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