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Primality Testing

The library implements a Miller-Rabin test for primality:

#include <boost/multiprecision/miller_rabin.hpp>

template <class Backend, expression_template_option ExpressionTemplates, class Engine>
bool miller_rabin_test(const number<Backend, ExpressionTemplates>& n, unsigned trials, Engine& gen);

template <class Backend, expression_template_option ExpressionTemplates, class Engine>
bool miller_rabin_test(const number<Backend, ExpressionTemplates>& n, unsigned trials);

These functions perform a Miller-Rabin test for primality, if the result is false then n is definitely composite, while if the result is true then n is prime with probability 0.25^trials. The algorithm used performs some trial divisions to exclude small prime factors, does one Fermat test to exclude many more composites, and then uses the Miller-Rabin algorithm straight out of Knuth Vol 2, which recommends 25 trials for a pretty strong likelihood that n is prime.

The third optional argument is for a Uniform Random Number Generator from Boost.Random. When not provided the mt19937 generator is used. Note that when producing random primes then you should probably use a different random number generator to produce candidate prime numbers for testing, than is used internally by miller_rabin_test for determining whether the value is prime. It also helps of course to seed the generators with some source of randomness.

The following example searches for a prime p for which (p-1)/2 is also probably prime:

#include <boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/miller_rabin.hpp>
#include <iostream>
#include <iomanip>

int main()
{
   using namespace boost::random;
   using namespace boost::multiprecision;

   typedef cpp_int int_type;
   mt11213b base_gen(clock());
   independent_bits_engine<mt11213b, 256, int_type> gen(base_gen);
   //
   // We must use a different generator for the tests and number generation, otherwise
   // we get false positives.
   //
   mt19937 gen2(clock());

   for(unsigned i = 0; i < 100000; ++i)
   {
      int_type n = gen();
      if(miller_rabin_test(n, 25, gen2))
      {
         // Value n is probably prime, see if (n-1)/2 is also prime:
         std::cout << "We have a probable prime with value: " << std::hex << std::showbase << n << std::endl;
         if(miller_rabin_test((n-1)/2, 25, gen2))
         {
            std::cout << "We have a safe prime with value: " << std::hex << std::showbase << n << std::endl;
            return 0;
         }
      }
   }
   std::cout << "Ooops, no safe primes were found" << std::endl;
   return 1;
}


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