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 
60 /* Returns 'ret' such that
61  *      ret^2 == a (mod p),
62  * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
63  * in Algebraic Computational Number Theory", algorithm 1.5.1).
64  * 'p' must be prime! */
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)65 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
66   BIGNUM *ret = in;
67   int err = 1;
68   int r;
69   BIGNUM *A, *b, *q, *t, *x, *y;
70   int e, i, j;
71 
72   if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
73     if (BN_abs_is_word(p, 2)) {
74       if (ret == NULL) {
75         ret = BN_new();
76       }
77       if (ret == NULL) {
78         goto end;
79       }
80       if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
81         if (ret != in) {
82           BN_free(ret);
83         }
84         return NULL;
85       }
86       return ret;
87     }
88 
89     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
90     return (NULL);
91   }
92 
93   if (BN_is_zero(a) || BN_is_one(a)) {
94     if (ret == NULL) {
95       ret = BN_new();
96     }
97     if (ret == NULL) {
98       goto end;
99     }
100     if (!BN_set_word(ret, BN_is_one(a))) {
101       if (ret != in) {
102         BN_free(ret);
103       }
104       return NULL;
105     }
106     return ret;
107   }
108 
109   BN_CTX_start(ctx);
110   A = BN_CTX_get(ctx);
111   b = BN_CTX_get(ctx);
112   q = BN_CTX_get(ctx);
113   t = BN_CTX_get(ctx);
114   x = BN_CTX_get(ctx);
115   y = BN_CTX_get(ctx);
116   if (y == NULL) {
117     goto end;
118   }
119 
120   if (ret == NULL) {
121     ret = BN_new();
122   }
123   if (ret == NULL) {
124     goto end;
125   }
126 
127   /* A = a mod p */
128   if (!BN_nnmod(A, a, p, ctx)) {
129     goto end;
130   }
131 
132   /* now write  |p| - 1  as  2^e*q  where  q  is odd */
133   e = 1;
134   while (!BN_is_bit_set(p, e)) {
135     e++;
136   }
137   /* we'll set  q  later (if needed) */
138 
139   if (e == 1) {
140     /* The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
141      * modulo  (|p|-1)/2,  and square roots can be computed
142      * directly by modular exponentiation.
143      * We have
144      *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
145      * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
146      */
147     if (!BN_rshift(q, p, 2)) {
148       goto end;
149     }
150     q->neg = 0;
151     if (!BN_add_word(q, 1) ||
152         !BN_mod_exp(ret, A, q, p, ctx)) {
153       goto end;
154     }
155     err = 0;
156     goto vrfy;
157   }
158 
159   if (e == 2) {
160     /* |p| == 5  (mod 8)
161      *
162      * In this case  2  is always a non-square since
163      * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
164      * So if  a  really is a square, then  2*a  is a non-square.
165      * Thus for
166      *      b := (2*a)^((|p|-5)/8),
167      *      i := (2*a)*b^2
168      * we have
169      *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
170      *         = (2*a)^((p-1)/2)
171      *         = -1;
172      * so if we set
173      *      x := a*b*(i-1),
174      * then
175      *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
176      *         = a^2 * b^2 * (-2*i)
177      *         = a*(-i)*(2*a*b^2)
178      *         = a*(-i)*i
179      *         = a.
180      *
181      * (This is due to A.O.L. Atkin,
182      * <URL:
183      *http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
184      * November 1992.)
185      */
186 
187     /* t := 2*a */
188     if (!BN_mod_lshift1_quick(t, A, p)) {
189       goto end;
190     }
191 
192     /* b := (2*a)^((|p|-5)/8) */
193     if (!BN_rshift(q, p, 3)) {
194       goto end;
195     }
196     q->neg = 0;
197     if (!BN_mod_exp(b, t, q, p, ctx)) {
198       goto end;
199     }
200 
201     /* y := b^2 */
202     if (!BN_mod_sqr(y, b, p, ctx)) {
203       goto end;
204     }
205 
206     /* t := (2*a)*b^2 - 1*/
207     if (!BN_mod_mul(t, t, y, p, ctx) ||
208         !BN_sub_word(t, 1)) {
209       goto end;
210     }
211 
212     /* x = a*b*t */
213     if (!BN_mod_mul(x, A, b, p, ctx) ||
214         !BN_mod_mul(x, x, t, p, ctx)) {
215       goto end;
216     }
217 
218     if (!BN_copy(ret, x)) {
219       goto end;
220     }
221     err = 0;
222     goto vrfy;
223   }
224 
225   /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
226    * First, find some  y  that is not a square. */
227   if (!BN_copy(q, p)) {
228     goto end; /* use 'q' as temp */
229   }
230   q->neg = 0;
231   i = 2;
232   do {
233     /* For efficiency, try small numbers first;
234      * if this fails, try random numbers.
235      */
236     if (i < 22) {
237       if (!BN_set_word(y, i)) {
238         goto end;
239       }
240     } else {
241       if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
242         goto end;
243       }
244       if (BN_ucmp(y, p) >= 0) {
245         if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
246           goto end;
247         }
248       }
249       /* now 0 <= y < |p| */
250       if (BN_is_zero(y)) {
251         if (!BN_set_word(y, i)) {
252           goto end;
253         }
254       }
255     }
256 
257     r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
258     if (r < -1) {
259       goto end;
260     }
261     if (r == 0) {
262       /* m divides p */
263       OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
264       goto end;
265     }
266   } while (r == 1 && ++i < 82);
267 
268   if (r != -1) {
269     /* Many rounds and still no non-square -- this is more likely
270      * a bug than just bad luck.
271      * Even if  p  is not prime, we should have found some  y
272      * such that r == -1.
273      */
274     OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
275     goto end;
276   }
277 
278   /* Here's our actual 'q': */
279   if (!BN_rshift(q, q, e)) {
280     goto end;
281   }
282 
283   /* Now that we have some non-square, we can find an element
284    * of order  2^e  by computing its q'th power. */
285   if (!BN_mod_exp(y, y, q, p, ctx)) {
286     goto end;
287   }
288   if (BN_is_one(y)) {
289     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
290     goto end;
291   }
292 
293   /* Now we know that (if  p  is indeed prime) there is an integer
294    * k,  0 <= k < 2^e,  such that
295    *
296    *      a^q * y^k == 1   (mod p).
297    *
298    * As  a^q  is a square and  y  is not,  k  must be even.
299    * q+1  is even, too, so there is an element
300    *
301    *     X := a^((q+1)/2) * y^(k/2),
302    *
303    * and it satisfies
304    *
305    *     X^2 = a^q * a     * y^k
306    *         = a,
307    *
308    * so it is the square root that we are looking for.
309    */
310 
311   /* t := (q-1)/2  (note that  q  is odd) */
312   if (!BN_rshift1(t, q)) {
313     goto end;
314   }
315 
316   /* x := a^((q-1)/2) */
317   if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
318   {
319     if (!BN_nnmod(t, A, p, ctx)) {
320       goto end;
321     }
322     if (BN_is_zero(t)) {
323       /* special case: a == 0  (mod p) */
324       BN_zero(ret);
325       err = 0;
326       goto end;
327     } else if (!BN_one(x)) {
328       goto end;
329     }
330   } else {
331     if (!BN_mod_exp(x, A, t, p, ctx)) {
332       goto end;
333     }
334     if (BN_is_zero(x)) {
335       /* special case: a == 0  (mod p) */
336       BN_zero(ret);
337       err = 0;
338       goto end;
339     }
340   }
341 
342   /* b := a*x^2  (= a^q) */
343   if (!BN_mod_sqr(b, x, p, ctx) ||
344       !BN_mod_mul(b, b, A, p, ctx)) {
345     goto end;
346   }
347 
348   /* x := a*x    (= a^((q+1)/2)) */
349   if (!BN_mod_mul(x, x, A, p, ctx)) {
350     goto end;
351   }
352 
353   while (1) {
354     /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
355      * where  E  refers to the original value of  e,  which we
356      * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
357      *
358      * We have  a*b = x^2,
359      *    y^2^(e-1) = -1,
360      *    b^2^(e-1) = 1.
361      */
362 
363     if (BN_is_one(b)) {
364       if (!BN_copy(ret, x)) {
365         goto end;
366       }
367       err = 0;
368       goto vrfy;
369     }
370 
371 
372     /* find smallest  i  such that  b^(2^i) = 1 */
373     i = 1;
374     if (!BN_mod_sqr(t, b, p, ctx)) {
375       goto end;
376     }
377     while (!BN_is_one(t)) {
378       i++;
379       if (i == e) {
380         OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
381         goto end;
382       }
383       if (!BN_mod_mul(t, t, t, p, ctx)) {
384         goto end;
385       }
386     }
387 
388 
389     /* t := y^2^(e - i - 1) */
390     if (!BN_copy(t, y)) {
391       goto end;
392     }
393     for (j = e - i - 1; j > 0; j--) {
394       if (!BN_mod_sqr(t, t, p, ctx)) {
395         goto end;
396       }
397     }
398     if (!BN_mod_mul(y, t, t, p, ctx) ||
399         !BN_mod_mul(x, x, t, p, ctx) ||
400         !BN_mod_mul(b, b, y, p, ctx)) {
401       goto end;
402     }
403     e = i;
404   }
405 
406 vrfy:
407   if (!err) {
408     /* verify the result -- the input might have been not a square
409      * (test added in 0.9.8) */
410 
411     if (!BN_mod_sqr(x, ret, p, ctx)) {
412       err = 1;
413     }
414 
415     if (!err && 0 != BN_cmp(x, A)) {
416       OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
417       err = 1;
418     }
419   }
420 
421 end:
422   if (err) {
423     if (ret != in) {
424       BN_clear_free(ret);
425     }
426     ret = NULL;
427   }
428   BN_CTX_end(ctx);
429   return ret;
430 }
431 
BN_sqrt(BIGNUM * out_sqrt,const BIGNUM * in,BN_CTX * ctx)432 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
433   BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
434   int ok = 0, last_delta_valid = 0;
435 
436   if (in->neg) {
437     OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
438     return 0;
439   }
440   if (BN_is_zero(in)) {
441     BN_zero(out_sqrt);
442     return 1;
443   }
444 
445   BN_CTX_start(ctx);
446   if (out_sqrt == in) {
447     estimate = BN_CTX_get(ctx);
448   } else {
449     estimate = out_sqrt;
450   }
451   tmp = BN_CTX_get(ctx);
452   last_delta = BN_CTX_get(ctx);
453   delta = BN_CTX_get(ctx);
454   if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
455     OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
456     goto err;
457   }
458 
459   /* We estimate that the square root of an n-bit number is 2^{n/2}. */
460   BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2);
461 
462   /* This is Newton's method for finding a root of the equation |estimate|^2 -
463    * |in| = 0. */
464   for (;;) {
465     /* |estimate| = 1/2 * (|estimate| + |in|/|estimate|) */
466     if (!BN_div(tmp, NULL, in, estimate, ctx) ||
467         !BN_add(tmp, tmp, estimate) ||
468         !BN_rshift1(estimate, tmp) ||
469         /* |tmp| = |estimate|^2 */
470         !BN_sqr(tmp, estimate, ctx) ||
471         /* |delta| = |in| - |tmp| */
472         !BN_sub(delta, in, tmp)) {
473       OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
474       goto err;
475     }
476 
477     delta->neg = 0;
478     /* The difference between |in| and |estimate| squared is required to always
479      * decrease. This ensures that the loop always terminates, but I don't have
480      * a proof that it always finds the square root for a given square. */
481     if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
482       break;
483     }
484 
485     last_delta_valid = 1;
486 
487     tmp2 = last_delta;
488     last_delta = delta;
489     delta = tmp2;
490   }
491 
492   if (BN_cmp(tmp, in) != 0) {
493     OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
494     goto err;
495   }
496 
497   ok = 1;
498 
499 err:
500   if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
501     ok = 0;
502   }
503   BN_CTX_end(ctx);
504   return ok;
505 }
506