# On homomorphic encryption using abelian groups: Classical security analysis

## TL;DR

This paper analyzes the classical and quantum security of a homomorphic encryption scheme based on abelian groups, focusing on the hardness assumptions and potential vulnerabilities in different group settings.

## Contribution

It provides a detailed security analysis of the scheme using abelian and solvable groups, clarifying its classical and quantum security properties.

## Key findings

- Classical security of the scheme with abelian groups is examined.
- Quantum attacks on instantiations with solvable groups are analyzed.
- The scheme's security depends on the hardness of the LHN problem.

## Abstract

In [15], Leonardi and Ruiz-Lopez propose an additively homomorphic public key encryption scheme whose security is expected to depend on the hardness of the learning homomorphism with noise problem (LHN). Choosing parameters for their primitive requires choosing three groups $G$, $H$, and $K$. In their paper, Leonardi and Ruiz-Lopez claim that, when $G$, $H$, and $K$ are abelian, then their public key cryptosystem is not quantum secure. In this paper, we study security for finite abelian groups $G$, $H$, and $K$ in the classical case. Moreover, we study quantum attacks on instantiations with solvable groups.