A strengthening of McConnel’s theorem on permutations 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.