A New Approach to Permutation Polynomials over Finite Fields, II

TL;DR

This paper introduces new techniques to identify when certain polynomials over finite fields are permutation polynomials, expanding the known classes of such polynomials with potential applications in cryptography and coding theory.

Contribution

It presents novel methods for proving permutation polynomial status and discovers many new classes of desirable triples previously unknown.

Findings

Identified new classes of permutation polynomials over finite fields.

Developed several new techniques for proving polynomials are permutation polynomials.

Expanded the understanding of when the polynomials g_{n,q} are permutations.

Abstract

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$ be the polynomial defined by the functional equation $\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$. When is $g_{n,q}$ a permutation polynomial (PP) of $\Bbb F_{q^e}$? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers $(n,e;q)$ {\em desirable} if $g_{n,q}$ is a PP of $\Bbb F_{q^e}$. In the present paper, we find many new classes of desirable triples whose corresponding PPs were previously unknown. Several new techniques are introduced for proving a given polynomial is a PP.