1 /* Written by Lenka Fibikova <[email protected]>
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 * [email protected].
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 * ([email protected]). This product includes software written by Tim
53 * Hudson ([email protected]). */
54
55 #include <openssl/bn.h>
56
57 #include <openssl/err.h>
58
59 #include "internal.h"
60
61
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)62 BIGNUM *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 !BN_set_word(ret, BN_is_bit_set(a, 0))) {
80 if (ret != in) {
81 BN_free(ret);
82 }
83 return NULL;
84 }
85 return ret;
86 }
87
88 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
89 return NULL;
90 }
91
92 if (BN_is_zero(a) || BN_is_one(a)) {
93 if (ret == NULL) {
94 ret = BN_new();
95 }
96 if (ret == NULL ||
97 !BN_set_word(ret, BN_is_one(a))) {
98 if (ret != in) {
99 BN_free(ret);
100 }
101 return NULL;
102 }
103 return ret;
104 }
105
106 BN_CTX_start(ctx);
107 A = BN_CTX_get(ctx);
108 b = BN_CTX_get(ctx);
109 q = BN_CTX_get(ctx);
110 t = BN_CTX_get(ctx);
111 x = BN_CTX_get(ctx);
112 y = BN_CTX_get(ctx);
113 if (y == NULL) {
114 goto end;
115 }
116
117 if (ret == NULL) {
118 ret = BN_new();
119 }
120 if (ret == NULL) {
121 goto end;
122 }
123
124 // A = a mod p
125 if (!BN_nnmod(A, a, p, ctx)) {
126 goto end;
127 }
128
129 // now write |p| - 1 as 2^e*q where q is odd
130 e = 1;
131 while (!BN_is_bit_set(p, e)) {
132 e++;
133 }
134 // we'll set q later (if needed)
135
136 if (e == 1) {
137 // The easy case: (|p|-1)/2 is odd, so 2 has an inverse
138 // modulo (|p|-1)/2, and square roots can be computed
139 // directly by modular exponentiation.
140 // We have
141 // 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
142 // so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
143 if (!BN_rshift(q, p, 2)) {
144 goto end;
145 }
146 q->neg = 0;
147 if (!BN_add_word(q, 1) ||
148 !BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
149 goto end;
150 }
151 err = 0;
152 goto vrfy;
153 }
154
155 if (e == 2) {
156 // |p| == 5 (mod 8)
157 //
158 // In this case 2 is always a non-square since
159 // Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
160 // So if a really is a square, then 2*a is a non-square.
161 // Thus for
162 // b := (2*a)^((|p|-5)/8),
163 // i := (2*a)*b^2
164 // we have
165 // i^2 = (2*a)^((1 + (|p|-5)/4)*2)
166 // = (2*a)^((p-1)/2)
167 // = -1;
168 // so if we set
169 // x := a*b*(i-1),
170 // then
171 // x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
172 // = a^2 * b^2 * (-2*i)
173 // = a*(-i)*(2*a*b^2)
174 // = a*(-i)*i
175 // = a.
176 //
177 // (This is due to A.O.L. Atkin,
178 // <URL:
179 //http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
180 // November 1992.)
181
182 // t := 2*a
183 if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
184 goto end;
185 }
186
187 // b := (2*a)^((|p|-5)/8)
188 if (!BN_rshift(q, p, 3)) {
189 goto end;
190 }
191 q->neg = 0;
192 if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
193 goto end;
194 }
195
196 // y := b^2
197 if (!BN_mod_sqr(y, b, p, ctx)) {
198 goto end;
199 }
200
201 // t := (2*a)*b^2 - 1
202 if (!BN_mod_mul(t, t, y, p, ctx) ||
203 !BN_sub_word(t, 1)) {
204 goto end;
205 }
206
207 // x = a*b*t
208 if (!BN_mod_mul(x, A, b, p, ctx) ||
209 !BN_mod_mul(x, x, t, p, ctx)) {
210 goto end;
211 }
212
213 if (!BN_copy(ret, x)) {
214 goto end;
215 }
216 err = 0;
217 goto vrfy;
218 }
219
220 // e > 2, so we really have to use the Tonelli/Shanks algorithm.
221 // First, find some y that is not a square.
222 if (!BN_copy(q, p)) {
223 goto end; // use 'q' as temp
224 }
225 q->neg = 0;
226 i = 2;
227 do {
228 // For efficiency, try small numbers first;
229 // if this fails, try random numbers.
230 if (i < 22) {
231 if (!BN_set_word(y, i)) {
232 goto end;
233 }
234 } else {
235 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
236 goto end;
237 }
238 if (BN_ucmp(y, p) >= 0) {
239 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
240 goto end;
241 }
242 }
243 // now 0 <= y < |p|
244 if (BN_is_zero(y)) {
245 if (!BN_set_word(y, i)) {
246 goto end;
247 }
248 }
249 }
250
251 r = bn_jacobi(y, q, ctx); // here 'q' is |p|
252 if (r < -1) {
253 goto end;
254 }
255 if (r == 0) {
256 // m divides p
257 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
258 goto end;
259 }
260 } while (r == 1 && ++i < 82);
261
262 if (r != -1) {
263 // Many rounds and still no non-square -- this is more likely
264 // a bug than just bad luck.
265 // Even if p is not prime, we should have found some y
266 // such that r == -1.
267 OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
268 goto end;
269 }
270
271 // Here's our actual 'q':
272 if (!BN_rshift(q, q, e)) {
273 goto end;
274 }
275
276 // Now that we have some non-square, we can find an element
277 // of order 2^e by computing its q'th power.
278 if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
279 goto end;
280 }
281 if (BN_is_one(y)) {
282 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
283 goto end;
284 }
285
286 // Now we know that (if p is indeed prime) there is an integer
287 // k, 0 <= k < 2^e, such that
288 //
289 // a^q * y^k == 1 (mod p).
290 //
291 // As a^q is a square and y is not, k must be even.
292 // q+1 is even, too, so there is an element
293 //
294 // X := a^((q+1)/2) * y^(k/2),
295 //
296 // and it satisfies
297 //
298 // X^2 = a^q * a * y^k
299 // = a,
300 //
301 // so it is the square root that we are looking for.
302
303 // t := (q-1)/2 (note that q is odd)
304 if (!BN_rshift1(t, q)) {
305 goto end;
306 }
307
308 // x := a^((q-1)/2)
309 if (BN_is_zero(t)) { // special case: p = 2^e + 1
310 if (!BN_nnmod(t, A, p, ctx)) {
311 goto end;
312 }
313 if (BN_is_zero(t)) {
314 // special case: a == 0 (mod p)
315 BN_zero(ret);
316 err = 0;
317 goto end;
318 } else if (!BN_one(x)) {
319 goto end;
320 }
321 } else {
322 if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
323 goto end;
324 }
325 if (BN_is_zero(x)) {
326 // special case: a == 0 (mod p)
327 BN_zero(ret);
328 err = 0;
329 goto end;
330 }
331 }
332
333 // b := a*x^2 (= a^q)
334 if (!BN_mod_sqr(b, x, p, ctx) ||
335 !BN_mod_mul(b, b, A, p, ctx)) {
336 goto end;
337 }
338
339 // x := a*x (= a^((q+1)/2))
340 if (!BN_mod_mul(x, x, A, p, ctx)) {
341 goto end;
342 }
343
344 while (1) {
345 // Now b is a^q * y^k for some even k (0 <= k < 2^E
346 // where E refers to the original value of e, which we
347 // don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
348 //
349 // We have a*b = x^2,
350 // y^2^(e-1) = -1,
351 // b^2^(e-1) = 1.
352 if (BN_is_one(b)) {
353 if (!BN_copy(ret, x)) {
354 goto end;
355 }
356 err = 0;
357 goto vrfy;
358 }
359
360 // Find the smallest i, 0 < i < e, such that b^(2^i) = 1
361 for (i = 1; i < e; i++) {
362 if (i == 1) {
363 if (!BN_mod_sqr(t, b, p, ctx)) {
364 goto end;
365 }
366 } else {
367 if (!BN_mod_mul(t, t, t, p, ctx)) {
368 goto end;
369 }
370 }
371 if (BN_is_one(t)) {
372 break;
373 }
374 }
375 // If not found, a is not a square or p is not a prime.
376 if (i >= e) {
377 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
378 goto end;
379 }
380
381 // t := y^2^(e - i - 1)
382 if (!BN_copy(t, y)) {
383 goto end;
384 }
385 for (j = e - i - 1; j > 0; j--) {
386 if (!BN_mod_sqr(t, t, p, ctx)) {
387 goto end;
388 }
389 }
390 if (!BN_mod_mul(y, t, t, p, ctx) ||
391 !BN_mod_mul(x, x, t, p, ctx) ||
392 !BN_mod_mul(b, b, y, p, ctx)) {
393 goto end;
394 }
395
396 // e decreases each iteration, so this loop will terminate.
397 assert(i < e);
398 e = i;
399 }
400
401 vrfy:
402 if (!err) {
403 // Verify the result. The input might have been not a square.
404 if (!BN_mod_sqr(x, ret, p, ctx)) {
405 err = 1;
406 }
407
408 if (!err && 0 != BN_cmp(x, A)) {
409 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
410 err = 1;
411 }
412 }
413
414 end:
415 if (err) {
416 if (ret != in) {
417 BN_clear_free(ret);
418 }
419 ret = NULL;
420 }
421 BN_CTX_end(ctx);
422 return ret;
423 }
424
BN_sqrt(BIGNUM * out_sqrt,const BIGNUM * in,BN_CTX * ctx)425 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
426 BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
427 int ok = 0, last_delta_valid = 0;
428
429 if (in->neg) {
430 OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
431 return 0;
432 }
433 if (BN_is_zero(in)) {
434 BN_zero(out_sqrt);
435 return 1;
436 }
437
438 BN_CTX_start(ctx);
439 if (out_sqrt == in) {
440 estimate = BN_CTX_get(ctx);
441 } else {
442 estimate = out_sqrt;
443 }
444 tmp = BN_CTX_get(ctx);
445 last_delta = BN_CTX_get(ctx);
446 delta = BN_CTX_get(ctx);
447 if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
448 goto err;
449 }
450
451 // We estimate that the square root of an n-bit number is 2^{n/2}.
452 if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
453 goto err;
454 }
455
456 // This is Newton's method for finding a root of the equation |estimate|^2 -
457 // |in| = 0.
458 for (;;) {
459 // |estimate| = 1/2 * (|estimate| + |in|/|estimate|)
460 if (!BN_div(tmp, NULL, in, estimate, ctx) ||
461 !BN_add(tmp, tmp, estimate) ||
462 !BN_rshift1(estimate, tmp) ||
463 // |tmp| = |estimate|^2
464 !BN_sqr(tmp, estimate, ctx) ||
465 // |delta| = |in| - |tmp|
466 !BN_sub(delta, in, tmp)) {
467 OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
468 goto err;
469 }
470
471 delta->neg = 0;
472 // The difference between |in| and |estimate| squared is required to always
473 // decrease. This ensures that the loop always terminates, but I don't have
474 // a proof that it always finds the square root for a given square.
475 if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
476 break;
477 }
478
479 last_delta_valid = 1;
480
481 tmp2 = last_delta;
482 last_delta = delta;
483 delta = tmp2;
484 }
485
486 if (BN_cmp(tmp, in) != 0) {
487 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
488 goto err;
489 }
490
491 ok = 1;
492
493 err:
494 if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
495 ok = 0;
496 }
497 BN_CTX_end(ctx);
498 return ok;
499 }
500