A Recursive Construction of Permutation Polynomials over $\mathbb{F}_{q^2}$ with Odd Characteristic from R\'{e}dei Functions

TL;DR

This paper introduces a recursive method to construct permutation polynomials over finite fields of odd characteristic using rational Rédéi functions, providing explicit inverses and easy permutation conditions.

Contribution

It presents a novel recursive construction of permutation polynomials over _{q^2} with explicit inverse characterization and broad applicability.

Findings

Large number of permutation polynomials identified through computer experiments.

Explicit conditions for permutation property are easy to verify.

Constructed polynomials include binomials and trinomials with wide practical use.

Abstract

In this paper, we construct two classes of permutation polynomials over $\mathbb{F}_{q^2}$ with odd characteristic from rational R\'{e}dei functions. A complete characterization of their compositional inverses is also given. These permutation polynomials can be generated recursively. As a consequence, we can generate recursively permutation polynomials with arbitrary number of terms. More importantly, the conditions of these polynomials being permutations are very easy to characterize. For wide applications in practice, several classes of permutation binomials and trinomials are given. With the help of a computer, we find that the number of permutation polynomials of these types is very large.