This paper extends the Kikuchi method to give algorithms for decisional $k$-sparse Learning With Errors (LWE) and $k$-sparse Learning Parity with Noise (LPN) problems for higher moduli $q$. We create a Kikuchi graph for a sparse LWE/LPN instance and use it to give two attacks for these problems. The first attack decides by computing the spectral norm of the adjacency matrix of the Kikuchi graph, which is a generalization of the attack for $q=2$ given by Wein et. al. (Journal of the ACM 2019). The second approach computes non-trivial closed walks of the graph, and then decides by computing a certain polynomial of edge labels in the walks. This is a generalization of the attack for $q=2$ given by Gupta et. al. (SODA 2026). Both the attacks yield new tradeoffs between sample complexity and time complexity of sparse LWE/LPN.

翻译：本文扩展了Kikuchi方法，针对更高模数$q$下的判定性$k$-稀疏带误差学习（LWE）问题与$k$-稀疏带噪声奇偶校验（LPN）问题提出了相应算法。我们为稀疏LWE/LPN实例构建了一个Kikuchi图，并利用该图给出了针对此类问题的两种攻击方法。第一种攻击通过计算Kikuchi图邻接矩阵的谱范数进行判定，这是Wein等人（Journal of the ACM，2019年）针对$q=2$情形所提出攻击的推广。第二种方法计算图中非平凡的闭路径，进而通过计算路径中边标签的特定多项式进行判定，这是Gupta等人（SODA，2026年）针对$q=2$情形所提出攻击的推广。两种攻击均给出了稀疏LWE/LPN在样本复杂度与时间复杂度之间的新权衡关系。