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git.ipfire.org Git - thirdparty/openssl.git/blob - crypto/bn/bn_sqrt.c
1 /* crypto/bn/bn_sqrt.c */
3 * Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> and Bodo
4 * Moeller for the OpenSSL project.
6 /* ====================================================================
7 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
9 * Redistribution and use in source and binary forms, with or without
10 * modification, are permitted provided that the following conditions
13 * 1. Redistributions of source code must retain the above copyright
14 * notice, this list of conditions and the following disclaimer.
16 * 2. Redistributions in binary form must reproduce the above copyright
17 * notice, this list of conditions and the following disclaimer in
18 * the documentation and/or other materials provided with the
21 * 3. All advertising materials mentioning features or use of this
22 * software must display the following acknowledgment:
23 * "This product includes software developed by the OpenSSL Project
24 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
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27 * endorse or promote products derived from this software without
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29 * openssl-core@openssl.org.
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32 * nor may "OpenSSL" appear in their names without prior written
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35 * 6. Redistributions of any form whatsoever must retain the following
37 * "This product includes software developed by the OpenSSL Project
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40 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
41 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
42 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
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49 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
50 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
51 * OF THE POSSIBILITY OF SUCH DAMAGE.
52 * ====================================================================
54 * This product includes cryptographic software written by Eric Young
55 * (eay@cryptsoft.com). This product includes software written by Tim
56 * Hudson (tjh@cryptsoft.com).
63 BIGNUM
*BN_mod_sqrt(BIGNUM
*in
, const BIGNUM
*a
, const BIGNUM
*p
, BN_CTX
*ctx
)
65 * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
66 * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
67 * Theory", algorithm 1.5.1). 'p' must be prime!
73 BIGNUM
*A
, *b
, *q
, *t
, *x
, *y
;
76 if (!BN_is_odd(p
) || BN_abs_is_word(p
, 1)) {
77 if (BN_abs_is_word(p
, 2)) {
82 if (!BN_set_word(ret
, BN_is_bit_set(a
, 0))) {
91 BNerr(BN_F_BN_MOD_SQRT
, BN_R_P_IS_NOT_PRIME
);
95 if (BN_is_zero(a
) || BN_is_one(a
)) {
100 if (!BN_set_word(ret
, BN_is_one(a
))) {
125 if (!BN_nnmod(A
, a
, p
, ctx
))
128 /* now write |p| - 1 as 2^e*q where q is odd */
130 while (!BN_is_bit_set(p
, e
))
132 /* we'll set q later (if needed) */
136 * The easy case: (|p|-1)/2 is odd, so 2 has an inverse
137 * modulo (|p|-1)/2, and square roots can be computed
138 * directly by modular exponentiation.
140 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
141 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
143 if (!BN_rshift(q
, p
, 2))
146 if (!BN_add_word(q
, 1))
148 if (!BN_mod_exp(ret
, A
, q
, p
, ctx
))
158 * In this case 2 is always a non-square since
159 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
160 * So if a really is a square, then 2*a is a non-square.
162 * b := (2*a)^((|p|-5)/8),
165 * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
171 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
172 * = a^2 * b^2 * (-2*i)
177 * (This is due to A.O.L. Atkin,
178 * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
183 if (!BN_mod_lshift1_quick(t
, A
, p
))
186 /* b := (2*a)^((|p|-5)/8) */
187 if (!BN_rshift(q
, p
, 3))
190 if (!BN_mod_exp(b
, t
, q
, p
, ctx
))
194 if (!BN_mod_sqr(y
, b
, p
, ctx
))
197 /* t := (2*a)*b^2 - 1 */
198 if (!BN_mod_mul(t
, t
, y
, p
, ctx
))
200 if (!BN_sub_word(t
, 1))
204 if (!BN_mod_mul(x
, A
, b
, p
, ctx
))
206 if (!BN_mod_mul(x
, x
, t
, p
, ctx
))
209 if (!BN_copy(ret
, x
))
216 * e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
217 * find some y that is not a square.
220 goto end
; /* use 'q' as temp */
225 * For efficiency, try small numbers first; if this fails, try random
229 if (!BN_set_word(y
, i
))
232 if (!BN_pseudo_rand(y
, BN_num_bits(p
), 0, 0))
234 if (BN_ucmp(y
, p
) >= 0) {
235 if (!(p
->neg
? BN_add
: BN_sub
) (y
, y
, p
))
238 /* now 0 <= y < |p| */
240 if (!BN_set_word(y
, i
))
244 r
= BN_kronecker(y
, q
, ctx
); /* here 'q' is |p| */
249 BNerr(BN_F_BN_MOD_SQRT
, BN_R_P_IS_NOT_PRIME
);
253 while (r
== 1 && ++i
< 82);
257 * Many rounds and still no non-square -- this is more likely a bug
258 * than just bad luck. Even if p is not prime, we should have found
259 * some y such that r == -1.
261 BNerr(BN_F_BN_MOD_SQRT
, BN_R_TOO_MANY_ITERATIONS
);
265 /* Here's our actual 'q': */
266 if (!BN_rshift(q
, q
, e
))
270 * Now that we have some non-square, we can find an element of order 2^e
271 * by computing its q'th power.
273 if (!BN_mod_exp(y
, y
, q
, p
, ctx
))
276 BNerr(BN_F_BN_MOD_SQRT
, BN_R_P_IS_NOT_PRIME
);
281 * Now we know that (if p is indeed prime) there is an integer
282 * k, 0 <= k < 2^e, such that
284 * a^q * y^k == 1 (mod p).
286 * As a^q is a square and y is not, k must be even.
287 * q+1 is even, too, so there is an element
289 * X := a^((q+1)/2) * y^(k/2),
293 * X^2 = a^q * a * y^k
296 * so it is the square root that we are looking for.
299 /* t := (q-1)/2 (note that q is odd) */
300 if (!BN_rshift1(t
, q
))
303 /* x := a^((q-1)/2) */
304 if (BN_is_zero(t
)) { /* special case: p = 2^e + 1 */
305 if (!BN_nnmod(t
, A
, p
, ctx
))
308 /* special case: a == 0 (mod p) */
312 } else if (!BN_one(x
))
315 if (!BN_mod_exp(x
, A
, t
, p
, ctx
))
318 /* special case: a == 0 (mod p) */
325 /* b := a*x^2 (= a^q) */
326 if (!BN_mod_sqr(b
, x
, p
, ctx
))
328 if (!BN_mod_mul(b
, b
, A
, p
, ctx
))
331 /* x := a*x (= a^((q+1)/2)) */
332 if (!BN_mod_mul(x
, x
, A
, p
, ctx
))
337 * Now b is a^q * y^k for some even k (0 <= k < 2^E
338 * where E refers to the original value of e, which we
339 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
347 if (!BN_copy(ret
, x
))
353 /* find smallest i such that b^(2^i) = 1 */
355 if (!BN_mod_sqr(t
, b
, p
, ctx
))
357 while (!BN_is_one(t
)) {
360 BNerr(BN_F_BN_MOD_SQRT
, BN_R_NOT_A_SQUARE
);
363 if (!BN_mod_mul(t
, t
, t
, p
, ctx
))
367 /* t := y^2^(e - i - 1) */
370 for (j
= e
- i
- 1; j
> 0; j
--) {
371 if (!BN_mod_sqr(t
, t
, p
, ctx
))
374 if (!BN_mod_mul(y
, t
, t
, p
, ctx
))
376 if (!BN_mod_mul(x
, x
, t
, p
, ctx
))
378 if (!BN_mod_mul(b
, b
, y
, p
, ctx
))
386 * verify the result -- the input might have been not a square (test
390 if (!BN_mod_sqr(x
, ret
, p
, ctx
))
393 if (!err
&& 0 != BN_cmp(x
, A
)) {
394 BNerr(BN_F_BN_MOD_SQRT
, BN_R_NOT_A_SQUARE
);