1 /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
2  * and Bodo Moeller for the OpenSSL project. */
3 /* ====================================================================
4  * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
5  *
6  * Redistribution and use in source and binary forms, with or without
7  * modification, are permitted provided that the following conditions
8  * are met:
9  *
10  * 1. Redistributions of source code must retain the above copyright
11  *    notice, this list of conditions and the following disclaimer.
12  *
13  * 2. Redistributions in binary form must reproduce the above copyright
14  *    notice, this list of conditions and the following disclaimer in
15  *    the documentation and/or other materials provided with the
16  *    distribution.
17  *
18  * 3. All advertising materials mentioning features or use of this
19  *    software must display the following acknowledgment:
20  *    "This product includes software developed by the OpenSSL Project
21  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
22  *
23  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
24  *    endorse or promote products derived from this software without
25  *    prior written permission. For written permission, please contact
26  *    openssl-core@openssl.org.
27  *
28  * 5. Products derived from this software may not be called "OpenSSL"
29  *    nor may "OpenSSL" appear in their names without prior written
30  *    permission of the OpenSSL Project.
31  *
32  * 6. Redistributions of any form whatsoever must retain the following
33  *    acknowledgment:
34  *    "This product includes software developed by the OpenSSL Project
35  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
36  *
37  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
38  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
39  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
40  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
41  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
42  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
43  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
44  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
45  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
46  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
47  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
48  * OF THE POSSIBILITY OF SUCH DAMAGE.
49  * ====================================================================
50  *
51  * This product includes cryptographic software written by Eric Young
52  * (eay@cryptsoft.com).  This product includes software written by Tim
53  * Hudson (tjh@cryptsoft.com). */
54 
55 #include <openssl/bn.h>
56 
57 #include <openssl/err.h>
58 
59 
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)60 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
61   /* Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
62    * (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
63    * algorithm 1.5.1). |p| is assumed to be a prime. */
64 
65   BIGNUM *ret = in;
66   int err = 1;
67   int r;
68   BIGNUM *A, *b, *q, *t, *x, *y;
69   int e, i, j;
70 
71   if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
72     if (BN_abs_is_word(p, 2)) {
73       if (ret == NULL) {
74         ret = BN_new();
75       }
76       if (ret == NULL) {
77         goto end;
78       }
79       if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
80         if (ret != in) {
81           BN_free(ret);
82         }
83         return NULL;
84       }
85       return ret;
86     }
87 
88     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
89     return (NULL);
90   }
91 
92   if (BN_is_zero(a) || BN_is_one(a)) {
93     if (ret == NULL) {
94       ret = BN_new();
95     }
96     if (ret == NULL) {
97       goto end;
98     }
99     if (!BN_set_word(ret, BN_is_one(a))) {
100       if (ret != in) {
101         BN_free(ret);
102       }
103       return NULL;
104     }
105     return ret;
106   }
107 
108   BN_CTX_start(ctx);
109   A = BN_CTX_get(ctx);
110   b = BN_CTX_get(ctx);
111   q = BN_CTX_get(ctx);
112   t = BN_CTX_get(ctx);
113   x = BN_CTX_get(ctx);
114   y = BN_CTX_get(ctx);
115   if (y == NULL) {
116     goto end;
117   }
118 
119   if (ret == NULL) {
120     ret = BN_new();
121   }
122   if (ret == NULL) {
123     goto end;
124   }
125 
126   /* A = a mod p */
127   if (!BN_nnmod(A, a, p, ctx)) {
128     goto end;
129   }
130 
131   /* now write  |p| - 1  as  2^e*q  where  q  is odd */
132   e = 1;
133   while (!BN_is_bit_set(p, e)) {
134     e++;
135   }
136   /* we'll set  q  later (if needed) */
137 
138   if (e == 1) {
139     /* The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
140      * modulo  (|p|-1)/2,  and square roots can be computed
141      * directly by modular exponentiation.
142      * We have
143      *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
144      * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
145      */
146     if (!BN_rshift(q, p, 2)) {
147       goto end;
148     }
149     q->neg = 0;
150     if (!BN_add_word(q, 1) ||
151         !BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
152       goto end;
153     }
154     err = 0;
155     goto vrfy;
156   }
157 
158   if (e == 2) {
159     /* |p| == 5  (mod 8)
160      *
161      * In this case  2  is always a non-square since
162      * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
163      * So if  a  really is a square, then  2*a  is a non-square.
164      * Thus for
165      *      b := (2*a)^((|p|-5)/8),
166      *      i := (2*a)*b^2
167      * we have
168      *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
169      *         = (2*a)^((p-1)/2)
170      *         = -1;
171      * so if we set
172      *      x := a*b*(i-1),
173      * then
174      *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
175      *         = a^2 * b^2 * (-2*i)
176      *         = a*(-i)*(2*a*b^2)
177      *         = a*(-i)*i
178      *         = a.
179      *
180      * (This is due to A.O.L. Atkin,
181      * <URL:
182      *http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
183      * November 1992.)
184      */
185 
186     /* t := 2*a */
187     if (!BN_mod_lshift1_quick(t, A, p)) {
188       goto end;
189     }
190 
191     /* b := (2*a)^((|p|-5)/8) */
192     if (!BN_rshift(q, p, 3)) {
193       goto end;
194     }
195     q->neg = 0;
196     if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
197       goto end;
198     }
199 
200     /* y := b^2 */
201     if (!BN_mod_sqr(y, b, p, ctx)) {
202       goto end;
203     }
204 
205     /* t := (2*a)*b^2 - 1*/
206     if (!BN_mod_mul(t, t, y, p, ctx) ||
207         !BN_sub_word(t, 1)) {
208       goto end;
209     }
210 
211     /* x = a*b*t */
212     if (!BN_mod_mul(x, A, b, p, ctx) ||
213         !BN_mod_mul(x, x, t, p, ctx)) {
214       goto end;
215     }
216 
217     if (!BN_copy(ret, x)) {
218       goto end;
219     }
220     err = 0;
221     goto vrfy;
222   }
223 
224   /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
225    * First, find some  y  that is not a square. */
226   if (!BN_copy(q, p)) {
227     goto end; /* use 'q' as temp */
228   }
229   q->neg = 0;
230   i = 2;
231   do {
232     /* For efficiency, try small numbers first;
233      * if this fails, try random numbers.
234      */
235     if (i < 22) {
236       if (!BN_set_word(y, i)) {
237         goto end;
238       }
239     } else {
240       if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
241         goto end;
242       }
243       if (BN_ucmp(y, p) >= 0) {
244         if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
245           goto end;
246         }
247       }
248       /* now 0 <= y < |p| */
249       if (BN_is_zero(y)) {
250         if (!BN_set_word(y, i)) {
251           goto end;
252         }
253       }
254     }
255 
256     r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
257     if (r < -1) {
258       goto end;
259     }
260     if (r == 0) {
261       /* m divides p */
262       OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
263       goto end;
264     }
265   } while (r == 1 && ++i < 82);
266 
267   if (r != -1) {
268     /* Many rounds and still no non-square -- this is more likely
269      * a bug than just bad luck.
270      * Even if  p  is not prime, we should have found some  y
271      * such that r == -1.
272      */
273     OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
274     goto end;
275   }
276 
277   /* Here's our actual 'q': */
278   if (!BN_rshift(q, q, e)) {
279     goto end;
280   }
281 
282   /* Now that we have some non-square, we can find an element
283    * of order  2^e  by computing its q'th power. */
284   if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
285     goto end;
286   }
287   if (BN_is_one(y)) {
288     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
289     goto end;
290   }
291 
292   /* Now we know that (if  p  is indeed prime) there is an integer
293    * k,  0 <= k < 2^e,  such that
294    *
295    *      a^q * y^k == 1   (mod p).
296    *
297    * As  a^q  is a square and  y  is not,  k  must be even.
298    * q+1  is even, too, so there is an element
299    *
300    *     X := a^((q+1)/2) * y^(k/2),
301    *
302    * and it satisfies
303    *
304    *     X^2 = a^q * a     * y^k
305    *         = a,
306    *
307    * so it is the square root that we are looking for.
308    */
309 
310   /* t := (q-1)/2  (note that  q  is odd) */
311   if (!BN_rshift1(t, q)) {
312     goto end;
313   }
314 
315   /* x := a^((q-1)/2) */
316   if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
317   {
318     if (!BN_nnmod(t, A, p, ctx)) {
319       goto end;
320     }
321     if (BN_is_zero(t)) {
322       /* special case: a == 0  (mod p) */
323       BN_zero(ret);
324       err = 0;
325       goto end;
326     } else if (!BN_one(x)) {
327       goto end;
328     }
329   } else {
330     if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
331       goto end;
332     }
333     if (BN_is_zero(x)) {
334       /* special case: a == 0  (mod p) */
335       BN_zero(ret);
336       err = 0;
337       goto end;
338     }
339   }
340 
341   /* b := a*x^2  (= a^q) */
342   if (!BN_mod_sqr(b, x, p, ctx) ||
343       !BN_mod_mul(b, b, A, p, ctx)) {
344     goto end;
345   }
346 
347   /* x := a*x    (= a^((q+1)/2)) */
348   if (!BN_mod_mul(x, x, A, p, ctx)) {
349     goto end;
350   }
351 
352   while (1) {
353     /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
354      * where  E  refers to the original value of  e,  which we
355      * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
356      *
357      * We have  a*b = x^2,
358      *    y^2^(e-1) = -1,
359      *    b^2^(e-1) = 1.
360      */
361 
362     if (BN_is_one(b)) {
363       if (!BN_copy(ret, x)) {
364         goto end;
365       }
366       err = 0;
367       goto vrfy;
368     }
369 
370 
371     /* find smallest  i  such that  b^(2^i) = 1 */
372     i = 1;
373     if (!BN_mod_sqr(t, b, p, ctx)) {
374       goto end;
375     }
376     while (!BN_is_one(t)) {
377       i++;
378       if (i == e) {
379         OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
380         goto end;
381       }
382       if (!BN_mod_mul(t, t, t, p, ctx)) {
383         goto end;
384       }
385     }
386 
387 
388     /* t := y^2^(e - i - 1) */
389     if (!BN_copy(t, y)) {
390       goto end;
391     }
392     for (j = e - i - 1; j > 0; j--) {
393       if (!BN_mod_sqr(t, t, p, ctx)) {
394         goto end;
395       }
396     }
397     if (!BN_mod_mul(y, t, t, p, ctx) ||
398         !BN_mod_mul(x, x, t, p, ctx) ||
399         !BN_mod_mul(b, b, y, p, ctx)) {
400       goto end;
401     }
402     e = i;
403   }
404 
405 vrfy:
406   if (!err) {
407     /* verify the result -- the input might have been not a square
408      * (test added in 0.9.8) */
409 
410     if (!BN_mod_sqr(x, ret, p, ctx)) {
411       err = 1;
412     }
413 
414     if (!err && 0 != BN_cmp(x, A)) {
415       OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
416       err = 1;
417     }
418   }
419 
420 end:
421   if (err) {
422     if (ret != in) {
423       BN_clear_free(ret);
424     }
425     ret = NULL;
426   }
427   BN_CTX_end(ctx);
428   return ret;
429 }
430 
BN_sqrt(BIGNUM * out_sqrt,const BIGNUM * in,BN_CTX * ctx)431 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
432   BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
433   int ok = 0, last_delta_valid = 0;
434 
435   if (in->neg) {
436     OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
437     return 0;
438   }
439   if (BN_is_zero(in)) {
440     BN_zero(out_sqrt);
441     return 1;
442   }
443 
444   BN_CTX_start(ctx);
445   if (out_sqrt == in) {
446     estimate = BN_CTX_get(ctx);
447   } else {
448     estimate = out_sqrt;
449   }
450   tmp = BN_CTX_get(ctx);
451   last_delta = BN_CTX_get(ctx);
452   delta = BN_CTX_get(ctx);
453   if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
454     OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
455     goto err;
456   }
457 
458   /* We estimate that the square root of an n-bit number is 2^{n/2}. */
459   if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
460     goto err;
461   }
462 
463   /* This is Newton's method for finding a root of the equation |estimate|^2 -
464    * |in| = 0. */
465   for (;;) {
466     /* |estimate| = 1/2 * (|estimate| + |in|/|estimate|) */
467     if (!BN_div(tmp, NULL, in, estimate, ctx) ||
468         !BN_add(tmp, tmp, estimate) ||
469         !BN_rshift1(estimate, tmp) ||
470         /* |tmp| = |estimate|^2 */
471         !BN_sqr(tmp, estimate, ctx) ||
472         /* |delta| = |in| - |tmp| */
473         !BN_sub(delta, in, tmp)) {
474       OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
475       goto err;
476     }
477 
478     delta->neg = 0;
479     /* The difference between |in| and |estimate| squared is required to always
480      * decrease. This ensures that the loop always terminates, but I don't have
481      * a proof that it always finds the square root for a given square. */
482     if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
483       break;
484     }
485 
486     last_delta_valid = 1;
487 
488     tmp2 = last_delta;
489     last_delta = delta;
490     delta = tmp2;
491   }
492 
493   if (BN_cmp(tmp, in) != 0) {
494     OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
495     goto err;
496   }
497 
498   ok = 1;
499 
500 err:
501   if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
502     ok = 0;
503   }
504   BN_CTX_end(ctx);
505   return ok;
506 }
507