1 /* crypto/ec/ec_mult.c */
3 * Originally written by Bodo Moeller and Nils Larsch for the OpenSSL project.
5 /* ====================================================================
6 * Copyright (c) 1998-2003 The OpenSSL Project. All rights reserved.
8 * Redistribution and use in source and binary forms, with or without
9 * modification, are permitted provided that the following conditions
12 * 1. Redistributions of source code must retain the above copyright
13 * notice, this list of conditions and the following disclaimer.
15 * 2. Redistributions in binary form must reproduce the above copyright
16 * notice, this list of conditions and the following disclaimer in
17 * the documentation and/or other materials provided with the
20 * 3. All advertising materials mentioning features or use of this
21 * software must display the following acknowledgment:
22 * "This product includes software developed by the OpenSSL Project
23 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
25 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
26 * endorse or promote products derived from this software without
27 * prior written permission. For written permission, please contact
28 * openssl-core@openssl.org.
30 * 5. Products derived from this software may not be called "OpenSSL"
31 * nor may "OpenSSL" appear in their names without prior written
32 * permission of the OpenSSL Project.
34 * 6. Redistributions of any form whatsoever must retain the following
36 * "This product includes software developed by the OpenSSL Project
37 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
39 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
40 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
41 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
42 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
43 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
44 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
45 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
46 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
47 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
48 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
49 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
50 * OF THE POSSIBILITY OF SUCH DAMAGE.
51 * ====================================================================
53 * This product includes cryptographic software written by Eric Young
54 * (eay@cryptsoft.com). This product includes software written by Tim
55 * Hudson (tjh@cryptsoft.com).
58 /* ====================================================================
59 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
60 * Portions of this software developed by SUN MICROSYSTEMS, INC.,
61 * and contributed to the OpenSSL project.
66 #include <openssl/err.h>
72 * This file implements the wNAF-based interleaving multi-exponentation method
73 * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp>);
74 * for multiplication with precomputation, we use wNAF splitting
75 * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#fastexp>).
81 /* structure for precomputed multiples of the generator */
82 typedef struct ec_pre_comp_st {
83 const EC_GROUP *group; /* parent EC_GROUP object */
84 size_t blocksize; /* block size for wNAF splitting */
85 size_t numblocks; /* max. number of blocks for which we have precomputation */
86 size_t w; /* window size */
87 EC_POINT **points; /* array with pre-calculated multiples of generator:
88 * 'num' pointers to EC_POINT objects followed by a NULL */
89 size_t num; /* numblocks * 2^(w-1) */
93 /* functions to manage EC_PRE_COMP within the EC_GROUP extra_data framework */
94 static void *ec_pre_comp_dup(void *);
95 static void ec_pre_comp_free(void *);
96 static void ec_pre_comp_clear_free(void *);
98 static EC_PRE_COMP *ec_pre_comp_new(const EC_GROUP *group)
100 EC_PRE_COMP *ret = NULL;
105 ret = (EC_PRE_COMP *)OPENSSL_malloc(sizeof(EC_PRE_COMP));
109 ret->blocksize = 8; /* default */
111 ret->w = 4; /* default */
118 static void *ec_pre_comp_dup(void *src_)
120 EC_PRE_COMP *src = src_;
122 /* no need to actually copy, these objects never change! */
124 CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
129 static void ec_pre_comp_free(void *pre_)
132 EC_PRE_COMP *pre = pre_;
137 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
145 for (p = pre->points; *p != NULL; p++)
147 OPENSSL_free(pre->points);
152 static void ec_pre_comp_clear_free(void *pre_)
155 EC_PRE_COMP *pre = pre_;
160 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
168 for (p = pre->points; *p != NULL; p++)
169 EC_POINT_clear_free(*p);
170 OPENSSL_cleanse(pre->points, sizeof pre->points);
171 OPENSSL_free(pre->points);
173 OPENSSL_cleanse(pre, sizeof pre);
180 /* Determine the modified width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'.
181 * This is an array r[] of values that are either zero or odd with an
182 * absolute value less than 2^w satisfying
183 * scalar = \sum_j r[j]*2^j
184 * where at most one of any w+1 consecutive digits is non-zero
185 * with the exception that the most significant digit may be only
186 * w-1 zeros away from that next non-zero digit.
188 static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len)
192 signed char *r = NULL;
194 int bit, next_bit, mask;
197 if (w <= 0 || w > 7) /* 'signed char' can represent integers with absolute values less than 2^7 */
199 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
202 bit = 1 << w; /* at most 128 */
203 next_bit = bit << 1; /* at most 256 */
204 mask = next_bit - 1; /* at most 255 */
206 if (BN_get_sign(scalar))
211 len = BN_num_bits(scalar);
212 r = OPENSSL_malloc(len + 1); /* modified wNAF may be one digit longer than binary representation
213 * (*ret_len will be set to the actual length, i.e. at most
214 * BN_num_bits(scalar) + 1) */
215 if (r == NULL) goto err;
217 if (scalar->d == NULL || scalar->top == 0)
219 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
222 window_val = scalar->d[0] & mask;
224 while ((window_val != 0) || (j + w + 1 < len)) /* if j+w+1 >= len, window_val will not increase */
228 /* 0 <= window_val <= 2^(w+1) */
232 /* 0 < window_val < 2^(w+1) */
234 if (window_val & bit)
236 digit = window_val - next_bit; /* -2^w < digit < 0 */
238 #if 1 /* modified wNAF */
239 if (j + w + 1 >= len)
241 /* special case for generating modified wNAFs:
242 * no new bits will be added into window_val,
243 * so using a positive digit here will decrease
244 * the total length of the representation */
246 digit = window_val & (mask >> 1); /* 0 < digit < 2^w */
252 digit = window_val; /* 0 < digit < 2^w */
255 if (digit <= -bit || digit >= bit || !(digit & 1))
257 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
263 /* now window_val is 0 or 2^(w+1) in standard wNAF generation;
264 * for modified window NAFs, it may also be 2^w
266 if (window_val != 0 && window_val != next_bit && window_val != bit)
268 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
273 r[j++] = sign * digit;
276 window_val += bit * BN_is_bit_set(scalar, j + w);
278 if (window_val > next_bit)
280 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
287 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
305 /* TODO: table should be optimised for the wNAF-based implementation,
306 * sometimes smaller windows will give better performance
307 * (thus the boundaries should be increased)
309 #define EC_window_bits_for_scalar_size(b) \
318 * \sum scalars[i]*points[i],
321 * in the addition if scalar != NULL
323 int ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
324 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
326 BN_CTX *new_ctx = NULL;
327 EC_POINT *generator = NULL;
328 EC_POINT *tmp = NULL;
330 size_t blocksize = 0, numblocks = 0; /* for wNAF splitting */
331 size_t pre_points_per_block = 0;
334 int r_is_inverted = 0;
335 int r_is_at_infinity = 1;
336 size_t *wsize = NULL; /* individual window sizes */
337 signed char **wNAF = NULL; /* individual wNAFs */
338 size_t *wNAF_len = NULL;
341 EC_POINT **val = NULL; /* precomputation */
343 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' or 'pre_comp->points' */
344 const EC_PRE_COMP *pre_comp = NULL;
345 int num_scalar = 0; /* flag: will be set to 1 if 'scalar' must be treated like other scalars,
346 * i.e. precomputation is not available */
349 if (group->meth != r->meth)
351 ECerr(EC_F_EC_WNAF_MUL, EC_R_INCOMPATIBLE_OBJECTS);
355 if ((scalar == NULL) && (num == 0))
357 return EC_POINT_set_to_infinity(group, r);
360 for (i = 0; i < num; i++)
362 if (group->meth != points[i]->meth)
364 ECerr(EC_F_EC_WNAF_MUL, EC_R_INCOMPATIBLE_OBJECTS);
371 ctx = new_ctx = BN_CTX_new();
378 generator = EC_GROUP_get0_generator(group);
379 if (generator == NULL)
381 ECerr(EC_F_EC_WNAF_MUL, EC_R_UNDEFINED_GENERATOR);
385 /* look if we can use precomputed multiples of generator */
387 pre_comp = EC_GROUP_get_extra_data(group, ec_pre_comp_dup, ec_pre_comp_free, ec_pre_comp_clear_free);
389 if (pre_comp && pre_comp->numblocks && (EC_POINT_cmp(group, generator, pre_comp->points[0], ctx) == 0))
391 blocksize = pre_comp->blocksize;
393 /* determine maximum number of blocks that wNAF splitting may yield
394 * (NB: maximum wNAF length is bit length plus one) */
395 numblocks = (BN_num_bits(scalar) / blocksize) + 1;
397 /* we cannot use more blocks than we have precomputation for */
398 if (numblocks > pre_comp->numblocks)
399 numblocks = pre_comp->numblocks;
401 pre_points_per_block = 1u << (pre_comp->w - 1);
403 /* check that pre_comp looks sane */
404 if (pre_comp->num != (pre_comp->numblocks * pre_points_per_block))
406 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
412 /* can't use precomputation */
415 num_scalar = 1; /* treat 'scalar' like 'num'-th element of 'scalars' */
419 totalnum = num + numblocks;
421 wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
422 wNAF_len = OPENSSL_malloc(totalnum * sizeof wNAF_len[0]);
423 wNAF = OPENSSL_malloc((totalnum + 1) * sizeof wNAF[0]); /* includes space for pivot */
424 val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
426 if (!wsize || !wNAF_len || !wNAF || !val_sub)
429 wNAF[0] = NULL; /* preliminary pivot */
431 /* num_val will be the total number of temporarily precomputed points */
434 for (i = 0; i < num + num_scalar; i++)
438 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
439 wsize[i] = EC_window_bits_for_scalar_size(bits);
440 num_val += 1u << (wsize[i] - 1);
441 wNAF[i + 1] = NULL; /* make sure we always have a pivot */
442 wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i]);
445 if (wNAF_len[i] > max_len)
446 max_len = wNAF_len[i];
451 /* we go here iff scalar != NULL */
453 if (pre_comp == NULL)
457 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
460 /* we have already generated a wNAF for 'scalar' */
464 signed char *tmp_wNAF = NULL;
469 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
473 /* use the window size for which we have precomputation */
474 wsize[num] = pre_comp->w;
475 tmp_wNAF = compute_wNAF(scalar, wsize[num], &tmp_len);
479 if (tmp_len <= max_len)
481 /* One of the other wNAFs is at least as long
482 * as the wNAF belonging to the generator,
483 * so wNAF splitting will not buy us anything. */
486 totalnum = num + 1; /* don't use wNAF splitting */
487 wNAF[num] = tmp_wNAF;
488 wNAF[num + 1] = NULL;
489 wNAF_len[num] = tmp_len;
490 if (tmp_len > max_len)
492 /* pre_comp->points starts with the points that we need here: */
493 val_sub[num] = pre_comp->points;
497 /* don't include tmp_wNAF directly into wNAF array
498 * - use wNAF splitting and include the blocks */
501 EC_POINT **tmp_points;
503 if (tmp_len < numblocks * blocksize)
505 /* possibly we can do with fewer blocks than estimated */
506 numblocks = (tmp_len + blocksize - 1) / blocksize;
507 if (numblocks > pre_comp->numblocks)
509 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
512 totalnum = num + numblocks;
515 /* split wNAF in 'numblocks' parts */
517 tmp_points = pre_comp->points;
519 for (i = num; i < totalnum; i++)
521 if (i < totalnum - 1)
523 wNAF_len[i] = blocksize;
524 if (tmp_len < blocksize)
526 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
529 tmp_len -= blocksize;
532 /* last block gets whatever is left
533 * (this could be more or less than 'blocksize'!) */
534 wNAF_len[i] = tmp_len;
537 wNAF[i] = OPENSSL_malloc(wNAF_len[i]);
540 OPENSSL_free(tmp_wNAF);
543 memcpy(wNAF[i], pp, wNAF_len[i]);
544 if (wNAF_len[i] > max_len)
545 max_len = wNAF_len[i];
547 if (*tmp_points == NULL)
549 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
550 OPENSSL_free(tmp_wNAF);
553 val_sub[i] = tmp_points;
554 tmp_points += pre_points_per_block;
557 OPENSSL_free(tmp_wNAF);
562 /* All points we precompute now go into a single array 'val'.
563 * 'val_sub[i]' is a pointer to the subarray for the i-th point,
564 * or to a subarray of 'pre_comp->points' if we already have precomputation. */
565 val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
566 if (val == NULL) goto err;
567 val[num_val] = NULL; /* pivot element */
569 /* allocate points for precomputation */
571 for (i = 0; i < num + num_scalar; i++)
574 for (j = 0; j < (1u << (wsize[i] - 1)); j++)
576 *v = EC_POINT_new(group);
577 if (*v == NULL) goto err;
581 if (!(v == val + num_val))
583 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
587 if (!(tmp = EC_POINT_new(group)))
590 /* prepare precomputed values:
591 * val_sub[i][0] := points[i]
592 * val_sub[i][1] := 3 * points[i]
593 * val_sub[i][2] := 5 * points[i]
596 for (i = 0; i < num + num_scalar; i++)
600 if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
604 if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
609 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
610 for (j = 1; j < (1u << (wsize[i] - 1)); j++)
612 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
617 #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
618 if (!EC_POINTs_make_affine(group, num_val, val, ctx))
622 r_is_at_infinity = 1;
624 for (k = max_len - 1; k >= 0; k--)
626 if (!r_is_at_infinity)
628 if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
631 for (i = 0; i < totalnum; i++)
633 if (wNAF_len[i] > (size_t)k)
635 int digit = wNAF[i][k];
645 if (is_neg != r_is_inverted)
647 if (!r_is_at_infinity)
649 if (!EC_POINT_invert(group, r, ctx)) goto err;
651 r_is_inverted = !r_is_inverted;
656 if (r_is_at_infinity)
658 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) goto err;
659 r_is_at_infinity = 0;
663 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) goto err;
670 if (r_is_at_infinity)
672 if (!EC_POINT_set_to_infinity(group, r)) goto err;
677 if (!EC_POINT_invert(group, r, ctx)) goto err;
684 BN_CTX_free(new_ctx);
689 if (wNAF_len != NULL)
690 OPENSSL_free(wNAF_len);
695 for (w = wNAF; *w != NULL; w++)
702 for (v = val; *v != NULL; v++)
703 EC_POINT_clear_free(*v);
709 OPENSSL_free(val_sub);
715 /* ec_wNAF_precompute_mult()
716 * creates an EC_PRE_COMP object with preprecomputed multiples of the generator
717 * for use with wNAF splitting as implemented in ec_wNAF_mul().
719 * 'pre_comp->points' is an array of multiples of the generator
720 * of the following form:
721 * points[0] = generator;
722 * points[1] = 3 * generator;
724 * points[2^(w-1)-1] = (2^(w-1)-1) * generator;
725 * points[2^(w-1)] = 2^blocksize * generator;
726 * points[2^(w-1)+1] = 3 * 2^blocksize * generator;
728 * points[2^(w-1)*(numblocks-1)-1] = (2^(w-1)) * 2^(blocksize*(numblocks-2)) * generator
729 * points[2^(w-1)*(numblocks-1)] = 2^(blocksize*(numblocks-1)) * generator
731 * points[2^(w-1)*numblocks-1] = (2^(w-1)) * 2^(blocksize*(numblocks-1)) * generator
732 * points[2^(w-1)*numblocks] = NULL
734 int ec_wNAF_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
736 const EC_POINT *generator;
737 EC_POINT *tmp_point = NULL, *base = NULL, **var;
738 BN_CTX *new_ctx = NULL;
740 size_t i, bits, w, pre_points_per_block, blocksize, numblocks, num;
741 EC_POINT **points = NULL;
742 EC_PRE_COMP *pre_comp;
745 /* if there is an old EC_PRE_COMP object, throw it away */
746 EC_GROUP_free_extra_data(group, ec_pre_comp_dup, ec_pre_comp_free, ec_pre_comp_clear_free);
748 if ((pre_comp = ec_pre_comp_new(group)) == NULL)
751 generator = EC_GROUP_get0_generator(group);
752 if (generator == NULL)
754 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
760 ctx = new_ctx = BN_CTX_new();
766 order = BN_CTX_get(ctx);
767 if (order == NULL) goto err;
769 if (!EC_GROUP_get_order(group, order, ctx)) goto err;
770 if (BN_is_zero(order))
772 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
776 bits = BN_num_bits(order);
777 /* The following parameters mean we precompute (approximately)
780 * TBD: The combination 8, 4 is perfect for 160 bits; for other
781 * bit lengths, other parameter combinations might provide better
786 if (EC_window_bits_for_scalar_size(bits) > w)
788 /* let's not make the window too small ... */
789 w = EC_window_bits_for_scalar_size(bits);
792 numblocks = (bits + blocksize - 1) / blocksize; /* max. number of blocks to use for wNAF splitting */
794 pre_points_per_block = 1u << (w - 1);
795 num = pre_points_per_block * numblocks; /* number of points to compute and store */
797 points = OPENSSL_malloc(sizeof (EC_POINT*)*(num + 1));
800 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_MALLOC_FAILURE);
805 var[num] = NULL; /* pivot */
806 for (i = 0; i < num; i++)
808 if ((var[i] = EC_POINT_new(group)) == NULL)
810 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_MALLOC_FAILURE);
815 if (!(tmp_point = EC_POINT_new(group)) || !(base = EC_POINT_new(group)))
817 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_MALLOC_FAILURE);
821 if (!EC_POINT_copy(base, generator))
824 /* do the precomputation */
825 for (i = 0; i < numblocks; i++)
829 if (!EC_POINT_dbl(group, tmp_point, base, ctx))
832 if (!EC_POINT_copy(*var++, base))
835 for (j = 1; j < pre_points_per_block; j++, var++)
837 /* calculate odd multiples of the current base point */
838 if (!EC_POINT_add(group, *var, tmp_point, *(var - 1), ctx))
842 if (i < numblocks - 1)
844 /* get the next base (multiply current one by 2^blocksize) */
849 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_INTERNAL_ERROR);
853 if (!EC_POINT_dbl(group, base, tmp_point, ctx))
855 for (k = 2; k < blocksize; k++)
857 if (!EC_POINT_dbl(group,base,base,ctx))
863 if (!EC_POINTs_make_affine(group, num, points, ctx))
866 pre_comp->group = group;
867 pre_comp->blocksize = blocksize;
868 pre_comp->numblocks = numblocks;
870 pre_comp->points = points;
874 if (!EC_GROUP_set_extra_data(group, pre_comp,
875 ec_pre_comp_dup, ec_pre_comp_free, ec_pre_comp_clear_free))
883 BN_CTX_free(new_ctx);
885 ec_pre_comp_free(pre_comp);
890 for (p = points; *p != NULL; p++)
892 OPENSSL_free(points);
895 EC_POINT_free(tmp_point);
902 int ec_wNAF_have_precompute_mult(const EC_GROUP *group)
904 if (EC_GROUP_get_extra_data(group, ec_pre_comp_dup, ec_pre_comp_free, ec_pre_comp_clear_free) != NULL)