410 lines
12 KiB
C
410 lines
12 KiB
C
/* crypto/bn/bn_sqrt.c */
|
|
/*
|
|
* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> and Bodo
|
|
* Moeller for the OpenSSL project.
|
|
*/
|
|
/* ====================================================================
|
|
* Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
|
|
*
|
|
* Redistribution and use in source and binary forms, with or without
|
|
* modification, are permitted provided that the following conditions
|
|
* are met:
|
|
*
|
|
* 1. Redistributions of source code must retain the above copyright
|
|
* notice, this list of conditions and the following disclaimer.
|
|
*
|
|
* 2. Redistributions in binary form must reproduce the above copyright
|
|
* notice, this list of conditions and the following disclaimer in
|
|
* the documentation and/or other materials provided with the
|
|
* distribution.
|
|
*
|
|
* 3. All advertising materials mentioning features or use of this
|
|
* software must display the following acknowledgment:
|
|
* "This product includes software developed by the OpenSSL Project
|
|
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
|
|
*
|
|
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
|
|
* endorse or promote products derived from this software without
|
|
* prior written permission. For written permission, please contact
|
|
* openssl-core@openssl.org.
|
|
*
|
|
* 5. Products derived from this software may not be called "OpenSSL"
|
|
* nor may "OpenSSL" appear in their names without prior written
|
|
* permission of the OpenSSL Project.
|
|
*
|
|
* 6. Redistributions of any form whatsoever must retain the following
|
|
* acknowledgment:
|
|
* "This product includes software developed by the OpenSSL Project
|
|
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
|
|
*
|
|
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
|
|
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
|
|
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
|
|
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
|
|
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
|
|
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
|
|
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
|
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
|
|
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
|
|
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
|
|
* OF THE POSSIBILITY OF SUCH DAMAGE.
|
|
* ====================================================================
|
|
*
|
|
* This product includes cryptographic software written by Eric Young
|
|
* (eay@cryptsoft.com). This product includes software written by Tim
|
|
* Hudson (tjh@cryptsoft.com).
|
|
*
|
|
*/
|
|
|
|
#include "cryptlib.h"
|
|
#include "bn_lcl.h"
|
|
|
|
BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
|
|
/*
|
|
* Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
|
|
* algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
|
|
* Theory", algorithm 1.5.1). 'p' must be prime!
|
|
*/
|
|
{
|
|
BIGNUM *ret = in;
|
|
int err = 1;
|
|
int r;
|
|
BIGNUM *A, *b, *q, *t, *x, *y;
|
|
int e, i, j;
|
|
|
|
if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
|
|
if (BN_abs_is_word(p, 2)) {
|
|
if (ret == NULL)
|
|
ret = BN_new();
|
|
if (ret == NULL)
|
|
goto end;
|
|
if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
|
|
if (ret != in)
|
|
BN_free(ret);
|
|
return NULL;
|
|
}
|
|
bn_check_top(ret);
|
|
return ret;
|
|
}
|
|
|
|
BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
|
|
return (NULL);
|
|
}
|
|
|
|
if (BN_is_zero(a) || BN_is_one(a)) {
|
|
if (ret == NULL)
|
|
ret = BN_new();
|
|
if (ret == NULL)
|
|
goto end;
|
|
if (!BN_set_word(ret, BN_is_one(a))) {
|
|
if (ret != in)
|
|
BN_free(ret);
|
|
return NULL;
|
|
}
|
|
bn_check_top(ret);
|
|
return ret;
|
|
}
|
|
|
|
BN_CTX_start(ctx);
|
|
A = BN_CTX_get(ctx);
|
|
b = BN_CTX_get(ctx);
|
|
q = BN_CTX_get(ctx);
|
|
t = BN_CTX_get(ctx);
|
|
x = BN_CTX_get(ctx);
|
|
y = BN_CTX_get(ctx);
|
|
if (y == NULL)
|
|
goto end;
|
|
|
|
if (ret == NULL)
|
|
ret = BN_new();
|
|
if (ret == NULL)
|
|
goto end;
|
|
|
|
/* A = a mod p */
|
|
if (!BN_nnmod(A, a, p, ctx))
|
|
goto end;
|
|
|
|
/* now write |p| - 1 as 2^e*q where q is odd */
|
|
e = 1;
|
|
while (!BN_is_bit_set(p, e))
|
|
e++;
|
|
/* we'll set q later (if needed) */
|
|
|
|
if (e == 1) {
|
|
/*-
|
|
* The easy case: (|p|-1)/2 is odd, so 2 has an inverse
|
|
* modulo (|p|-1)/2, and square roots can be computed
|
|
* directly by modular exponentiation.
|
|
* We have
|
|
* 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2),
|
|
* so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1.
|
|
*/
|
|
if (!BN_rshift(q, p, 2))
|
|
goto end;
|
|
q->neg = 0;
|
|
if (!BN_add_word(q, 1))
|
|
goto end;
|
|
if (!BN_mod_exp(ret, A, q, p, ctx))
|
|
goto end;
|
|
err = 0;
|
|
goto vrfy;
|
|
}
|
|
|
|
if (e == 2) {
|
|
/*-
|
|
* |p| == 5 (mod 8)
|
|
*
|
|
* In this case 2 is always a non-square since
|
|
* Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
|
|
* So if a really is a square, then 2*a is a non-square.
|
|
* Thus for
|
|
* b := (2*a)^((|p|-5)/8),
|
|
* i := (2*a)*b^2
|
|
* we have
|
|
* i^2 = (2*a)^((1 + (|p|-5)/4)*2)
|
|
* = (2*a)^((p-1)/2)
|
|
* = -1;
|
|
* so if we set
|
|
* x := a*b*(i-1),
|
|
* then
|
|
* x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
|
|
* = a^2 * b^2 * (-2*i)
|
|
* = a*(-i)*(2*a*b^2)
|
|
* = a*(-i)*i
|
|
* = a.
|
|
*
|
|
* (This is due to A.O.L. Atkin,
|
|
* <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
|
|
* November 1992.)
|
|
*/
|
|
|
|
/* t := 2*a */
|
|
if (!BN_mod_lshift1_quick(t, A, p))
|
|
goto end;
|
|
|
|
/* b := (2*a)^((|p|-5)/8) */
|
|
if (!BN_rshift(q, p, 3))
|
|
goto end;
|
|
q->neg = 0;
|
|
if (!BN_mod_exp(b, t, q, p, ctx))
|
|
goto end;
|
|
|
|
/* y := b^2 */
|
|
if (!BN_mod_sqr(y, b, p, ctx))
|
|
goto end;
|
|
|
|
/* t := (2*a)*b^2 - 1 */
|
|
if (!BN_mod_mul(t, t, y, p, ctx))
|
|
goto end;
|
|
if (!BN_sub_word(t, 1))
|
|
goto end;
|
|
|
|
/* x = a*b*t */
|
|
if (!BN_mod_mul(x, A, b, p, ctx))
|
|
goto end;
|
|
if (!BN_mod_mul(x, x, t, p, ctx))
|
|
goto end;
|
|
|
|
if (!BN_copy(ret, x))
|
|
goto end;
|
|
err = 0;
|
|
goto vrfy;
|
|
}
|
|
|
|
/*
|
|
* e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
|
|
* find some y that is not a square.
|
|
*/
|
|
if (!BN_copy(q, p))
|
|
goto end; /* use 'q' as temp */
|
|
q->neg = 0;
|
|
i = 2;
|
|
do {
|
|
/*
|
|
* For efficiency, try small numbers first; if this fails, try random
|
|
* numbers.
|
|
*/
|
|
if (i < 22) {
|
|
if (!BN_set_word(y, i))
|
|
goto end;
|
|
} else {
|
|
if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
|
|
goto end;
|
|
if (BN_ucmp(y, p) >= 0) {
|
|
if (!(p->neg ? BN_add : BN_sub) (y, y, p))
|
|
goto end;
|
|
}
|
|
/* now 0 <= y < |p| */
|
|
if (BN_is_zero(y))
|
|
if (!BN_set_word(y, i))
|
|
goto end;
|
|
}
|
|
|
|
r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
|
|
if (r < -1)
|
|
goto end;
|
|
if (r == 0) {
|
|
/* m divides p */
|
|
BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
|
|
goto end;
|
|
}
|
|
}
|
|
while (r == 1 && ++i < 82);
|
|
|
|
if (r != -1) {
|
|
/*
|
|
* Many rounds and still no non-square -- this is more likely a bug
|
|
* than just bad luck. Even if p is not prime, we should have found
|
|
* some y such that r == -1.
|
|
*/
|
|
BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
|
|
goto end;
|
|
}
|
|
|
|
/* Here's our actual 'q': */
|
|
if (!BN_rshift(q, q, e))
|
|
goto end;
|
|
|
|
/*
|
|
* Now that we have some non-square, we can find an element of order 2^e
|
|
* by computing its q'th power.
|
|
*/
|
|
if (!BN_mod_exp(y, y, q, p, ctx))
|
|
goto end;
|
|
if (BN_is_one(y)) {
|
|
BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
|
|
goto end;
|
|
}
|
|
|
|
/*-
|
|
* Now we know that (if p is indeed prime) there is an integer
|
|
* k, 0 <= k < 2^e, such that
|
|
*
|
|
* a^q * y^k == 1 (mod p).
|
|
*
|
|
* As a^q is a square and y is not, k must be even.
|
|
* q+1 is even, too, so there is an element
|
|
*
|
|
* X := a^((q+1)/2) * y^(k/2),
|
|
*
|
|
* and it satisfies
|
|
*
|
|
* X^2 = a^q * a * y^k
|
|
* = a,
|
|
*
|
|
* so it is the square root that we are looking for.
|
|
*/
|
|
|
|
/* t := (q-1)/2 (note that q is odd) */
|
|
if (!BN_rshift1(t, q))
|
|
goto end;
|
|
|
|
/* x := a^((q-1)/2) */
|
|
if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
|
|
if (!BN_nnmod(t, A, p, ctx))
|
|
goto end;
|
|
if (BN_is_zero(t)) {
|
|
/* special case: a == 0 (mod p) */
|
|
BN_zero(ret);
|
|
err = 0;
|
|
goto end;
|
|
} else if (!BN_one(x))
|
|
goto end;
|
|
} else {
|
|
if (!BN_mod_exp(x, A, t, p, ctx))
|
|
goto end;
|
|
if (BN_is_zero(x)) {
|
|
/* special case: a == 0 (mod p) */
|
|
BN_zero(ret);
|
|
err = 0;
|
|
goto end;
|
|
}
|
|
}
|
|
|
|
/* b := a*x^2 (= a^q) */
|
|
if (!BN_mod_sqr(b, x, p, ctx))
|
|
goto end;
|
|
if (!BN_mod_mul(b, b, A, p, ctx))
|
|
goto end;
|
|
|
|
/* x := a*x (= a^((q+1)/2)) */
|
|
if (!BN_mod_mul(x, x, A, p, ctx))
|
|
goto end;
|
|
|
|
while (1) {
|
|
/*-
|
|
* Now b is a^q * y^k for some even k (0 <= k < 2^E
|
|
* where E refers to the original value of e, which we
|
|
* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
|
|
*
|
|
* We have a*b = x^2,
|
|
* y^2^(e-1) = -1,
|
|
* b^2^(e-1) = 1.
|
|
*/
|
|
|
|
if (BN_is_one(b)) {
|
|
if (!BN_copy(ret, x))
|
|
goto end;
|
|
err = 0;
|
|
goto vrfy;
|
|
}
|
|
|
|
/* find smallest i such that b^(2^i) = 1 */
|
|
i = 1;
|
|
if (!BN_mod_sqr(t, b, p, ctx))
|
|
goto end;
|
|
while (!BN_is_one(t)) {
|
|
i++;
|
|
if (i == e) {
|
|
BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
|
|
goto end;
|
|
}
|
|
if (!BN_mod_mul(t, t, t, p, ctx))
|
|
goto end;
|
|
}
|
|
|
|
/* t := y^2^(e - i - 1) */
|
|
if (!BN_copy(t, y))
|
|
goto end;
|
|
for (j = e - i - 1; j > 0; j--) {
|
|
if (!BN_mod_sqr(t, t, p, ctx))
|
|
goto end;
|
|
}
|
|
if (!BN_mod_mul(y, t, t, p, ctx))
|
|
goto end;
|
|
if (!BN_mod_mul(x, x, t, p, ctx))
|
|
goto end;
|
|
if (!BN_mod_mul(b, b, y, p, ctx))
|
|
goto end;
|
|
e = i;
|
|
}
|
|
|
|
vrfy:
|
|
if (!err) {
|
|
/*
|
|
* verify the result -- the input might have been not a square (test
|
|
* added in 0.9.8)
|
|
*/
|
|
|
|
if (!BN_mod_sqr(x, ret, p, ctx))
|
|
err = 1;
|
|
|
|
if (!err && 0 != BN_cmp(x, A)) {
|
|
BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
|
|
err = 1;
|
|
}
|
|
}
|
|
|
|
end:
|
|
if (err) {
|
|
if (ret != NULL && ret != in) {
|
|
BN_clear_free(ret);
|
|
}
|
|
ret = NULL;
|
|
}
|
|
BN_CTX_end(ctx);
|
|
bn_check_top(ret);
|
|
return ret;
|
|
}
|