1 /* $OpenBSD: bn_sqrt.c,v 1.11 2022/06/20 15:02:21 tb Exp $ */
2 /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
3 * and Bodo Moeller for the OpenSSL project. */
4 /* ====================================================================
5 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 *
11 * 1. Redistributions of source code must retain the above copyright
12 * notice, this list of conditions and the following disclaimer.
13 *
14 * 2. Redistributions in binary form must reproduce the above copyright
15 * notice, this list of conditions and the following disclaimer in
16 * the documentation and/or other materials provided with the
17 * distribution.
18 *
19 * 3. All advertising materials mentioning features or use of this
20 * software must display the following acknowledgment:
21 * "This product includes software developed by the OpenSSL Project
22 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
23 *
24 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
25 * endorse or promote products derived from this software without
26 * prior written permission. For written permission, please contact
27 * openssl-core@openssl.org.
28 *
29 * 5. Products derived from this software may not be called "OpenSSL"
30 * nor may "OpenSSL" appear in their names without prior written
31 * permission of the OpenSSL Project.
32 *
33 * 6. Redistributions of any form whatsoever must retain the following
34 * acknowledgment:
35 * "This product includes software developed by the OpenSSL Project
36 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
37 *
38 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
39 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
40 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
41 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
42 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
43 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
44 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
45 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
46 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
47 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
48 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
49 * OF THE POSSIBILITY OF SUCH DAMAGE.
50 * ====================================================================
51 *
52 * This product includes cryptographic software written by Eric Young
53 * (eay@cryptsoft.com). This product includes software written by Tim
54 * Hudson (tjh@cryptsoft.com).
55 *
56 */
57
58 #include <openssl/err.h>
59
60 #include "bn_lcl.h"
61
62 BIGNUM *
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)63 BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
64 /* Returns 'ret' such that
65 * ret^2 == a (mod p),
66 * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
67 * in Algebraic Computational Number Theory", algorithm 1.5.1).
68 * 'p' must be prime!
69 */
70 {
71 BIGNUM *ret = in;
72 int err = 1;
73 int r;
74 BIGNUM *A, *b, *q, *t, *x, *y;
75 int e, i, j;
76
77 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
78 if (BN_abs_is_word(p, 2)) {
79 if (ret == NULL)
80 ret = BN_new();
81 if (ret == NULL)
82 goto end;
83 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
84 if (ret != in)
85 BN_free(ret);
86 return NULL;
87 }
88 bn_check_top(ret);
89 return ret;
90 }
91
92 BNerror(BN_R_P_IS_NOT_PRIME);
93 return (NULL);
94 }
95
96 if (BN_is_zero(a) || BN_is_one(a)) {
97 if (ret == NULL)
98 ret = BN_new();
99 if (ret == NULL)
100 goto end;
101 if (!BN_set_word(ret, BN_is_one(a))) {
102 if (ret != in)
103 BN_free(ret);
104 return NULL;
105 }
106 bn_check_top(ret);
107 return ret;
108 }
109
110 BN_CTX_start(ctx);
111 if ((A = BN_CTX_get(ctx)) == NULL)
112 goto end;
113 if ((b = BN_CTX_get(ctx)) == NULL)
114 goto end;
115 if ((q = BN_CTX_get(ctx)) == NULL)
116 goto end;
117 if ((t = BN_CTX_get(ctx)) == NULL)
118 goto end;
119 if ((x = BN_CTX_get(ctx)) == NULL)
120 goto end;
121 if ((y = BN_CTX_get(ctx)) == NULL)
122 goto end;
123
124 if (ret == NULL)
125 ret = BN_new();
126 if (ret == NULL)
127 goto end;
128
129 /* A = a mod p */
130 if (!BN_nnmod(A, a, p, ctx))
131 goto end;
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 /* we'll set q later (if needed) */
138
139 if (e == 1) {
140 /* The easy case: (|p|-1)/2 is odd, so 2 has an inverse
141 * modulo (|p|-1)/2, and square roots can be computed
142 * directly by modular exponentiation.
143 * We have
144 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
145 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
146 */
147 if (!BN_rshift(q, p, 2))
148 goto end;
149 q->neg = 0;
150 if (!BN_add_word(q, 1))
151 goto end;
152 if (!BN_mod_exp_ct(ret, A, q, p, ctx))
153 goto end;
154 err = 0;
155 goto vrfy;
156 }
157
158 if (e == 2) {
159 /* |p| == 5 (mod 8)
160 *
161 * In this case 2 is always a non-square since
162 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
163 * So if a really is a square, then 2*a is a non-square.
164 * Thus for
165 * b := (2*a)^((|p|-5)/8),
166 * i := (2*a)*b^2
167 * we have
168 * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
169 * = (2*a)^((p-1)/2)
170 * = -1;
171 * so if we set
172 * x := a*b*(i-1),
173 * then
174 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
175 * = a^2 * b^2 * (-2*i)
176 * = a*(-i)*(2*a*b^2)
177 * = a*(-i)*i
178 * = a.
179 *
180 * (This is due to A.O.L. Atkin,
181 * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
182 * November 1992.)
183 */
184
185 /* t := 2*a */
186 if (!BN_mod_lshift1_quick(t, A, p))
187 goto end;
188
189 /* b := (2*a)^((|p|-5)/8) */
190 if (!BN_rshift(q, p, 3))
191 goto end;
192 q->neg = 0;
193 if (!BN_mod_exp_ct(b, t, q, p, ctx))
194 goto end;
195
196 /* y := b^2 */
197 if (!BN_mod_sqr(y, b, p, ctx))
198 goto end;
199
200 /* t := (2*a)*b^2 - 1*/
201 if (!BN_mod_mul(t, t, y, p, ctx))
202 goto end;
203 if (!BN_sub_word(t, 1))
204 goto end;
205
206 /* x = a*b*t */
207 if (!BN_mod_mul(x, A, b, p, ctx))
208 goto end;
209 if (!BN_mod_mul(x, x, t, p, ctx))
210 goto end;
211
212 if (!BN_copy(ret, x))
213 goto end;
214 err = 0;
215 goto vrfy;
216 }
217
218 /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
219 * First, find some y that is not a square. */
220 if (!BN_copy(q, p)) /* use 'q' as temp */
221 goto end;
222 q->neg = 0;
223 i = 2;
224 do {
225 /* For efficiency, try small numbers first;
226 * if this fails, try random numbers.
227 */
228 if (i < 22) {
229 if (!BN_set_word(y, i))
230 goto end;
231 } else {
232 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
233 goto end;
234 if (BN_ucmp(y, p) >= 0) {
235 if (p->neg) {
236 if (!BN_add(y, y, p))
237 goto end;
238 } else {
239 if (!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 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
250 if (r < -1)
251 goto end;
252 if (r == 0) {
253 /* m divides p */
254 BNerror(BN_R_P_IS_NOT_PRIME);
255 goto end;
256 }
257 } while (r == 1 && ++i < 82);
258
259 if (r != -1) {
260 /* Many rounds and still no non-square -- this is more likely
261 * a bug than just bad luck.
262 * Even if p is not prime, we should have found some y
263 * such that r == -1.
264 */
265 BNerror(BN_R_TOO_MANY_ITERATIONS);
266 goto end;
267 }
268
269 /* Here's our actual 'q': */
270 if (!BN_rshift(q, q, e))
271 goto end;
272
273 /* Now that we have some non-square, we can find an element
274 * of order 2^e by computing its q'th power. */
275 if (!BN_mod_exp_ct(y, y, q, p, ctx))
276 goto end;
277 if (BN_is_one(y)) {
278 BNerror(BN_R_P_IS_NOT_PRIME);
279 goto end;
280 }
281
282 /* Now we know that (if p is indeed prime) there is an integer
283 * k, 0 <= k < 2^e, such that
284 *
285 * a^q * y^k == 1 (mod p).
286 *
287 * As a^q is a square and y is not, k must be even.
288 * q+1 is even, too, so there is an element
289 *
290 * X := a^((q+1)/2) * y^(k/2),
291 *
292 * and it satisfies
293 *
294 * X^2 = a^q * a * y^k
295 * = a,
296 *
297 * so it is the square root that we are looking for.
298 */
299
300 /* t := (q-1)/2 (note that q is odd) */
301 if (!BN_rshift1(t, q))
302 goto end;
303
304 /* x := a^((q-1)/2) */
305 if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
306 if (!BN_nnmod(t, A, p, ctx))
307 goto end;
308 if (BN_is_zero(t)) {
309 /* special case: a == 0 (mod p) */
310 BN_zero(ret);
311 err = 0;
312 goto end;
313 } else if (!BN_one(x))
314 goto end;
315 } else {
316 if (!BN_mod_exp_ct(x, A, t, p, ctx))
317 goto end;
318 if (BN_is_zero(x)) {
319 /* special case: a == 0 (mod p) */
320 BN_zero(ret);
321 err = 0;
322 goto end;
323 }
324 }
325
326 /* b := a*x^2 (= a^q) */
327 if (!BN_mod_sqr(b, x, p, ctx))
328 goto end;
329 if (!BN_mod_mul(b, b, A, p, ctx))
330 goto end;
331
332 /* x := a*x (= a^((q+1)/2)) */
333 if (!BN_mod_mul(x, x, A, p, ctx))
334 goto end;
335
336 while (1) {
337 /* Now b is a^q * y^k for some even k (0 <= k < 2^E
338 * where E refers to the original value of e, which we
339 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
340 *
341 * We have a*b = x^2,
342 * y^2^(e-1) = -1,
343 * b^2^(e-1) = 1.
344 */
345
346 if (BN_is_one(b)) {
347 if (!BN_copy(ret, x))
348 goto end;
349 err = 0;
350 goto vrfy;
351 }
352
353 /* Find the smallest i with 0 < i < e such that b^(2^i) = 1. */
354 for (i = 1; i < e; i++) {
355 if (i == 1) {
356 if (!BN_mod_sqr(t, b, p, ctx))
357 goto end;
358 } else {
359 if (!BN_mod_sqr(t, t, p, ctx))
360 goto end;
361 }
362 if (BN_is_one(t))
363 break;
364 }
365 if (i >= e) {
366 BNerror(BN_R_NOT_A_SQUARE);
367 goto end;
368 }
369
370 /* t := y^2^(e - i - 1) */
371 if (!BN_copy(t, y))
372 goto end;
373 for (j = e - i - 1; j > 0; j--) {
374 if (!BN_mod_sqr(t, t, p, ctx))
375 goto end;
376 }
377 if (!BN_mod_mul(y, t, t, p, ctx))
378 goto end;
379 if (!BN_mod_mul(x, x, t, p, ctx))
380 goto end;
381 if (!BN_mod_mul(b, b, y, p, ctx))
382 goto end;
383 e = i;
384 }
385
386 vrfy:
387 if (!err) {
388 /* verify the result -- the input might have been not a square
389 * (test added in 0.9.8) */
390
391 if (!BN_mod_sqr(x, ret, p, ctx))
392 err = 1;
393
394 if (!err && 0 != BN_cmp(x, A)) {
395 BNerror(BN_R_NOT_A_SQUARE);
396 err = 1;
397 }
398 }
399
400 end:
401 if (err) {
402 if (ret != NULL && ret != in) {
403 BN_clear_free(ret);
404 }
405 ret = NULL;
406 }
407 BN_CTX_end(ctx);
408 bn_check_top(ret);
409 return ret;
410 }
411