A strengthening of McConnel's theorem on permutations over finite fields

TL;DR

This paper extends McConnel's theorem on permutation functions over finite fields by providing a new sufficient condition involving small doubling of the subset D, broadening the class of functions characterized.

Contribution

It introduces a new condition on the subset D, specifically small doubling, under which McConnel's theorem still holds, enhancing the understanding of permutation functions.

Findings

McConnel's theorem extended to subsets D with small doubling

Characterization of functions with derivatives in D under new conditions

Broader class of permutation functions over finite fields

Abstract

Let $p$ be a prime, $q=p^n$, and $D \subset \mathbb{F}_q^*$. A celebrated result of McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^*$, and $f:\mathbb{F}_q \to \mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on $D$ to obtain the same conclusion on $f$. In particular, we show that McConnel's theorem extends if $D$ has small doubling.