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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: optional precomputation of multiples of the generator */
62
63
64
65/*
66 * wNAF-based interleaving multi-exponentation method
67 * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp>)
68 */
69
70
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.
76 */
77static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len, BN_CTX *ctx)
78 {
79 BIGNUM *c;
80 int ok = 0;
81 signed char *r = NULL;
82 int sign = 1;
83 int bit, next_bit, mask;
84 size_t len = 0, j;
85
86 BN_CTX_start(ctx);
87 c = BN_CTX_get(ctx);
88 if (c == NULL) goto err;
89
90 if (w <= 0 || w > 7) /* 'signed char' can represent integers with absolute values less than 2^7 */
91 {
92 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
93 goto err;
94 }
95 bit = 1 << w; /* at most 128 */
96 next_bit = bit << 1; /* at most 256 */
97 mask = next_bit - 1; /* at most 255 */
98
99 if (!BN_copy(c, scalar)) goto err;
100 if (c->neg)
101 {
102 sign = -1;
103 c->neg = 0;
104 }
105
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;
109
110 j = 0;
111 while (!BN_is_zero(c))
112 {
113 int u = 0;
114
115 if (BN_is_odd(c))
116 {
117 if (c->d == NULL || c->top == 0)
118 {
119 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
120 goto err;
121 }
122 u = c->d[0] & mask;
123 if (u & bit)
124 {
125 u -= next_bit;
126 /* u < 0 */
127 if (!BN_add_word(c, -u)) goto err;
128 }
129 else
130 {
131 /* u > 0 */
132 if (!BN_sub_word(c, u)) goto err;
133 }
134
135 if (u <= -bit || u >= bit || !(u & 1) || c->neg)
136 {
137 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
138 goto err;
139 }
140 }
141
142 r[j++] = sign * u;
143
144 if (BN_is_odd(c))
145 {
146 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
147 goto err;
148 }
149 if (!BN_rshift1(c, c)) goto err;
150 }
151
152 if (j > len)
153 {
154 ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
155 goto err;
156 }
157 len = j;
158 ok = 1;
159
160 err:
161 BN_CTX_end(ctx);
162 if (!ok)
163 {
164 OPENSSL_free(r);
165 r = NULL;
166 }
167 if (ok)
168 *ret_len = len;
169 return r;
170 }
171
172
173/* TODO: table should be optimised for the wNAF-based implementation,
174 * sometimes smaller windows will give better performance
175 * (thus the boundaries should be increased)
176 */
177#define EC_window_bits_for_scalar_size(b) \
178 ((b) >= 2000 ? 6 : \
179 (b) >= 800 ? 5 : \
180 (b) >= 300 ? 4 : \
181 (b) >= 70 ? 3 : \
182 (b) >= 20 ? 2 : \
183 1)
184
185/* Compute
186 * \sum scalars[i]*points[i],
187 * also including
188 * scalar*generator
189 * in the addition if scalar != NULL
190 */
191int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
192 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
193 {
194 BN_CTX *new_ctx = NULL;
195 EC_POINT *generator = NULL;
196 EC_POINT *tmp = NULL;
197 size_t totalnum;
198 size_t i, j;
199 int k;
200 int r_is_inverted = 0;
201 int r_is_at_infinity = 1;
202 size_t *wsize = NULL; /* individual window sizes */
203 signed char **wNAF = NULL; /* individual wNAFs */
204 size_t *wNAF_len = NULL;
205 size_t max_len = 0;
206 size_t num_val;
207 EC_POINT **val = NULL; /* precomputation */
208 EC_POINT **v;
209 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
210 int ret = 0;
211
212 if (scalar != NULL)
213 {
214 generator = EC_GROUP_get0_generator(group);
215 if (generator == NULL)
216 {
217 ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
218 return 0;
219 }
220 }
221
222 for (i = 0; i < num; i++)
223 {
224 if (group->meth != points[i]->meth)
225 {
226 ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
227 return 0;
228 }
229 }
230
231 totalnum = num + (scalar != NULL);
232
233 wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
234 wNAF_len = OPENSSL_malloc(totalnum * sizeof wNAF_len[0]);
235 wNAF = OPENSSL_malloc((totalnum + 1) * sizeof wNAF[0]);
236 if (wNAF != NULL)
237 {
238 wNAF[0] = NULL; /* preliminary pivot */
239 }
240 if (wsize == NULL || wNAF_len == NULL || wNAF == NULL) goto err;
241
242 /* num_val := total number of points to precompute */
243 num_val = 0;
244 for (i = 0; i < totalnum; i++)
245 {
246 size_t bits;
247
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 += 1u << (wsize[i] - 1);
251 }
252
253 /* all precomputed points go into a single array 'val',
254 * 'val_sub[i]' is a pointer to the subarray for the i-th point */
255 val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
256 if (val == NULL) goto err;
257 val[num_val] = NULL; /* pivot element */
258
259 val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
260 if (val_sub == NULL) goto err;
261
262 /* allocate points for precomputation */
263 v = val;
264 for (i = 0; i < totalnum; i++)
265 {
266 val_sub[i] = v;
267 for (j = 0; j < (1u << (wsize[i] - 1)); j++)
268 {
269 *v = EC_POINT_new(group);
270 if (*v == NULL) goto err;
271 v++;
272 }
273 }
274 if (!(v == val + num_val))
275 {
276 ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
277 goto err;
278 }
279
280 if (ctx == NULL)
281 {
282 ctx = new_ctx = BN_CTX_new();
283 if (ctx == NULL)
284 goto err;
285 }
286
287 tmp = EC_POINT_new(group);
288 if (tmp == NULL) goto err;
289
290 /* prepare precomputed values:
291 * val_sub[i][0] := points[i]
292 * val_sub[i][1] := 3 * points[i]
293 * val_sub[i][2] := 5 * points[i]
294 * ...
295 */
296 for (i = 0; i < totalnum; i++)
297 {
298 if (i < num)
299 {
300 if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
301 }
302 else
303 {
304 if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
305 }
306
307 if (wsize[i] > 1)
308 {
309 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
310 for (j = 1; j < (1u << (wsize[i] - 1)); j++)
311 {
312 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
313 }
314 }
315
316 wNAF[i + 1] = NULL; /* make sure we always have a pivot */
317 wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i], ctx);
318 if (wNAF[i] == NULL) goto err;
319 if (wNAF_len[i] > max_len)
320 max_len = wNAF_len[i];
321 }
322
323#if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
324 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
325#endif
326
327 r_is_at_infinity = 1;
328
329 for (k = max_len - 1; k >= 0; k--)
330 {
331 if (!r_is_at_infinity)
332 {
333 if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
334 }
335
336 for (i = 0; i < totalnum; i++)
337 {
338 if (wNAF_len[i] > (size_t)k)
339 {
340 int digit = wNAF[i][k];
341 int is_neg;
342
343 if (digit)
344 {
345 is_neg = digit < 0;
346
347 if (is_neg)
348 digit = -digit;
349
350 if (is_neg != r_is_inverted)
351 {
352 if (!r_is_at_infinity)
353 {
354 if (!EC_POINT_invert(group, r, ctx)) goto err;
355 }
356 r_is_inverted = !r_is_inverted;
357 }
358
359 /* digit > 0 */
360
361 if (r_is_at_infinity)
362 {
363 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) goto err;
364 r_is_at_infinity = 0;
365 }
366 else
367 {
368 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) goto err;
369 }
370 }
371 }
372 }
373 }
374
375 if (r_is_at_infinity)
376 {
377 if (!EC_POINT_set_to_infinity(group, r)) goto err;
378 }
379 else
380 {
381 if (r_is_inverted)
382 if (!EC_POINT_invert(group, r, ctx)) goto err;
383 }
384
385 ret = 1;
386
387 err:
388 if (new_ctx != NULL)
389 BN_CTX_free(new_ctx);
390 if (tmp != NULL)
391 EC_POINT_free(tmp);
392 if (wsize != NULL)
393 OPENSSL_free(wsize);
394 if (wNAF_len != NULL)
395 OPENSSL_free(wNAF_len);
396 if (wNAF != NULL)
397 {
398 signed char **w;
399
400 for (w = wNAF; *w != NULL; w++)
401 OPENSSL_free(*w);
402
403 OPENSSL_free(wNAF);
404 }
405 if (val != NULL)
406 {
407 for (v = val; *v != NULL; v++)
408 EC_POINT_clear_free(*v);
409
410 OPENSSL_free(val);
411 }
412 if (val_sub != NULL)
413 {
414 OPENSSL_free(val_sub);
415 }
416 return ret;
417 }
418
419
420int 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)
421 {
422 const EC_POINT *points[1];
423 const BIGNUM *scalars[1];
424
425 points[0] = point;
426 scalars[0] = p_scalar;
427
428 return EC_POINTs_mul(group, r, g_scalar, (point != NULL && p_scalar != NULL), points, scalars, ctx);
429 }
430
431
432int EC_GROUP_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
433 {
434 const EC_POINT *generator;
435 BN_CTX *new_ctx = NULL;
436 BIGNUM *order;
437 int ret = 0;
438
439 generator = EC_GROUP_get0_generator(group);
440 if (generator == NULL)
441 {
442 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
443 return 0;
444 }
445
446 if (ctx == NULL)
447 {
448 ctx = new_ctx = BN_CTX_new();
449 if (ctx == NULL)
450 return 0;
451 }
452
453 BN_CTX_start(ctx);
454 order = BN_CTX_get(ctx);
455 if (order == NULL) goto err;
456
457 if (!EC_GROUP_get_order(group, order, ctx)) return 0;
458 if (BN_is_zero(order))
459 {
460 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
461 goto err;
462 }
463
464 /* TODO */
465
466 ret = 1;
467
468 err:
469 BN_CTX_end(ctx);
470 if (new_ctx != NULL)
471 BN_CTX_free(new_ctx);
472 return ret;
473 }
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