On Compression Functions over Groups with Applications to Homomorphic Encryption

Fully homomorphic encryption (FHE) enables an entity to perform arbitrary computation on encrypted data without decrypting the ciphertexts. An ongoing group-theoretical approach to construct an FHE scheme uses a certain "compression" function $F(x)$ implemented by group operations on a given finite group $G$, which satisfies that $F(1) = 1$ and $F(σ) = F(σ^2) = σ$ where $σ\in G$ is some element of order $3$. The previous work gave an example of such a function over the symmetric group $G = S_5$ by just a heuristic approach. In this paper, we systematically study the possibilities of such a function over various groups. We show that such a function does not exist over any solvable group $G$ (such as an Abelian group and a smaller symmetric group $S_n$ with $n \leq 4$). We also construct such a function over the alternating group $G = A_5$ that has a shortest possible expression. Moreover, by using this new function, we give a reduction of a construction of an FHE scheme to a construction of a homomorphic encryption scheme over the group $A_5$, which is more efficient than the previously known reductions.