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

Abstract

Recently, there has been a lot of work on constructions of permutation polynomials of the form $(x^{2^m}+x+\hat{I}´{)}^s+x$ 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 $(x^{2^m}+x+\hat{I}´{)}^{i(2^m-1)+1}+x$ permutes $\F_{2^{2m}}$, that is, $(2^k+1)i\equiv 1\text{or}{2}^k(\text{mod}{2}^m+1)$, where $1\le k\le m-1$ 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.