1 /* Originally written by Bodo Moeller for the OpenSSL project.
2 * ====================================================================
3 * Copyright (c) 1998-2005 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 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
57 *
58 * Portions of the attached software ("Contribution") are developed by
59 * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
60 *
61 * The Contribution is licensed pursuant to the OpenSSL open source
62 * license provided above.
63 *
64 * The elliptic curve binary polynomial software is originally written by
65 * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
66 * Laboratories. */
67
68 #include <openssl/ec.h>
69
70 #include <string.h>
71
72 #include <openssl/bn.h>
73 #include <openssl/err.h>
74 #include <openssl/mem.h>
75 #include <openssl/thread.h>
76
77 #include "internal.h"
78 #include "../../internal.h"
79
80
81 /* This file implements the wNAF-based interleaving multi-exponentiation method
82 * at:
83 * http://link.springer.com/chapter/10.1007%2F3-540-45537-X_13
84 * http://www.bmoeller.de/pdf/TI-01-08.multiexp.pdf */
85
86 /* Determine the modified width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'.
87 * This is an array r[] of values that are either zero or odd with an
88 * absolute value less than 2^w satisfying
89 * scalar = \sum_j r[j]*2^j
90 * where at most one of any w+1 consecutive digits is non-zero
91 * with the exception that the most significant digit may be only
92 * w-1 zeros away from that next non-zero digit.
93 */
compute_wNAF(const BIGNUM * scalar,int w,size_t * ret_len)94 static int8_t *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len) {
95 int window_val;
96 int ok = 0;
97 int8_t *r = NULL;
98 int sign = 1;
99 int bit, next_bit, mask;
100 size_t len = 0, j;
101
102 if (BN_is_zero(scalar)) {
103 r = OPENSSL_malloc(1);
104 if (!r) {
105 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
106 goto err;
107 }
108 r[0] = 0;
109 *ret_len = 1;
110 return r;
111 }
112
113 /* 'int8_t' can represent integers with absolute values less than 2^7. */
114 if (w <= 0 || w > 7) {
115 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
116 goto err;
117 }
118 bit = 1 << w; /* at most 128 */
119 next_bit = bit << 1; /* at most 256 */
120 mask = next_bit - 1; /* at most 255 */
121
122 if (BN_is_negative(scalar)) {
123 sign = -1;
124 }
125
126 if (scalar->d == NULL || scalar->top == 0) {
127 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
128 goto err;
129 }
130
131 len = BN_num_bits(scalar);
132 /* The modified wNAF may be one digit longer than binary representation
133 * (*ret_len will be set to the actual length, i.e. at most
134 * BN_num_bits(scalar) + 1). */
135 r = OPENSSL_malloc(len + 1);
136 if (r == NULL) {
137 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
138 goto err;
139 }
140 window_val = scalar->d[0] & mask;
141 j = 0;
142 /* If j+w+1 >= len, window_val will not increase. */
143 while (window_val != 0 || j + w + 1 < len) {
144 int digit = 0;
145
146 /* 0 <= window_val <= 2^(w+1) */
147
148 if (window_val & 1) {
149 /* 0 < window_val < 2^(w+1) */
150
151 if (window_val & bit) {
152 digit = window_val - next_bit; /* -2^w < digit < 0 */
153
154 #if 1 /* modified wNAF */
155 if (j + w + 1 >= len) {
156 /* special case for generating modified wNAFs:
157 * no new bits will be added into window_val,
158 * so using a positive digit here will decrease
159 * the total length of the representation */
160
161 digit = window_val & (mask >> 1); /* 0 < digit < 2^w */
162 }
163 #endif
164 } else {
165 digit = window_val; /* 0 < digit < 2^w */
166 }
167
168 if (digit <= -bit || digit >= bit || !(digit & 1)) {
169 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
170 goto err;
171 }
172
173 window_val -= digit;
174
175 /* Now window_val is 0 or 2^(w+1) in standard wNAF generation;
176 * for modified window NAFs, it may also be 2^w. */
177 if (window_val != 0 && window_val != next_bit && window_val != bit) {
178 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
179 goto err;
180 }
181 }
182
183 r[j++] = sign * digit;
184
185 window_val >>= 1;
186 window_val += bit * BN_is_bit_set(scalar, j + w);
187
188 if (window_val > next_bit) {
189 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
190 goto err;
191 }
192 }
193
194 if (j > len + 1) {
195 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
196 goto err;
197 }
198 len = j;
199 ok = 1;
200
201 err:
202 if (!ok) {
203 OPENSSL_free(r);
204 r = NULL;
205 }
206 if (ok) {
207 *ret_len = len;
208 }
209 return r;
210 }
211
212
213 /* TODO: table should be optimised for the wNAF-based implementation,
214 * sometimes smaller windows will give better performance
215 * (thus the boundaries should be increased)
216 */
window_bits_for_scalar_size(size_t b)217 static size_t window_bits_for_scalar_size(size_t b) {
218 if (b >= 2000) {
219 return 6;
220 }
221
222 if (b >= 800) {
223 return 5;
224 }
225
226 if (b >= 300) {
227 return 4;
228 }
229
230 if (b >= 70) {
231 return 3;
232 }
233
234 if (b >= 20) {
235 return 2;
236 }
237
238 return 1;
239 }
240
ec_wNAF_mul(const EC_GROUP * group,EC_POINT * r,const BIGNUM * g_scalar,const EC_POINT * p,const BIGNUM * p_scalar,BN_CTX * ctx)241 int ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar,
242 const EC_POINT *p, const BIGNUM *p_scalar, BN_CTX *ctx) {
243 BN_CTX *new_ctx = NULL;
244 const EC_POINT *generator = NULL;
245 EC_POINT *tmp = NULL;
246 size_t total_num = 0;
247 size_t i, j;
248 int k;
249 int r_is_inverted = 0;
250 int r_is_at_infinity = 1;
251 size_t *wsize = NULL; /* individual window sizes */
252 int8_t **wNAF = NULL; /* individual wNAFs */
253 size_t *wNAF_len = NULL;
254 size_t max_len = 0;
255 size_t num_val = 0;
256 EC_POINT **val = NULL; /* precomputation */
257 EC_POINT **v;
258 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
259 int ret = 0;
260
261 if (ctx == NULL) {
262 ctx = new_ctx = BN_CTX_new();
263 if (ctx == NULL) {
264 goto err;
265 }
266 }
267
268 /* TODO: This function used to take |points| and |scalars| as arrays of
269 * |num| elements. The code below should be simplified to work in terms of |p|
270 * and |p_scalar|. */
271 size_t num = p != NULL ? 1 : 0;
272 const EC_POINT **points = p != NULL ? &p : NULL;
273 const BIGNUM **scalars = p != NULL ? &p_scalar : NULL;
274
275 total_num = num;
276
277 if (g_scalar != NULL) {
278 generator = EC_GROUP_get0_generator(group);
279 if (generator == NULL) {
280 OPENSSL_PUT_ERROR(EC, EC_R_UNDEFINED_GENERATOR);
281 goto err;
282 }
283
284 ++total_num; /* treat 'g_scalar' like 'num'-th element of 'scalars' */
285 }
286
287
288 wsize = OPENSSL_malloc(total_num * sizeof(wsize[0]));
289 wNAF_len = OPENSSL_malloc(total_num * sizeof(wNAF_len[0]));
290 wNAF = OPENSSL_malloc(total_num * sizeof(wNAF[0]));
291 val_sub = OPENSSL_malloc(total_num * sizeof(val_sub[0]));
292
293 /* Ensure wNAF is initialised in case we end up going to err. */
294 if (wNAF != NULL) {
295 OPENSSL_memset(wNAF, 0, total_num * sizeof(wNAF[0]));
296 }
297
298 if (!wsize || !wNAF_len || !wNAF || !val_sub) {
299 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
300 goto err;
301 }
302
303 /* num_val will be the total number of temporarily precomputed points */
304 num_val = 0;
305
306 for (i = 0; i < total_num; i++) {
307 size_t bits;
308
309 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(g_scalar);
310 wsize[i] = window_bits_for_scalar_size(bits);
311 num_val += (size_t)1 << (wsize[i] - 1);
312 wNAF[i] =
313 compute_wNAF((i < num ? scalars[i] : g_scalar), wsize[i], &wNAF_len[i]);
314 if (wNAF[i] == NULL) {
315 goto err;
316 }
317 if (wNAF_len[i] > max_len) {
318 max_len = wNAF_len[i];
319 }
320 }
321
322 /* All points we precompute now go into a single array 'val'. 'val_sub[i]' is
323 * a pointer to the subarray for the i-th point. */
324 val = OPENSSL_malloc(num_val * sizeof(val[0]));
325 if (val == NULL) {
326 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
327 goto err;
328 }
329 OPENSSL_memset(val, 0, num_val * sizeof(val[0]));
330
331 /* allocate points for precomputation */
332 v = val;
333 for (i = 0; i < total_num; i++) {
334 val_sub[i] = v;
335 for (j = 0; j < ((size_t)1 << (wsize[i] - 1)); j++) {
336 *v = EC_POINT_new(group);
337 if (*v == NULL) {
338 goto err;
339 }
340 v++;
341 }
342 }
343 if (!(v == val + num_val)) {
344 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
345 goto err;
346 }
347
348 if (!(tmp = EC_POINT_new(group))) {
349 goto err;
350 }
351
352 /* prepare precomputed values:
353 * val_sub[i][0] := points[i]
354 * val_sub[i][1] := 3 * points[i]
355 * val_sub[i][2] := 5 * points[i]
356 * ...
357 */
358 for (i = 0; i < total_num; i++) {
359 if (i < num) {
360 if (!EC_POINT_copy(val_sub[i][0], points[i])) {
361 goto err;
362 }
363 } else if (!EC_POINT_copy(val_sub[i][0], generator)) {
364 goto err;
365 }
366
367 if (wsize[i] > 1) {
368 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) {
369 goto err;
370 }
371 for (j = 1; j < ((size_t)1 << (wsize[i] - 1)); j++) {
372 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) {
373 goto err;
374 }
375 }
376 }
377 }
378
379 #if 1 /* optional; window_bits_for_scalar_size assumes we do this step */
380 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) {
381 goto err;
382 }
383 #endif
384
385 r_is_at_infinity = 1;
386
387 for (k = max_len - 1; k >= 0; k--) {
388 if (!r_is_at_infinity && !EC_POINT_dbl(group, r, r, ctx)) {
389 goto err;
390 }
391
392 for (i = 0; i < total_num; i++) {
393 if (wNAF_len[i] > (size_t)k) {
394 int digit = wNAF[i][k];
395 int is_neg;
396
397 if (digit) {
398 is_neg = digit < 0;
399
400 if (is_neg) {
401 digit = -digit;
402 }
403
404 if (is_neg != r_is_inverted) {
405 if (!r_is_at_infinity && !EC_POINT_invert(group, r, ctx)) {
406 goto err;
407 }
408 r_is_inverted = !r_is_inverted;
409 }
410
411 /* digit > 0 */
412
413 if (r_is_at_infinity) {
414 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) {
415 goto err;
416 }
417 r_is_at_infinity = 0;
418 } else {
419 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) {
420 goto err;
421 }
422 }
423 }
424 }
425 }
426 }
427
428 if (r_is_at_infinity) {
429 if (!EC_POINT_set_to_infinity(group, r)) {
430 goto err;
431 }
432 } else if (r_is_inverted && !EC_POINT_invert(group, r, ctx)) {
433 goto err;
434 }
435
436 ret = 1;
437
438 err:
439 BN_CTX_free(new_ctx);
440 EC_POINT_free(tmp);
441 OPENSSL_free(wsize);
442 OPENSSL_free(wNAF_len);
443 if (wNAF != NULL) {
444 for (i = 0; i < total_num; i++) {
445 OPENSSL_free(wNAF[i]);
446 }
447
448 OPENSSL_free(wNAF);
449 }
450 if (val != NULL) {
451 for (i = 0; i < num_val; i++) {
452 EC_POINT_clear_free(val[i]);
453 }
454
455 OPENSSL_free(val);
456 }
457 OPENSSL_free(val_sub);
458 return ret;
459 }
460