Givaro
examples/Integer/probable_primroot.C

NO DOC

// Copyright(c)'1994-2009 by The Givaro group
// This file is part of Givaro.
// Givaro is governed by the CeCILL-B license under French law
// and abiding by the rules of distribution of free software.
// see the COPYRIGHT file for more details.
#include <iostream>
#include <givaro/givintnumtheo.h>
#include <givaro/givtimer.h>
#include <cmath>
using namespace Givaro;
// Polynomial-time generation of primitive roots
// L is number of loops of Pollard partial factorization of n-1
// 10,000,000 gives at least 1-2^{-40} probability of success
// [Dubrois & Dumas, Industrial-strength primitive roots]
// Returns the probable primitive root and the probability of error.
int main(int argc, char** argv)
{
#ifdef __GMP_PLUSPLUS__
IP.seeding( (unsigned long)BaseTimer::seed() );
#endif
double error;
if (argc > 1) a = IntNumTheoDom<>::Element(argv[1]); else std::cin >> a;
bool comp ; if ( (comp=(! IP.isprime(a))) ) std::cerr << a << " is not prime, primitive root will have no sense and may loop forever ..." << std::endl;
double epsilon = argc > 2 ? atof(argv[2]) : 0.0000001;
Timer tim; tim.clear();
tim.start();
//======================================================================
// Default is partial factorization up to factors of at least 12 digits.
// with probability of error much less than 2^{-40}
// IP.probable_prim_root(pr, error, a );
//======================================================================
// Choosing L to be O(log^2(p))
// gives the best probability with O(log^4(p)) complexity
// Probability of error is approximately O(1/log^4(p))
// IP.probable_prim_root(pr, error, a, (unsigned long)power(logtwo(a),2));
//======================================================================
// Choosing L to be O( \sqrt(epsilon) )
// gives probability of error at most epsilon
// Newton-Raphson iteration is used for
// 1-epsilon = (1+2/(p-1))*(1-1/B)^(ln( (p-1)/2 )/ln(B))
// So that no factor less than B can be avoided
// With Pollard's rho factorization, L is chosen to be sqrt(B)
// Might not be polynomial if epsilon is too big
#define GIVARO_POLLARD
IP.probable_prim_root(pr, error, a, epsilon );
tim.stop();
if (comp) std::cerr << IP.gcd(g,a,pr) << " is a factor of " << a << std::endl;
IntegerDom().write( std::cout << "Prim root : ", pr );
if (error > 0) {
std::cout << ", correct with probability at least : 1-" << error << std::endl;
std::cerr << tim << std::endl;
std::cerr << "Now checking primitivity, this may take some time (complete factorization of n-1) ...";
#define GIVARO_LENSTRA
Timer verif; verif.clear(); verif.start();
if ( IP.isorder(a-1, pr, a) ) {
verif.stop();
std::cerr << "... Pimitivity checked" << std::endl;
std::cerr << verif << std::endl;
}
else {
verif.stop();
std::cerr << "... WARNING : FAILURE" << std::endl;
std::cerr << verif << std::endl;
}
} else {
std::cout << ", deterministically correct" << std::endl;
std::cerr << tim << std::endl;
}
return 0;
}