boost/multiprecision/miller_rabin.hpp
/////////////////////////////////////////////////////////////// // Copyright 2012 John Maddock. Distributed under the Boost // Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at https://www.boost.org/LICENSE_1_0.txt #ifndef BOOST_MP_MR_HPP #define BOOST_MP_MR_HPP #include <random> #include <cstdint> #include <type_traits> #include <boost/multiprecision/detail/standalone_config.hpp> #include <boost/multiprecision/integer.hpp> #include <boost/multiprecision/detail/uniform_int_distribution.hpp> #include <boost/multiprecision/detail/assert.hpp> namespace boost { namespace multiprecision { namespace detail { template <class I> bool check_small_factors(const I& n) { constexpr std::uint32_t small_factors1[] = { 3u, 5u, 7u, 11u, 13u, 17u, 19u, 23u}; constexpr std::uint32_t pp1 = 223092870u; std::uint32_t m1 = integer_modulus(n, pp1); for (std::size_t i = 0; i < sizeof(small_factors1) / sizeof(small_factors1[0]); ++i) { BOOST_MP_ASSERT(pp1 % small_factors1[i] == 0); if (m1 % small_factors1[i] == 0) return false; } constexpr std::uint32_t small_factors2[] = { 29u, 31u, 37u, 41u, 43u, 47u}; constexpr std::uint32_t pp2 = 2756205443u; m1 = integer_modulus(n, pp2); for (std::size_t i = 0; i < sizeof(small_factors2) / sizeof(small_factors2[0]); ++i) { BOOST_MP_ASSERT(pp2 % small_factors2[i] == 0); if (m1 % small_factors2[i] == 0) return false; } constexpr std::uint32_t small_factors3[] = { 53u, 59u, 61u, 67u, 71u}; constexpr std::uint32_t pp3 = 907383479u; m1 = integer_modulus(n, pp3); for (std::size_t i = 0; i < sizeof(small_factors3) / sizeof(small_factors3[0]); ++i) { BOOST_MP_ASSERT(pp3 % small_factors3[i] == 0); if (m1 % small_factors3[i] == 0) return false; } constexpr std::uint32_t small_factors4[] = { 73u, 79u, 83u, 89u, 97u}; constexpr std::uint32_t pp4 = 4132280413u; m1 = integer_modulus(n, pp4); for (std::size_t i = 0; i < sizeof(small_factors4) / sizeof(small_factors4[0]); ++i) { BOOST_MP_ASSERT(pp4 % small_factors4[i] == 0); if (m1 % small_factors4[i] == 0) return false; } constexpr std::uint32_t small_factors5[6][4] = { {101u, 103u, 107u, 109u}, {113u, 127u, 131u, 137u}, {139u, 149u, 151u, 157u}, {163u, 167u, 173u, 179u}, {181u, 191u, 193u, 197u}, {199u, 211u, 223u, 227u}}; constexpr std::uint32_t pp5[6] = { 121330189u, 113u * 127u * 131u * 137u, 139u * 149u * 151u * 157u, 163u * 167u * 173u * 179u, 181u * 191u * 193u * 197u, 199u * 211u * 223u * 227u}; for (std::size_t k = 0; k < sizeof(pp5) / sizeof(*pp5); ++k) { m1 = integer_modulus(n, pp5[k]); for (std::size_t i = 0; i < 4; ++i) { BOOST_MP_ASSERT(pp5[k] % small_factors5[k][i] == 0); if (m1 % small_factors5[k][i] == 0) return false; } } return true; } inline bool is_small_prime(std::size_t n) { constexpr unsigned char p[] = { 3u, 5u, 7u, 11u, 13u, 17u, 19u, 23u, 29u, 31u, 37u, 41u, 43u, 47u, 53u, 59u, 61u, 67u, 71u, 73u, 79u, 83u, 89u, 97u, 101u, 103u, 107u, 109u, 113u, 127u, 131u, 137u, 139u, 149u, 151u, 157u, 163u, 167u, 173u, 179u, 181u, 191u, 193u, 197u, 199u, 211u, 223u, 227u}; for (std::size_t i = 0; i < sizeof(p) / sizeof(*p); ++i) { if (n == p[i]) return true; } return false; } template <class I> typename std::enable_if<std::is_convertible<I, unsigned>::value, unsigned>::type cast_to_unsigned(const I& val) { return static_cast<unsigned>(val); } template <class I> typename std::enable_if<!std::is_convertible<I, unsigned>::value, unsigned>::type cast_to_unsigned(const I& val) { return val.template convert_to<unsigned>(); } } // namespace detail template <class I, class Engine> typename std::enable_if<number_category<I>::value == number_kind_integer, bool>::type miller_rabin_test(const I& n, std::size_t trials, Engine& gen) { using number_type = I; if (n == 2) return true; // Trivial special case. if (bit_test(n, 0) == 0) return false; // n is even if (n <= 227) return detail::is_small_prime(detail::cast_to_unsigned(n)); if (!detail::check_small_factors(n)) return false; number_type nm1 = n - 1; // // Begin with a single Fermat test - it excludes a lot of candidates: // number_type q(228), x, y; // We know n is greater than this, as we've excluded small factors x = powm(q, nm1, n); if (x != 1u) return false; q = n - 1; std::size_t k = lsb(q); q >>= k; // Declare our random number generator: boost::multiprecision::uniform_int_distribution<number_type> dist(2, n - 2); // // Execute the trials: // for (std::size_t i = 0; i < trials; ++i) { x = dist(gen); y = powm(x, q, n); std::size_t j = 0; while (true) { if (y == nm1) break; if (y == 1) { if (j == 0) break; return false; // test failed } if (++j == k) return false; // failed y = powm(y, 2, n); } } return true; // Yeheh! probably prime. } template <class I> typename std::enable_if<number_category<I>::value == number_kind_integer, bool>::type miller_rabin_test(const I& x, std::size_t trials) { static std::mt19937 gen; return miller_rabin_test(x, trials, gen); } template <class tag, class Arg1, class Arg2, class Arg3, class Arg4, class Engine> bool miller_rabin_test(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& n, std::size_t trials, Engine& gen) { using number_type = typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type; return miller_rabin_test(number_type(n), trials, gen); } template <class tag, class Arg1, class Arg2, class Arg3, class Arg4> bool miller_rabin_test(const detail::expression<tag, Arg1, Arg2, Arg3, Arg4>& n, std::size_t trials) { using number_type = typename detail::expression<tag, Arg1, Arg2, Arg3, Arg4>::result_type; return miller_rabin_test(number_type(n), trials); } }} // namespace boost::multiprecision #endif