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.

翻译：在文献[15]中，Leonardi和Ruiz-Lopez提出了一种加法同态公钥加密方案，其安全性预期依赖于学习同态噪声问题（LHN）的难度。选择该原语的参数需要选择三个群$G$、$H$和$K$。在他们的论文中，Leonardi和Ruiz-Lopez声称，当$G$、$H$和$K$为阿贝尔群时，其公钥密码系统不具备量子安全性。本文研究了有限阿贝尔群$G$、$H$和$K$在经典情形下的安全性。此外，我们还研究了针对可解群实例化的量子攻击。