On the Inversion Modulo a Power of an Integer

TL;DR

This paper introduces a flexible and efficient algorithm for computing modular inverses modulo powers of any integer, extending existing methods and optimizing for computer architecture, with experimental validation.

Contribution

It generalizes modular inverse algorithms to any integer base, improving efficiency and flexibility, and explores implementation benefits using computer architecture features.

Findings

Significant performance improvements in modular inverse computations.

Successful generalization of existing algorithms to broader classes of moduli.

Experimental results demonstrate practical efficiency gains.

Abstract

Recently, Ko\c{c} proposed a neat and efficient algorithm for computing \[ x = a^{-1} \pmod {p^k} \] for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right shift per step. In the first part of this paper, we design an algorithm that computes \[ x = a^{-1} \pmod {n^k} \] for any integers $a, n>1$ with $\gcd(a, n)=1$. The algorithm has a motivation from the schoolbook multiplication and achieves both efficiency and generality. The greater flexibility of our algorithm is explored by utilizing the built-in arithmetic of computer architecture, e.g., $n=2^{64}$, and experimental results show significant improvements. This paper also contains some results on modular inverse based on an alternative proof of correctness of Ko\c{c} algorithm. For the computation of modular inverses when the modulus is a special power of a prime $p$ (i.e., of the form $p^{2^s}$), an efficient algorithm was developed by Dumas and later improved by Hurchalla. These methods are based on Hensel lifting and perform particularly well when $p=2$ and $2^s$ matches the native bit width of a computer. In the second part of the paper, we present a generalization of these methods to moduli of the form $n^{2^s}$ for any integer $n>1$. The derivation of our algorithm follows from a simple algebraic manipulation.