An algebraic approach to the Rank Support Learning problem

TL;DR

This paper introduces a new algebraic attack on the Rank Support Learning problem in rank-metric cryptography, demonstrating more efficient key recovery methods for the Durandal signature scheme than previously known.

Contribution

It presents a simpler, more elementary algebraic attack on RSL that outperforms prior methods and challenges assumptions about the security of Durandal.

Findings

The attack is more efficient than previous methods.

It simplifies the algebraic approach using elementary assumptions.

Key recovery on Durandal is easier than previously believed.

Abstract

Rank-metric code-based cryptography relies on the hardness of decoding a random linear code in the rank metric. The Rank Support Learning problem (RSL) is a variant where an attacker has access to N decoding instances whose errors have the same support and wants to solve one of them. This problem is for instance used in the Durandal signature scheme. In this paper, we propose an algebraic attack on RSL which clearly outperforms the previous attacks to solve this problem. We build upon Bardet et al., Asiacrypt 2020, where similar techniques are used to solve MinRank and RD. However, our analysis is simpler and overall our attack relies on very elementary assumptions compared to standard Gr{\"o}bner bases attacks. In particular, our results show that key recovery attacks on Durandal are more efficient than was previously thought.