Algorithms for Inversion Mod $p^k$

This article describes and analyzes all existing algorithms for computing x = a
^-1
(mod p
^k
) for a prime p, and also introduces a new algorithm based on the exact solution of linear equations using p-adic expansions. The algorithm starts with the initial value c = a
^-1
(mod p) and iterativelycomputes the digits of the inverse x = a
^-1
(mod p
^k
) in base p. The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of k. Moreover, it stands out as being the only one that works for any p, any k, and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms.