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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.
#include <boost/integer/mod_inverse.hpp> namespace boost { namespace integer { template<class Z> Z mod_inverse(Z a, Z m); }}
int x = mod_inverse(2, 5); // prints x = 3: std::cout << "x = " << x << "\n"; int y = mod_inverse(2, 4); if (y == 0) { std::cout << "There is no inverse of 2 mod 4\n"; }
Multiplicative modular inverses exist if and only if a
and m are coprime. If a and m
share a common factor, then mod_inverse(a,
m)
returns zero.
Wagstaff, Samuel S., The Joy of Factoring, Vol. 68. American Mathematical Soc., 2013.