Collection of Polynomials over Finite Fields Providing Involutary Permutations

TL;DR

This paper constructs specific permutation polynomials over finite fields that are involutions with a precise number of fixed points, including many trinomials, expanding the understanding of permutation polynomials in finite field theory.

Contribution

It introduces a new family of permutation polynomials over finite fields with involutory properties and detailed fixed point counts, including explicit constructions of trinomials and 6-term polynomials.

Findings

Constructed 2(q-1) involutory permutation polynomials for certain finite fields.

All constructed polynomials have exactly 1 + (q-1)/3 fixed points.

Included are (q-1) trinomials and additional 6-term polynomials.

Abstract

For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally $2(q-1) $ permutation polyomials, all giving involutory permutations with exactly $ 1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are trinomials, and the rest are 6-term polynomials.