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git.ipfire.org Git - thirdparty/openssl.git/blob - crypto/bn/bn_sqrt.c
1 /* crypto/bn/bn_mod.c */
2 /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
3 * and Bodo Moeller for the OpenSSL project. */
4 /* ====================================================================
5 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
11 * 1. Redistributions of source code must retain the above copyright
12 * notice, this list of conditions and the following disclaimer.
14 * 2. Redistributions in binary form must reproduce the above copyright
15 * notice, this list of conditions and the following disclaimer in
16 * the documentation and/or other materials provided with the
19 * 3. All advertising materials mentioning features or use of this
20 * software must display the following acknowledgment:
21 * "This product includes software developed by the OpenSSL Project
22 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
24 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
25 * endorse or promote products derived from this software without
26 * prior written permission. For written permission, please contact
27 * openssl-core@openssl.org.
29 * 5. Products derived from this software may not be called "OpenSSL"
30 * nor may "OpenSSL" appear in their names without prior written
31 * permission of the OpenSSL Project.
33 * 6. Redistributions of any form whatsoever must retain the following
35 * "This product includes software developed by the OpenSSL Project
36 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
38 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
39 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
40 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
41 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
42 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
43 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
44 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
45 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
46 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
47 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
48 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
49 * OF THE POSSIBILITY OF SUCH DAMAGE.
50 * ====================================================================
52 * This product includes cryptographic software written by Eric Young
53 * (eay@cryptsoft.com). This product includes software written by Tim
54 * Hudson (tjh@cryptsoft.com).
62 BIGNUM
*BN_mod_sqrt(BIGNUM
*in
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
63 /* Returns 'ret' such that
65 * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
66 * in Algebraic Computational Number Theory", algorithm 1.5.1).
73 BIGNUM
*b
, *q
, *t
, *x
, *y
;
76 if (!BN_is_odd(p
) || BN_abs_is_word(p
, 1))
78 if (BN_abs_is_word(p
, 2))
84 if (!BN_set_word(ret
, BN_is_bit_set(a
, 0)))
92 BNerr(BN_F_BN_MOD_SQRT
, BN_R_P_IS_NOT_PRIME
);
96 if (BN_is_zero(a
) || BN_is_one(a
))
102 if (!BN_set_word(ret
, BN_is_one(a
)))
110 #if 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
111 r
= BN_kronecker(a
, p
, ctx
);
112 if (r
< -1) return NULL
;
115 BNerr(BN_F_BN_MOD_SQRT
, BN_R_NOT_A_SQUARE
);
126 if (y
== NULL
) goto end
;
130 if (ret
== NULL
) goto end
;
132 /* now write |p| - 1 as 2^e*q where q is odd */
134 while (!BN_is_bit_set(p
, e
))
136 /* we'll set q later (if needed) */
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.
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.
147 if (!BN_rshift(q
, p
, 2)) goto end
;
149 if (!BN_add_word(q
, 1)) goto end
;
150 if (!BN_mod_exp(ret
, a
, q
, p
, ctx
)) goto end
;
159 * In this case 2 is always a non-square since
160 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
161 * So if a really is a square, then 2*a is a non-square.
163 * b := (2*a)^((|p|-5)/8),
166 * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
172 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
173 * = a^2 * b^2 * (-2*i)
178 * (This is due to A.O.L. Atkin,
179 * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
183 /* make sure that a is reduced modulo p */
184 if (a
->neg
|| BN_ucmp(a
, p
) >= 0)
186 if (!BN_nnmod(x
, a
, p
, ctx
)) goto end
;
187 a
= x
; /* use x as temporary variable */
191 if (!BN_mod_lshift1_quick(t
, a
, p
)) goto end
;
193 /* b := (2*a)^((|p|-5)/8) */
194 if (!BN_rshift(q
, p
, 3)) goto end
;
196 if (!BN_mod_exp(b
, t
, q
, p
, ctx
)) goto end
;
199 if (!BN_mod_sqr(y
, b
, p
, ctx
)) goto end
;
201 /* t := (2*a)*b^2 - 1*/
202 if (!BN_mod_mul(t
, t
, y
, p
, ctx
)) goto end
;
203 if (!BN_sub_word(t
, 1)) goto end
;
206 if (!BN_mod_mul(x
, a
, b
, p
, ctx
)) goto end
;
207 if (!BN_mod_mul(x
, x
, t
, p
, ctx
)) goto end
;
209 if (!BN_copy(ret
, x
)) goto end
;
214 /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
215 * First, find some y that is not a square. */
216 if (!BN_copy(q
, p
)) goto end
; /* use 'q' as temp */
221 /* For efficiency, try small numbers first;
222 * if this fails, try random numbers.
226 if (!BN_set_word(y
, i
)) goto end
;
230 if (!BN_pseudo_rand(y
, BN_num_bits(p
), 0, 0)) goto end
;
231 if (BN_ucmp(y
, p
) >= 0)
233 if (!(p
->neg
? BN_add
: BN_sub
)(y
, y
, p
)) goto end
;
235 /* now 0 <= y < |p| */
237 if (!BN_set_word(y
, i
)) goto end
;
240 r
= BN_kronecker(y
, q
, ctx
); /* here 'q' is |p| */
241 if (r
< -1) goto end
;
245 BNerr(BN_F_BN_MOD_SQRT
, BN_R_P_IS_NOT_PRIME
);
249 while (r
== 1 && ++i
< 82);
253 /* Many rounds and still no non-square -- this is more likely
254 * a bug than just bad luck.
255 * Even if p is not prime, we should have found some y
258 BNerr(BN_F_BN_MOD_SQRT
, BN_R_TOO_MANY_ITERATIONS
);
262 /* Here's our actual 'q': */
263 if (!BN_rshift(q
, q
, e
)) goto end
;
265 /* Now that we have some non-square, we can find an element
266 * of order 2^e by computing its q'th power. */
267 if (!BN_mod_exp(y
, y
, q
, p
, ctx
)) goto end
;
270 BNerr(BN_F_BN_MOD_SQRT
, BN_R_P_IS_NOT_PRIME
);
274 /* Now we know that (if p is indeed prime) there is an integer
275 * k, 0 <= k < 2^e, such that
277 * a^q * y^k == 1 (mod p).
279 * As a^q is a square and y is not, k must be even.
280 * q+1 is even, too, so there is an element
282 * X := a^((q+1)/2) * y^(k/2),
286 * X^2 = a^q * a * y^k
289 * so it is the square root that we are looking for.
292 /* t := (q-1)/2 (note that q is odd) */
293 if (!BN_rshift1(t
, q
)) goto end
;
295 /* x := a^((q-1)/2) */
296 if (BN_is_zero(t
)) /* special case: p = 2^e + 1 */
298 if (!BN_nnmod(t
, a
, p
, ctx
)) goto end
;
301 /* special case: a == 0 (mod p) */
302 if (!BN_zero(ret
)) goto end
;
307 if (!BN_one(x
)) goto end
;
311 if (!BN_mod_exp(x
, a
, t
, p
, ctx
)) goto end
;
314 /* special case: a == 0 (mod p) */
315 if (!BN_zero(ret
)) goto end
;
321 /* b := a*x^2 (= a^q) */
322 if (!BN_mod_sqr(b
, x
, p
, ctx
)) goto end
;
323 if (!BN_mod_mul(b
, b
, a
, p
, ctx
)) goto end
;
325 /* x := a*x (= a^((q+1)/2)) */
326 if (!BN_mod_mul(x
, x
, a
, p
, ctx
)) goto end
;
330 /* Now b is a^q * y^k for some even k (0 <= k < 2^E
331 * where E refers to the original value of e, which we
332 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
341 if (!BN_copy(ret
, x
)) goto end
;
347 /* find smallest i such that b^(2^i) = 1 */
349 if (!BN_mod_sqr(t
, b
, p
, ctx
)) goto end
;
350 while (!BN_is_one(t
))
355 BNerr(BN_F_BN_MOD_SQRT
, BN_R_NOT_A_SQUARE
);
358 if (!BN_mod_mul(t
, t
, t
, p
, ctx
)) goto end
;
362 /* t := y^2^(e - i - 1) */
363 if (!BN_copy(t
, y
)) goto end
;
364 for (j
= e
- i
- 1; j
> 0; j
--)
366 if (!BN_mod_sqr(t
, t
, p
, ctx
)) goto end
;
368 if (!BN_mod_mul(y
, t
, t
, p
, ctx
)) goto end
;
369 if (!BN_mod_mul(x
, x
, t
, p
, ctx
)) goto end
;
370 if (!BN_mod_mul(b
, b
, y
, p
, ctx
)) goto end
;
377 if (ret
!= NULL
&& ret
!= in
)