  In the area of cryptography, fully homomorphic encryption (FHE) enables any
entity to perform arbitrary computation on encrypted data without decrypting
the ciphertexts. An ongoing group-theoretic approach to construct FHE schemes
uses a certain "compression" function $F(x)$ implemented by group operators on
a given finite group $G$ (i.e., it is given by a sequence of elements of $G$
and variable $x$), which satisfies that $F(1) = 1$ and $F(\sigma) = F(\sigma^2)
= \sigma$ where $\sigma \in G$ is some element of order three. The previous
work gave an example of such $F$ over $G = S_5$ by just a heuristic approach.
In this paper, we systematically study the possibilities of such $F$. We
construct a shortest possible $F$ over smaller group $G = A_5$, and prove that
no such $F$ exists over other groups $G$ of order up to $60 = |A_5|$.
