Some classes of permutation polynomials over finite fields with odd characteristic

Abstract

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. In this paper, for an integer s satisfying \(s=\frac{q^n-1}{2}+q^r\), we give six classes of permutation polynomials of the form \((ax^{q^m}-bx+\delta )^s+L(x)\) over \(\mathbb {F}_{q^n}\), and for s satisfying \(s(p^m-1)\equiv p^m-1\ (mod\ p^n-1)\) or \(s(p^{{\frac{k}{2}}m}-1)\equiv p^{km}-1 (mod\ p^n-1)\), we propose three classes of permutation polynomials of the form \((aTr_m^n(x)+\delta )^s+L(x)\) over \(\mathbb {F}_{p^n}\), respectively.