Permutation Polynomial Representatives and their Matrices

Abstract

Permutation polynomials are a topic of research due to their applications in various areas like coding theory, cryptography and combinatorial designs. The seminal paper [1] lists many open problems in this area. There are q^qpolynomials of degree < $q$ over $\mathbb{F}_{q}$ and $q!$ among them are the permutation polynomials. Therefore as $q$ increases it becomes more difficult to find a permutation polynomial. In this paper, we define a notion of a “Permutation Polynomial Representative (PPR)” which can be used to reduce the search space for permutation polynomials. We give some properties of a PPR. Then we give matrix representation of a PPR; which can be used to construct the ‘compositional inverse’ of the PPR. In every application compositional inverses are required to invert the permutation established by the permutation polynomial, but finding the compositional inverse of a given permutation polynomial is not a straightforward problem. Further, we introduce a product of two vectors over $\mathbb{F}_{q}$ which we call as the ‘Butterfly Product’, use it to define a $\mathcal{H}$ matrix’ and provide a necessary and sufficient condition for any (q - 2) × (q - 2) matrix over $\mathbb{F}_{q}$ to be the matrix representation of a permutation of non-zero elements of $\mathbb{F}_{q}$. In the end we give a theorem about finding more permutation polynomials from the matrix of a PPR.