New classes of permutation polynomials of special form

Source abstract

In this paper, we propose six new classes of permutation polynomials. Precisely, for $p=3$ or $p=5$, three new classes of permutation polynomials over $\mathbb{F}_{p^{2m}}$ of the form $(x^{p^m}-x+\delta)^{s}+L(x)$, and three new classes of permutation polynomials over $\mathbb{F}_{p^{2m}}$ of the form $(x^{p^m}-x+\delta)^{s_1}+(x^{p^m}-x+\delta)^{s_2}+L(x)$, are given where $L(x)=ax^{p^m}+ax$ with $a \in \mathbb{F}_{p^m}^*$. We use a method based on the AGW criterion and the determination of the number of solutions of certain type of equations. The sufficient conditions on $\delta\in \mathbb{F}_{p^{2m}}$ are shown to be necessary. Received: 03 July 2024 | Accepted: 13 October 2024