integer/doc/modular_arithmetic/mod_inverse.qbk

[section:mod_inverse Modular Multiplicative Inverse]

[section Introduction]

The modular multiplicative inverse of a number /a/ is that number /x/ which satisfies /ax/ = 1 mod /p/.
A fast algorithm for computing modular multiplicative inverses based on the extended Euclidean algorithm exists and is provided by Boost.

[endsect]

[section Synopsis]

    #include <boost/integer/mod_inverse.hpp>

    namespace boost { namespace integer {

    template<class Z>
    boost::optional<Z> mod_inverse(Z a, Z m);

    }}

[endsect]

[section Usage]

Multiplicative modular inverses exist if and only if /a/ and /m/ are coprime.
So for example

    auto x = mod_inverse(2, 5);
    if (x)
    {
        int should_be_three = x.value();
    }
    auto y = mod_inverse(2, 4);
    if (!y)
    {
        std::cout << "There is no inverse of 2 mod 4\n";
    }

[endsect]

[section References]
Wagstaff, Samuel S., ['The Joy of Factoring], Vol. 68. American Mathematical Soc., 2013.

[endsect]
[endsect]
[/
Copyright 2018 Nick Thompson.
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt).
]