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
13BIGNUM *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