The Permuted Kernel Problem (PKP) asks to find a permutation of a given vector belonging to the kernel of a given matrix. The PKP is at the basis of PKP-DSS, a post-quantum signature scheme deriving from the identification scheme proposed by Shamir in 1989. The most efficient solver for PKP is due to a recent paper by Koussa et al. In this paper we propose an improvement of such an algorithm, which we achieve by considering an additional collision search step applied on kernel equations involving a small number of coordinates. We study the conditions for such equations to exist from a coding theory perspective, and we describe how to efficiently find them with methods borrowed from coding theory, such as information set decoding. We assess the complexity of the resulting algorithm and show that it outperforms previous approaches in several cases. We also show that, taking the new solver into account, the security level of some instances of PKP-DSS turns out to be slightly overestimated.

翻译：内核问题( PKP) 要求找到属于一个特定矩阵内核的矢量的变异。 PKP 处于 PKP- DSS 的基础,这是1989年Shamir 提议的识别方法产生的一个量子后签名办法。 PKP 最有效的解答器是由于Koussa 等人最近撰写的论文。 本文中我们建议改进这种算法。 我们通过考虑在涉及少数坐标的内核方程式上应用一个额外的碰撞搜索步骤来实现这一目标。 我们从编码理论的角度来研究这些方程式存在的条件,我们描述如何有效地找到这些方程式,从编码理论中借用的方法,例如资料解码。 我们评估由此产生的算法的复杂性,并表明它在若干情况下已经超越了先前的方法。 我们还表明,考虑到新的解算器,一些PKP- DSS 案例的安全程度被略微高估。