The Learning-With-Errors (LWE) problem (and its variants including Ring-LWE and Module-LWE), whose security are based on hard ideal lattice problems, has proven to be a promising primitive with diverse applications in cryptography. For the sake of expanding sources for constructing LWE, we study the LWE problem on group rings in this work. One can regard the Ring-LWE on cyclotomic integers as a special case when the underlying group is cyclic, while our proposal utilizes non-commutative groups. In particular, we show how to build public key encryption schemes from dihedral group rings, while maintaining the efficiency of the Ring-LWE. We prove that the PKC system is semantically secure, by providing a reduction from the SIVP problem of group ring ideal lattice to the decisional group ring LWE problem. It turns out that irreducible representations of groups play important roles here. We believe that the introduction of the representation view point enriches the tool set for studying the Ring-LWE problem.

错误学习 (LWE) 问题（及其变体，包括 Ring-LWE 和 Module-LWE），其安全性基于硬理想格问题，已被证明是一种有前途的原语，在密码学中具有多种应用。为了扩大构建LWE的来源，我们在这项工作中研究了群环上的LWE问题。当基础群是循环的，而我们的提议利用非交换群时，可以将循环整数上的 Ring-LWE 视为一种特殊情况。特别是，我们展示了如何从二面体群环构建公钥加密方案，同时保持 Ring-LWE 的效率。我们通过将群环理想格的 SIVP 问题简化为决策群环 LWE 问题，证明 PKC 系统在语义上是安全的。事实证明，群体的不可约表示在这里起着重要作用。我们相信表示观点的引入丰富了研究 Ring-LWE 问题的工具集。