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：噪声学习奇偶性（LPN）问题是一个公认的计算难题，其难度对于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，这些方法在相关参数化下的复杂度是指数级的。本文探究量子方法是否可能提供替代途径。这一研究思路受到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Simon承诺的函数，其差值等于秘密奇偶向量。虽然直接恢复秘密奇偶向量的可能性不大，但运行Simon算法本质上能生成新的LPN样本。这使我们有望生成足够多的新样本，从而忽略一个或多个变量并通过迭代方式简化问题。我们并不主张这些方法必然会优于现有方案，仅认为其值得更深入的探究。