Cryptanalysis of a system based on twisted Reed–Solomon codes

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.

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Notes

  1. 1.

    Since the parameters k and \(h_1,\ldots ,h_{\ell }\) are public, an attacker knows the set \(\mathcal {I}\).

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Acknowledgements

This work was done while the second author was visiting the Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1, France. The first author is funded by the French Direction Générale l’Armement, through the Pôle d’excellence cyber. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 801434). We would like to thank Antonia Wachter-Zeh (TUM) for fruitful discussions and Oliver De Candido (TUM) for his comments that helped to improve the manuscript. We would further like to thank the authors of the proposed cryptosystem [5] for validating our attack and pointing out a possible repair of the system with respect to our attack.

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Correspondence to Julien Lavauzelle.

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Lavauzelle, J., Renner, J. Cryptanalysis of a system based on twisted Reed–Solomon codes. Des. Codes Cryptogr. 88, 1285–1300 (2020). https://doi.org/10.1007/s10623-020-00747-6

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Keywords

  • Code-Based Cryptography
  • McEliece Cryptosystem
  • Subfield Subcodes
  • Twisted Reed–Solomon Codes

Mathematics Subject Classification

  • 11T71