On a Type of Permutation Rational Functions over Finite Fields

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 $\Bbb F_{p^n}$.