A class of permutation polynomials over the group ring $\mathbb{F}_pC_{p^{n}}$

Abstract

Let $p$ be a prime number and $\mathbb{F}_pC_{p^n}$ denote the group ring of a cyclic group of order $p^n$ over $\mathbb{F}_{p}$. We study the permutation property of the polynomial $a_0 + a_1 x + a_p x^p + \cdots + a_{p^n} x^{p^n}$ with coefficients $a_i \in \mathbb{F}_pC_{p^n}$ where $i=0, 1, p, \ldots, p^n$. Necessary and sufficient conditions on the coefficients have been obtained so that it becomes a permutation polynomial.

Received: 22 February 2022 | Accepted: 18 June 2022