On semiregular digraphs of the congruence $x^k\equiv y \pmod n$

summary:We assign to each pair of positive integers $n$ and $k\geq 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\dots,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b\pmod n$. The digraph $G(n,k)$ is semiregular if there exists a positive integer $d$ such that each vertex of the digraph has indegree $d$ or 0. Generalizing earlier results of the authors for the case in which $k=2$, we characterize all semiregular digraphs $G(n,k)$ when $k\geq 2$ is arbitrary