crypto/ec/ec_mult.c

   1 /* crypto/ec/ec_mult.c */
   2 /* ====================================================================
   3  * Copyright (c) 1998-2001 The OpenSSL Project.  All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  *
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  *
  12  * 2. Redistributions in binary form must reproduce the above copyright
  13  *    notice, this list of conditions and the following disclaimer in
  14  *    the documentation and/or other materials provided with the
  15  *    distribution.
  16  *
  17  * 3. All advertising materials mentioning features or use of this
  18  *    software must display the following acknowledgment:
  19  *    "This product includes software developed by the OpenSSL Project
  20  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
  21  *
  22  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
  23  *    endorse or promote products derived from this software without
  24  *    prior written permission. For written permission, please contact
  25  *    openssl-core@openssl.org.
  26  *
  27  * 5. Products derived from this software may not be called "OpenSSL"
  28  *    nor may "OpenSSL" appear in their names without prior written
  29  *    permission of the OpenSSL Project.
  30  *
  31  * 6. Redistributions of any form whatsoever must retain the following
  32  *    acknowledgment:
  33  *    "This product includes software developed by the OpenSSL Project
  34  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
  35  *
  36  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
  37  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  38  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
  39  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
  40  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  41  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
  42  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
  43  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  44  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
  45  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  46  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
  47  * OF THE POSSIBILITY OF SUCH DAMAGE.
  48  * ====================================================================
  49  *
  50  * This product includes cryptographic software written by Eric Young
  51  * (eay@cryptsoft.com).  This product includes software written by Tim
  52  * Hudson (tjh@cryptsoft.com).
  53  *
  54  */
  55
  56 #include <openssl/err.h>
  57
  58 #include "ec_lcl.h"
  59
  60
  61 /* TODO: width-m NAFs */
  62
  63 /* TODO: optional precomputation of multiples of the generator */
  64
  65
  66 #define EC_window_bits_for_scalar_size(b) \
  67                 ((b) >= 2000 ? 6 : \
  68                  (b) >=  800 ? 5 : \
  69                  (b) >=  300 ? 4 : \
  70                  (b) >=   70 ? 3 : \
  71                  (b) >=   20 ? 2 : \
  72                   1)
  73 /* For window size 'w' (w >= 2), we compute the odd multiples
  74  *      1*P .. (2^w-1)*P.
  75  * This accounts for  2^(w-1)  point additions (neglecting constants),
  76  * each of which requires 16 field multiplications (4 squarings
  77  * and 12 general multiplications) in the case of curves defined
  78  * over GF(p), which are the only curves we have so far.
  79  *
  80  * Converting these precomputed points into affine form takes
  81  * three field multiplications for inverting Z and one squaring
  82  * and three multiplications for adjusting X and Y, i.e.
  83  * 7 multiplications in total (1 squaring and 6 general multiplications),
  84  * again except for constants.
  85  *
  86  * The average number of windows for a 'b' bit scalar is roughly
  87  *          b/(w+1).
  88  * Each of these windows (except possibly for the first one, but
  89  * we are ignoring constants anyway) requires one point addition.
  90  * As the precomputed table stores points in affine form, these
  91  * additions take only 11 field multiplications each (3 squarings
  92  * and 8 general multiplications).
  93  *
  94  * So the total workload, except for constants, is
  95  *
  96  *        2^(w-1)*[5 squarings + 18 multiplications]
  97  *      + (b/(w+1))*[3 squarings + 8 multiplications]
  98  *
  99  * If we assume that 10 squarings are as costly as 9 multiplications,
 100  * our task is to find the 'w' that, given 'b', minimizes
 101  *
 102  *        2^(w-1)*(5*9 + 18*10) + (b/(w+1))*(3*9 + 8*10)
 103  *      = 2^(w-1)*225 +           (b/(w+1))*107.
 104  *
 105  * Thus optimal window sizes should be roughly as follows:
 106  *
 107  *    w >= 6  if         b >= 1414
 108  *     w = 5  if 1413 >= b >=  505
 109  *     w = 4  if  504 >= b >=  169
 110  *     w = 3  if  168 >= b >=   51
 111  *     w = 2  if   50 >= b >=   13
 112  *     w = 1  if   12 >= b
 113  *
 114  * If we assume instead that squarings are exactly as costly as
 115  * multiplications, we have to minimize
 116  *      2^(w-1)*23 + (b/(w+1))*11.
 117  *
 118  * This gives us the following (nearly unchanged) table of optimal
 119  * windows sizes:
 120  *
 121  *    w >= 6  if         b >= 1406
 122  *     w = 5  if 1405 >= b >=  502
 123  *     w = 4  if  501 >= b >=  168
 124  *     w = 3  if  167 >= b >=   51
 125  *     w = 2  if   50 >= b >=   13
 126  *     w = 1  if   12 >= b
 127  *
 128  * Note that neither table tries to take into account memory usage
 129  * (allocation overhead, code locality etc.).  Actual timings with
 130  * NIST curves P-192, P-224, and P-256 with scalars of 192, 224,
 131  * and 256 bits, respectively, show that  w = 3  (instead of 4) is
 132  * preferrable; timings with NIST curve P-384 and 384-bit scalars
 133  * confirm that  w = 4  is optimal for this case; and timings with
 134  * NIST curve P-521 and 521-bit scalars show that  w = 4  (instead
 135  * of 5) is preferrable.  So we generously round up all the
 136  * boundaries and use the following table:
 137  *
 138  *    w >= 6  if         b >= 2000
 139  *     w = 5  if 1999 >= b >=  800
 140  *     w = 4  if  799 >= b >=  300
 141  *     w = 3  if  299 >= b >=   70
 142  *     w = 2  if   69 >= b >=   20
 143  *     w = 1  if   19 >= b
 144  */
 145
 146
 147
 148 /* Compute
 149  *      \sum scalars[i]*points[i],
 150  * also including
 151  *      scalar*generator
 152  * in the addition if scalar != NULL
 153  */
 154 int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
 155         size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
 156         {
 157         BN_CTX *new_ctx = NULL;
 158         EC_POINT *generator = NULL;
 159         EC_POINT *tmp = NULL;
 160         size_t totalnum;
 161         size_t i, j;
 162         int k, t;
 163         int r_is_at_infinity = 1;
 164         size_t max_bits = 0;
 165         size_t *wsize = NULL; /* individual window sizes */
 166         unsigned long *wbits = NULL; /* individual window contents */
 167         int *wpos = NULL; /* position of bottom bit of current individual windows
 168                            * (wpos[i] is valid if wbits[i] != 0) */
 169         size_t num_val;
 170         EC_POINT **val = NULL; /* precomputation */
 171         EC_POINT **v;
 172         EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
 173         int ret = 0;
 174
 175         if (scalar != NULL)
 176                 {
 177                 generator = EC_GROUP_get0_generator(group);
 178                 if (generator == NULL)
 179                         {
 180                         ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
 181                         return 0;
 182                         }
 183                 }
 184
 185         for (i = 0; i < num; i++)
 186                 {
 187                 if (group->meth != points[i]->meth)
 188                         {
 189                         ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
 190                         return 0;
 191                         }
 192                 }
 193
 194         totalnum = num + (scalar != NULL);
 195
 196         wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
 197         wbits = OPENSSL_malloc(totalnum * sizeof wbits[0]);
 198         wpos = OPENSSL_malloc(totalnum * sizeof wpos[0]);
 199         if (wsize == NULL || wbits == NULL || wpos == NULL) goto err;
 200
 201         /* num_val := total number of points to precompute */
 202         num_val = 0;
 203         for (i = 0; i < totalnum; i++)
 204                 {
 205                 size_t bits;
 206
 207                 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
 208                 wsize[i] = EC_window_bits_for_scalar_size(bits);
 209                 num_val += 1u << (wsize[i] - 1);
 210                 if (bits > max_bits)
 211                         max_bits = bits;
 212                 wbits[i] = 0;
 213                 wpos[i] = 0;
 214                 }
 215
 216         /* all precomputed points go into a single array 'val',
 217          * 'val_sub[i]' is a pointer to the subarray for the i-th point */
 218         val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
 219         if (val == NULL) goto err;
 220         val[num_val] = NULL; /* pivot element */
 221
 222         val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
 223         if (val_sub == NULL) goto err;
 224
 225         /* allocate points for precomputation */
 226         v = val;
 227         for (i = 0; i < totalnum; i++)
 228                 {
 229                 val_sub[i] = v;
 230                 for (j = 0; j < (1u << (wsize[i] - 1)); j++)
 231                         {
 232                         *v = EC_POINT_new(group);
 233                         if (*v == NULL) goto err;
 234                         v++;
 235                         }
 236                 }
 237         if (!(v == val + num_val))
 238                 {
 239                 ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
 240                 goto err;
 241                 }
 242
 243         if (ctx == NULL)
 244                 {
 245                 ctx = new_ctx = BN_CTX_new();
 246                 if (ctx == NULL)
 247                         goto err;
 248                 }
 249
 250         tmp = EC_POINT_new(group);
 251         if (tmp == NULL) goto err;
 252
 253         /* prepare precomputed values:
 254          *    val_sub[i][0] :=     points[i]
 255          *    val_sub[i][1] := 3 * points[i]
 256          *    val_sub[i][2] := 5 * points[i]
 257          *    ...
 258          */
 259         for (i = 0; i < totalnum; i++)
 260                 {
 261                 if (i < num)
 262                         {
 263                         if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
 264                         if (scalars[i]->neg)
 265                                 {
 266                                 if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
 267                                 }
 268                         }
 269                 else
 270                         {
 271                         if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
 272                         if (scalar->neg)
 273                                 {
 274                                 if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
 275                                 }
 276                         }
 277
 278                 if (wsize[i] > 1)
 279                         {
 280                         if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
 281                         for (j = 1; j < (1u << (wsize[i] - 1)); j++)
 282                                 {
 283                                 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
 284                                 }
 285                         }
 286                 }
 287
 288 #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
 289         if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
 290 #endif
 291
 292         r_is_at_infinity = 1;
 293
 294         for (k = max_bits - 1; k >= 0; k--)
 295                 {
 296                 if (!r_is_at_infinity)
 297                         {
 298                         if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
 299                         }
 300
 301                 for (i = 0; i < totalnum; i++)
 302                         {
 303                         if (wbits[i] == 0)
 304                                 {
 305                                 const BIGNUM *s;
 306
 307                                 s = i < num ? scalars[i] : scalar;
 308
 309                                 if (BN_is_bit_set(s, k))
 310                                         {
 311                                         /* look at bits  k - wsize[i] + 1 .. k  for this window */
 312                                         t = k - wsize[i] + 1;
 313                                         while (!BN_is_bit_set(s, t)) /* BN_is_bit_set is false for t < 0 */
 314                                                 t++;
 315                                         wpos[i] = t;
 316                                         wbits[i] = 1;
 317                                         for (t = k - 1; t >= wpos[i]; t--)
 318                                                 {
 319                                                 wbits[i] <<= 1;
 320                                                 if (BN_is_bit_set(s, t))
 321                                                         wbits[i]++;
 322                                                 }
 323                                         /* now wbits[i] is the odd bit pattern at bits wpos[i] .. k */
 324                                         }
 325                                 }
 326
 327                         if ((wbits[i] != 0) && (wpos[i] == k))
 328                                 {
 329                                 if (r_is_at_infinity)
 330                                         {
 331                                         if (!EC_POINT_copy(r, val_sub[i][wbits[i] >> 1])) goto err;
 332                                         r_is_at_infinity = 0;
 333                                         }
 334                                 else
 335                                         {
 336                                         if (!EC_POINT_add(group, r, r, val_sub[i][wbits[i] >> 1], ctx)) goto err;
 337                                         }
 338                                 wbits[i] = 0;
 339                                 }
 340                         }
 341                 }
 342
 343         if (r_is_at_infinity)
 344                 if (!EC_POINT_set_to_infinity(group, r)) goto err;
 345
 346         ret = 1;
 347
 348  err:
 349         if (new_ctx != NULL)
 350                 BN_CTX_free(new_ctx);
 351         if (tmp != NULL)
 352                 EC_POINT_free(tmp);
 353         if (wsize != NULL)
 354                 OPENSSL_free(wsize);
 355         if (wbits != NULL)
 356                 OPENSSL_free(wbits);
 357         if (wpos != NULL)
 358                 OPENSSL_free(wpos);
 359         if (val != NULL)
 360                 {
 361                 for (v = val; *v != NULL; v++)
 362                         EC_POINT_clear_free(*v);
 363
 364                 OPENSSL_free(val);
 365                 }
 366         if (val_sub != NULL)
 367                 {
 368                 OPENSSL_free(val_sub);
 369                 }
 370         return ret;
 371         }
 372
 373
 374 int EC_POINT_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar, const EC_POINT *point, const BIGNUM *p_scalar, BN_CTX *ctx)
 375         {
 376         const EC_POINT *points[1];
 377         const BIGNUM *scalars[1];
 378
 379         points[0] = point;
 380         scalars[0] = p_scalar;
 381
 382         return EC_POINTs_mul(group, r, g_scalar, (point != NULL && p_scalar != NULL), points, scalars, ctx);
 383         }
 384
 385
 386 int EC_GROUP_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
 387         {
 388         const EC_POINT *generator;
 389         BN_CTX *new_ctx = NULL;
 390         BIGNUM *order;
 391         int ret = 0;
 392
 393         generator = EC_GROUP_get0_generator(group);
 394         if (generator == NULL)
 395                 {
 396                 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
 397                 return 0;
 398                 }
 399
 400         if (ctx == NULL)
 401                 {
 402                 ctx = new_ctx = BN_CTX_new();
 403                 if (ctx == NULL)
 404                         return 0;
 405                 }
 406
 407         BN_CTX_start(ctx);
 408         order = BN_CTX_get(ctx);
 409         if (order == NULL) goto err;
 410
 411         if (!EC_GROUP_get_order(group, order, ctx)) return 0;
 412         if (BN_is_zero(order))
 413                 {
 414                 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
 415                 goto err;
 416                 }
 417
 418         /* TODO */
 419
 420         ret = 1;
 421
 422  err:
 423         BN_CTX_end(ctx);
 424         if (new_ctx != NULL)
 425                 BN_CTX_free(new_ctx);
 426         return ret;
 427         }