Linear Permutations and their Compositional Inverses over $\mathbb{F}_{q^n}$

TL;DR

This paper introduces a method to construct linear permutation polynomials and their inverses over finite fields using algebraic decomposition, enabling the creation of involutions useful in cryptography.

Contribution

It presents a novel algebraic approach to construct linear permutation polynomials and their inverses over finite fields, including involutions, via decomposition based on primitive idempotents.

Findings

Constructed several linear permutation polynomials over _{q^n}

Provided explicit methods for their compositional inverses

Enabled immediate construction of involutions

Abstract

The use of permutation polynomials has appeared, along to their compositional inverses, as a good choice in the implementation of cryptographic systems. Hence, there has been a demand for constructions of these polynomials which coefficients belong to a finite field. As a particular case of permutation polynomial, involution is highly desired since its compositional inverse is itself. In this work, we present an effective way of how to construct several linear permutation polynomials over $\mathbb{F}_{q^n}$ as well as their compositional inverses using a decomposition of $\displaystyle{\frac{\mathbb{F}_q[x]}{\left\langle x^n -1 \right\rangle}}$ based on its primitive idempotents. As a consequence, an immediate construction of involutions is presented.