1 /* crypto/ec/ec_mult.c */
2 /* ====================================================================
3 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
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
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/)"
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.
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.
31 * 6. Redistributions of any form whatsoever must retain the following
33 * "This product includes software developed by the OpenSSL Project
34 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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 * ====================================================================
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).
56 #include <openssl/err.h>
61 /* TODO: optional precomputation of multiples of the generator */
66 * wNAF-based interleaving multi-exponentation method
71 /* Determine the width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'.
72 * This is an array r[] of values that are either zero or odd with an
73 * absolute value less than 2^w satisfying
74 * scalar = \sum_j r[j]*2^j
75 * where at most one of any w+1 consecutive digits is non-zero.
77 static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len, BN_CTX *ctx)
81 signed char *r = NULL;
83 int bit, next_bit, mask;
88 if (c == NULL) goto err;
90 if (w <= 0 || w > 7) /* 'signed char' can represent integers with absolute values less than 2^7 */
92 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
95 bit = 1 << w; /* at most 128 */
96 next_bit = bit << 1; /* at most 256 */
97 mask = next_bit - 1; /* at most 255 */
99 if (!BN_copy(c, scalar)) goto err;
106 len = BN_num_bits(c) + 1; /* wNAF may be one digit longer than binary representation */
107 r = OPENSSL_malloc(len);
108 if (r == NULL) goto err;
111 while (!BN_is_zero(c))
117 if (c->d == NULL || c->top == 0)
119 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
127 if (!BN_add_word(c, -u)) goto err;
132 if (!BN_sub_word(c, u)) goto err;
135 if (u <= -bit || u >= bit || !(u & 1) || c->neg)
137 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
146 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
149 if (!BN_rshift1(c, c)) goto err;
154 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
173 /* TODO: table should be optimised for the wNAF-based implementation */
174 #define EC_window_bits_for_scalar_size(b) \
183 * \sum scalars[i]*points[i],
186 * in the addition if scalar != NULL
188 int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
189 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
191 BN_CTX *new_ctx = NULL;
192 EC_POINT *generator = NULL;
193 EC_POINT *tmp = NULL;
197 int r_is_inverted = 0;
198 int r_is_at_infinity = 1;
199 size_t *wsize = NULL; /* individual window sizes */
200 signed char **wNAF = NULL; /* individual wNAFs */
201 size_t *wNAF_len = NULL;
204 EC_POINT **val = NULL; /* precomputation */
206 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
211 generator = EC_GROUP_get0_generator(group);
212 if (generator == NULL)
214 ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
219 for (i = 0; i < num; i++)
221 if (group->meth != points[i]->meth)
223 ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
228 totalnum = num + (scalar != NULL);
230 wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
231 wNAF_len = OPENSSL_malloc(totalnum * sizeof wNAF_len[0]);
232 wNAF = OPENSSL_malloc(totalnum * sizeof wNAF[0] + 1);
235 wNAF[0] = NULL; /* preliminary pivot */
237 if (wsize == NULL || wNAF_len == NULL || wNAF == NULL) goto err;
239 /* num_val := total number of points to precompute */
241 for (i = 0; i < totalnum; i++)
245 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
246 wsize[i] = EC_window_bits_for_scalar_size(bits);
247 num_val += 1u << (wsize[i] - 1);
250 /* all precomputed points go into a single array 'val',
251 * 'val_sub[i]' is a pointer to the subarray for the i-th point */
252 val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
253 if (val == NULL) goto err;
254 val[num_val] = NULL; /* pivot element */
256 val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
257 if (val_sub == NULL) goto err;
259 /* allocate points for precomputation */
261 for (i = 0; i < totalnum; i++)
264 for (j = 0; j < (1u << (wsize[i] - 1)); j++)
266 *v = EC_POINT_new(group);
267 if (*v == NULL) goto err;
271 if (!(v == val + num_val))
273 ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
279 ctx = new_ctx = BN_CTX_new();
284 tmp = EC_POINT_new(group);
285 if (tmp == NULL) goto err;
287 /* prepare precomputed values:
288 * val_sub[i][0] := points[i]
289 * val_sub[i][1] := 3 * points[i]
290 * val_sub[i][2] := 5 * points[i]
293 for (i = 0; i < totalnum; i++)
297 if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
301 if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
306 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
307 for (j = 1; j < (1u << (wsize[i] - 1)); j++)
309 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
313 wNAF[i + 1] = NULL; /* make sure we always have a pivot */
314 wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i], ctx);
315 if (wNAF[i] == NULL) goto err;
316 if (wNAF_len[i] > max_len)
317 max_len = wNAF_len[i];
320 #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
321 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
324 r_is_at_infinity = 1;
326 for (k = max_len - 1; k >= 0; k--)
328 if (!r_is_at_infinity)
330 if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
333 for (i = 0; i < totalnum; i++)
337 int digit = wNAF[i][k];
347 if (is_neg != r_is_inverted)
349 if (!r_is_at_infinity)
351 if (!EC_POINT_invert(group, r, ctx)) goto err;
353 r_is_inverted = !r_is_inverted;
358 if (r_is_at_infinity)
360 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) goto err;
361 r_is_at_infinity = 0;
365 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) goto err;
372 if (r_is_at_infinity)
374 if (!EC_POINT_set_to_infinity(group, r)) goto err;
379 if (!EC_POINT_invert(group, r, ctx)) goto err;
386 BN_CTX_free(new_ctx);
391 if (wNAF_len != NULL)
392 OPENSSL_free(wNAF_len);
397 for (w = wNAF; *w != NULL; w++)
404 for (v = val; *v != NULL; v++)
405 EC_POINT_clear_free(*v);
411 OPENSSL_free(val_sub);
419 * Basic interleaving multi-exponentation method
424 #define EC_window_bits_for_scalar_size(b) \
431 /* For window size 'w' (w >= 2), we compute the odd multiples
433 * This accounts for 2^(w-1) point additions (neglecting constants),
434 * each of which requires 16 field multiplications (4 squarings
435 * and 12 general multiplications) in the case of curves defined
436 * over GF(p), which are the only curves we have so far.
438 * Converting these precomputed points into affine form takes
439 * three field multiplications for inverting Z and one squaring
440 * and three multiplications for adjusting X and Y, i.e.
441 * 7 multiplications in total (1 squaring and 6 general multiplications),
442 * again except for constants.
444 * The average number of windows for a 'b' bit scalar is roughly
446 * Each of these windows (except possibly for the first one, but
447 * we are ignoring constants anyway) requires one point addition.
448 * As the precomputed table stores points in affine form, these
449 * additions take only 11 field multiplications each (3 squarings
450 * and 8 general multiplications).
452 * So the total workload, except for constants, is
454 * 2^(w-1)*[5 squarings + 18 multiplications]
455 * + (b/(w+1))*[3 squarings + 8 multiplications]
457 * If we assume that 10 squarings are as costly as 9 multiplications,
458 * our task is to find the 'w' that, given 'b', minimizes
460 * 2^(w-1)*(5*9 + 18*10) + (b/(w+1))*(3*9 + 8*10)
461 * = 2^(w-1)*225 + (b/(w+1))*107.
463 * Thus optimal window sizes should be roughly as follows:
465 * w >= 6 if b >= 1414
466 * w = 5 if 1413 >= b >= 505
467 * w = 4 if 504 >= b >= 169
468 * w = 3 if 168 >= b >= 51
469 * w = 2 if 50 >= b >= 13
472 * If we assume instead that squarings are exactly as costly as
473 * multiplications, we have to minimize
474 * 2^(w-1)*23 + (b/(w+1))*11.
476 * This gives us the following (nearly unchanged) table of optimal
479 * w >= 6 if b >= 1406
480 * w = 5 if 1405 >= b >= 502
481 * w = 4 if 501 >= b >= 168
482 * w = 3 if 167 >= b >= 51
483 * w = 2 if 50 >= b >= 13
486 * Note that neither table tries to take into account memory usage
487 * (allocation overhead, code locality etc.). Actual timings with
488 * NIST curves P-192, P-224, and P-256 with scalars of 192, 224,
489 * and 256 bits, respectively, show that w = 3 (instead of 4) is
490 * preferrable; timings with NIST curve P-384 and 384-bit scalars
491 * confirm that w = 4 is optimal for this case; and timings with
492 * NIST curve P-521 and 521-bit scalars show that w = 4 (instead
493 * of 5) is preferrable. So we generously round up all the
494 * boundaries and use the following table:
496 * w >= 6 if b >= 2000
497 * w = 5 if 1999 >= b >= 800
498 * w = 4 if 799 >= b >= 300
499 * w = 3 if 299 >= b >= 70
500 * w = 2 if 69 >= b >= 20
504 int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
505 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
507 BN_CTX *new_ctx = NULL;
508 EC_POINT *generator = NULL;
509 EC_POINT *tmp = NULL;
513 int r_is_at_infinity = 1;
515 size_t *wsize = NULL; /* individual window sizes */
516 unsigned long *wbits = NULL; /* individual window contents */
517 int *wpos = NULL; /* position of bottom bit of current individual windows
518 * (wpos[i] is valid if wbits[i] != 0) */
520 EC_POINT **val = NULL; /* precomputation */
522 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
527 generator = EC_GROUP_get0_generator(group);
528 if (generator == NULL)
530 ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
535 for (i = 0; i < num; i++)
537 if (group->meth != points[i]->meth)
539 ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
544 totalnum = num + (scalar != NULL);
546 wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
547 wbits = OPENSSL_malloc(totalnum * sizeof wbits[0]);
548 wpos = OPENSSL_malloc(totalnum * sizeof wpos[0]);
549 if (wsize == NULL || wbits == NULL || wpos == NULL) goto err;
551 /* num_val := total number of points to precompute */
553 for (i = 0; i < totalnum; i++)
557 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
558 wsize[i] = EC_window_bits_for_scalar_size(bits);
559 num_val += 1u << (wsize[i] - 1);
566 /* all precomputed points go into a single array 'val',
567 * 'val_sub[i]' is a pointer to the subarray for the i-th point */
568 val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
569 if (val == NULL) goto err;
570 val[num_val] = NULL; /* pivot element */
572 val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
573 if (val_sub == NULL) goto err;
575 /* allocate points for precomputation */
577 for (i = 0; i < totalnum; i++)
580 for (j = 0; j < (1u << (wsize[i] - 1)); j++)
582 *v = EC_POINT_new(group);
583 if (*v == NULL) goto err;
587 if (!(v == val + num_val))
589 ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
595 ctx = new_ctx = BN_CTX_new();
600 tmp = EC_POINT_new(group);
601 if (tmp == NULL) goto err;
603 /* prepare precomputed values:
604 * val_sub[i][0] := points[i]
605 * val_sub[i][1] := 3 * points[i]
606 * val_sub[i][2] := 5 * points[i]
609 for (i = 0; i < totalnum; i++)
613 if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
616 if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
621 if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
624 if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
630 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
631 for (j = 1; j < (1u << (wsize[i] - 1)); j++)
633 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
638 #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
639 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
642 r_is_at_infinity = 1;
644 for (k = max_bits - 1; k >= 0; k--)
646 if (!r_is_at_infinity)
648 if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
651 for (i = 0; i < totalnum; i++)
657 s = i < num ? scalars[i] : scalar;
659 if (BN_is_bit_set(s, k))
661 /* look at bits k - wsize[i] + 1 .. k for this window */
662 t = k - wsize[i] + 1;
663 while (!BN_is_bit_set(s, t)) /* BN_is_bit_set is false for t < 0 */
667 for (t = k - 1; t >= wpos[i]; t--)
670 if (BN_is_bit_set(s, t))
673 /* now wbits[i] is the odd bit pattern at bits wpos[i] .. k */
677 if ((wbits[i] != 0) && (wpos[i] == k))
679 if (r_is_at_infinity)
681 if (!EC_POINT_copy(r, val_sub[i][wbits[i] >> 1])) goto err;
682 r_is_at_infinity = 0;
686 if (!EC_POINT_add(group, r, r, val_sub[i][wbits[i] >> 1], ctx)) goto err;
693 if (r_is_at_infinity)
694 if (!EC_POINT_set_to_infinity(group, r)) goto err;
700 BN_CTX_free(new_ctx);
711 for (v = val; *v != NULL; v++)
712 EC_POINT_clear_free(*v);
718 OPENSSL_free(val_sub);
725 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)
727 const EC_POINT *points[1];
728 const BIGNUM *scalars[1];
731 scalars[0] = p_scalar;
733 return EC_POINTs_mul(group, r, g_scalar, (point != NULL && p_scalar != NULL), points, scalars, ctx);
737 int EC_GROUP_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
739 const EC_POINT *generator;
740 BN_CTX *new_ctx = NULL;
744 generator = EC_GROUP_get0_generator(group);
745 if (generator == NULL)
747 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
753 ctx = new_ctx = BN_CTX_new();
759 order = BN_CTX_get(ctx);
760 if (order == NULL) goto err;
762 if (!EC_GROUP_get_order(group, order, ctx)) return 0;
763 if (BN_is_zero(order))
765 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
776 BN_CTX_free(new_ctx);