255 	bn_wexpand(x2, group->field.top);
    256 	bn_wexpand(z2, group->field.top);
    257 
    258 	if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */
    259 	if (!BN_one(z1)) goto err; /* z1 = 1 */
    260 	if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
    261 	if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
    262 	if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */
    263 
    264 	/* find top most bit and go one past it */
    265 	i = scalar->top - 1;
    266 	mask = BN_TBIT;
    267 	word = scalar->d[i];
    268 	while (!(word & mask)) mask >>= 1;
    269 	mask >>= 1;
    270 	/* if top most bit was at word break, go to next word */
    271 	if (!mask)
    272 		{
    273 		i--;
    274 		mask = BN_TBIT;
    275 		}
    276 
    277 	for (; i >= 0; i--)
    278 		{
    279 		word = scalar->d[i];
    280 		while (mask)
    281 			{
    282 			BN_consttime_swap(word & mask, x1, x2, group->field.top);
    283 			BN_consttime_swap(word & mask, z1, z2, group->field.top);
    284 			if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err;
    285 			if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
    286 			BN_consttime_swap(word & mask, x1, x2, group->field.top);
    287 			BN_consttime_swap(word & mask, z1, z2, group->field.top);
    288 			mask >>= 1;
    289 			}
    290 		mask = BN_TBIT;
    291 		}
    292 
    293 	/* convert out of "projective" coordinates */
    294 	i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
    295 	if (i == 0) goto err;
    296 	else if (i == 1)
    297 		{
    298 		if (!EC_POINT_set_to_infinity(group, r)) goto err;
    299 		}
    300 	else
    301 		{
    302 		if (!BN_one(&r->Z)) goto err;
    303 		r->Z_is_one = 1;
    304 		}
    305 
    306 	/* GF(2^m) field elements should always have BIGNUM::neg = 0 */
    307 	BN_set_negative(&r->X, 0);
    308 	BN_set_negative(&r->Y, 0);
    309 
    310 	ret = 1;
    311 
    312  err:
    313 	BN_CTX_end(ctx);
    314 	return ret;
    315 	}
    316 
    317 
    318 /* Computes the sum
    319  *     scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
    320  * gracefully ignoring NULL scalar values.
    321  */
    322 int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
    323 	size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
    324 	{
    325 	BN_CTX *new_ctx = NULL;
    326 	int ret = 0;
    327 	size_t i;
    328 	EC_POINT *p=NULL;
    329 	EC_POINT *acc = NULL;
    330 
    331 	if (ctx == NULL)
    332 		{
    333 		ctx = new_ctx = BN_CTX_new();
    334 		if (ctx == NULL)
    335 			return 0;
    336 		}
    337 
    338 	/* This implementation is more efficient than the wNAF implementation for 2
    339 	 * or fewer points.  Use the ec_wNAF_mul implementation for 3 or more points,
    340 	 * or if we can perform a fast multiplication based on precomputation.
    341 	 */
    342 	if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
    343 		{
    344 		ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
    345 		goto err;
    346 		}
    347 
    348 	if ((p = EC_POINT_new(group)) == NULL) goto err;
    349 	if ((acc = EC_POINT_new(group)) == NULL) goto err;
    350 
    351 	if (!EC_POINT_set_to_infinity(group, acc)) goto err;
    352 
    353 	if (scalar)
    354 		{
    355 		if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
    356 		if (BN_is_negative(scalar))
    357 			if (!group->meth->invert(group, p, ctx)) goto err;
    358 		if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
    359 		}
    360 
    361 	for (i = 0; i < num; i++)
    362 		{
    363 		if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
    364 		if (BN_is_negative(scalars[i]))
    365 			if (!group->meth->invert(group, p, ctx)) goto err;
    366 		if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
    367 		}
    368 
    369 	if (!EC_POINT_copy(r, acc)) goto err;
    370 
    371 	ret = 1;
    372 
    373   err:
    374 	if (p) EC_POINT_free(p);
    375 	if (acc) EC_POINT_free(acc);
    376 	if (new_ctx != NULL)
    377 		BN_CTX_free(new_ctx);
    378 	return ret;
    379 	}
    380 
    381 
    382 /* Precomputation for point multiplication: fall back to wNAF methods
    383  * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
    384 
    385 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
    386 	{
    387 	return ec_wNAF_precompute_mult(group, ctx);
    388  	}
    389 
    390 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
    391 	{
    392 	return ec_wNAF_have_precompute_mult(group);
    393  	}
    394 
    395 #endif
    396