1 /* crypto/bn/bn_sqrt.c */
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 "cryptlib.h"
59 #include "bn_lcl.h"
60
61
62 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
63 /* Returns 'ret' such that
64 * ret^2 == a (mod p),
65 * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
66 * in Algebraic Computational Number Theory", algorithm 1.5.1).
67 * '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 if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
77 {
78 if (BN_abs_is_word(p, 2))
79 {
80 if (ret == NULL)
81 ret = BN_new();
82 if (ret == NULL)
83 goto end;
84 if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
85 {
86 if (ret != in)
87 BN_free(ret);
88 return NULL;
89 }
90 bn_check_top(ret);
91 return ret;
92 }
93
94 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
95 return(NULL);
96 }
97
98 if (BN_is_zero(a) || BN_is_one(a))
99 {
100 if (ret == NULL)
101 ret = BN_new();
102 if (ret == NULL)
103 goto end;
104 if (!BN_set_word(ret, BN_is_one(a)))
105 {
106 if (ret != in)
107 BN_free(ret);
108 return NULL;
109 }
110 bn_check_top(ret);
111 return ret;
112 }
113
114 BN_CTX_start(ctx);
115 A = BN_CTX_get(ctx);
116 b = BN_CTX_get(ctx);
117 q = BN_CTX_get(ctx);
118 t = BN_CTX_get(ctx);
119 x = BN_CTX_get(ctx);
120 y = BN_CTX_get(ctx);
121 if (y == NULL) goto end;
122
123 if (ret == NULL)
124 ret = BN_new();
125 if (ret == NULL) goto end;
126
127 /* A = a mod p */
128 if (!BN_nnmod(A, a, p, ctx)) goto end;
129
130 /* now write |p| - 1 as 2^e*q where q is odd */
131 e = 1;
132 while (!BN_is_bit_set(p, e))
133 e++;
134 /* we'll set q later (if needed) */
135
136 if (e == 1)
137 {
138 /* The easy case: (|p|-1)/2 is odd, so 2 has an inverse
139 * modulo (|p|-1)/2, and square roots can be computed
140 * directly by modular exponentiation.
141 * We have
142 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
143 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
144 */
145 if (!BN_rshift(q, p, 2)) goto end;
146 q->neg = 0;
147 if (!BN_add_word(q, 1)) goto end;
148 if (!BN_mod_exp(ret, A, q, p, ctx)) goto end;
149 err = 0;
150 goto vrfy;
151 }
152
153 if (e == 2)
154 {
155 /* |p| == 5 (mod 8)
156 *
157 * In this case 2 is always a non-square since
158 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
159 * So if a really is a square, then 2*a is a non-square.
160 * Thus for
161 * b := (2*a)^((|p|-5)/8),
162 * i := (2*a)*b^2
163 * we have
164 * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
165 * = (2*a)^((p-1)/2)
166 * = -1;
167 * so if we set
168 * x := a*b*(i-1),
169 * then
170 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
171 * = a^2 * b^2 * (-2*i)
172 * = a*(-i)*(2*a*b^2)
173 * = a*(-i)*i
174 * = a.
175 *
176 * (This is due to A.O.L. Atkin,
177 * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
178 * November 1992.)
179 */
180
181 /* t := 2*a */
182 if (!BN_mod_lshift1_quick(t, A, p)) goto end;
183
184 /* b := (2*a)^((|p|-5)/8) */
185 if (!BN_rshift(q, p, 3)) goto end;
186 q->neg = 0;
187 if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
188
189 /* y := b^2 */
190 if (!BN_mod_sqr(y, b, p, ctx)) goto end;
191
192 /* t := (2*a)*b^2 - 1*/
193 if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
194 if (!BN_sub_word(t, 1)) goto end;
195
196 /* x = a*b*t */
197 if (!BN_mod_mul(x, A, b, p, ctx)) goto end;
198 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
199
200 if (!BN_copy(ret, x)) goto end;
201 err = 0;
202 goto vrfy;
203 }
204
205 /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
206 * First, find some y that is not a square. */
207 if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
208 q->neg = 0;
209 i = 2;
210 do
211 {
212 /* For efficiency, try small numbers first;
213 * if this fails, try random numbers.
214 */
215 if (i < 22)
216 {
217 if (!BN_set_word(y, i)) goto end;
218 }
219 else
220 {
221 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
222 if (BN_ucmp(y, p) >= 0)
223 {
224 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
225 }
226 /* now 0 <= y < |p| */
227 if (BN_is_zero(y))
228 if (!BN_set_word(y, i)) goto end;
229 }
230
231 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
232 if (r < -1) goto end;
233 if (r == 0)
234 {
235 /* m divides p */
236 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
237 goto end;
238 }
239 }
240 while (r == 1 && ++i < 82);
241
242 if (r != -1)
243 {
244 /* Many rounds and still no non-square -- this is more likely
245 * a bug than just bad luck.
246 * Even if p is not prime, we should have found some y
247 * such that r == -1.
248 */
249 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
250 goto end;
251 }
252
253 /* Here's our actual 'q': */
254 if (!BN_rshift(q, q, e)) goto end;
255
256 /* Now that we have some non-square, we can find an element
257 * of order 2^e by computing its q'th power. */
258 if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
259 if (BN_is_one(y))
260 {
261 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
262 goto end;
263 }
264
265 /* Now we know that (if p is indeed prime) there is an integer
266 * k, 0 <= k < 2^e, such that
267 *
268 * a^q * y^k == 1 (mod p).
269 *
270 * As a^q is a square and y is not, k must be even.
271 * q+1 is even, too, so there is an element
272 *
273 * X := a^((q+1)/2) * y^(k/2),
274 *
275 * and it satisfies
276 *
277 * X^2 = a^q * a * y^k
278 * = a,
279 *
280 * so it is the square root that we are looking for.
281 */
282
283 /* t := (q-1)/2 (note that q is odd) */
284 if (!BN_rshift1(t, q)) goto end;
285
286 /* x := a^((q-1)/2) */
287 if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
288 {
289 if (!BN_nnmod(t, A, p, ctx)) goto end;
290 if (BN_is_zero(t))
291 {
292 /* special case: a == 0 (mod p) */
293 BN_zero(ret);
294 err = 0;
295 goto end;
296 }
297 else
298 if (!BN_one(x)) goto end;
299 }
300 else
301 {
302 if (!BN_mod_exp(x, A, t, p, ctx)) goto end;
303 if (BN_is_zero(x))
304 {
305 /* special case: a == 0 (mod p) */
306 BN_zero(ret);
307 err = 0;
308 goto end;
309 }
310 }
311
312 /* b := a*x^2 (= a^q) */
313 if (!BN_mod_sqr(b, x, p, ctx)) goto end;
314 if (!BN_mod_mul(b, b, A, p, ctx)) goto end;
315
316 /* x := a*x (= a^((q+1)/2)) */
317 if (!BN_mod_mul(x, x, A, p, ctx)) goto end;
318
319 while (1)
320 {
321 /* Now b is a^q * y^k for some even k (0 <= k < 2^E
322 * where E refers to the original value of e, which we
323 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
324 *
325 * We have a*b = x^2,
326 * y^2^(e-1) = -1,
327 * b^2^(e-1) = 1.
328 */
329
330 if (BN_is_one(b))
331 {
332 if (!BN_copy(ret, x)) goto end;
333 err = 0;
334 goto vrfy;
335 }
336
337
338 /* find smallest i such that b^(2^i) = 1 */
339 i = 1;
340 if (!BN_mod_sqr(t, b, p, ctx)) goto end;
341 while (!BN_is_one(t))
342 {
343 i++;
344 if (i == e)
345 {
346 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
347 goto end;
348 }
349 if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
350 }
351
352
353 /* t := y^2^(e - i - 1) */
354 if (!BN_copy(t, y)) goto end;
355 for (j = e - i - 1; j > 0; j--)
356 {
357 if (!BN_mod_sqr(t, t, p, ctx)) goto end;
358 }
359 if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
360 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
361 if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
362 e = i;
363 }
364
365 vrfy:
366 if (!err)
367 {
368 /* verify the result -- the input might have been not a square
369 * (test added in 0.9.8) */
370
371 if (!BN_mod_sqr(x, ret, p, ctx))
372 err = 1;
373
374 if (!err && 0 != BN_cmp(x, A))
375 {
376 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
377 err = 1;
378 }
379 }
380
381 end:
382 if (err)
383 {
384 if (ret != NULL && ret != in)
385 {
386 BN_clear_free(ret);
387 }
388 ret = NULL;
389 }
390 BN_CTX_end(ctx);
391 bn_check_top(ret);
392 return ret;
393 }