2 * Copyright 2001-2017 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved
5 * Licensed under the OpenSSL license (the "License"). You may not use
6 * this file except in compliance with the License. You can obtain a copy
7 * in the file LICENSE in the source distribution or at
8 * https://www.openssl.org/source/license.html
12 #include <openssl/err.h>
14 #include "internal/cryptlib.h"
15 #include "internal/bn_int.h"
17 #include "internal/refcount.h"
20 * This file implements the wNAF-based interleaving multi-exponentiation method
22 * http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp
23 * You might now find it here:
24 * http://link.springer.com/chapter/10.1007%2F3-540-45537-X_13
25 * http://www.bmoeller.de/pdf/TI-01-08.multiexp.pdf
26 * For multiplication with precomputation, we use wNAF splitting, formerly at:
27 * http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#fastexp
30 /* structure for precomputed multiples of the generator */
31 struct ec_pre_comp_st {
32 const EC_GROUP *group; /* parent EC_GROUP object */
33 size_t blocksize; /* block size for wNAF splitting */
34 size_t numblocks; /* max. number of blocks for which we have
36 size_t w; /* window size */
37 EC_POINT **points; /* array with pre-calculated multiples of
38 * generator: 'num' pointers to EC_POINT
39 * objects followed by a NULL */
40 size_t num; /* numblocks * 2^(w-1) */
41 CRYPTO_REF_COUNT references;
45 static EC_PRE_COMP *ec_pre_comp_new(const EC_GROUP *group)
47 EC_PRE_COMP *ret = NULL;
52 ret = OPENSSL_zalloc(sizeof(*ret));
54 ECerr(EC_F_EC_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
59 ret->blocksize = 8; /* default */
60 ret->w = 4; /* default */
63 ret->lock = CRYPTO_THREAD_lock_new();
64 if (ret->lock == NULL) {
65 ECerr(EC_F_EC_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
72 EC_PRE_COMP *EC_ec_pre_comp_dup(EC_PRE_COMP *pre)
76 CRYPTO_UP_REF(&pre->references, &i, pre->lock);
80 void EC_ec_pre_comp_free(EC_PRE_COMP *pre)
87 CRYPTO_DOWN_REF(&pre->references, &i, pre->lock);
88 REF_PRINT_COUNT("EC_ec", pre);
91 REF_ASSERT_ISNT(i < 0);
93 if (pre->points != NULL) {
96 for (pts = pre->points; *pts != NULL; pts++)
98 OPENSSL_free(pre->points);
100 CRYPTO_THREAD_lock_free(pre->lock);
104 #define EC_POINT_BN_set_flags(P, flags) do { \
105 BN_set_flags((P)->X, (flags)); \
106 BN_set_flags((P)->Y, (flags)); \
107 BN_set_flags((P)->Z, (flags)); \
111 * This functions computes (in constant time) a point multiplication over the
114 * It performs either a fixed scalar point multiplication
115 * (scalar * generator)
116 * when point is NULL, or a generic scalar point multiplication
118 * when point is not NULL.
120 * scalar should be in the range [0,n) otherwise all constant time bets are off.
122 * NB: This says nothing about EC_POINT_add and EC_POINT_dbl,
123 * which of course are not constant time themselves.
125 * The product is stored in r.
127 * Returns 1 on success, 0 otherwise.
129 static int ec_mul_consttime(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
130 const EC_POINT *point, BN_CTX *ctx)
132 int i, order_bits, group_top, kbit, pbit, Z_is_one;
135 BIGNUM *lambda = NULL;
136 BN_CTX *new_ctx = NULL;
139 if (ctx == NULL && (ctx = new_ctx = BN_CTX_secure_new()) == NULL)
142 if ((group->order == NULL) || (group->field == NULL))
145 order_bits = BN_num_bits(group->order);
147 s = EC_POINT_new(group);
152 if (group->generator == NULL)
154 if (!EC_POINT_copy(s, group->generator))
157 if (!EC_POINT_copy(s, point))
161 EC_POINT_BN_set_flags(s, BN_FLG_CONSTTIME);
164 lambda = BN_CTX_get(ctx);
170 * Group orders are often on a word boundary.
171 * So when we pad the scalar, some timing diff might
172 * pop if it needs to be expanded due to carries.
173 * So expand ahead of time.
175 group_top = bn_get_top(group->order);
176 if ((bn_wexpand(k, group_top + 1) == NULL)
177 || (bn_wexpand(lambda, group_top + 1) == NULL))
180 if (!BN_copy(k, scalar))
183 BN_set_flags(k, BN_FLG_CONSTTIME);
185 if ((BN_num_bits(k) > order_bits) || (BN_is_negative(k))) {
187 * this is an unusual input, and we don't guarantee
190 if(!BN_nnmod(k, k, group->order, ctx))
194 if (!BN_add(lambda, k, group->order))
196 BN_set_flags(lambda, BN_FLG_CONSTTIME);
197 if (!BN_add(k, lambda, group->order))
200 * lambda := scalar + order
201 * k := scalar + 2*order
203 kbit = BN_is_bit_set(lambda, order_bits);
204 BN_consttime_swap(kbit, k, lambda, group_top + 1);
206 group_top = bn_get_top(group->field);
207 if ((bn_wexpand(s->X, group_top) == NULL)
208 || (bn_wexpand(s->Y, group_top) == NULL)
209 || (bn_wexpand(s->Z, group_top) == NULL)
210 || (bn_wexpand(r->X, group_top) == NULL)
211 || (bn_wexpand(r->Y, group_top) == NULL)
212 || (bn_wexpand(r->Z, group_top) == NULL))
215 /* top bit is a 1, in a fixed pos */
216 if (!EC_POINT_copy(r, s))
219 EC_POINT_BN_set_flags(r, BN_FLG_CONSTTIME);
221 if (!EC_POINT_dbl(group, s, s, ctx))
226 #define EC_POINT_CSWAP(c, a, b, w, t) do { \
227 BN_consttime_swap(c, (a)->X, (b)->X, w); \
228 BN_consttime_swap(c, (a)->Y, (b)->Y, w); \
229 BN_consttime_swap(c, (a)->Z, (b)->Z, w); \
230 t = ((a)->Z_is_one ^ (b)->Z_is_one) & (c); \
231 (a)->Z_is_one ^= (t); \
232 (b)->Z_is_one ^= (t); \
235 for (i = order_bits - 1; i >= 0; i--) {
236 kbit = BN_is_bit_set(k, i) ^ pbit;
237 EC_POINT_CSWAP(kbit, r, s, group_top, Z_is_one);
238 if (!EC_POINT_add(group, s, r, s, ctx))
240 if (!EC_POINT_dbl(group, r, r, ctx))
243 * pbit logic merges this cswap with that of the
248 /* one final cswap to move the right value into r */
249 EC_POINT_CSWAP(pbit, r, s, group_top, Z_is_one);
250 #undef EC_POINT_CSWAP
257 BN_CTX_free(new_ctx);
261 #undef EC_POINT_BN_set_flags
264 * TODO: table should be optimised for the wNAF-based implementation,
265 * sometimes smaller windows will give better performance (thus the
266 * boundaries should be increased)
268 #define EC_window_bits_for_scalar_size(b) \
279 * \sum scalars[i]*points[i],
282 * in the addition if scalar != NULL
284 int ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
285 size_t num, const EC_POINT *points[], const BIGNUM *scalars[],
288 BN_CTX *new_ctx = NULL;
289 const EC_POINT *generator = NULL;
290 EC_POINT *tmp = NULL;
292 size_t blocksize = 0, numblocks = 0; /* for wNAF splitting */
293 size_t pre_points_per_block = 0;
296 int r_is_inverted = 0;
297 int r_is_at_infinity = 1;
298 size_t *wsize = NULL; /* individual window sizes */
299 signed char **wNAF = NULL; /* individual wNAFs */
300 size_t *wNAF_len = NULL;
303 EC_POINT **val = NULL; /* precomputation */
305 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' or
306 * 'pre_comp->points' */
307 const EC_PRE_COMP *pre_comp = NULL;
308 int num_scalar = 0; /* flag: will be set to 1 if 'scalar' must be
309 * treated like other scalars, i.e.
310 * precomputation is not available */
313 /* Handle the common cases where the scalar is secret, enforcing a
314 * constant time scalar multiplication algorithm.
316 if ((scalar != NULL) && (num == 0)) {
317 /* In this case we want to compute scalar * GeneratorPoint:
318 * this codepath is reached most prominently by (ephemeral) key
319 * generation of EC cryptosystems (i.e. ECDSA keygen and sign setup,
320 * ECDH keygen/first half), where the scalar is always secret.
321 * This is why we ignore if BN_FLG_CONSTTIME is actually set and we
322 * always call the constant time version.
324 return ec_mul_consttime(group, r, scalar, NULL, ctx);
326 if ((scalar == NULL) && (num == 1)) {
327 /* In this case we want to compute scalar * GenericPoint:
328 * this codepath is reached most prominently by the second half of
329 * ECDH, where the secret scalar is multiplied by the peer's public
331 * To protect the secret scalar, we ignore if BN_FLG_CONSTTIME is
332 * actually set and we always call the constant time version.
334 return ec_mul_consttime(group, r, scalars[0], points[0], ctx);
338 if (group->meth != r->meth) {
339 ECerr(EC_F_EC_WNAF_MUL, EC_R_INCOMPATIBLE_OBJECTS);
343 if ((scalar == NULL) && (num == 0)) {
344 return EC_POINT_set_to_infinity(group, r);
347 for (i = 0; i < num; i++) {
348 if (group->meth != points[i]->meth) {
349 ECerr(EC_F_EC_WNAF_MUL, EC_R_INCOMPATIBLE_OBJECTS);
355 ctx = new_ctx = BN_CTX_new();
360 if (scalar != NULL) {
361 generator = EC_GROUP_get0_generator(group);
362 if (generator == NULL) {
363 ECerr(EC_F_EC_WNAF_MUL, EC_R_UNDEFINED_GENERATOR);
367 /* look if we can use precomputed multiples of generator */
369 pre_comp = group->pre_comp.ec;
370 if (pre_comp && pre_comp->numblocks
371 && (EC_POINT_cmp(group, generator, pre_comp->points[0], ctx) ==
373 blocksize = pre_comp->blocksize;
376 * determine maximum number of blocks that wNAF splitting may
377 * yield (NB: maximum wNAF length is bit length plus one)
379 numblocks = (BN_num_bits(scalar) / blocksize) + 1;
382 * we cannot use more blocks than we have precomputation for
384 if (numblocks > pre_comp->numblocks)
385 numblocks = pre_comp->numblocks;
387 pre_points_per_block = (size_t)1 << (pre_comp->w - 1);
389 /* check that pre_comp looks sane */
390 if (pre_comp->num != (pre_comp->numblocks * pre_points_per_block)) {
391 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
395 /* can't use precomputation */
398 num_scalar = 1; /* treat 'scalar' like 'num'-th element of
403 totalnum = num + numblocks;
405 wsize = OPENSSL_malloc(totalnum * sizeof(wsize[0]));
406 wNAF_len = OPENSSL_malloc(totalnum * sizeof(wNAF_len[0]));
407 /* include space for pivot */
408 wNAF = OPENSSL_malloc((totalnum + 1) * sizeof(wNAF[0]));
409 val_sub = OPENSSL_malloc(totalnum * sizeof(val_sub[0]));
411 /* Ensure wNAF is initialised in case we end up going to err */
413 wNAF[0] = NULL; /* preliminary pivot */
415 if (wsize == NULL || wNAF_len == NULL || wNAF == NULL || val_sub == NULL) {
416 ECerr(EC_F_EC_WNAF_MUL, ERR_R_MALLOC_FAILURE);
421 * num_val will be the total number of temporarily precomputed points
425 for (i = 0; i < num + num_scalar; i++) {
428 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
429 wsize[i] = EC_window_bits_for_scalar_size(bits);
430 num_val += (size_t)1 << (wsize[i] - 1);
431 wNAF[i + 1] = NULL; /* make sure we always have a pivot */
433 bn_compute_wNAF((i < num ? scalars[i] : scalar), wsize[i],
437 if (wNAF_len[i] > max_len)
438 max_len = wNAF_len[i];
442 /* we go here iff scalar != NULL */
444 if (pre_comp == NULL) {
445 if (num_scalar != 1) {
446 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
449 /* we have already generated a wNAF for 'scalar' */
451 signed char *tmp_wNAF = NULL;
454 if (num_scalar != 0) {
455 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
460 * use the window size for which we have precomputation
462 wsize[num] = pre_comp->w;
463 tmp_wNAF = bn_compute_wNAF(scalar, wsize[num], &tmp_len);
467 if (tmp_len <= max_len) {
469 * One of the other wNAFs is at least as long as the wNAF
470 * belonging to the generator, so wNAF splitting will not buy
475 totalnum = num + 1; /* don't use wNAF splitting */
476 wNAF[num] = tmp_wNAF;
477 wNAF[num + 1] = NULL;
478 wNAF_len[num] = tmp_len;
480 * pre_comp->points starts with the points that we need here:
482 val_sub[num] = pre_comp->points;
485 * don't include tmp_wNAF directly into wNAF array - use wNAF
486 * splitting and include the blocks
490 EC_POINT **tmp_points;
492 if (tmp_len < numblocks * blocksize) {
494 * possibly we can do with fewer blocks than estimated
496 numblocks = (tmp_len + blocksize - 1) / blocksize;
497 if (numblocks > pre_comp->numblocks) {
498 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
499 OPENSSL_free(tmp_wNAF);
502 totalnum = num + numblocks;
505 /* split wNAF in 'numblocks' parts */
507 tmp_points = pre_comp->points;
509 for (i = num; i < totalnum; i++) {
510 if (i < totalnum - 1) {
511 wNAF_len[i] = blocksize;
512 if (tmp_len < blocksize) {
513 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
514 OPENSSL_free(tmp_wNAF);
517 tmp_len -= blocksize;
520 * last block gets whatever is left (this could be
521 * more or less than 'blocksize'!)
523 wNAF_len[i] = tmp_len;
526 wNAF[i] = OPENSSL_malloc(wNAF_len[i]);
527 if (wNAF[i] == NULL) {
528 ECerr(EC_F_EC_WNAF_MUL, ERR_R_MALLOC_FAILURE);
529 OPENSSL_free(tmp_wNAF);
532 memcpy(wNAF[i], pp, wNAF_len[i]);
533 if (wNAF_len[i] > max_len)
534 max_len = wNAF_len[i];
536 if (*tmp_points == NULL) {
537 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
538 OPENSSL_free(tmp_wNAF);
541 val_sub[i] = tmp_points;
542 tmp_points += pre_points_per_block;
545 OPENSSL_free(tmp_wNAF);
551 * All points we precompute now go into a single array 'val'.
552 * 'val_sub[i]' is a pointer to the subarray for the i-th point, or to a
553 * subarray of 'pre_comp->points' if we already have precomputation.
555 val = OPENSSL_malloc((num_val + 1) * sizeof(val[0]));
557 ECerr(EC_F_EC_WNAF_MUL, ERR_R_MALLOC_FAILURE);
560 val[num_val] = NULL; /* pivot element */
562 /* allocate points for precomputation */
564 for (i = 0; i < num + num_scalar; i++) {
566 for (j = 0; j < ((size_t)1 << (wsize[i] - 1)); j++) {
567 *v = EC_POINT_new(group);
573 if (!(v == val + num_val)) {
574 ECerr(EC_F_EC_WNAF_MUL, ERR_R_INTERNAL_ERROR);
578 if ((tmp = EC_POINT_new(group)) == NULL)
582 * prepare precomputed values:
583 * val_sub[i][0] := points[i]
584 * val_sub[i][1] := 3 * points[i]
585 * val_sub[i][2] := 5 * points[i]
588 for (i = 0; i < num + num_scalar; i++) {
590 if (!EC_POINT_copy(val_sub[i][0], points[i]))
593 if (!EC_POINT_copy(val_sub[i][0], generator))
598 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx))
600 for (j = 1; j < ((size_t)1 << (wsize[i] - 1)); j++) {
602 (group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx))
608 if (!EC_POINTs_make_affine(group, num_val, val, ctx))
611 r_is_at_infinity = 1;
613 for (k = max_len - 1; k >= 0; k--) {
614 if (!r_is_at_infinity) {
615 if (!EC_POINT_dbl(group, r, r, ctx))
619 for (i = 0; i < totalnum; i++) {
620 if (wNAF_len[i] > (size_t)k) {
621 int digit = wNAF[i][k];
630 if (is_neg != r_is_inverted) {
631 if (!r_is_at_infinity) {
632 if (!EC_POINT_invert(group, r, ctx))
635 r_is_inverted = !r_is_inverted;
640 if (r_is_at_infinity) {
641 if (!EC_POINT_copy(r, val_sub[i][digit >> 1]))
643 r_is_at_infinity = 0;
646 (group, r, r, val_sub[i][digit >> 1], ctx))
654 if (r_is_at_infinity) {
655 if (!EC_POINT_set_to_infinity(group, r))
659 if (!EC_POINT_invert(group, r, ctx))
666 BN_CTX_free(new_ctx);
669 OPENSSL_free(wNAF_len);
673 for (w = wNAF; *w != NULL; w++)
679 for (v = val; *v != NULL; v++)
680 EC_POINT_clear_free(*v);
684 OPENSSL_free(val_sub);
689 * ec_wNAF_precompute_mult()
690 * creates an EC_PRE_COMP object with preprecomputed multiples of the generator
691 * for use with wNAF splitting as implemented in ec_wNAF_mul().
693 * 'pre_comp->points' is an array of multiples of the generator
694 * of the following form:
695 * points[0] = generator;
696 * points[1] = 3 * generator;
698 * points[2^(w-1)-1] = (2^(w-1)-1) * generator;
699 * points[2^(w-1)] = 2^blocksize * generator;
700 * points[2^(w-1)+1] = 3 * 2^blocksize * generator;
702 * points[2^(w-1)*(numblocks-1)-1] = (2^(w-1)) * 2^(blocksize*(numblocks-2)) * generator
703 * points[2^(w-1)*(numblocks-1)] = 2^(blocksize*(numblocks-1)) * generator
705 * points[2^(w-1)*numblocks-1] = (2^(w-1)) * 2^(blocksize*(numblocks-1)) * generator
706 * points[2^(w-1)*numblocks] = NULL
708 int ec_wNAF_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
710 const EC_POINT *generator;
711 EC_POINT *tmp_point = NULL, *base = NULL, **var;
712 BN_CTX *new_ctx = NULL;
714 size_t i, bits, w, pre_points_per_block, blocksize, numblocks, num;
715 EC_POINT **points = NULL;
716 EC_PRE_COMP *pre_comp;
719 /* if there is an old EC_PRE_COMP object, throw it away */
720 EC_pre_comp_free(group);
721 if ((pre_comp = ec_pre_comp_new(group)) == NULL)
724 generator = EC_GROUP_get0_generator(group);
725 if (generator == NULL) {
726 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
731 ctx = new_ctx = BN_CTX_new();
738 order = EC_GROUP_get0_order(group);
741 if (BN_is_zero(order)) {
742 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
746 bits = BN_num_bits(order);
748 * The following parameters mean we precompute (approximately) one point
749 * per bit. TBD: The combination 8, 4 is perfect for 160 bits; for other
750 * bit lengths, other parameter combinations might provide better
755 if (EC_window_bits_for_scalar_size(bits) > w) {
756 /* let's not make the window too small ... */
757 w = EC_window_bits_for_scalar_size(bits);
760 numblocks = (bits + blocksize - 1) / blocksize; /* max. number of blocks
764 pre_points_per_block = (size_t)1 << (w - 1);
765 num = pre_points_per_block * numblocks; /* number of points to compute
768 points = OPENSSL_malloc(sizeof(*points) * (num + 1));
769 if (points == NULL) {
770 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_MALLOC_FAILURE);
775 var[num] = NULL; /* pivot */
776 for (i = 0; i < num; i++) {
777 if ((var[i] = EC_POINT_new(group)) == NULL) {
778 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_MALLOC_FAILURE);
783 if ((tmp_point = EC_POINT_new(group)) == NULL
784 || (base = EC_POINT_new(group)) == NULL) {
785 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_MALLOC_FAILURE);
789 if (!EC_POINT_copy(base, generator))
792 /* do the precomputation */
793 for (i = 0; i < numblocks; i++) {
796 if (!EC_POINT_dbl(group, tmp_point, base, ctx))
799 if (!EC_POINT_copy(*var++, base))
802 for (j = 1; j < pre_points_per_block; j++, var++) {
804 * calculate odd multiples of the current base point
806 if (!EC_POINT_add(group, *var, tmp_point, *(var - 1), ctx))
810 if (i < numblocks - 1) {
812 * get the next base (multiply current one by 2^blocksize)
816 if (blocksize <= 2) {
817 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT, ERR_R_INTERNAL_ERROR);
821 if (!EC_POINT_dbl(group, base, tmp_point, ctx))
823 for (k = 2; k < blocksize; k++) {
824 if (!EC_POINT_dbl(group, base, base, ctx))
830 if (!EC_POINTs_make_affine(group, num, points, ctx))
833 pre_comp->group = group;
834 pre_comp->blocksize = blocksize;
835 pre_comp->numblocks = numblocks;
837 pre_comp->points = points;
840 SETPRECOMP(group, ec, pre_comp);
847 BN_CTX_free(new_ctx);
848 EC_ec_pre_comp_free(pre_comp);
852 for (p = points; *p != NULL; p++)
854 OPENSSL_free(points);
856 EC_POINT_free(tmp_point);
861 int ec_wNAF_have_precompute_mult(const EC_GROUP *group)
863 return HAVEPRECOMP(group, ec);