1/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
2 * and Bodo Moeller for the OpenSSL project. */
3/* ====================================================================
4 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 *
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 *
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in
15 * the documentation and/or other materials provided with the
16 * distribution.
17 *
18 * 3. All advertising materials mentioning features or use of this
19 * software must display the following acknowledgment:
20 * "This product includes software developed by the OpenSSL Project
21 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
22 *
23 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
24 * endorse or promote products derived from this software without
25 * prior written permission. For written permission, please contact
26 * openssl-core@openssl.org.
27 *
28 * 5. Products derived from this software may not be called "OpenSSL"
29 * nor may "OpenSSL" appear in their names without prior written
30 * permission of the OpenSSL Project.
31 *
32 * 6. Redistributions of any form whatsoever must retain the following
33 * acknowledgment:
34 * "This product includes software developed by the OpenSSL Project
35 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
36 *
37 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
38 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
39 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
40 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
41 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
42 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
43 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
44 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
45 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
46 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
47 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
48 * OF THE POSSIBILITY OF SUCH DAMAGE.
49 * ====================================================================
50 *
51 * This product includes cryptographic software written by Eric Young
52 * (eay@cryptsoft.com). This product includes software written by Tim
53 * Hudson (tjh@cryptsoft.com). */
54
55#include <openssl/bn.h>
56
57#include <openssl/err.h>
58
59#include "internal.h"
60
61
62BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
63 // Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
64 // (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
65 // algorithm 1.5.1). |p| is assumed to be a prime.
66
67 BIGNUM *ret = in;
68 int err = 1;
69 int r;
70 BIGNUM *A, *b, *q, *t, *x, *y;
71 int e, i, j;
72
73 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
74 if (BN_abs_is_word(p, 2)) {
75 if (ret == NULL) {
76 ret = BN_new();
77 }
78 if (ret == NULL) {
79 goto end;
80 }
81 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
82 if (ret != in) {
83 BN_free(ret);
84 }
85 return NULL;
86 }
87 return ret;
88 }
89
90 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
91 return (NULL);
92 }
93
94 if (BN_is_zero(a) || BN_is_one(a)) {
95 if (ret == NULL) {
96 ret = BN_new();
97 }
98 if (ret == NULL) {
99 goto end;
100 }
101 if (!BN_set_word(ret, BN_is_one(a))) {
102 if (ret != in) {
103 BN_free(ret);
104 }
105 return NULL;
106 }
107 return ret;
108 }
109
110 BN_CTX_start(ctx);
111 A = BN_CTX_get(ctx);
112 b = BN_CTX_get(ctx);
113 q = BN_CTX_get(ctx);
114 t = BN_CTX_get(ctx);
115 x = BN_CTX_get(ctx);
116 y = BN_CTX_get(ctx);
117 if (y == NULL) {
118 goto end;
119 }
120
121 if (ret == NULL) {
122 ret = BN_new();
123 }
124 if (ret == NULL) {
125 goto end;
126 }
127
128 // A = a mod p
129 if (!BN_nnmod(A, a, p, ctx)) {
130 goto end;
131 }
132
133 // now write |p| - 1 as 2^e*q where q is odd
134 e = 1;
135 while (!BN_is_bit_set(p, e)) {
136 e++;
137 }
138 // we'll set q later (if needed)
139
140 if (e == 1) {
141 // The easy case: (|p|-1)/2 is odd, so 2 has an inverse
142 // modulo (|p|-1)/2, and square roots can be computed
143 // directly by modular exponentiation.
144 // We have
145 // 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
146 // so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
147 if (!BN_rshift(q, p, 2)) {
148 goto end;
149 }
150 q->neg = 0;
151 if (!BN_add_word(q, 1) ||
152 !BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
153 goto end;
154 }
155 err = 0;
156 goto vrfy;
157 }
158
159 if (e == 2) {
160 // |p| == 5 (mod 8)
161 //
162 // In this case 2 is always a non-square since
163 // Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
164 // So if a really is a square, then 2*a is a non-square.
165 // Thus for
166 // b := (2*a)^((|p|-5)/8),
167 // i := (2*a)*b^2
168 // we have
169 // i^2 = (2*a)^((1 + (|p|-5)/4)*2)
170 // = (2*a)^((p-1)/2)
171 // = -1;
172 // so if we set
173 // x := a*b*(i-1),
174 // then
175 // x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
176 // = a^2 * b^2 * (-2*i)
177 // = a*(-i)*(2*a*b^2)
178 // = a*(-i)*i
179 // = a.
180 //
181 // (This is due to A.O.L. Atkin,
182 // <URL:
183 //http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
184 // November 1992.)
185
186 // t := 2*a
187 if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
188 goto end;
189 }
190
191 // b := (2*a)^((|p|-5)/8)
192 if (!BN_rshift(q, p, 3)) {
193 goto end;
194 }
195 q->neg = 0;
196 if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
197 goto end;
198 }
199
200 // y := b^2
201 if (!BN_mod_sqr(y, b, p, ctx)) {
202 goto end;
203 }
204
205 // t := (2*a)*b^2 - 1
206 if (!BN_mod_mul(t, t, y, p, ctx) ||
207 !BN_sub_word(t, 1)) {
208 goto end;
209 }
210
211 // x = a*b*t
212 if (!BN_mod_mul(x, A, b, p, ctx) ||
213 !BN_mod_mul(x, x, t, p, ctx)) {
214 goto end;
215 }
216
217 if (!BN_copy(ret, x)) {
218 goto end;
219 }
220 err = 0;
221 goto vrfy;
222 }
223
224 // e > 2, so we really have to use the Tonelli/Shanks algorithm.
225 // First, find some y that is not a square.
226 if (!BN_copy(q, p)) {
227 goto end; // use 'q' as temp
228 }
229 q->neg = 0;
230 i = 2;
231 do {
232 // For efficiency, try small numbers first;
233 // if this fails, try random numbers.
234 if (i < 22) {
235 if (!BN_set_word(y, i)) {
236 goto end;
237 }
238 } else {
239 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
240 goto end;
241 }
242 if (BN_ucmp(y, p) >= 0) {
243 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
244 goto end;
245 }
246 }
247 // now 0 <= y < |p|
248 if (BN_is_zero(y)) {
249 if (!BN_set_word(y, i)) {
250 goto end;
251 }
252 }
253 }
254
255 r = bn_jacobi(y, q, ctx); // here 'q' is |p|
256 if (r < -1) {
257 goto end;
258 }
259 if (r == 0) {
260 // m divides p
261 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
262 goto end;
263 }
264 } while (r == 1 && ++i < 82);
265
266 if (r != -1) {
267 // Many rounds and still no non-square -- this is more likely
268 // a bug than just bad luck.
269 // Even if p is not prime, we should have found some y
270 // such that r == -1.
271 OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
272 goto end;
273 }
274
275 // Here's our actual 'q':
276 if (!BN_rshift(q, q, e)) {
277 goto end;
278 }
279
280 // Now that we have some non-square, we can find an element
281 // of order 2^e by computing its q'th power.
282 if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
283 goto end;
284 }
285 if (BN_is_one(y)) {
286 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
287 goto end;
288 }
289
290 // Now we know that (if p is indeed prime) there is an integer
291 // k, 0 <= k < 2^e, such that
292 //
293 // a^q * y^k == 1 (mod p).
294 //
295 // As a^q is a square and y is not, k must be even.
296 // q+1 is even, too, so there is an element
297 //
298 // X := a^((q+1)/2) * y^(k/2),
299 //
300 // and it satisfies
301 //
302 // X^2 = a^q * a * y^k
303 // = a,
304 //
305 // so it is the square root that we are looking for.
306
307 // t := (q-1)/2 (note that q is odd)
308 if (!BN_rshift1(t, q)) {
309 goto end;
310 }
311
312 // x := a^((q-1)/2)
313 if (BN_is_zero(t)) // special case: p = 2^e + 1
314 {
315 if (!BN_nnmod(t, A, p, ctx)) {
316 goto end;
317 }
318 if (BN_is_zero(t)) {
319 // special case: a == 0 (mod p)
320 BN_zero(ret);
321 err = 0;
322 goto end;
323 } else if (!BN_one(x)) {
324 goto end;
325 }
326 } else {
327 if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
328 goto end;
329 }
330 if (BN_is_zero(x)) {
331 // special case: a == 0 (mod p)
332 BN_zero(ret);
333 err = 0;
334 goto end;
335 }
336 }
337
338 // b := a*x^2 (= a^q)
339 if (!BN_mod_sqr(b, x, p, ctx) ||
340 !BN_mod_mul(b, b, A, p, ctx)) {
341 goto end;
342 }
343
344 // x := a*x (= a^((q+1)/2))
345 if (!BN_mod_mul(x, x, A, p, ctx)) {
346 goto end;
347 }
348
349 while (1) {
350 // Now b is a^q * y^k for some even k (0 <= k < 2^E
351 // where E refers to the original value of e, which we
352 // don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
353 //
354 // We have a*b = x^2,
355 // y^2^(e-1) = -1,
356 // b^2^(e-1) = 1.
357
358 if (BN_is_one(b)) {
359 if (!BN_copy(ret, x)) {
360 goto end;
361 }
362 err = 0;
363 goto vrfy;
364 }
365
366
367 // find smallest i such that b^(2^i) = 1
368 i = 1;
369 if (!BN_mod_sqr(t, b, p, ctx)) {
370 goto end;
371 }
372 while (!BN_is_one(t)) {
373 i++;
374 if (i == e) {
375 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
376 goto end;
377 }
378 if (!BN_mod_mul(t, t, t, p, ctx)) {
379 goto end;
380 }
381 }
382
383
384 // t := y^2^(e - i - 1)
385 if (!BN_copy(t, y)) {
386 goto end;
387 }
388 for (j = e - i - 1; j > 0; j--) {
389 if (!BN_mod_sqr(t, t, p, ctx)) {
390 goto end;
391 }
392 }
393 if (!BN_mod_mul(y, t, t, p, ctx) ||
394 !BN_mod_mul(x, x, t, p, ctx) ||
395 !BN_mod_mul(b, b, y, p, ctx)) {
396 goto end;
397 }
398 e = i;
399 }
400
401vrfy:
402 if (!err) {
403 // verify the result -- the input might have been not a square
404 // (test added in 0.9.8)
405
406 if (!BN_mod_sqr(x, ret, p, ctx)) {
407 err = 1;
408 }
409
410 if (!err && 0 != BN_cmp(x, A)) {
411 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
412 err = 1;
413 }
414 }
415
416end:
417 if (err) {
418 if (ret != in) {
419 BN_clear_free(ret);
420 }
421 ret = NULL;
422 }
423 BN_CTX_end(ctx);
424 return ret;
425}
426
427int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
428 BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
429 int ok = 0, last_delta_valid = 0;
430
431 if (in->neg) {
432 OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
433 return 0;
434 }
435 if (BN_is_zero(in)) {
436 BN_zero(out_sqrt);
437 return 1;
438 }
439
440 BN_CTX_start(ctx);
441 if (out_sqrt == in) {
442 estimate = BN_CTX_get(ctx);
443 } else {
444 estimate = out_sqrt;
445 }
446 tmp = BN_CTX_get(ctx);
447 last_delta = BN_CTX_get(ctx);
448 delta = BN_CTX_get(ctx);
449 if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
450 OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
451 goto err;
452 }
453
454 // We estimate that the square root of an n-bit number is 2^{n/2}.
455 if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
456 goto err;
457 }
458
459 // This is Newton's method for finding a root of the equation |estimate|^2 -
460 // |in| = 0.
461 for (;;) {
462 // |estimate| = 1/2 * (|estimate| + |in|/|estimate|)
463 if (!BN_div(tmp, NULL, in, estimate, ctx) ||
464 !BN_add(tmp, tmp, estimate) ||
465 !BN_rshift1(estimate, tmp) ||
466 // |tmp| = |estimate|^2
467 !BN_sqr(tmp, estimate, ctx) ||
468 // |delta| = |in| - |tmp|
469 !BN_sub(delta, in, tmp)) {
470 OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
471 goto err;
472 }
473
474 delta->neg = 0;
475 // The difference between |in| and |estimate| squared is required to always
476 // decrease. This ensures that the loop always terminates, but I don't have
477 // a proof that it always finds the square root for a given square.
478 if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
479 break;
480 }
481
482 last_delta_valid = 1;
483
484 tmp2 = last_delta;
485 last_delta = delta;
486 delta = tmp2;
487 }
488
489 if (BN_cmp(tmp, in) != 0) {
490 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
491 goto err;
492 }
493
494 ok = 1;
495
496err:
497 if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
498 ok = 0;
499 }
500 BN_CTX_end(ctx);
501 return ok;
502}
503