Polynomial-Time Key Recovery Attack on the Lau-Tan Cryptosystem Based on Gabidulin Codes

TL;DR

This paper introduces a polynomial-time key recovery attack on the Lau-Tan cryptosystem that uses Gabidulin codes, revealing a critical vulnerability that breaks the cryptosystem's security.

Contribution

The paper demonstrates that the private key can be recovered efficiently by reducing the problem to solving a linear system, exposing a flaw in the cryptosystem's design.

Findings

The attack costs polynomial time.

The private key can be recovered from public information.

The cryptosystem is completely broken by this attack.

Abstract

This paper presents a key recovery attack on the cryptosystem proposed by Lau and Tan in a talk at ACISP 2018. The Lau-Tan cryptosystem uses Gabidulin codes as the underlying decodable code. To hide the algebraic structure of Gabidulin codes, the authors chose a matrix of column rank $n$ to mix with a generator matrix of the secret Gabidulin code. The other part of the public key, however, reveals crucial information about the private key. Our analysis shows that the problem of recovering the private key can be reduced to solving a multivariate linear system over the base field, rather than solving a multivariate quadratic system as claimed by the authors. Solving the linear system for any nonzero solution permits us to recover the private key. Apparently, this attack costs polynomial time, and therefore completely breaks the cryptosystem.