Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups

An orthomorphism is a permutation $\sigma$ of $\{1,\dots ,n-1\}$ for which $x+\sigma (x)\mathrm{m}\mathrm{o}\mathrm{d}n$ is also a permutation on $\{1,\dots ,n-1\}$. Eberhard, Manners, Mrazovi'c, 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 $x+\sigma (x)$ is replaced by $x\sigma (x)$ (a “multiplicative orthomorphism”) or with $x^{\sigma (x)}$ (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.