]>
Commit | Line | Data |
---|---|---|
fea4280a | 1 | /* crypto/bn/bn_sqrt.c */ |
ae5c8664 MC |
2 | /* |
3 | * Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> and Bodo | |
4 | * Moeller for the OpenSSL project. | |
5 | */ | |
cd2eebfd BM |
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 | |
ae5c8664 | 14 | * notice, this list of conditions and the following disclaimer. |
cd2eebfd BM |
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 | ||
ae5c8664 MC |
63 | 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! | |
cd2eebfd | 68 | */ |
ae5c8664 MC |
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 | if (BN_abs_is_word(p, 2)) { | |
78 | if (ret == NULL) | |
79 | ret = BN_new(); | |
80 | if (ret == NULL) | |
81 | goto end; | |
82 | if (!BN_set_word(ret, BN_is_bit_set(a, 0))) { | |
83 | if (ret != in) | |
84 | BN_free(ret); | |
85 | return NULL; | |
86 | } | |
87 | bn_check_top(ret); | |
88 | return ret; | |
89 | } | |
90 | ||
91 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
92 | return (NULL); | |
93 | } | |
94 | ||
95 | if (BN_is_zero(a) || BN_is_one(a)) { | |
96 | if (ret == NULL) | |
97 | ret = BN_new(); | |
98 | if (ret == NULL) | |
99 | goto end; | |
100 | if (!BN_set_word(ret, BN_is_one(a))) { | |
101 | if (ret != in) | |
102 | BN_free(ret); | |
103 | return NULL; | |
104 | } | |
105 | bn_check_top(ret); | |
106 | return ret; | |
107 | } | |
108 | ||
109 | BN_CTX_start(ctx); | |
110 | A = BN_CTX_get(ctx); | |
111 | b = BN_CTX_get(ctx); | |
112 | q = BN_CTX_get(ctx); | |
113 | t = BN_CTX_get(ctx); | |
114 | x = BN_CTX_get(ctx); | |
115 | y = BN_CTX_get(ctx); | |
116 | if (y == NULL) | |
117 | goto end; | |
118 | ||
119 | if (ret == NULL) | |
120 | ret = BN_new(); | |
121 | if (ret == NULL) | |
122 | goto end; | |
123 | ||
124 | /* A = a mod p */ | |
125 | 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 | while (!BN_is_bit_set(p, e)) | |
131 | e++; | |
132 | /* we'll set q later (if needed) */ | |
133 | ||
134 | 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 | if (!BN_rshift(q, p, 2)) | |
144 | goto end; | |
145 | q->neg = 0; | |
146 | if (!BN_add_word(q, 1)) | |
147 | goto end; | |
148 | if (!BN_mod_exp(ret, A, q, p, ctx)) | |
149 | goto end; | |
150 | err = 0; | |
151 | goto vrfy; | |
152 | } | |
153 | ||
154 | 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 | if (!BN_mod_lshift1_quick(t, A, p)) | |
184 | goto end; | |
185 | ||
186 | /* b := (2*a)^((|p|-5)/8) */ | |
187 | if (!BN_rshift(q, p, 3)) | |
188 | goto end; | |
189 | q->neg = 0; | |
190 | if (!BN_mod_exp(b, t, q, p, ctx)) | |
191 | goto end; | |
192 | ||
193 | /* y := b^2 */ | |
194 | if (!BN_mod_sqr(y, b, p, ctx)) | |
195 | goto end; | |
196 | ||
197 | /* t := (2*a)*b^2 - 1 */ | |
198 | if (!BN_mod_mul(t, t, y, p, ctx)) | |
199 | goto end; | |
200 | if (!BN_sub_word(t, 1)) | |
201 | goto end; | |
202 | ||
203 | /* x = a*b*t */ | |
204 | if (!BN_mod_mul(x, A, b, p, ctx)) | |
205 | goto end; | |
206 | if (!BN_mod_mul(x, x, t, p, ctx)) | |
207 | goto end; | |
208 | ||
209 | 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 | if (!BN_copy(q, p)) | |
220 | goto end; /* use 'q' as temp */ | |
221 | 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 | 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 ? BN_add : BN_sub) (y, y, p)) | |
236 | goto end; | |
237 | } | |
238 | /* now 0 <= y < |p| */ | |
239 | if (BN_is_zero(y)) | |
240 | if (!BN_set_word(y, i)) | |
241 | goto end; | |
242 | } | |
243 | ||
244 | r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ | |
245 | if (r < -1) | |
246 | goto end; | |
247 | if (r == 0) { | |
248 | /* m divides p */ | |
249 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
250 | goto end; | |
251 | } | |
252 | } | |
253 | while (r == 1 && ++i < 82); | |
254 | ||
255 | 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 | BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS); | |
262 | goto end; | |
263 | } | |
264 | ||
265 | /* Here's our actual 'q': */ | |
266 | 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 | if (!BN_mod_exp(y, y, q, p, ctx)) | |
274 | goto end; | |
275 | if (BN_is_one(y)) { | |
276 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
277 | 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 | if (!BN_rshift1(t, q)) | |
301 | goto end; | |
302 | ||
303 | /* x := a^((q-1)/2) */ | |
304 | if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */ | |
305 | if (!BN_nnmod(t, A, p, ctx)) | |
306 | goto end; | |
307 | if (BN_is_zero(t)) { | |
308 | /* special case: a == 0 (mod p) */ | |
309 | BN_zero(ret); | |
310 | err = 0; | |
311 | goto end; | |
312 | } else if (!BN_one(x)) | |
313 | goto end; | |
314 | } else { | |
315 | if (!BN_mod_exp(x, A, t, p, ctx)) | |
316 | goto end; | |
317 | if (BN_is_zero(x)) { | |
318 | /* special case: a == 0 (mod p) */ | |
319 | BN_zero(ret); | |
320 | err = 0; | |
321 | goto end; | |
322 | } | |
323 | } | |
324 | ||
325 | /* b := a*x^2 (= a^q) */ | |
326 | if (!BN_mod_sqr(b, x, p, ctx)) | |
327 | goto end; | |
328 | if (!BN_mod_mul(b, b, A, p, ctx)) | |
329 | goto end; | |
330 | ||
331 | /* x := a*x (= a^((q+1)/2)) */ | |
332 | 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 | if (BN_is_one(b)) { | |
347 | 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 | if (!BN_mod_sqr(t, b, p, ctx)) | |
356 | goto end; | |
357 | while (!BN_is_one(t)) { | |
358 | i++; | |
359 | if (i == e) { | |
360 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); | |
361 | goto end; | |
362 | } | |
363 | if (!BN_mod_mul(t, t, t, p, ctx)) | |
364 | goto end; | |
365 | } | |
366 | ||
367 | /* t := y^2^(e - i - 1) */ | |
368 | if (!BN_copy(t, y)) | |
369 | goto end; | |
370 | for (j = e - i - 1; j > 0; j--) { | |
371 | if (!BN_mod_sqr(t, t, p, ctx)) | |
372 | goto end; | |
373 | } | |
374 | if (!BN_mod_mul(y, t, t, p, ctx)) | |
375 | goto end; | |
376 | if (!BN_mod_mul(x, x, t, p, ctx)) | |
377 | goto end; | |
378 | if (!BN_mod_mul(b, b, y, p, ctx)) | |
379 | goto end; | |
380 | e = i; | |
381 | } | |
cd2eebfd | 382 | |
6fb60a84 | 383 | vrfy: |
ae5c8664 MC |
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 | if (!BN_mod_sqr(x, ret, p, ctx)) | |
391 | err = 1; | |
392 | ||
393 | if (!err && 0 != BN_cmp(x, A)) { | |
394 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); | |
395 | err = 1; | |
396 | } | |
397 | } | |
6fb60a84 | 398 | |
cd2eebfd | 399 | end: |
ae5c8664 MC |
400 | if (err) { |
401 | if (ret != NULL && ret != in) { | |
402 | BN_clear_free(ret); | |
403 | } | |
404 | ret = NULL; | |
405 | } | |
406 | BN_CTX_end(ctx); | |
407 | bn_check_top(ret); | |
408 | return ret; | |
409 | } |