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/*
* Shanks-Tonnelli (RESSOL)
* (C) 2007-2008 Falko Strenzke, FlexSecure GmbH
* (C) 2008 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
#include <botan/numthry.h>
#include <botan/reducer.h>
namespace Botan {
/*
* Shanks-Tonnelli algorithm
*/
BigInt ressol(const BigInt& a, const BigInt& p)
{
if(a < 0)
throw Invalid_Argument("ressol(): a to solve for must be positive");
if(p <= 1)
throw Invalid_Argument("ressol(): prime must be > 1");
if(a == 0)
return 0;
if(p == 2)
return a;
if(jacobi(a, p) != 1) // not a quadratic residue
return -BigInt(1);
if(p % 4 == 3)
return power_mod(a, ((p+1) >> 2), p);
size_t s = low_zero_bits(p - 1);
BigInt q = p >> s;
q -= 1;
q >>= 1;
Modular_Reducer mod_p(p);
BigInt r = power_mod(a, q, p);
BigInt n = mod_p.multiply(a, mod_p.square(r));
r = mod_p.multiply(r, a);
if(n == 1)
return r;
// find random non quadratic residue z
BigInt z = 2;
while(jacobi(z, p) == 1) // while z quadratic residue
++z;
BigInt c = power_mod(z, (q << 1) + 1, p);
while(n > 1)
{
q = n;
size_t i = 0;
while(q != 1)
{
q = mod_p.square(q);
++i;
}
if(s <= i)
return -BigInt(1);
c = power_mod(c, BigInt::power_of_2(s-i-1), p);
r = mod_p.multiply(r, c);
c = mod_p.square(c);
n = mod_p.multiply(n, c);
s = i;
}
return r;
}
}
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