1 | /* |
2 | * Copyright 2000-2018 The OpenSSL Project Authors. All Rights Reserved. |
3 | * |
4 | * Licensed under the Apache License 2.0 (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 |
8 | */ |
9 | |
10 | #include "internal/cryptlib.h" |
11 | #include "bn_local.h" |
12 | |
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! |
18 | */ |
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) { |
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 | */ |
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) { |
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 | * Subject: Square Roots and Cognate Matters modulo p=8n+5. |
129 | * URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026 |
130 | * November 1992.) |
131 | */ |
132 | |
133 | /* t := 2*a */ |
134 | if (!BN_mod_lshift1_quick(t, A, p)) |
135 | goto end; |
136 | |
137 | /* b := (2*a)^((|p|-5)/8) */ |
138 | if (!BN_rshift(q, p, 3)) |
139 | goto end; |
140 | q->neg = 0; |
141 | if (!BN_mod_exp(b, t, q, p, ctx)) |
142 | goto end; |
143 | |
144 | /* y := b^2 */ |
145 | if (!BN_mod_sqr(y, b, p, ctx)) |
146 | goto end; |
147 | |
148 | /* t := (2*a)*b^2 - 1 */ |
149 | if (!BN_mod_mul(t, t, y, p, ctx)) |
150 | goto end; |
151 | if (!BN_sub_word(t, 1)) |
152 | goto end; |
153 | |
154 | /* x = a*b*t */ |
155 | if (!BN_mod_mul(x, A, b, p, ctx)) |
156 | goto end; |
157 | if (!BN_mod_mul(x, x, t, p, ctx)) |
158 | goto end; |
159 | |
160 | if (!BN_copy(ret, x)) |
161 | goto end; |
162 | err = 0; |
163 | goto vrfy; |
164 | } |
165 | |
166 | /* |
167 | * e > 2, so we really have to use the Tonelli/Shanks algorithm. First, |
168 | * find some y that is not a square. |
169 | */ |
170 | if (!BN_copy(q, p)) |
171 | goto end; /* use 'q' as temp */ |
172 | q->neg = 0; |
173 | i = 2; |
174 | do { |
175 | /* |
176 | * For efficiency, try small numbers first; if this fails, try random |
177 | * numbers. |
178 | */ |
179 | if (i < 22) { |
180 | if (!BN_set_word(y, i)) |
181 | goto end; |
182 | } else { |
183 | if (!BN_priv_rand_ex(y, BN_num_bits(p), 0, 0, ctx)) |
184 | goto end; |
185 | if (BN_ucmp(y, p) >= 0) { |
186 | if (!(p->neg ? BN_add : BN_sub) (y, y, p)) |
187 | goto end; |
188 | } |
189 | /* now 0 <= y < |p| */ |
190 | if (BN_is_zero(y)) |
191 | if (!BN_set_word(y, i)) |
192 | goto end; |
193 | } |
194 | |
195 | r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ |
196 | if (r < -1) |
197 | goto end; |
198 | if (r == 0) { |
199 | /* m divides p */ |
200 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); |
201 | goto end; |
202 | } |
203 | } |
204 | while (r == 1 && ++i < 82); |
205 | |
206 | if (r != -1) { |
207 | /* |
208 | * Many rounds and still no non-square -- this is more likely a bug |
209 | * than just bad luck. Even if p is not prime, we should have found |
210 | * some y such that r == -1. |
211 | */ |
212 | BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS); |
213 | goto end; |
214 | } |
215 | |
216 | /* Here's our actual 'q': */ |
217 | if (!BN_rshift(q, q, e)) |
218 | goto end; |
219 | |
220 | /* |
221 | * Now that we have some non-square, we can find an element of order 2^e |
222 | * by computing its q'th power. |
223 | */ |
224 | if (!BN_mod_exp(y, y, q, p, ctx)) |
225 | goto end; |
226 | if (BN_is_one(y)) { |
227 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); |
228 | goto end; |
229 | } |
230 | |
231 | /*- |
232 | * Now we know that (if p is indeed prime) there is an integer |
233 | * k, 0 <= k < 2^e, such that |
234 | * |
235 | * a^q * y^k == 1 (mod p). |
236 | * |
237 | * As a^q is a square and y is not, k must be even. |
238 | * q+1 is even, too, so there is an element |
239 | * |
240 | * X := a^((q+1)/2) * y^(k/2), |
241 | * |
242 | * and it satisfies |
243 | * |
244 | * X^2 = a^q * a * y^k |
245 | * = a, |
246 | * |
247 | * so it is the square root that we are looking for. |
248 | */ |
249 | |
250 | /* t := (q-1)/2 (note that q is odd) */ |
251 | if (!BN_rshift1(t, q)) |
252 | goto end; |
253 | |
254 | /* x := a^((q-1)/2) */ |
255 | if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */ |
256 | if (!BN_nnmod(t, A, p, ctx)) |
257 | goto end; |
258 | if (BN_is_zero(t)) { |
259 | /* special case: a == 0 (mod p) */ |
260 | BN_zero(ret); |
261 | err = 0; |
262 | goto end; |
263 | } else if (!BN_one(x)) |
264 | goto end; |
265 | } else { |
266 | if (!BN_mod_exp(x, A, t, p, ctx)) |
267 | goto end; |
268 | if (BN_is_zero(x)) { |
269 | /* special case: a == 0 (mod p) */ |
270 | BN_zero(ret); |
271 | err = 0; |
272 | goto end; |
273 | } |
274 | } |
275 | |
276 | /* b := a*x^2 (= a^q) */ |
277 | if (!BN_mod_sqr(b, x, p, ctx)) |
278 | goto end; |
279 | if (!BN_mod_mul(b, b, A, p, ctx)) |
280 | goto end; |
281 | |
282 | /* x := a*x (= a^((q+1)/2)) */ |
283 | if (!BN_mod_mul(x, x, A, p, ctx)) |
284 | goto end; |
285 | |
286 | while (1) { |
287 | /*- |
288 | * Now b is a^q * y^k for some even k (0 <= k < 2^E |
289 | * where E refers to the original value of e, which we |
290 | * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). |
291 | * |
292 | * We have a*b = x^2, |
293 | * y^2^(e-1) = -1, |
294 | * b^2^(e-1) = 1. |
295 | */ |
296 | |
297 | if (BN_is_one(b)) { |
298 | if (!BN_copy(ret, x)) |
299 | goto end; |
300 | err = 0; |
301 | goto vrfy; |
302 | } |
303 | |
304 | /* find smallest i such that b^(2^i) = 1 */ |
305 | i = 1; |
306 | if (!BN_mod_sqr(t, b, p, ctx)) |
307 | goto end; |
308 | while (!BN_is_one(t)) { |
309 | i++; |
310 | if (i == e) { |
311 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); |
312 | goto end; |
313 | } |
314 | if (!BN_mod_mul(t, t, t, p, ctx)) |
315 | goto end; |
316 | } |
317 | |
318 | /* t := y^2^(e - i - 1) */ |
319 | if (!BN_copy(t, y)) |
320 | goto end; |
321 | for (j = e - i - 1; j > 0; j--) { |
322 | if (!BN_mod_sqr(t, t, p, ctx)) |
323 | goto end; |
324 | } |
325 | if (!BN_mod_mul(y, t, t, p, ctx)) |
326 | goto end; |
327 | if (!BN_mod_mul(x, x, t, p, ctx)) |
328 | goto end; |
329 | if (!BN_mod_mul(b, b, y, p, ctx)) |
330 | goto end; |
331 | e = i; |
332 | } |
333 | |
334 | vrfy: |
335 | if (!err) { |
336 | /* |
337 | * verify the result -- the input might have been not a square (test |
338 | * added in 0.9.8) |
339 | */ |
340 | |
341 | if (!BN_mod_sqr(x, ret, p, ctx)) |
342 | err = 1; |
343 | |
344 | if (!err && 0 != BN_cmp(x, A)) { |
345 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); |
346 | err = 1; |
347 | } |
348 | } |
349 | |
350 | end: |
351 | if (err) { |
352 | if (ret != in) |
353 | BN_clear_free(ret); |
354 | ret = NULL; |
355 | } |
356 | BN_CTX_end(ctx); |
357 | bn_check_top(ret); |
358 | return ret; |
359 | } |
360 | |