Line data Source code
1 : /* crypto/bn/bn_sqrt.c */
2 : /*
3 : * Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> and Bodo
4 : * Moeller for the OpenSSL project.
5 : */
6 : /* ====================================================================
7 : * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
8 : *
9 : * Redistribution and use in source and binary forms, with or without
10 : * modification, are permitted provided that the following conditions
11 : * are met:
12 : *
13 : * 1. Redistributions of source code must retain the above copyright
14 : * notice, this list of conditions and the following disclaimer.
15 : *
16 : * 2. Redistributions in binary form must reproduce the above copyright
17 : * notice, this list of conditions and the following disclaimer in
18 : * the documentation and/or other materials provided with the
19 : * distribution.
20 : *
21 : * 3. All advertising materials mentioning features or use of this
22 : * software must display the following acknowledgment:
23 : * "This product includes software developed by the OpenSSL Project
24 : * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
25 : *
26 : * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
27 : * endorse or promote products derived from this software without
28 : * prior written permission. For written permission, please contact
29 : * openssl-core@openssl.org.
30 : *
31 : * 5. Products derived from this software may not be called "OpenSSL"
32 : * nor may "OpenSSL" appear in their names without prior written
33 : * permission of the OpenSSL Project.
34 : *
35 : * 6. Redistributions of any form whatsoever must retain the following
36 : * acknowledgment:
37 : * "This product includes software developed by the OpenSSL Project
38 : * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
39 : *
40 : * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
41 : * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
42 : * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
43 : * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
44 : * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
45 : * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
46 : * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
47 : * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
48 : * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
49 : * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
50 : * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
51 : * OF THE POSSIBILITY OF SUCH DAMAGE.
52 : * ====================================================================
53 : *
54 : * This product includes cryptographic software written by Eric Young
55 : * (eay@cryptsoft.com). This product includes software written by Tim
56 : * Hudson (tjh@cryptsoft.com).
57 : *
58 : */
59 :
60 : #include "cryptlib.h"
61 : #include "bn_lcl.h"
62 :
63 0 : BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
64 : /*
65 : * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
66 : * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
67 : * Theory", algorithm 1.5.1). 'p' must be prime!
68 : */
69 : {
70 : BIGNUM *ret = in;
71 : int err = 1;
72 : int r;
73 : BIGNUM *A, *b, *q, *t, *x, *y;
74 : int e, i, j;
75 :
76 0 : if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
77 0 : if (BN_abs_is_word(p, 2)) {
78 0 : if (ret == NULL)
79 0 : ret = BN_new();
80 0 : if (ret == NULL)
81 : goto end;
82 0 : if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
83 0 : if (ret != in)
84 0 : BN_free(ret);
85 : return NULL;
86 : }
87 : bn_check_top(ret);
88 : return ret;
89 : }
90 :
91 0 : BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
92 0 : return (NULL);
93 : }
94 :
95 0 : if (BN_is_zero(a) || BN_is_one(a)) {
96 0 : if (ret == NULL)
97 0 : ret = BN_new();
98 0 : if (ret == NULL)
99 : goto end;
100 0 : if (!BN_set_word(ret, BN_is_one(a))) {
101 0 : if (ret != in)
102 0 : BN_free(ret);
103 : return NULL;
104 : }
105 : bn_check_top(ret);
106 : return ret;
107 : }
108 :
109 0 : BN_CTX_start(ctx);
110 0 : A = BN_CTX_get(ctx);
111 0 : b = BN_CTX_get(ctx);
112 0 : q = BN_CTX_get(ctx);
113 0 : t = BN_CTX_get(ctx);
114 0 : x = BN_CTX_get(ctx);
115 0 : y = BN_CTX_get(ctx);
116 0 : if (y == NULL)
117 : goto end;
118 :
119 0 : if (ret == NULL)
120 0 : ret = BN_new();
121 0 : if (ret == NULL)
122 : goto end;
123 :
124 : /* A = a mod p */
125 0 : if (!BN_nnmod(A, a, p, ctx))
126 : goto end;
127 :
128 : /* now write |p| - 1 as 2^e*q where q is odd */
129 : e = 1;
130 0 : while (!BN_is_bit_set(p, e))
131 0 : e++;
132 : /* we'll set q later (if needed) */
133 :
134 0 : if (e == 1) {
135 : /*-
136 : * The easy case: (|p|-1)/2 is odd, so 2 has an inverse
137 : * modulo (|p|-1)/2, and square roots can be computed
138 : * directly by modular exponentiation.
139 : * We have
140 : * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
141 : * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
142 : */
143 0 : if (!BN_rshift(q, p, 2))
144 : goto end;
145 0 : q->neg = 0;
146 0 : if (!BN_add_word(q, 1))
147 : goto end;
148 0 : if (!BN_mod_exp(ret, A, q, p, ctx))
149 : goto end;
150 : err = 0;
151 : goto vrfy;
152 : }
153 :
154 0 : if (e == 2) {
155 : /*-
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: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
179 : * November 1992.)
180 : */
181 :
182 : /* t := 2*a */
183 0 : if (!BN_mod_lshift1_quick(t, A, p))
184 : goto end;
185 :
186 : /* b := (2*a)^((|p|-5)/8) */
187 0 : if (!BN_rshift(q, p, 3))
188 : goto end;
189 0 : q->neg = 0;
190 0 : if (!BN_mod_exp(b, t, q, p, ctx))
191 : goto end;
192 :
193 : /* y := b^2 */
194 0 : if (!BN_mod_sqr(y, b, p, ctx))
195 : goto end;
196 :
197 : /* t := (2*a)*b^2 - 1 */
198 0 : if (!BN_mod_mul(t, t, y, p, ctx))
199 : goto end;
200 0 : if (!BN_sub_word(t, 1))
201 : goto end;
202 :
203 : /* x = a*b*t */
204 0 : if (!BN_mod_mul(x, A, b, p, ctx))
205 : goto end;
206 0 : if (!BN_mod_mul(x, x, t, p, ctx))
207 : goto end;
208 :
209 0 : if (!BN_copy(ret, x))
210 : goto end;
211 : err = 0;
212 : goto vrfy;
213 : }
214 :
215 : /*
216 : * e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
217 : * find some y that is not a square.
218 : */
219 0 : if (!BN_copy(q, p))
220 : goto end; /* use 'q' as temp */
221 0 : q->neg = 0;
222 : i = 2;
223 : do {
224 : /*
225 : * For efficiency, try small numbers first; if this fails, try random
226 : * numbers.
227 : */
228 0 : if (i < 22) {
229 0 : if (!BN_set_word(y, i))
230 : goto end;
231 : } else {
232 0 : if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
233 : goto end;
234 0 : if (BN_ucmp(y, p) >= 0) {
235 0 : if (!(p->neg ? BN_add : BN_sub) (y, y, p))
236 : goto end;
237 : }
238 : /* now 0 <= y < |p| */
239 0 : if (BN_is_zero(y))
240 0 : if (!BN_set_word(y, i))
241 : goto end;
242 : }
243 :
244 0 : r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
245 0 : if (r < -1)
246 : goto end;
247 0 : if (r == 0) {
248 : /* m divides p */
249 0 : BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
250 0 : goto end;
251 : }
252 : }
253 0 : while (r == 1 && ++i < 82);
254 :
255 0 : if (r != -1) {
256 : /*
257 : * Many rounds and still no non-square -- this is more likely a bug
258 : * than just bad luck. Even if p is not prime, we should have found
259 : * some y such that r == -1.
260 : */
261 0 : BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
262 0 : goto end;
263 : }
264 :
265 : /* Here's our actual 'q': */
266 0 : if (!BN_rshift(q, q, e))
267 : goto end;
268 :
269 : /*
270 : * Now that we have some non-square, we can find an element of order 2^e
271 : * by computing its q'th power.
272 : */
273 0 : if (!BN_mod_exp(y, y, q, p, ctx))
274 : goto end;
275 0 : if (BN_is_one(y)) {
276 0 : BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
277 0 : goto end;
278 : }
279 :
280 : /*-
281 : * Now we know that (if p is indeed prime) there is an integer
282 : * k, 0 <= k < 2^e, such that
283 : *
284 : * a^q * y^k == 1 (mod p).
285 : *
286 : * As a^q is a square and y is not, k must be even.
287 : * q+1 is even, too, so there is an element
288 : *
289 : * X := a^((q+1)/2) * y^(k/2),
290 : *
291 : * and it satisfies
292 : *
293 : * X^2 = a^q * a * y^k
294 : * = a,
295 : *
296 : * so it is the square root that we are looking for.
297 : */
298 :
299 : /* t := (q-1)/2 (note that q is odd) */
300 0 : if (!BN_rshift1(t, q))
301 : goto end;
302 :
303 : /* x := a^((q-1)/2) */
304 0 : if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
305 0 : if (!BN_nnmod(t, A, p, ctx))
306 : goto end;
307 0 : if (BN_is_zero(t)) {
308 : /* special case: a == 0 (mod p) */
309 0 : BN_zero(ret);
310 : err = 0;
311 0 : goto end;
312 0 : } else if (!BN_one(x))
313 : goto end;
314 : } else {
315 0 : if (!BN_mod_exp(x, A, t, p, ctx))
316 : goto end;
317 0 : if (BN_is_zero(x)) {
318 : /* special case: a == 0 (mod p) */
319 0 : BN_zero(ret);
320 : err = 0;
321 0 : goto end;
322 : }
323 : }
324 :
325 : /* b := a*x^2 (= a^q) */
326 0 : if (!BN_mod_sqr(b, x, p, ctx))
327 : goto end;
328 0 : if (!BN_mod_mul(b, b, A, p, ctx))
329 : goto end;
330 :
331 : /* x := a*x (= a^((q+1)/2)) */
332 0 : if (!BN_mod_mul(x, x, A, p, ctx))
333 : goto end;
334 :
335 : while (1) {
336 : /*-
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 0 : if (BN_is_one(b)) {
347 0 : if (!BN_copy(ret, x))
348 : goto end;
349 : err = 0;
350 : goto vrfy;
351 : }
352 :
353 : /* find smallest i such that b^(2^i) = 1 */
354 : i = 1;
355 0 : if (!BN_mod_sqr(t, b, p, ctx))
356 : goto end;
357 0 : while (!BN_is_one(t)) {
358 0 : i++;
359 0 : if (i == e) {
360 0 : BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
361 0 : goto end;
362 : }
363 0 : if (!BN_mod_mul(t, t, t, p, ctx))
364 : goto end;
365 : }
366 :
367 : /* t := y^2^(e - i - 1) */
368 0 : if (!BN_copy(t, y))
369 : goto end;
370 0 : for (j = e - i - 1; j > 0; j--) {
371 0 : if (!BN_mod_sqr(t, t, p, ctx))
372 : goto end;
373 : }
374 0 : if (!BN_mod_mul(y, t, t, p, ctx))
375 : goto end;
376 0 : if (!BN_mod_mul(x, x, t, p, ctx))
377 : goto end;
378 0 : if (!BN_mod_mul(b, b, y, p, ctx))
379 : goto end;
380 : e = i;
381 : }
382 :
383 : vrfy:
384 : if (!err) {
385 : /*
386 : * verify the result -- the input might have been not a square (test
387 : * added in 0.9.8)
388 : */
389 :
390 0 : if (!BN_mod_sqr(x, ret, p, ctx))
391 : err = 1;
392 :
393 0 : if (!err && 0 != BN_cmp(x, A)) {
394 0 : BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
395 : err = 1;
396 : }
397 : }
398 :
399 : end:
400 0 : if (err) {
401 0 : if (ret != NULL && ret != in) {
402 0 : BN_clear_free(ret);
403 : }
404 : ret = NULL;
405 : }
406 0 : BN_CTX_end(ctx);
407 : bn_check_top(ret);
408 0 : return ret;
409 : }
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