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0f113f3e | 1 | /* |
4f22f405 | 2 | * Copyright 2000-2016 The OpenSSL Project Authors. All Rights Reserved. |
cd2eebfd | 3 | * |
4f22f405 RS |
4 | * Licensed under the OpenSSL license (the "License"). You may not use |
5 | * this file except in compliance with the License. You can obtain a copy | |
6 | * in the file LICENSE in the source distribution or at | |
7 | * https://www.openssl.org/source/license.html | |
cd2eebfd BM |
8 | */ |
9 | ||
b39fc560 | 10 | #include "internal/cryptlib.h" |
cd2eebfd BM |
11 | #include "bn_lcl.h" |
12 | ||
0f113f3e MC |
13 | BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) |
14 | /* | |
15 | * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks | |
16 | * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number | |
17 | * Theory", algorithm 1.5.1). 'p' must be prime! | |
cd2eebfd | 18 | */ |
0f113f3e MC |
19 | { |
20 | BIGNUM *ret = in; | |
21 | int err = 1; | |
22 | int r; | |
23 | BIGNUM *A, *b, *q, *t, *x, *y; | |
24 | int e, i, j; | |
25 | ||
26 | if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) { | |
27 | if (BN_abs_is_word(p, 2)) { | |
28 | if (ret == NULL) | |
29 | ret = BN_new(); | |
30 | if (ret == NULL) | |
31 | goto end; | |
32 | if (!BN_set_word(ret, BN_is_bit_set(a, 0))) { | |
33 | if (ret != in) | |
34 | BN_free(ret); | |
35 | return NULL; | |
36 | } | |
37 | bn_check_top(ret); | |
38 | return ret; | |
39 | } | |
40 | ||
41 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
42 | return (NULL); | |
43 | } | |
44 | ||
45 | if (BN_is_zero(a) || BN_is_one(a)) { | |
46 | if (ret == NULL) | |
47 | ret = BN_new(); | |
48 | if (ret == NULL) | |
49 | goto end; | |
50 | if (!BN_set_word(ret, BN_is_one(a))) { | |
51 | if (ret != in) | |
52 | BN_free(ret); | |
53 | return NULL; | |
54 | } | |
55 | bn_check_top(ret); | |
56 | return ret; | |
57 | } | |
58 | ||
59 | BN_CTX_start(ctx); | |
60 | A = BN_CTX_get(ctx); | |
61 | b = BN_CTX_get(ctx); | |
62 | q = BN_CTX_get(ctx); | |
63 | t = BN_CTX_get(ctx); | |
64 | x = BN_CTX_get(ctx); | |
65 | y = BN_CTX_get(ctx); | |
66 | if (y == NULL) | |
67 | goto end; | |
68 | ||
69 | if (ret == NULL) | |
70 | ret = BN_new(); | |
71 | if (ret == NULL) | |
72 | goto end; | |
73 | ||
74 | /* A = a mod p */ | |
75 | if (!BN_nnmod(A, a, p, ctx)) | |
76 | goto end; | |
77 | ||
78 | /* now write |p| - 1 as 2^e*q where q is odd */ | |
79 | e = 1; | |
80 | while (!BN_is_bit_set(p, e)) | |
81 | e++; | |
82 | /* we'll set q later (if needed) */ | |
83 | ||
84 | if (e == 1) { | |
50e735f9 MC |
85 | /*- |
86 | * The easy case: (|p|-1)/2 is odd, so 2 has an inverse | |
87 | * modulo (|p|-1)/2, and square roots can be computed | |
88 | * directly by modular exponentiation. | |
89 | * We have | |
90 | * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2), | |
91 | * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1. | |
92 | */ | |
0f113f3e MC |
93 | if (!BN_rshift(q, p, 2)) |
94 | goto end; | |
95 | q->neg = 0; | |
96 | if (!BN_add_word(q, 1)) | |
97 | goto end; | |
98 | if (!BN_mod_exp(ret, A, q, p, ctx)) | |
99 | goto end; | |
100 | err = 0; | |
101 | goto vrfy; | |
102 | } | |
103 | ||
104 | if (e == 2) { | |
35a1cc90 MC |
105 | /*- |
106 | * |p| == 5 (mod 8) | |
107 | * | |
108 | * In this case 2 is always a non-square since | |
109 | * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. | |
110 | * So if a really is a square, then 2*a is a non-square. | |
111 | * Thus for | |
112 | * b := (2*a)^((|p|-5)/8), | |
113 | * i := (2*a)*b^2 | |
114 | * we have | |
115 | * i^2 = (2*a)^((1 + (|p|-5)/4)*2) | |
116 | * = (2*a)^((p-1)/2) | |
117 | * = -1; | |
118 | * so if we set | |
119 | * x := a*b*(i-1), | |
120 | * then | |
121 | * x^2 = a^2 * b^2 * (i^2 - 2*i + 1) | |
122 | * = a^2 * b^2 * (-2*i) | |
123 | * = a*(-i)*(2*a*b^2) | |
124 | * = a*(-i)*i | |
125 | * = a. | |
126 | * | |
127 | * (This is due to A.O.L. Atkin, | |
128 | * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, | |
129 | * November 1992.) | |
130 | */ | |
0f113f3e MC |
131 | |
132 | /* t := 2*a */ | |
133 | if (!BN_mod_lshift1_quick(t, A, p)) | |
134 | goto end; | |
135 | ||
136 | /* b := (2*a)^((|p|-5)/8) */ | |
137 | if (!BN_rshift(q, p, 3)) | |
138 | goto end; | |
139 | q->neg = 0; | |
140 | if (!BN_mod_exp(b, t, q, p, ctx)) | |
141 | goto end; | |
142 | ||
143 | /* y := b^2 */ | |
144 | if (!BN_mod_sqr(y, b, p, ctx)) | |
145 | goto end; | |
146 | ||
147 | /* t := (2*a)*b^2 - 1 */ | |
148 | if (!BN_mod_mul(t, t, y, p, ctx)) | |
149 | goto end; | |
150 | if (!BN_sub_word(t, 1)) | |
151 | goto end; | |
152 | ||
153 | /* x = a*b*t */ | |
154 | if (!BN_mod_mul(x, A, b, p, ctx)) | |
155 | goto end; | |
156 | if (!BN_mod_mul(x, x, t, p, ctx)) | |
157 | goto end; | |
158 | ||
159 | if (!BN_copy(ret, x)) | |
160 | goto end; | |
161 | err = 0; | |
162 | goto vrfy; | |
163 | } | |
164 | ||
165 | /* | |
166 | * e > 2, so we really have to use the Tonelli/Shanks algorithm. First, | |
167 | * find some y that is not a square. | |
168 | */ | |
169 | if (!BN_copy(q, p)) | |
170 | goto end; /* use 'q' as temp */ | |
171 | q->neg = 0; | |
172 | i = 2; | |
173 | do { | |
174 | /* | |
175 | * For efficiency, try small numbers first; if this fails, try random | |
176 | * numbers. | |
177 | */ | |
178 | if (i < 22) { | |
179 | if (!BN_set_word(y, i)) | |
180 | goto end; | |
181 | } else { | |
182 | if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) | |
183 | goto end; | |
184 | if (BN_ucmp(y, p) >= 0) { | |
185 | if (!(p->neg ? BN_add : BN_sub) (y, y, p)) | |
186 | goto end; | |
187 | } | |
188 | /* now 0 <= y < |p| */ | |
189 | if (BN_is_zero(y)) | |
190 | if (!BN_set_word(y, i)) | |
191 | goto end; | |
192 | } | |
193 | ||
194 | r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ | |
195 | if (r < -1) | |
196 | goto end; | |
197 | if (r == 0) { | |
198 | /* m divides p */ | |
199 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
200 | goto end; | |
201 | } | |
202 | } | |
203 | while (r == 1 && ++i < 82); | |
204 | ||
205 | if (r != -1) { | |
206 | /* | |
207 | * Many rounds and still no non-square -- this is more likely a bug | |
208 | * than just bad luck. Even if p is not prime, we should have found | |
209 | * some y such that r == -1. | |
210 | */ | |
211 | BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS); | |
212 | goto end; | |
213 | } | |
214 | ||
215 | /* Here's our actual 'q': */ | |
216 | if (!BN_rshift(q, q, e)) | |
217 | goto end; | |
218 | ||
219 | /* | |
220 | * Now that we have some non-square, we can find an element of order 2^e | |
221 | * by computing its q'th power. | |
222 | */ | |
223 | if (!BN_mod_exp(y, y, q, p, ctx)) | |
224 | goto end; | |
225 | if (BN_is_one(y)) { | |
226 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); | |
227 | goto end; | |
228 | } | |
229 | ||
50e735f9 MC |
230 | /*- |
231 | * Now we know that (if p is indeed prime) there is an integer | |
232 | * k, 0 <= k < 2^e, such that | |
233 | * | |
234 | * a^q * y^k == 1 (mod p). | |
235 | * | |
236 | * As a^q is a square and y is not, k must be even. | |
237 | * q+1 is even, too, so there is an element | |
238 | * | |
239 | * X := a^((q+1)/2) * y^(k/2), | |
240 | * | |
241 | * and it satisfies | |
242 | * | |
243 | * X^2 = a^q * a * y^k | |
244 | * = a, | |
245 | * | |
246 | * so it is the square root that we are looking for. | |
247 | */ | |
0f113f3e MC |
248 | |
249 | /* t := (q-1)/2 (note that q is odd) */ | |
250 | if (!BN_rshift1(t, q)) | |
251 | goto end; | |
252 | ||
253 | /* x := a^((q-1)/2) */ | |
254 | if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */ | |
255 | if (!BN_nnmod(t, A, p, ctx)) | |
256 | goto end; | |
257 | if (BN_is_zero(t)) { | |
258 | /* special case: a == 0 (mod p) */ | |
259 | BN_zero(ret); | |
260 | err = 0; | |
261 | goto end; | |
262 | } else if (!BN_one(x)) | |
263 | goto end; | |
264 | } else { | |
265 | if (!BN_mod_exp(x, A, t, p, ctx)) | |
266 | goto end; | |
267 | if (BN_is_zero(x)) { | |
268 | /* special case: a == 0 (mod p) */ | |
269 | BN_zero(ret); | |
270 | err = 0; | |
271 | goto end; | |
272 | } | |
273 | } | |
274 | ||
275 | /* b := a*x^2 (= a^q) */ | |
276 | if (!BN_mod_sqr(b, x, p, ctx)) | |
277 | goto end; | |
278 | if (!BN_mod_mul(b, b, A, p, ctx)) | |
279 | goto end; | |
280 | ||
281 | /* x := a*x (= a^((q+1)/2)) */ | |
282 | if (!BN_mod_mul(x, x, A, p, ctx)) | |
283 | goto end; | |
284 | ||
285 | while (1) { | |
50e735f9 MC |
286 | /*- |
287 | * Now b is a^q * y^k for some even k (0 <= k < 2^E | |
288 | * where E refers to the original value of e, which we | |
289 | * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). | |
290 | * | |
291 | * We have a*b = x^2, | |
292 | * y^2^(e-1) = -1, | |
293 | * b^2^(e-1) = 1. | |
294 | */ | |
0f113f3e MC |
295 | |
296 | if (BN_is_one(b)) { | |
297 | if (!BN_copy(ret, x)) | |
298 | goto end; | |
299 | err = 0; | |
300 | goto vrfy; | |
301 | } | |
302 | ||
303 | /* find smallest i such that b^(2^i) = 1 */ | |
304 | i = 1; | |
305 | if (!BN_mod_sqr(t, b, p, ctx)) | |
306 | goto end; | |
307 | while (!BN_is_one(t)) { | |
308 | i++; | |
309 | if (i == e) { | |
310 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); | |
311 | goto end; | |
312 | } | |
313 | if (!BN_mod_mul(t, t, t, p, ctx)) | |
314 | goto end; | |
315 | } | |
316 | ||
317 | /* t := y^2^(e - i - 1) */ | |
318 | if (!BN_copy(t, y)) | |
319 | goto end; | |
320 | for (j = e - i - 1; j > 0; j--) { | |
321 | if (!BN_mod_sqr(t, t, p, ctx)) | |
322 | goto end; | |
323 | } | |
324 | if (!BN_mod_mul(y, t, t, p, ctx)) | |
325 | goto end; | |
326 | if (!BN_mod_mul(x, x, t, p, ctx)) | |
327 | goto end; | |
328 | if (!BN_mod_mul(b, b, y, p, ctx)) | |
329 | goto end; | |
330 | e = i; | |
331 | } | |
cd2eebfd | 332 | |
6fb60a84 | 333 | vrfy: |
0f113f3e MC |
334 | if (!err) { |
335 | /* | |
336 | * verify the result -- the input might have been not a square (test | |
337 | * added in 0.9.8) | |
338 | */ | |
339 | ||
340 | if (!BN_mod_sqr(x, ret, p, ctx)) | |
341 | err = 1; | |
342 | ||
343 | if (!err && 0 != BN_cmp(x, A)) { | |
344 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); | |
345 | err = 1; | |
346 | } | |
347 | } | |
6fb60a84 | 348 | |
cd2eebfd | 349 | end: |
0f113f3e | 350 | if (err) { |
23a1d5e9 | 351 | if (ret != in) |
0f113f3e | 352 | BN_clear_free(ret); |
0f113f3e MC |
353 | ret = NULL; |
354 | } | |
355 | BN_CTX_end(ctx); | |
356 | bn_check_top(ret); | |
357 | return ret; | |
358 | } |