Some classes of permutation pentanomials

TL;DR

This paper introduces new large classes of permutation polynomials over finite fields with specific forms, generalizing previous results for p=2 and solving an open problem for p>2 with concise, non-computational proofs.

Contribution

It presents two broad classes of permutation pentanomials over finite fields for all primes p≠3, including a significant generalization for p=2 and novel constructions for p>2.

Findings

Generalization of 76 recent results for p=2

First known permutation polynomials of this form for p>2

Short, non-computational proofs of the main results

Abstract

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The special case p=2 of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case p>2, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.