# Boost C++ Libraries

...one of the most highly regarded and expertly designed C++ library projects in the world.

### boost/integer/mod_inverse.hpp

```/*
*  (C) Copyright Nick Thompson 2018.
*  Use, modification and distribution are subject to the
*  Boost Software License, Version 1.0. (See accompanying file
*/
#ifndef BOOST_INTEGER_MOD_INVERSE_HPP
#define BOOST_INTEGER_MOD_INVERSE_HPP
#include <stdexcept>
#include <boost/throw_exception.hpp>
#include <boost/integer/extended_euclidean.hpp>

namespace boost { namespace integer {

// From "The Joy of Factoring", Algorithm 2.7.
// Here's some others names I've found for this function:
// PowerMod[a, -1, m] (Mathematica)
// mpz_invert (gmplib)
// modinv (some dude on stackoverflow)
// Would mod_inverse be sometimes mistaken as the modular *additive* inverse?
// In any case, I think this is the best name we can get for this function without agonizing.
template<class Z>
Z mod_inverse(Z a, Z modulus)
{
if (modulus < Z(2))
{
BOOST_THROW_EXCEPTION(std::domain_error("mod_inverse: modulus must be > 1"));
}
// make sure a < modulus:
a = a % modulus;
if (a == Z(0))
{
// a doesn't have a modular multiplicative inverse:
return Z(0);
}
boost::integer::euclidean_result_t<Z> u = boost::integer::extended_euclidean(a, modulus);
if (u.gcd > Z(1))
{
return Z(0);
}
// x might not be in the range 0 < x < m, let's fix that:
while (u.x <= Z(0))
{
u.x += modulus;
}
// While indeed this is an inexpensive and comforting check,
// the multiplication overflows and hence makes the check itself buggy.
//BOOST_ASSERT(u.x*a % modulus == 1);
return u.x;
}

}}
#endif
```