A general construction of permutation polynomials of the form $ (x^{2^m}+x+\delta)^{i(2^m-1)+1}+x$ over $\F_{2^{2m}}$

TL;DR

This paper provides a general criterion for constructing permutation polynomials of a specific form over finite fields, extending previous results and introducing many new classes of such polynomials.

Contribution

It establishes a unified sufficient condition on the parameter i for permutation polynomials of a given form over finite fields, generalizing prior work and broadening the class of known permutations.

Findings

Derived a general congruence condition for i

Unified previous sporadic constructions under a common framework

Generated many new classes of permutation polynomials

Abstract

Recently, there has been a lot of work on constructions of permutation polynomials of the form $(x^{2^m}+x+\delta)^{s}+x$ over the finite field $\F_{2^{2m}}$, especially in the case when $s$ is of the form $s=i(2^m-1)+1$ (Niho exponent). In this paper, we further investigate permutation polynomials with this form. Instead of seeking for sporadic constructions of the parameter $i$, we give a general sufficient condition on $i$ such that $(x^{2^m}+x+\delta)^{i(2^m-1)+1}+x$ permutes $\F_{2^{2m}}$, that is, $(2^k+1)i \equiv 1 ~\textrm{or}~ 2^k~(\textrm{mod}~ 2^m+1)$, where $1 \leq k \leq m-1$ is any integer. This generalizes a recent result obtained by Gupta and Sharma who actually dealt with the case $k=2$. It turns out that most of previous constructions of the parameter $i$ are covered by our result, and it yields many new classes of permutation polynomials as well.