Some families of permutation polynomials over finite fields

TL;DR

This paper establishes precise conditions under which certain structured polynomials permute finite field elements, simplifying the identification of permutation polynomials especially when specific divisibility conditions are met.

Contribution

It provides necessary and sufficient criteria for permutation behavior of polynomials of a specific form over finite fields, including simplified conditions for particular cases.

Findings

Derived explicit permutation criteria for polynomials of the form x^r*(1+x^v+...+x^{kv})^t

Simplified permutation conditions when (q-1)/gcd(q-1,v) is a small prime

Enhanced understanding of permutation polynomials with structured exponents

Abstract

We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small prime.