Permutation Polynomials of $\mathbb{F}_{q^2}$ : A Linear Algebraic Approach

In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over $\mathbb{F}_{q^2}$, in the context of rank deficient and full rank linear maps over $\mathbb{F}_{q^2}$. We derive necessary and sufficient conditions under which the given class of polynomials are permutation polynomials. We further show that the number of such permutation polynomials can be easily enumerated. Only a subset of these permutation polynomials have been reported in literature earlier. It turns out that this class of permutation polynomials have compositional inverses of the same kind and we provide algorithms to evaluate the compositional inverses of most of these permutation polynomials.