# Cryptanalysis of a System Based on Twisted Reed-Solomon Codes

## TL;DR

This paper presents an efficient key-recovery attack on a McEliece cryptosystem variant using twisted Reed-Solomon codes, demonstrating that the scheme is insecure under practical parameters and highlighting the need for careful analysis of subfield subcodes.

## Contribution

It introduces a novel attack exploiting subfield subcode structure, breaking the cryptosystem and challenging the assumed security of twisted Reed-Solomon code-based cryptography.

## Key findings

- The attack always succeeds for practical parameters.
- Private keys can be recovered within minutes.
- The scheme's security claims are invalidated by the attack.

## Abstract

Twisted Reed-Solomon (TRS) codes are a family of codes that contains a large number of maximum distance separable codes that are non-equivalent to Reed--Solomon codes. TRS codes were recently proposed as an alternative to Goppa codes for the McEliece code-based cryptosystem, resulting in a potential reduction of key sizes. The use of TRS codes in the McEliece cryptosystem has been motivated by the fact that a large subfamily of TRS codes is resilient to a direct use of known algebraic key-recovery methods. In this paper, an efficient key-recovery attack on the TRS variant that was used in the McEliece cryptosystem is presented. The algorithm exploits a new approach based on recovering the structure of a well-chosen subfield subcode of the public code. It is proved that the attack always succeeds and breaks the system for all practical parameters in $O(n^4)$ field operations. A software implementation of the algorithm retrieves a valid private key from the public key within a few minutes, for parameters claiming a security level of 128 bits. The success of the attack also indicates that, contrary to common beliefs, subfield subcodes of the public code need to be precisely analyzed when proposing a McEliece-type code-based cryptosystem. Finally, the paper discusses an attempt to repair the scheme and a modification of the attack aiming at Gabidulin-Paramonov-Tretjakov cryptosystems based on twisted Gabidulin codes.