On Compression Functions over Groups with Applications to Homomorphic Encryption

TL;DR

This paper investigates the existence and construction of specific compression functions over finite groups for use in homomorphic encryption, demonstrating non-existence over many groups and providing a new efficient construction over A_5.

Contribution

It systematically analyzes the existence of compression functions over various groups, proves non-existence over solvable groups, and constructs an efficient function over A_5 for FHE schemes.

Findings

No such function exists over any solvable group.

Constructed a shortest possible expression of the function over A_5.

Reduced FHE construction to homomorphic encryption over A_5, improving efficiency.

Abstract

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 $F(\sigma) = F(\sigma^2) = \sigma$ where $\sigma \in G$ 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.