Paper 2017/411

A New Algorithm for Inversion mod $p^k$

Çetin Kaya Koç

Abstract

A new algorithm for computing $x=a^{-1}\pmod{p^k}$ is introduced. It is based on the exact solution of linear equations using $p$-adic expansions. It starts with the initial value $c=a^{-1}\pmod{p}$ and iteratively computes the digits of the inverse $x=a^{-1}\pmod{p^k}$ in base $p$. The mod 2 version of the algorithm is significantly more efficient than the algorithm given by Dussé and Kaliski for computing $a^{-1}\pmod{2^k}$.