The learning parity with noise (LPN) problem is a well-established computational challenge whose difficulty is critical to the security of several post-quantum cryptographic primitives such as HQC and Classic McEliece. Classically, the best-known attacks involve information set decoding methods which are exponential in complexity for parameterisations of interest. In this paper we investigate whether quantum methods might offer alternative approaches. The line of inquiry is inspired by Regev's relating of certain lattice problems to the hidden dihedral subgroup problem. We use neighbourhoods of binary fields to produce a function close to fulfilling Simon's promise with difference equal to the secret parity vector. Although unlikely to recover the secret parity vector directly, running Simon's algorithm essentially produces new LPN samples. This gives the hope that we might be able to produce enough new samples to ignore one or more variables and iteratively reduce the problem. We make no claim that these methods will necessarily be competitive with existing approaches, merely that they warrant deeper investigation.

翻译：带噪声奇偶学习（LPN）问题是一个公认的计算难题，其求解难度对HQC和Classic McEliece等后量子密码原语的安全性至关重要。在经典计算领域，最著名的攻击方法涉及信息集解码技术，该技术在相关参数化场景下具有指数级复杂度。本文旨在探究量子方法是否可能提供替代解决路径。该研究思路受到Regev将特定格问题与隐藏二面体子群问题相关联的启发。我们利用二元域的邻域结构构造了一个近似满足西蒙承诺的函数，其差值等于秘密奇偶向量。尽管直接恢复秘密奇偶向量的可能性较低，但运行西蒙算法本质上能生成新的LPN样本。这为通过生成足够新样本来忽略部分变量并迭代简化问题提供了可能。需要说明的是，本文并不主张这些方法必然优于现有方案，仅认为其值得更深入的探究。