  An orthomorphism is a permutation $\sigma$ of $\{1, \dots, n-1\}$ for which
$x + \sigma(x) \mod n$ 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.
