crypto/ec/ec2_mult.c

   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  */
  82 static 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  */
 123 static 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  */
 171 static 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  */
 261 static 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  */
 376 int 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
 455 int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
 456 {
 457     return ec_wNAF_precompute_mult(group, ctx);
 458 }
 459
 460 int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
 461 {
 462     return ec_wNAF_have_precompute_mult(group);
 463 }