Some classes of permutation binomials and trinomials of index \(q-1\) over \(\mathbb{F}_{q^n}\) (Q6542669)

Let \(\mathbb{F}_q\) be a finite field with \(q\) elements, where \(q\) is a prime power. A permutation polynomial (PP) \(f(x)\) over \(\mathbb{F}_q\) defines a permutation \(f:c\to f(c)\) on \(\mathbb{F}_q\). Permutation polynomials have applications in coding theory, cryptography and combinatorial designs. In this paper, the authors classify sparse permutation polynomials of the form \(x^rf(x^s)\) of \(\mathbb{F}_{q^n}\), where \(1\le r\le 5\) is a positive integer, \(s=\frac{q^n-1}{q-1}\), \(n=2\) and \(3\), for two cases: \(f(x)=x+a\) and \(f(x)=x^2+ax+b\). In particular, they give necessary and sufficient conditions for the polynomial \(x(x^{2s}+ax^s+b)\) in \(\mathbb{F}_{q^n}[x]\) with \(n=3\) and \(q>409\) to be a permutation polynomial of \(\mathbb{F}_{q^n}\).