1 /*
2 * Copyright 2000-2022 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
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)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, otherwise an error or
18 * an incorrect "result" will be returned.
19 */
20 {
21 BIGNUM *ret = in;
22 int err = 1;
23 int r;
24 BIGNUM *A, *b, *q, *t, *x, *y;
25 int e, i, j;
26 int used_ctx = 0;
27
28 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
29 if (BN_abs_is_word(p, 2)) {
30 if (ret == NULL)
31 ret = BN_new();
32 if (ret == NULL)
33 goto end;
34 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
35 if (ret != in)
36 BN_free(ret);
37 return NULL;
38 }
39 bn_check_top(ret);
40 return ret;
41 }
42
43 ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME);
44 return NULL;
45 }
46
47 if (BN_is_zero(a) || BN_is_one(a)) {
48 if (ret == NULL)
49 ret = BN_new();
50 if (ret == NULL)
51 goto end;
52 if (!BN_set_word(ret, BN_is_one(a))) {
53 if (ret != in)
54 BN_free(ret);
55 return NULL;
56 }
57 bn_check_top(ret);
58 return ret;
59 }
60
61 BN_CTX_start(ctx);
62 used_ctx = 1;
63 A = BN_CTX_get(ctx);
64 b = BN_CTX_get(ctx);
65 q = BN_CTX_get(ctx);
66 t = BN_CTX_get(ctx);
67 x = BN_CTX_get(ctx);
68 y = BN_CTX_get(ctx);
69 if (y == NULL)
70 goto end;
71
72 if (ret == NULL)
73 ret = BN_new();
74 if (ret == NULL)
75 goto end;
76
77 /* A = a mod p */
78 if (!BN_nnmod(A, a, p, ctx))
79 goto end;
80
81 /* now write |p| - 1 as 2^e*q where q is odd */
82 e = 1;
83 while (!BN_is_bit_set(p, e))
84 e++;
85 /* we'll set q later (if needed) */
86
87 if (e == 1) {
88 /*-
89 * The easy case: (|p|-1)/2 is odd, so 2 has an inverse
90 * modulo (|p|-1)/2, and square roots can be computed
91 * directly by modular exponentiation.
92 * We have
93 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
94 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
95 */
96 if (!BN_rshift(q, p, 2))
97 goto end;
98 q->neg = 0;
99 if (!BN_add_word(q, 1))
100 goto end;
101 if (!BN_mod_exp(ret, A, q, p, ctx))
102 goto end;
103 err = 0;
104 goto vrfy;
105 }
106
107 if (e == 2) {
108 /*-
109 * |p| == 5 (mod 8)
110 *
111 * In this case 2 is always a non-square since
112 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
113 * So if a really is a square, then 2*a is a non-square.
114 * Thus for
115 * b := (2*a)^((|p|-5)/8),
116 * i := (2*a)*b^2
117 * we have
118 * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
119 * = (2*a)^((p-1)/2)
120 * = -1;
121 * so if we set
122 * x := a*b*(i-1),
123 * then
124 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
125 * = a^2 * b^2 * (-2*i)
126 * = a*(-i)*(2*a*b^2)
127 * = a*(-i)*i
128 * = a.
129 *
130 * (This is due to A.O.L. Atkin,
131 * Subject: Square Roots and Cognate Matters modulo p=8n+5.
132 * URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026
133 * November 1992.)
134 */
135
136 /* t := 2*a */
137 if (!BN_mod_lshift1_quick(t, A, p))
138 goto end;
139
140 /* b := (2*a)^((|p|-5)/8) */
141 if (!BN_rshift(q, p, 3))
142 goto end;
143 q->neg = 0;
144 if (!BN_mod_exp(b, t, q, p, ctx))
145 goto end;
146
147 /* y := b^2 */
148 if (!BN_mod_sqr(y, b, p, ctx))
149 goto end;
150
151 /* t := (2*a)*b^2 - 1 */
152 if (!BN_mod_mul(t, t, y, p, ctx))
153 goto end;
154 if (!BN_sub_word(t, 1))
155 goto end;
156
157 /* x = a*b*t */
158 if (!BN_mod_mul(x, A, b, p, ctx))
159 goto end;
160 if (!BN_mod_mul(x, x, t, p, ctx))
161 goto end;
162
163 if (!BN_copy(ret, x))
164 goto end;
165 err = 0;
166 goto vrfy;
167 }
168
169 /*
170 * e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
171 * find some y that is not a square.
172 */
173 if (!BN_copy(q, p))
174 goto end; /* use 'q' as temp */
175 q->neg = 0;
176 i = 2;
177 do {
178 /*
179 * For efficiency, try small numbers first; if this fails, try random
180 * numbers.
181 */
182 if (i < 22) {
183 if (!BN_set_word(y, i))
184 goto end;
185 } else {
186 if (!BN_priv_rand_ex(y, BN_num_bits(p), 0, 0, 0, ctx))
187 goto end;
188 if (BN_ucmp(y, p) >= 0) {
189 if (!(p->neg ? BN_add : BN_sub) (y, y, p))
190 goto end;
191 }
192 /* now 0 <= y < |p| */
193 if (BN_is_zero(y))
194 if (!BN_set_word(y, i))
195 goto end;
196 }
197
198 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
199 if (r < -1)
200 goto end;
201 if (r == 0) {
202 /* m divides p */
203 ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME);
204 goto end;
205 }
206 }
207 while (r == 1 && ++i < 82);
208
209 if (r != -1) {
210 /*
211 * Many rounds and still no non-square -- this is more likely a bug
212 * than just bad luck. Even if p is not prime, we should have found
213 * some y such that r == -1.
214 */
215 ERR_raise(ERR_LIB_BN, BN_R_TOO_MANY_ITERATIONS);
216 goto end;
217 }
218
219 /* Here's our actual 'q': */
220 if (!BN_rshift(q, q, e))
221 goto end;
222
223 /*
224 * Now that we have some non-square, we can find an element of order 2^e
225 * by computing its q'th power.
226 */
227 if (!BN_mod_exp(y, y, q, p, ctx))
228 goto end;
229 if (BN_is_one(y)) {
230 ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME);
231 goto end;
232 }
233
234 /*-
235 * Now we know that (if p is indeed prime) there is an integer
236 * k, 0 <= k < 2^e, such that
237 *
238 * a^q * y^k == 1 (mod p).
239 *
240 * As a^q is a square and y is not, k must be even.
241 * q+1 is even, too, so there is an element
242 *
243 * X := a^((q+1)/2) * y^(k/2),
244 *
245 * and it satisfies
246 *
247 * X^2 = a^q * a * y^k
248 * = a,
249 *
250 * so it is the square root that we are looking for.
251 */
252
253 /* t := (q-1)/2 (note that q is odd) */
254 if (!BN_rshift1(t, q))
255 goto end;
256
257 /* x := a^((q-1)/2) */
258 if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
259 if (!BN_nnmod(t, A, p, ctx))
260 goto end;
261 if (BN_is_zero(t)) {
262 /* special case: a == 0 (mod p) */
263 BN_zero(ret);
264 err = 0;
265 goto end;
266 } else if (!BN_one(x))
267 goto end;
268 } else {
269 if (!BN_mod_exp(x, A, t, p, ctx))
270 goto end;
271 if (BN_is_zero(x)) {
272 /* special case: a == 0 (mod p) */
273 BN_zero(ret);
274 err = 0;
275 goto end;
276 }
277 }
278
279 /* b := a*x^2 (= a^q) */
280 if (!BN_mod_sqr(b, x, p, ctx))
281 goto end;
282 if (!BN_mod_mul(b, b, A, p, ctx))
283 goto end;
284
285 /* x := a*x (= a^((q+1)/2)) */
286 if (!BN_mod_mul(x, x, A, p, ctx))
287 goto end;
288
289 while (1) {
290 /*-
291 * Now b is a^q * y^k for some even k (0 <= k < 2^E
292 * where E refers to the original value of e, which we
293 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
294 *
295 * We have a*b = x^2,
296 * y^2^(e-1) = -1,
297 * b^2^(e-1) = 1.
298 */
299
300 if (BN_is_one(b)) {
301 if (!BN_copy(ret, x))
302 goto end;
303 err = 0;
304 goto vrfy;
305 }
306
307 /* Find the smallest i, 0 < i < e, such that b^(2^i) = 1. */
308 for (i = 1; i < e; i++) {
309 if (i == 1) {
310 if (!BN_mod_sqr(t, b, p, ctx))
311 goto end;
312
313 } else {
314 if (!BN_mod_mul(t, t, t, p, ctx))
315 goto end;
316 }
317 if (BN_is_one(t))
318 break;
319 }
320 /* If not found, a is not a square or p is not prime. */
321 if (i >= e) {
322 ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE);
323 goto end;
324 }
325
326 /* t := y^2^(e - i - 1) */
327 if (!BN_copy(t, y))
328 goto end;
329 for (j = e - i - 1; j > 0; j--) {
330 if (!BN_mod_sqr(t, t, p, ctx))
331 goto end;
332 }
333 if (!BN_mod_mul(y, t, t, p, ctx))
334 goto end;
335 if (!BN_mod_mul(x, x, t, p, ctx))
336 goto end;
337 if (!BN_mod_mul(b, b, y, p, ctx))
338 goto end;
339 e = i;
340 }
341
342 vrfy:
343 if (!err) {
344 /*
345 * verify the result -- the input might have been not a square (test
346 * added in 0.9.8)
347 */
348
349 if (!BN_mod_sqr(x, ret, p, ctx))
350 err = 1;
351
352 if (!err && 0 != BN_cmp(x, A)) {
353 ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE);
354 err = 1;
355 }
356 }
357
358 end:
359 if (err) {
360 if (ret != in)
361 BN_clear_free(ret);
362 ret = NULL;
363 }
364 if (used_ctx)
365 BN_CTX_end(ctx);
366 bn_check_top(ret);
367 return ret;
368 }
369