Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups

Abstract

An orthomorphism is a permutation $σ$ of $\{1, \dots, n-1\}$ for which $x + σ(x) \mod n$ is also a permutation on $\{1, \dots, n-1\}$. Eberhard, Manners, Mrazović, showed that the number of such orthomorphisms is $(\sqrt{e} + o(1)) \cdot \frac{n!^2}{n^n}$ for odd $n$ and zero otherwise. In this paper we prove two analogs of these results where $x+σ(x)$ is replaced by $x σ(x)$ (a "multiplicative orthomorphism") or with $x^{σ(x)}$ (an "exponential orthomorphism"). Namely, we show that no multiplicative orthomorphisms exist for $n > 2$ but that exponential orthomorphisms exist whenever $n$ is twice a prime $p$ such that $p-1$ is squarefree. In the latter case we then estimate the number of exponential orthomorphisms.