On a Type of Permutation Rational Functions over Finite Fields

TL;DR

This paper investigates a specific class of permutation rational functions over finite fields, establishing conditions under which they permute the fields and demonstrating that for larger fields, they generally do not.

Contribution

The paper extends previous results by analyzing permutation properties of the functions for larger prime powers and provides new conditions for permutation behavior in specific cases.

Findings

For p=2,3, the functions permute all finite fields.

For p>3 and n≥5, the functions do not permute the fields.

For p>3 and n=2, permutation occurs iff the trace condition is met.

Abstract

Let $p$ be a prime and $n$ be a positive integer. Let $f_b(X)=X+(X^p-X+b)^{-1}$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. In 2008, Yuan et al. \cite{Yuan-Ding-Wang-Pieprzyk-FFA-2008} showed that for $p=2,3$, $f_b$ permutes $\Bbb F_{p^n}$ for all $n\ge 1$. Using the Hasse-Weil bound, we show that when $p>3$ and $n\ge 5$, $f$ does not permute $\Bbb F_{p^n}$. For $p>3$ and $n=2$, we prove that $f_b$ permutes $\Bbb F_{p^2}$ if and only if $\text{Tr}_{p^2/p}(b)=\pm 1$. We conjecture that for $p>3$ and $n=3,4$, $f_b$ does not permute $\Bbb F_{p^n}$.