ressol.cpp

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/*
* Shanks-Tonnelli (RESSOL)
* (C) 2007-2008 Falko Strenzke, FlexSecure GmbH
* (C) 2008 Jack Lloyd
*
* Distributed under the terms of the Botan license
*/
 
#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(i >= s)
            return -BigInt(1);
         }
 
      c = power_mod(c, BigInt(BigInt::Power2, 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;
   }
 
}