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,
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44 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
45 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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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 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
97 r = BN_kronecker(a, p, ctx);
98 if (r < -1) return NULL;
101 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
112 if (y == NULL) goto end;
116 if (ret == NULL) goto end;
118 /* now write |p| - 1 as 2^e*q where q is odd */
120 while (!BN_is_bit_set(p, e))
122 if (!BN_rshift(q, p, e)) goto end;
127 /* The easy case: (p-1)/2 is odd, so 2 has an inverse
128 * modulo (p-1)/2, and square roots can be computed
129 * directly by modular exponentiation.
131 * 2 * (p+1)/4 == 1 (mod (p-1)/2),
132 * so we can use exponent (p+1)/4, i.e. (q+1)/2.
134 if (!BN_add_word(q,1)) goto end;
135 if (!BN_rshift1(q,q)) goto end;
136 if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
141 /* e > 1, so we really have to use the Tonelli/Shanks algorithm.
142 * First, find some y that is not a square. */
146 /* For efficiency, try small numbers first;
147 * if this fails, try random numbers.
151 if (!BN_set_word(y, i)) goto end;
155 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
156 if (BN_ucmp(y, p) >= 0)
158 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
160 /* now 0 <= y < |p| */
162 if (!BN_set_word(y, i)) goto end;
165 r = BN_kronecker(y, p, ctx);
166 if (r < -1) goto end;
170 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
174 while (r == 1 && i++ < 80);
178 /* Many rounds and still no non-square -- this is more likely
179 * a bug than just bad luck.
180 * Even if p is not prime, we should have found some y
183 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
188 /* Now that we have some non-square, we can find an element
189 * of order 2^e by computing its q'th power. */
190 if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
193 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
197 /* Now we know that (if p is indeed prime) there is an integer
198 * k, 0 <= k < 2^e, such that
200 * a^q * y^k == 1 (mod p).
202 * As a^q is a square and y is not, k must be even.
203 * q+1 is even, too, so there is an element
205 * X := a^((q+1)/2) * y^(k/2),
209 * X^2 = a^q * a * y^k
212 * so it is the square root that we are looking for.
215 /* t := (q-1)/2 (note that q is odd) */
216 if (!BN_rshift1(t, q)) goto end;
218 /* x := a^((q-1)/2) */
219 if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
221 if (!BN_nnmod(t, a, p, ctx)) goto end;
224 /* special case: a == 0 (mod p) */
225 if (!BN_zero(ret)) goto end;
230 if (!BN_one(x)) goto end;
234 if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
237 /* special case: a == 0 (mod p) */
238 if (!BN_zero(ret)) goto end;
244 /* b := a*x^2 (= a^q) */
245 if (!BN_mod_sqr(b, x, p, ctx)) goto end;
246 if (!BN_mod_mul(b, b, a, p, ctx)) goto end;
248 /* x := a*x (= a^((q+1)/2)) */
249 if (!BN_mod_mul(x, x, a, p, ctx)) goto end;
253 /* Now b is a^q * y^k for some even k (0 <= k < 2^E
254 * where E refers to the original value of e, which we
255 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
264 if (!BN_copy(ret, x)) goto end;
270 /* find smallest i such that b^(2^i) = 1 */
272 if (!BN_mod_sqr(t, b, p, ctx)) goto end;
273 while (!BN_is_one(t))
278 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
281 if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
285 /* t := y^2^(e - i - 1) */
286 if (!BN_copy(t, y)) goto end;
287 for (j = e - i - 1; j > 0; j--)
289 if (!BN_mod_sqr(t, t, p, ctx)) goto end;
291 if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
292 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
293 if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
300 if (ret != NULL && ret != in)