/* 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 <openssl/bn.h>
#include <openssl/err.h>
#include "internal.h"
BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
// Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
// (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
// algorithm 1.5.1). |p| is assumed to be a 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;
}
return ret;
}
OPENSSL_PUT_ERROR(BN, 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;
}
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) ||
!BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
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_consttime(t, A, p, ctx)) {
goto end;
}
// b := (2*a)^((|p|-5)/8)
if (!BN_rshift(q, p, 3)) {
goto end;
}
q->neg = 0;
if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
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) ||
!BN_sub_word(t, 1)) {
goto end;
}
// x = a*b*t
if (!BN_mod_mul(x, A, b, p, ctx) ||
!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_jacobi(y, q, ctx); // here 'q' is |p|
if (r < -1) {
goto end;
}
if (r == 0) {
// m divides p
OPENSSL_PUT_ERROR(BN, 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.
OPENSSL_PUT_ERROR(BN, 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_mont(y, y, q, p, ctx, NULL)) {
goto end;
}
if (BN_is_one(y)) {
OPENSSL_PUT_ERROR(BN, 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_mont(x, A, t, p, ctx, NULL)) {
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) ||
!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) {
OPENSSL_PUT_ERROR(BN, 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) ||
!BN_mod_mul(x, x, t, p, ctx) ||
!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)) {
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
err = 1;
}
}
end:
if (err) {
if (ret != in) {
BN_clear_free(ret);
}
ret = NULL;
}
BN_CTX_end(ctx);
return ret;
}
int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
int ok = 0, last_delta_valid = 0;
if (in->neg) {
OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
return 0;
}
if (BN_is_zero(in)) {
BN_zero(out_sqrt);
return 1;
}
BN_CTX_start(ctx);
if (out_sqrt == in) {
estimate = BN_CTX_get(ctx);
} else {
estimate = out_sqrt;
}
tmp = BN_CTX_get(ctx);
last_delta = BN_CTX_get(ctx);
delta = BN_CTX_get(ctx);
if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
goto err;
}
// We estimate that the square root of an n-bit number is 2^{n/2}.
if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
goto err;
}
// This is Newton's method for finding a root of the equation |estimate|^2 -
// |in| = 0.
for (;;) {
// |estimate| = 1/2 * (|estimate| + |in|/|estimate|)
if (!BN_div(tmp, NULL, in, estimate, ctx) ||
!BN_add(tmp, tmp, estimate) ||
!BN_rshift1(estimate, tmp) ||
// |tmp| = |estimate|^2
!BN_sqr(tmp, estimate, ctx) ||
// |delta| = |in| - |tmp|
!BN_sub(delta, in, tmp)) {
OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
goto err;
}
delta->neg = 0;
// The difference between |in| and |estimate| squared is required to always
// decrease. This ensures that the loop always terminates, but I don't have
// a proof that it always finds the square root for a given square.
if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
break;
}
last_delta_valid = 1;
tmp2 = last_delta;
last_delta = delta;
delta = tmp2;
}
if (BN_cmp(tmp, in) != 0) {
OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
goto err;
}
ok = 1;
err:
if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
ok = 0;
}
BN_CTX_end(ctx);
return ok;
}