2 * Copyright 2001-2016 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* ====================================================================
11 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
12 * Portions of this software developed by SUN MICROSYSTEMS, INC.,
13 * and contributed to the OpenSSL project.
17 #include <openssl/err.h>
19 #include "internal/cryptlib.h"
20 #include "internal/bn_int.h"
24 * This file implements the wNAF-based interleaving multi-exponentiation method
26 * http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp
27 * You might now find it here:
28 * http://link.springer.com/chapter/10.1007%2F3-540-45537-X_13
29 * http://www.bmoeller.de/pdf/TI-01-08.multiexp.pdf
30 * For multiplication with precomputation, we use wNAF splitting, formerly at:
31 * http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#fastexp
34 /* structure for precomputed multiples of the generator */
35 struct ec_pre_comp_st
{
36 const EC_GROUP
*group
; /* parent EC_GROUP object */
37 size_t blocksize
; /* block size for wNAF splitting */
38 size_t numblocks
; /* max. number of blocks for which we have
40 size_t w
; /* window size */
41 EC_POINT
**points
; /* array with pre-calculated multiples of
42 * generator: 'num' pointers to EC_POINT
43 * objects followed by a NULL */
44 size_t num
; /* numblocks * 2^(w-1) */
45 CRYPTO_REF_COUNT references
;
49 static EC_PRE_COMP
*ec_pre_comp_new(const EC_GROUP
*group
)
51 EC_PRE_COMP
*ret
= NULL
;
56 ret
= OPENSSL_zalloc(sizeof(*ret
));
58 ECerr(EC_F_EC_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
63 ret
->blocksize
= 8; /* default */
64 ret
->w
= 4; /* default */
67 ret
->lock
= CRYPTO_THREAD_lock_new();
68 if (ret
->lock
== NULL
) {
69 ECerr(EC_F_EC_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
76 EC_PRE_COMP
*EC_ec_pre_comp_dup(EC_PRE_COMP
*pre
)
80 CRYPTO_UP_REF(&pre
->references
, &i
, pre
->lock
);
84 void EC_ec_pre_comp_free(EC_PRE_COMP
*pre
)
91 CRYPTO_DOWN_REF(&pre
->references
, &i
, pre
->lock
);
92 REF_PRINT_COUNT("EC_ec", pre
);
95 REF_ASSERT_ISNT(i
< 0);
97 if (pre
->points
!= NULL
) {
100 for (pts
= pre
->points
; *pts
!= NULL
; pts
++)
102 OPENSSL_free(pre
->points
);
104 CRYPTO_THREAD_lock_free(pre
->lock
);
109 * TODO: table should be optimised for the wNAF-based implementation,
110 * sometimes smaller windows will give better performance (thus the
111 * boundaries should be increased)
113 #define EC_window_bits_for_scalar_size(b) \
124 * \sum scalars[i]*points[i],
127 * in the addition if scalar != NULL
129 int ec_wNAF_mul(const EC_GROUP
*group
, EC_POINT
*r
, const BIGNUM
*scalar
,
130 size_t num
, const EC_POINT
*points
[], const BIGNUM
*scalars
[],
133 BN_CTX
*new_ctx
= NULL
;
134 const EC_POINT
*generator
= NULL
;
135 EC_POINT
*tmp
= NULL
;
137 size_t blocksize
= 0, numblocks
= 0; /* for wNAF splitting */
138 size_t pre_points_per_block
= 0;
141 int r_is_inverted
= 0;
142 int r_is_at_infinity
= 1;
143 size_t *wsize
= NULL
; /* individual window sizes */
144 signed char **wNAF
= NULL
; /* individual wNAFs */
145 size_t *wNAF_len
= NULL
;
148 EC_POINT
**val
= NULL
; /* precomputation */
150 EC_POINT
***val_sub
= NULL
; /* pointers to sub-arrays of 'val' or
151 * 'pre_comp->points' */
152 const EC_PRE_COMP
*pre_comp
= NULL
;
153 int num_scalar
= 0; /* flag: will be set to 1 if 'scalar' must be
154 * treated like other scalars, i.e.
155 * precomputation is not available */
158 if (group
->meth
!= r
->meth
) {
159 ECerr(EC_F_EC_WNAF_MUL
, EC_R_INCOMPATIBLE_OBJECTS
);
163 if ((scalar
== NULL
) && (num
== 0)) {
164 return EC_POINT_set_to_infinity(group
, r
);
167 for (i
= 0; i
< num
; i
++) {
168 if (group
->meth
!= points
[i
]->meth
) {
169 ECerr(EC_F_EC_WNAF_MUL
, EC_R_INCOMPATIBLE_OBJECTS
);
175 ctx
= new_ctx
= BN_CTX_new();
180 if (scalar
!= NULL
) {
181 generator
= EC_GROUP_get0_generator(group
);
182 if (generator
== NULL
) {
183 ECerr(EC_F_EC_WNAF_MUL
, EC_R_UNDEFINED_GENERATOR
);
187 /* look if we can use precomputed multiples of generator */
189 pre_comp
= group
->pre_comp
.ec
;
190 if (pre_comp
&& pre_comp
->numblocks
191 && (EC_POINT_cmp(group
, generator
, pre_comp
->points
[0], ctx
) ==
193 blocksize
= pre_comp
->blocksize
;
196 * determine maximum number of blocks that wNAF splitting may
197 * yield (NB: maximum wNAF length is bit length plus one)
199 numblocks
= (BN_num_bits(scalar
) / blocksize
) + 1;
202 * we cannot use more blocks than we have precomputation for
204 if (numblocks
> pre_comp
->numblocks
)
205 numblocks
= pre_comp
->numblocks
;
207 pre_points_per_block
= (size_t)1 << (pre_comp
->w
- 1);
209 /* check that pre_comp looks sane */
210 if (pre_comp
->num
!= (pre_comp
->numblocks
* pre_points_per_block
)) {
211 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
215 /* can't use precomputation */
218 num_scalar
= 1; /* treat 'scalar' like 'num'-th element of
223 totalnum
= num
+ numblocks
;
225 wsize
= OPENSSL_malloc(totalnum
* sizeof wsize
[0]);
226 wNAF_len
= OPENSSL_malloc(totalnum
* sizeof wNAF_len
[0]);
227 wNAF
= OPENSSL_malloc((totalnum
+ 1) * sizeof wNAF
[0]); /* includes space
229 val_sub
= OPENSSL_malloc(totalnum
* sizeof val_sub
[0]);
231 /* Ensure wNAF is initialised in case we end up going to err */
233 wNAF
[0] = NULL
; /* preliminary pivot */
235 if (wsize
== NULL
|| wNAF_len
== NULL
|| wNAF
== NULL
|| val_sub
== NULL
) {
236 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_MALLOC_FAILURE
);
241 * num_val will be the total number of temporarily precomputed points
245 for (i
= 0; i
< num
+ num_scalar
; i
++) {
248 bits
= i
< num
? BN_num_bits(scalars
[i
]) : BN_num_bits(scalar
);
249 wsize
[i
] = EC_window_bits_for_scalar_size(bits
);
250 num_val
+= (size_t)1 << (wsize
[i
] - 1);
251 wNAF
[i
+ 1] = NULL
; /* make sure we always have a pivot */
253 bn_compute_wNAF((i
< num
? scalars
[i
] : scalar
), wsize
[i
],
257 if (wNAF_len
[i
] > max_len
)
258 max_len
= wNAF_len
[i
];
262 /* we go here iff scalar != NULL */
264 if (pre_comp
== NULL
) {
265 if (num_scalar
!= 1) {
266 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
269 /* we have already generated a wNAF for 'scalar' */
271 signed char *tmp_wNAF
= NULL
;
274 if (num_scalar
!= 0) {
275 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
280 * use the window size for which we have precomputation
282 wsize
[num
] = pre_comp
->w
;
283 tmp_wNAF
= bn_compute_wNAF(scalar
, wsize
[num
], &tmp_len
);
287 if (tmp_len
<= max_len
) {
289 * One of the other wNAFs is at least as long as the wNAF
290 * belonging to the generator, so wNAF splitting will not buy
295 totalnum
= num
+ 1; /* don't use wNAF splitting */
296 wNAF
[num
] = tmp_wNAF
;
297 wNAF
[num
+ 1] = NULL
;
298 wNAF_len
[num
] = tmp_len
;
300 * pre_comp->points starts with the points that we need here:
302 val_sub
[num
] = pre_comp
->points
;
305 * don't include tmp_wNAF directly into wNAF array - use wNAF
306 * splitting and include the blocks
310 EC_POINT
**tmp_points
;
312 if (tmp_len
< numblocks
* blocksize
) {
314 * possibly we can do with fewer blocks than estimated
316 numblocks
= (tmp_len
+ blocksize
- 1) / blocksize
;
317 if (numblocks
> pre_comp
->numblocks
) {
318 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
319 OPENSSL_free(tmp_wNAF
);
322 totalnum
= num
+ numblocks
;
325 /* split wNAF in 'numblocks' parts */
327 tmp_points
= pre_comp
->points
;
329 for (i
= num
; i
< totalnum
; i
++) {
330 if (i
< totalnum
- 1) {
331 wNAF_len
[i
] = blocksize
;
332 if (tmp_len
< blocksize
) {
333 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
334 OPENSSL_free(tmp_wNAF
);
337 tmp_len
-= blocksize
;
340 * last block gets whatever is left (this could be
341 * more or less than 'blocksize'!)
343 wNAF_len
[i
] = tmp_len
;
346 wNAF
[i
] = OPENSSL_malloc(wNAF_len
[i
]);
347 if (wNAF
[i
] == NULL
) {
348 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_MALLOC_FAILURE
);
349 OPENSSL_free(tmp_wNAF
);
352 memcpy(wNAF
[i
], pp
, wNAF_len
[i
]);
353 if (wNAF_len
[i
] > max_len
)
354 max_len
= wNAF_len
[i
];
356 if (*tmp_points
== NULL
) {
357 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
358 OPENSSL_free(tmp_wNAF
);
361 val_sub
[i
] = tmp_points
;
362 tmp_points
+= pre_points_per_block
;
365 OPENSSL_free(tmp_wNAF
);
371 * All points we precompute now go into a single array 'val'.
372 * 'val_sub[i]' is a pointer to the subarray for the i-th point, or to a
373 * subarray of 'pre_comp->points' if we already have precomputation.
375 val
= OPENSSL_malloc((num_val
+ 1) * sizeof val
[0]);
377 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_MALLOC_FAILURE
);
380 val
[num_val
] = NULL
; /* pivot element */
382 /* allocate points for precomputation */
384 for (i
= 0; i
< num
+ num_scalar
; i
++) {
386 for (j
= 0; j
< ((size_t)1 << (wsize
[i
] - 1)); j
++) {
387 *v
= EC_POINT_new(group
);
393 if (!(v
== val
+ num_val
)) {
394 ECerr(EC_F_EC_WNAF_MUL
, ERR_R_INTERNAL_ERROR
);
398 if ((tmp
= EC_POINT_new(group
)) == NULL
)
402 * prepare precomputed values:
403 * val_sub[i][0] := points[i]
404 * val_sub[i][1] := 3 * points[i]
405 * val_sub[i][2] := 5 * points[i]
408 for (i
= 0; i
< num
+ num_scalar
; i
++) {
410 if (!EC_POINT_copy(val_sub
[i
][0], points
[i
]))
413 if (!EC_POINT_copy(val_sub
[i
][0], generator
))
418 if (!EC_POINT_dbl(group
, tmp
, val_sub
[i
][0], ctx
))
420 for (j
= 1; j
< ((size_t)1 << (wsize
[i
] - 1)); j
++) {
422 (group
, val_sub
[i
][j
], val_sub
[i
][j
- 1], tmp
, ctx
))
428 if (!EC_POINTs_make_affine(group
, num_val
, val
, ctx
))
431 r_is_at_infinity
= 1;
433 for (k
= max_len
- 1; k
>= 0; k
--) {
434 if (!r_is_at_infinity
) {
435 if (!EC_POINT_dbl(group
, r
, r
, ctx
))
439 for (i
= 0; i
< totalnum
; i
++) {
440 if (wNAF_len
[i
] > (size_t)k
) {
441 int digit
= wNAF
[i
][k
];
450 if (is_neg
!= r_is_inverted
) {
451 if (!r_is_at_infinity
) {
452 if (!EC_POINT_invert(group
, r
, ctx
))
455 r_is_inverted
= !r_is_inverted
;
460 if (r_is_at_infinity
) {
461 if (!EC_POINT_copy(r
, val_sub
[i
][digit
>> 1]))
463 r_is_at_infinity
= 0;
466 (group
, r
, r
, val_sub
[i
][digit
>> 1], ctx
))
474 if (r_is_at_infinity
) {
475 if (!EC_POINT_set_to_infinity(group
, r
))
479 if (!EC_POINT_invert(group
, r
, ctx
))
486 BN_CTX_free(new_ctx
);
489 OPENSSL_free(wNAF_len
);
493 for (w
= wNAF
; *w
!= NULL
; w
++)
499 for (v
= val
; *v
!= NULL
; v
++)
500 EC_POINT_clear_free(*v
);
504 OPENSSL_free(val_sub
);
509 * ec_wNAF_precompute_mult()
510 * creates an EC_PRE_COMP object with preprecomputed multiples of the generator
511 * for use with wNAF splitting as implemented in ec_wNAF_mul().
513 * 'pre_comp->points' is an array of multiples of the generator
514 * of the following form:
515 * points[0] = generator;
516 * points[1] = 3 * generator;
518 * points[2^(w-1)-1] = (2^(w-1)-1) * generator;
519 * points[2^(w-1)] = 2^blocksize * generator;
520 * points[2^(w-1)+1] = 3 * 2^blocksize * generator;
522 * points[2^(w-1)*(numblocks-1)-1] = (2^(w-1)) * 2^(blocksize*(numblocks-2)) * generator
523 * points[2^(w-1)*(numblocks-1)] = 2^(blocksize*(numblocks-1)) * generator
525 * points[2^(w-1)*numblocks-1] = (2^(w-1)) * 2^(blocksize*(numblocks-1)) * generator
526 * points[2^(w-1)*numblocks] = NULL
528 int ec_wNAF_precompute_mult(EC_GROUP
*group
, BN_CTX
*ctx
)
530 const EC_POINT
*generator
;
531 EC_POINT
*tmp_point
= NULL
, *base
= NULL
, **var
;
532 BN_CTX
*new_ctx
= NULL
;
534 size_t i
, bits
, w
, pre_points_per_block
, blocksize
, numblocks
, num
;
535 EC_POINT
**points
= NULL
;
536 EC_PRE_COMP
*pre_comp
;
539 /* if there is an old EC_PRE_COMP object, throw it away */
540 EC_pre_comp_free(group
);
541 if ((pre_comp
= ec_pre_comp_new(group
)) == NULL
)
544 generator
= EC_GROUP_get0_generator(group
);
545 if (generator
== NULL
) {
546 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT
, EC_R_UNDEFINED_GENERATOR
);
551 ctx
= new_ctx
= BN_CTX_new();
558 order
= EC_GROUP_get0_order(group
);
561 if (BN_is_zero(order
)) {
562 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT
, EC_R_UNKNOWN_ORDER
);
566 bits
= BN_num_bits(order
);
568 * The following parameters mean we precompute (approximately) one point
569 * per bit. TBD: The combination 8, 4 is perfect for 160 bits; for other
570 * bit lengths, other parameter combinations might provide better
575 if (EC_window_bits_for_scalar_size(bits
) > w
) {
576 /* let's not make the window too small ... */
577 w
= EC_window_bits_for_scalar_size(bits
);
580 numblocks
= (bits
+ blocksize
- 1) / blocksize
; /* max. number of blocks
584 pre_points_per_block
= (size_t)1 << (w
- 1);
585 num
= pre_points_per_block
* numblocks
; /* number of points to compute
588 points
= OPENSSL_malloc(sizeof(*points
) * (num
+ 1));
589 if (points
== NULL
) {
590 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT
, ERR_R_MALLOC_FAILURE
);
595 var
[num
] = NULL
; /* pivot */
596 for (i
= 0; i
< num
; i
++) {
597 if ((var
[i
] = EC_POINT_new(group
)) == NULL
) {
598 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT
, ERR_R_MALLOC_FAILURE
);
603 if ((tmp_point
= EC_POINT_new(group
)) == NULL
604 || (base
= EC_POINT_new(group
)) == NULL
) {
605 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT
, ERR_R_MALLOC_FAILURE
);
609 if (!EC_POINT_copy(base
, generator
))
612 /* do the precomputation */
613 for (i
= 0; i
< numblocks
; i
++) {
616 if (!EC_POINT_dbl(group
, tmp_point
, base
, ctx
))
619 if (!EC_POINT_copy(*var
++, base
))
622 for (j
= 1; j
< pre_points_per_block
; j
++, var
++) {
624 * calculate odd multiples of the current base point
626 if (!EC_POINT_add(group
, *var
, tmp_point
, *(var
- 1), ctx
))
630 if (i
< numblocks
- 1) {
632 * get the next base (multiply current one by 2^blocksize)
636 if (blocksize
<= 2) {
637 ECerr(EC_F_EC_WNAF_PRECOMPUTE_MULT
, ERR_R_INTERNAL_ERROR
);
641 if (!EC_POINT_dbl(group
, base
, tmp_point
, ctx
))
643 for (k
= 2; k
< blocksize
; k
++) {
644 if (!EC_POINT_dbl(group
, base
, base
, ctx
))
650 if (!EC_POINTs_make_affine(group
, num
, points
, ctx
))
653 pre_comp
->group
= group
;
654 pre_comp
->blocksize
= blocksize
;
655 pre_comp
->numblocks
= numblocks
;
657 pre_comp
->points
= points
;
660 SETPRECOMP(group
, ec
, pre_comp
);
667 BN_CTX_free(new_ctx
);
668 EC_ec_pre_comp_free(pre_comp
);
672 for (p
= points
; *p
!= NULL
; p
++)
674 OPENSSL_free(points
);
676 EC_POINT_free(tmp_point
);
681 int ec_wNAF_have_precompute_mult(const EC_GROUP
*group
)
683 return HAVEPRECOMP(group
, ec
);