A General Construction of Permutation Polynomials of ${\mathbb{F}}_{q^2}$

Abstract

Let $r$ be a positive integer, $h(X)\in {\mathbb{F}}_{q^2}[X]$, and ${\mu }_{q+1}$ be the subgroup of order $q+1$ of ${\mathbb{F}}_{q^2}^{\ast }$. It is well known that $X^{r}h(X^{q-1})$ permutes ${\mathbb{F}}_{q^2}$ if and only if $\text{gcd}(r,q-1)=1$ and $X^{r}h(X{)}^{q-1}$ permutes ${\mu }_{q+1}$. There are many ad hoc constructions of permutation polynomials of ${\mathbb{F}}_{q^2}$ of this type such that $h(X{)}^{q-1}$ induces monomial functions on the cosets of a subgroup of ${\mu }_{q+1}$. We give a general construction that can generate, through an algorithm, {\em all} permutation polynomials of ${\mathbb{F}}_{q^2}$ with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.

30 pages, 2 figures