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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 #include "internal.h"
60 
61 
BN_mod_sqrt(BIGNUM * in,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx)62 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
63   // Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
64   // (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
65   // algorithm 1.5.1). |p| is assumed to be a prime.
66 
67   BIGNUM *ret = in;
68   int err = 1;
69   int r;
70   BIGNUM *A, *b, *q, *t, *x, *y;
71   int e, i, j;
72 
73   if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
74     if (BN_abs_is_word(p, 2)) {
75       if (ret == NULL) {
76         ret = BN_new();
77       }
78       if (ret == NULL) {
79         goto end;
80       }
81       if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
82         if (ret != in) {
83           BN_free(ret);
84         }
85         return NULL;
86       }
87       return ret;
88     }
89 
90     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
91     return (NULL);
92   }
93 
94   if (BN_is_zero(a) || BN_is_one(a)) {
95     if (ret == NULL) {
96       ret = BN_new();
97     }
98     if (ret == NULL) {
99       goto end;
100     }
101     if (!BN_set_word(ret, BN_is_one(a))) {
102       if (ret != in) {
103         BN_free(ret);
104       }
105       return NULL;
106     }
107     return ret;
108   }
109 
110   BN_CTX_start(ctx);
111   A = BN_CTX_get(ctx);
112   b = BN_CTX_get(ctx);
113   q = BN_CTX_get(ctx);
114   t = BN_CTX_get(ctx);
115   x = BN_CTX_get(ctx);
116   y = BN_CTX_get(ctx);
117   if (y == NULL) {
118     goto end;
119   }
120 
121   if (ret == NULL) {
122     ret = BN_new();
123   }
124   if (ret == NULL) {
125     goto end;
126   }
127 
128   // A = a mod p
129   if (!BN_nnmod(A, a, p, ctx)) {
130     goto end;
131   }
132 
133   // now write  |p| - 1  as  2^e*q  where  q  is odd
134   e = 1;
135   while (!BN_is_bit_set(p, e)) {
136     e++;
137   }
138   // we'll set  q  later (if needed)
139 
140   if (e == 1) {
141     // The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
142     // modulo  (|p|-1)/2,  and square roots can be computed
143     // directly by modular exponentiation.
144     // We have
145     //     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
146     // so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
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_mont(ret, A, q, p, ctx, NULL)) {
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     // t := 2*a
187     if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
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     if (i < 22) {
235       if (!BN_set_word(y, i)) {
236         goto end;
237       }
238     } else {
239       if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
240         goto end;
241       }
242       if (BN_ucmp(y, p) >= 0) {
243         if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
244           goto end;
245         }
246       }
247       // now 0 <= y < |p|
248       if (BN_is_zero(y)) {
249         if (!BN_set_word(y, i)) {
250           goto end;
251         }
252       }
253     }
254 
255     r = bn_jacobi(y, q, ctx);  // here 'q' is |p|
256     if (r < -1) {
257       goto end;
258     }
259     if (r == 0) {
260       // m divides p
261       OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
262       goto end;
263     }
264   } while (r == 1 && ++i < 82);
265 
266   if (r != -1) {
267     // Many rounds and still no non-square -- this is more likely
268     // a bug than just bad luck.
269     // Even if  p  is not prime, we should have found some  y
270     // such that r == -1.
271     OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
272     goto end;
273   }
274 
275   // Here's our actual 'q':
276   if (!BN_rshift(q, q, e)) {
277     goto end;
278   }
279 
280   // Now that we have some non-square, we can find an element
281   // of order  2^e  by computing its q'th power.
282   if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
283     goto end;
284   }
285   if (BN_is_one(y)) {
286     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
287     goto end;
288   }
289 
290   // Now we know that (if  p  is indeed prime) there is an integer
291   // k,  0 <= k < 2^e,  such that
292   //
293   //      a^q * y^k == 1   (mod p).
294   //
295   // As  a^q  is a square and  y  is not,  k  must be even.
296   // q+1  is even, too, so there is an element
297   //
298   //     X := a^((q+1)/2) * y^(k/2),
299   //
300   // and it satisfies
301   //
302   //     X^2 = a^q * a     * y^k
303   //         = a,
304   //
305   // so it is the square root that we are looking for.
306 
307   // t := (q-1)/2  (note that  q  is odd)
308   if (!BN_rshift1(t, q)) {
309     goto end;
310   }
311 
312   // x := a^((q-1)/2)
313   if (BN_is_zero(t))  // special case: p = 2^e + 1
314   {
315     if (!BN_nnmod(t, A, p, ctx)) {
316       goto end;
317     }
318     if (BN_is_zero(t)) {
319       // special case: a == 0  (mod p)
320       BN_zero(ret);
321       err = 0;
322       goto end;
323     } else if (!BN_one(x)) {
324       goto end;
325     }
326   } else {
327     if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
328       goto end;
329     }
330     if (BN_is_zero(x)) {
331       // special case: a == 0  (mod p)
332       BN_zero(ret);
333       err = 0;
334       goto end;
335     }
336   }
337 
338   // b := a*x^2  (= a^q)
339   if (!BN_mod_sqr(b, x, p, ctx) ||
340       !BN_mod_mul(b, b, A, p, ctx)) {
341     goto end;
342   }
343 
344   // x := a*x    (= a^((q+1)/2))
345   if (!BN_mod_mul(x, x, A, p, ctx)) {
346     goto end;
347   }
348 
349   while (1) {
350     // Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
351     // where  E  refers to the original value of  e,  which we
352     // don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
353     //
354     // We have  a*b = x^2,
355     //    y^2^(e-1) = -1,
356     //    b^2^(e-1) = 1.
357 
358     if (BN_is_one(b)) {
359       if (!BN_copy(ret, x)) {
360         goto end;
361       }
362       err = 0;
363       goto vrfy;
364     }
365 
366 
367     // find smallest  i  such that  b^(2^i) = 1
368     i = 1;
369     if (!BN_mod_sqr(t, b, p, ctx)) {
370       goto end;
371     }
372     while (!BN_is_one(t)) {
373       i++;
374       if (i == e) {
375         OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
376         goto end;
377       }
378       if (!BN_mod_mul(t, t, t, p, ctx)) {
379         goto end;
380       }
381     }
382 
383 
384     // t := y^2^(e - i - 1)
385     if (!BN_copy(t, y)) {
386       goto end;
387     }
388     for (j = e - i - 1; j > 0; j--) {
389       if (!BN_mod_sqr(t, t, p, ctx)) {
390         goto end;
391       }
392     }
393     if (!BN_mod_mul(y, t, t, p, ctx) ||
394         !BN_mod_mul(x, x, t, p, ctx) ||
395         !BN_mod_mul(b, b, y, p, ctx)) {
396       goto end;
397     }
398     e = i;
399   }
400 
401 vrfy:
402   if (!err) {
403     // verify the result -- the input might have been not a square
404     // (test added in 0.9.8)
405 
406     if (!BN_mod_sqr(x, ret, p, ctx)) {
407       err = 1;
408     }
409 
410     if (!err && 0 != BN_cmp(x, A)) {
411       OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
412       err = 1;
413     }
414   }
415 
416 end:
417   if (err) {
418     if (ret != in) {
419       BN_clear_free(ret);
420     }
421     ret = NULL;
422   }
423   BN_CTX_end(ctx);
424   return ret;
425 }
426 
BN_sqrt(BIGNUM * out_sqrt,const BIGNUM * in,BN_CTX * ctx)427 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
428   BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
429   int ok = 0, last_delta_valid = 0;
430 
431   if (in->neg) {
432     OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
433     return 0;
434   }
435   if (BN_is_zero(in)) {
436     BN_zero(out_sqrt);
437     return 1;
438   }
439 
440   BN_CTX_start(ctx);
441   if (out_sqrt == in) {
442     estimate = BN_CTX_get(ctx);
443   } else {
444     estimate = out_sqrt;
445   }
446   tmp = BN_CTX_get(ctx);
447   last_delta = BN_CTX_get(ctx);
448   delta = BN_CTX_get(ctx);
449   if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
450     OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
451     goto err;
452   }
453 
454   // We estimate that the square root of an n-bit number is 2^{n/2}.
455   if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
456     goto err;
457   }
458 
459   // This is Newton's method for finding a root of the equation |estimate|^2 -
460   // |in| = 0.
461   for (;;) {
462     // |estimate| = 1/2 * (|estimate| + |in|/|estimate|)
463     if (!BN_div(tmp, NULL, in, estimate, ctx) ||
464         !BN_add(tmp, tmp, estimate) ||
465         !BN_rshift1(estimate, tmp) ||
466         // |tmp| = |estimate|^2
467         !BN_sqr(tmp, estimate, ctx) ||
468         // |delta| = |in| - |tmp|
469         !BN_sub(delta, in, tmp)) {
470       OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
471       goto err;
472     }
473 
474     delta->neg = 0;
475     // The difference between |in| and |estimate| squared is required to always
476     // decrease. This ensures that the loop always terminates, but I don't have
477     // a proof that it always finds the square root for a given square.
478     if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
479       break;
480     }
481 
482     last_delta_valid = 1;
483 
484     tmp2 = last_delta;
485     last_delta = delta;
486     delta = tmp2;
487   }
488 
489   if (BN_cmp(tmp, in) != 0) {
490     OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
491     goto err;
492   }
493 
494   ok = 1;
495 
496 err:
497   if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
498     ok = 0;
499   }
500   BN_CTX_end(ctx);
501   return ok;
502 }
503