xref: /openssl/crypto/bn/bn_sqrt.c (revision fecb3aae)
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