A New Approach to Permutation Polynomials over Finite Fields, II

Abstract

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in {\mathbb{F}}_p[\mathtt{x}]$ be the polynomial defined by the functional equation ${\sum }_{a\in {\mathbb{F}}_q}(\mathtt{x}+a{)}^n=g_{n,q}({\mathtt{x}}^q-\mathtt{x})$. When is $g_{n,q}$ a permutation polynomial (PP) of ${\mathbb{F}}_{q^e}$? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers $(n,e;q)$ {\em desirable} if $g_{n,q}$ is a PP of ${\mathbb{F}}_{q^e}$. In the present paper, we find many new classes of desirable triples whose corresponding PPs were previously unknown. Several new techniques are introduced for proving a given polynomial is a PP.

47 pages, 3 tables