2 * Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> and Bodo
3 * Moeller for the OpenSSL project.
5 /* ====================================================================
6 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
8 * Redistribution and use in source and binary forms, with or without
9 * modification, are permitted provided that the following conditions
12 * 1. Redistributions of source code must retain the above copyright
13 * notice, this list of conditions and the following disclaimer.
15 * 2. Redistributions in binary form must reproduce the above copyright
16 * notice, this list of conditions and the following disclaimer in
17 * the documentation and/or other materials provided with the
20 * 3. All advertising materials mentioning features or use of this
21 * software must display the following acknowledgment:
22 * "This product includes software developed by the OpenSSL Project
23 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
25 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
26 * endorse or promote products derived from this software without
27 * prior written permission. For written permission, please contact
28 * openssl-core@openssl.org.
30 * 5. Products derived from this software may not be called "OpenSSL"
31 * nor may "OpenSSL" appear in their names without prior written
32 * permission of the OpenSSL Project.
34 * 6. Redistributions of any form whatsoever must retain the following
36 * "This product includes software developed by the OpenSSL Project
37 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
39 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
40 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
41 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
42 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
43 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
44 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
45 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
46 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
47 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
48 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
49 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
50 * OF THE POSSIBILITY OF SUCH DAMAGE.
51 * ====================================================================
53 * This product includes cryptographic software written by Eric Young
54 * (eay@cryptsoft.com). This product includes software written by Tim
55 * Hudson (tjh@cryptsoft.com).
59 #include "internal/cryptlib.h"
62 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
64 * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
65 * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
66 * Theory", algorithm 1.5.1). 'p' must be prime!
72 BIGNUM *A, *b, *q, *t, *x, *y;
75 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
76 if (BN_abs_is_word(p, 2)) {
81 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
90 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
94 if (BN_is_zero(a) || BN_is_one(a)) {
99 if (!BN_set_word(ret, BN_is_one(a))) {
124 if (!BN_nnmod(A, a, p, ctx))
127 /* now write |p| - 1 as 2^e*q where q is odd */
129 while (!BN_is_bit_set(p, e))
131 /* we'll set q later (if needed) */
135 * The easy case: (|p|-1)/2 is odd, so 2 has an inverse
136 * modulo (|p|-1)/2, and square roots can be computed
137 * directly by modular exponentiation.
139 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
140 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
142 if (!BN_rshift(q, p, 2))
145 if (!BN_add_word(q, 1))
147 if (!BN_mod_exp(ret, A, q, p, ctx))
157 * In this case 2 is always a non-square since
158 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
159 * So if a really is a square, then 2*a is a non-square.
161 * b := (2*a)^((|p|-5)/8),
164 * i^2 = (2*a)^((1 + (|p|-5)/4)*2)
170 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
171 * = a^2 * b^2 * (-2*i)
176 * (This is due to A.O.L. Atkin,
177 * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
182 if (!BN_mod_lshift1_quick(t, A, p))
185 /* b := (2*a)^((|p|-5)/8) */
186 if (!BN_rshift(q, p, 3))
189 if (!BN_mod_exp(b, t, q, p, ctx))
193 if (!BN_mod_sqr(y, b, p, ctx))
196 /* t := (2*a)*b^2 - 1 */
197 if (!BN_mod_mul(t, t, y, p, ctx))
199 if (!BN_sub_word(t, 1))
203 if (!BN_mod_mul(x, A, b, p, ctx))
205 if (!BN_mod_mul(x, x, t, p, ctx))
208 if (!BN_copy(ret, x))
215 * e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
216 * find some y that is not a square.
219 goto end; /* use 'q' as temp */
224 * For efficiency, try small numbers first; if this fails, try random
228 if (!BN_set_word(y, i))
231 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
233 if (BN_ucmp(y, p) >= 0) {
234 if (!(p->neg ? BN_add : BN_sub) (y, y, p))
237 /* now 0 <= y < |p| */
239 if (!BN_set_word(y, i))
243 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
248 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
252 while (r == 1 && ++i < 82);
256 * Many rounds and still no non-square -- this is more likely a bug
257 * than just bad luck. Even if p is not prime, we should have found
258 * some y such that r == -1.
260 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
264 /* Here's our actual 'q': */
265 if (!BN_rshift(q, q, e))
269 * Now that we have some non-square, we can find an element of order 2^e
270 * by computing its q'th power.
272 if (!BN_mod_exp(y, y, q, p, ctx))
275 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
280 * Now we know that (if p is indeed prime) there is an integer
281 * k, 0 <= k < 2^e, such that
283 * a^q * y^k == 1 (mod p).
285 * As a^q is a square and y is not, k must be even.
286 * q+1 is even, too, so there is an element
288 * X := a^((q+1)/2) * y^(k/2),
292 * X^2 = a^q * a * y^k
295 * so it is the square root that we are looking for.
298 /* t := (q-1)/2 (note that q is odd) */
299 if (!BN_rshift1(t, q))
302 /* x := a^((q-1)/2) */
303 if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
304 if (!BN_nnmod(t, A, p, ctx))
307 /* special case: a == 0 (mod p) */
311 } else if (!BN_one(x))
314 if (!BN_mod_exp(x, A, t, p, ctx))
317 /* special case: a == 0 (mod p) */
324 /* b := a*x^2 (= a^q) */
325 if (!BN_mod_sqr(b, x, p, ctx))
327 if (!BN_mod_mul(b, b, A, p, ctx))
330 /* x := a*x (= a^((q+1)/2)) */
331 if (!BN_mod_mul(x, x, A, p, ctx))
336 * Now b is a^q * y^k for some even k (0 <= k < 2^E
337 * where E refers to the original value of e, which we
338 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
346 if (!BN_copy(ret, x))
352 /* find smallest i such that b^(2^i) = 1 */
354 if (!BN_mod_sqr(t, b, p, ctx))
356 while (!BN_is_one(t)) {
359 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
362 if (!BN_mod_mul(t, t, t, p, ctx))
366 /* t := y^2^(e - i - 1) */
369 for (j = e - i - 1; j > 0; j--) {
370 if (!BN_mod_sqr(t, t, p, ctx))
373 if (!BN_mod_mul(y, t, t, p, ctx))
375 if (!BN_mod_mul(x, x, t, p, ctx))
377 if (!BN_mod_mul(b, b, y, p, ctx))
385 * verify the result -- the input might have been not a square (test
389 if (!BN_mod_sqr(x, ret, p, ctx))
392 if (!err && 0 != BN_cmp(x, A)) {
393 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);