On Homomorphic Encryption Using Abelian Groups: Classical Security Analysis

Leonardi and Ruiz-Lopez recently proposed an additively homomorphic public key encryption scheme based on combining group homomorphisms with noise. 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 chapter, 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.