# [Resource Topic] 2017/411: A New Algorithm for Inversion mod $p^k$

Welcome to the resource topic for 2017/411

Title:
A New Algorithm for Inversion mod p^k

Authors: 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 existing algorithms for small values of k. We also describe and analyze all existing algorithms, and compare them to the proposed algorithm. Our algorithm stands out as being the only one that works for any p, any k, and digit-by-digit. Moreover it requires the minimal number of arithmetic operations (just a single addition) per step.

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