A Novel Attack to the Permuted Kernel Problem

TL;DR

This paper introduces an improved algorithm for solving the Permuted Kernel Problem, which enhances the security analysis of PKP-DSS, a post-quantum signature scheme, by leveraging coding theory techniques to outperform previous methods.

Contribution

It proposes a novel collision search step for PKP solving, integrating coding theory methods like information set decoding to improve efficiency.

Findings

The new algorithm outperforms previous approaches in several cases.

Security levels of some PKP-DSS instances are slightly overestimated.

The approach links coding theory with cryptanalysis of PKP.

Abstract

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.