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cd2eebfd BM |
1 | /* crypto/bn/bn_mod.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; | |
6fb60a84 | 73 | BIGNUM *A, *b, *q, *t, *x, *y; |
cd2eebfd BM |
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 | BN_free(ret); | |
87 | return NULL; | |
88 | } | |
d870740c | 89 | bn_check_top(ret); |
cd2eebfd BM |
90 | return ret; |
91 | } | |
92 | ||
93 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
94 | return(NULL); | |
95 | } | |
96 | ||
bac68541 BM |
97 | if (BN_is_zero(a) || BN_is_one(a)) |
98 | { | |
99 | if (ret == NULL) | |
100 | ret = BN_new(); | |
101 | if (ret == NULL) | |
102 | goto end; | |
103 | if (!BN_set_word(ret, BN_is_one(a))) | |
104 | { | |
105 | BN_free(ret); | |
106 | return NULL; | |
107 | } | |
d870740c | 108 | bn_check_top(ret); |
bac68541 BM |
109 | return ret; |
110 | } | |
111 | ||
cd2eebfd | 112 | BN_CTX_start(ctx); |
6fb60a84 | 113 | A = BN_CTX_get(ctx); |
cd2eebfd BM |
114 | b = BN_CTX_get(ctx); |
115 | q = BN_CTX_get(ctx); | |
116 | t = BN_CTX_get(ctx); | |
117 | x = BN_CTX_get(ctx); | |
118 | y = BN_CTX_get(ctx); | |
119 | if (y == NULL) goto end; | |
120 | ||
121 | if (ret == NULL) | |
122 | ret = BN_new(); | |
123 | if (ret == NULL) goto end; | |
124 | ||
6fb60a84 BM |
125 | /* A = a mod p */ |
126 | if (!BN_nnmod(A, a, p, ctx)) goto end; | |
127 | ||
cd2eebfd BM |
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++; | |
80d89e6a | 132 | /* we'll set q later (if needed) */ |
cd2eebfd BM |
133 | |
134 | if (e == 1) | |
135 | { | |
80d89e6a BM |
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 | |
cd2eebfd BM |
138 | * directly by modular exponentiation. |
139 | * We have | |
80d89e6a BM |
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. | |
cd2eebfd | 142 | */ |
bac68541 | 143 | if (!BN_rshift(q, p, 2)) goto end; |
bc5f2740 | 144 | q->neg = 0; |
bac68541 | 145 | if (!BN_add_word(q, 1)) goto end; |
6fb60a84 | 146 | if (!BN_mod_exp(ret, A, q, p, ctx)) goto end; |
cd2eebfd | 147 | err = 0; |
6fb60a84 | 148 | goto vrfy; |
cd2eebfd BM |
149 | } |
150 | ||
bac68541 BM |
151 | if (e == 2) |
152 | { | |
80d89e6a | 153 | /* |p| == 5 (mod 8) |
bac68541 BM |
154 | * |
155 | * In this case 2 is always a non-square since | |
156 | * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. | |
157 | * So if a really is a square, then 2*a is a non-square. | |
158 | * Thus for | |
80d89e6a | 159 | * b := (2*a)^((|p|-5)/8), |
bac68541 BM |
160 | * i := (2*a)*b^2 |
161 | * we have | |
80d89e6a | 162 | * i^2 = (2*a)^((1 + (|p|-5)/4)*2) |
bac68541 BM |
163 | * = (2*a)^((p-1)/2) |
164 | * = -1; | |
165 | * so if we set | |
166 | * x := a*b*(i-1), | |
167 | * then | |
168 | * x^2 = a^2 * b^2 * (i^2 - 2*i + 1) | |
169 | * = a^2 * b^2 * (-2*i) | |
170 | * = a*(-i)*(2*a*b^2) | |
171 | * = a*(-i)*i | |
172 | * = a. | |
173 | * | |
174 | * (This is due to A.O.L. Atkin, | |
175 | * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, | |
176 | * November 1992.) | |
177 | */ | |
178 | ||
bac68541 | 179 | /* t := 2*a */ |
6fb60a84 | 180 | if (!BN_mod_lshift1_quick(t, A, p)) goto end; |
bac68541 | 181 | |
80d89e6a | 182 | /* b := (2*a)^((|p|-5)/8) */ |
bac68541 | 183 | if (!BN_rshift(q, p, 3)) goto end; |
bc5f2740 | 184 | q->neg = 0; |
bac68541 BM |
185 | if (!BN_mod_exp(b, t, q, p, ctx)) goto end; |
186 | ||
187 | /* y := b^2 */ | |
188 | if (!BN_mod_sqr(y, b, p, ctx)) goto end; | |
189 | ||
190 | /* t := (2*a)*b^2 - 1*/ | |
191 | if (!BN_mod_mul(t, t, y, p, ctx)) goto end; | |
aa66eba7 | 192 | if (!BN_sub_word(t, 1)) goto end; |
bac68541 BM |
193 | |
194 | /* x = a*b*t */ | |
6fb60a84 | 195 | if (!BN_mod_mul(x, A, b, p, ctx)) goto end; |
bac68541 BM |
196 | if (!BN_mod_mul(x, x, t, p, ctx)) goto end; |
197 | ||
198 | if (!BN_copy(ret, x)) goto end; | |
199 | err = 0; | |
6fb60a84 | 200 | goto vrfy; |
bac68541 BM |
201 | } |
202 | ||
203 | /* e > 2, so we really have to use the Tonelli/Shanks algorithm. | |
cd2eebfd | 204 | * First, find some y that is not a square. */ |
80d89e6a BM |
205 | if (!BN_copy(q, p)) goto end; /* use 'q' as temp */ |
206 | q->neg = 0; | |
25439b76 | 207 | i = 2; |
cd2eebfd BM |
208 | do |
209 | { | |
210 | /* For efficiency, try small numbers first; | |
211 | * if this fails, try random numbers. | |
212 | */ | |
25439b76 | 213 | if (i < 22) |
cd2eebfd BM |
214 | { |
215 | if (!BN_set_word(y, i)) goto end; | |
216 | } | |
217 | else | |
218 | { | |
219 | if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end; | |
220 | if (BN_ucmp(y, p) >= 0) | |
221 | { | |
222 | if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end; | |
223 | } | |
224 | /* now 0 <= y < |p| */ | |
225 | if (BN_is_zero(y)) | |
226 | if (!BN_set_word(y, i)) goto end; | |
227 | } | |
228 | ||
80d89e6a | 229 | r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ |
cd2eebfd BM |
230 | if (r < -1) goto end; |
231 | if (r == 0) | |
232 | { | |
233 | /* m divides p */ | |
234 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
235 | goto end; | |
236 | } | |
237 | } | |
25439b76 | 238 | while (r == 1 && ++i < 82); |
cd2eebfd BM |
239 | |
240 | if (r != -1) | |
241 | { | |
242 | /* Many rounds and still no non-square -- this is more likely | |
243 | * a bug than just bad luck. | |
244 | * Even if p is not prime, we should have found some y | |
245 | * such that r == -1. | |
246 | */ | |
247 | BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS); | |
248 | goto end; | |
249 | } | |
250 | ||
80d89e6a BM |
251 | /* Here's our actual 'q': */ |
252 | if (!BN_rshift(q, q, e)) goto end; | |
cd2eebfd BM |
253 | |
254 | /* Now that we have some non-square, we can find an element | |
255 | * of order 2^e by computing its q'th power. */ | |
256 | if (!BN_mod_exp(y, y, q, p, ctx)) goto end; | |
257 | if (BN_is_one(y)) | |
258 | { | |
259 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
260 | goto end; | |
261 | } | |
262 | ||
263 | /* Now we know that (if p is indeed prime) there is an integer | |
264 | * k, 0 <= k < 2^e, such that | |
265 | * | |
266 | * a^q * y^k == 1 (mod p). | |
267 | * | |
268 | * As a^q is a square and y is not, k must be even. | |
269 | * q+1 is even, too, so there is an element | |
270 | * | |
271 | * X := a^((q+1)/2) * y^(k/2), | |
272 | * | |
273 | * and it satisfies | |
274 | * | |
275 | * X^2 = a^q * a * y^k | |
276 | * = a, | |
277 | * | |
278 | * so it is the square root that we are looking for. | |
279 | */ | |
280 | ||
281 | /* t := (q-1)/2 (note that q is odd) */ | |
282 | if (!BN_rshift1(t, q)) goto end; | |
283 | ||
284 | /* x := a^((q-1)/2) */ | |
285 | if (BN_is_zero(t)) /* special case: p = 2^e + 1 */ | |
286 | { | |
6fb60a84 | 287 | if (!BN_nnmod(t, A, p, ctx)) goto end; |
cd2eebfd BM |
288 | if (BN_is_zero(t)) |
289 | { | |
290 | /* special case: a == 0 (mod p) */ | |
b6358c89 | 291 | BN_zero(ret); |
cd2eebfd BM |
292 | err = 0; |
293 | goto end; | |
294 | } | |
295 | else | |
296 | if (!BN_one(x)) goto end; | |
297 | } | |
298 | else | |
299 | { | |
6fb60a84 | 300 | if (!BN_mod_exp(x, A, t, p, ctx)) goto end; |
cd2eebfd BM |
301 | if (BN_is_zero(x)) |
302 | { | |
303 | /* special case: a == 0 (mod p) */ | |
b6358c89 | 304 | BN_zero(ret); |
cd2eebfd BM |
305 | err = 0; |
306 | goto end; | |
307 | } | |
308 | } | |
309 | ||
310 | /* b := a*x^2 (= a^q) */ | |
311 | if (!BN_mod_sqr(b, x, p, ctx)) goto end; | |
6fb60a84 | 312 | if (!BN_mod_mul(b, b, A, p, ctx)) goto end; |
cd2eebfd BM |
313 | |
314 | /* x := a*x (= a^((q+1)/2)) */ | |
6fb60a84 | 315 | if (!BN_mod_mul(x, x, A, p, ctx)) goto end; |
cd2eebfd BM |
316 | |
317 | while (1) | |
318 | { | |
319 | /* Now b is a^q * y^k for some even k (0 <= k < 2^E | |
320 | * where E refers to the original value of e, which we | |
321 | * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). | |
322 | * | |
323 | * We have a*b = x^2, | |
324 | * y^2^(e-1) = -1, | |
325 | * b^2^(e-1) = 1. | |
326 | */ | |
327 | ||
328 | if (BN_is_one(b)) | |
329 | { | |
330 | if (!BN_copy(ret, x)) goto end; | |
331 | err = 0; | |
6fb60a84 | 332 | goto vrfy; |
cd2eebfd BM |
333 | } |
334 | ||
335 | ||
336 | /* find smallest i such that b^(2^i) = 1 */ | |
337 | i = 1; | |
338 | if (!BN_mod_sqr(t, b, p, ctx)) goto end; | |
339 | while (!BN_is_one(t)) | |
340 | { | |
341 | i++; | |
342 | if (i == e) | |
343 | { | |
344 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); | |
345 | goto end; | |
346 | } | |
347 | if (!BN_mod_mul(t, t, t, p, ctx)) goto end; | |
348 | } | |
349 | ||
350 | ||
351 | /* t := y^2^(e - i - 1) */ | |
352 | if (!BN_copy(t, y)) goto end; | |
353 | for (j = e - i - 1; j > 0; j--) | |
354 | { | |
355 | if (!BN_mod_sqr(t, t, p, ctx)) goto end; | |
356 | } | |
357 | if (!BN_mod_mul(y, t, t, p, ctx)) goto end; | |
358 | if (!BN_mod_mul(x, x, t, p, ctx)) goto end; | |
359 | if (!BN_mod_mul(b, b, y, p, ctx)) goto end; | |
360 | e = i; | |
361 | } | |
362 | ||
6fb60a84 BM |
363 | vrfy: |
364 | if (!err) | |
365 | { | |
366 | /* verify the result -- the input might have been not a square | |
367 | * (test added in 0.9.8) */ | |
368 | ||
369 | if (!BN_mod_sqr(x, ret, p, ctx)) | |
370 | err = 1; | |
371 | ||
372 | if (!err && 0 != BN_cmp(x, A)) | |
373 | { | |
374 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); | |
375 | err = 1; | |
376 | } | |
377 | } | |
378 | ||
cd2eebfd BM |
379 | end: |
380 | if (err) | |
381 | { | |
382 | if (ret != NULL && ret != in) | |
383 | { | |
384 | BN_clear_free(ret); | |
385 | } | |
386 | ret = NULL; | |
387 | } | |
388 | BN_CTX_end(ctx); | |
d870740c | 389 | bn_check_top(ret); |
cd2eebfd BM |
390 | return ret; |
391 | } |