# Necessities and sufficiencies of a class of permutation polynomials over finite fields

## TL;DR

This paper investigates specific permutation polynomials over finite fields of characteristic two, providing necessary and sufficient conditions for their permutation properties by analyzing field element structures.

## Contribution

It offers a complete characterization of when certain polynomial forms permute elements in finite fields of the form _{2^{3m}}.

## Key findings

- Derived necessary and sufficient conditions for permutation polynomials
- Analyzed structures and properties of field elements involved
- Established criteria for permutation behavior in finite fields

## Abstract

For the finite field $\mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+\delta)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the structures and properties of the field elements are analyzed.