bn_sqrt.c
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/*
* Copyright 2000-2018 The OpenSSL Project Authors. All Rights Reserved.
*
* Licensed under the OpenSSL license (the "License"). You may not use
* this file except in compliance with the License. You can obtain a copy
* in the file LICENSE in the source distribution or at
* https://www.openssl.org/source/license.html
*/
#include "internal/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_priv_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 != in)
BN_clear_free(ret);
ret = NULL;
}
BN_CTX_end(ctx);
bn_check_top(ret);
return ret;
}