On the inverses of some classes of permutations of finite fields

TL;DR

This paper investigates the inverses of certain permutation polynomials over finite fields, providing explicit formulas and methods to compute their compositional inverses, especially involving linearized polynomials and subspace bijections.

Contribution

It introduces new techniques to explicitly compute inverses of permutation polynomials over finite fields using linearized polynomials and subspace bijections.

Findings

Explicit inverses for specific classes of permutation polynomials.

Method to compute linearized polynomials inducing inverse maps.

Generalization of previous inverse formulas for permutation polynomials.

Abstract

We study the compositional inverses of some general classes of permutation polynomials over finite fields. We show that we can write these inverses in terms of the inverses of two other polynomials bijecting subspaces of the finite field, where one of these is a linearized polynomial. In some cases we are able to explicitly obtain these inverses, thus obtaining the compositional inverse of the permutation in question. In addition we show how to compute a linearized polynomial inducing the inverse map over subspaces on which a prescribed linearized polynomial induces a bijection. We also obtain the explicit compositional inverses of two classes of permutation polynomials generalizing those whose compositional inverses were recently obtained in [22] and [24], respectively.