Skip to main content
SpringerLink
Log in

Extension of Overbeck’s attack for Gabidulin-based cryptosystems

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Cryptosystems based on codes in the rank metric were introduced in 1991 by Gabidulin, Paramanov, and Tretjakov (GPT) and have been studied as a promising alternative to cryptosystems based on codes in the Hamming metric. In particular, it was observed that the combinatorial solution for solving the rank analogy of the syndrome decoding problem appears significantly harder. Early proposals were often made with an underlying Gabidulin code structure. Gibson, in 1995, made a promising attack which was later extended by Overbeck in 2008 to cryptanalyze many of the systems in the literature. Improved systems were then designed to resist the attack of Overbeck and yet continue to use Gabidulin codes. In this paper, we generalize Overbeck’s attack to break the GPT cryptosystem for all possible parameter sets, and then extend the attack to cryptanalyze particular variants which explicitly resist the attack of Overbeck.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The attack needs \(O(k^2nm^2 (s^2+k))\) operations over \(\mathbb {F}_q\), plus the operations needed for the Gabidulin code decoding algorithm. E.g., the decoding algorithm of [23] needs \(O(m^3 \log m)\) operations over \(\mathbb {F}_q\).

  2. The attack needs \(O(k^2nm^2 (\hat{t}^2+k))\) operations over \(\mathbb {F}_q\), plus the operations needed for the Gabidulin code decoding algorithm.

References

  1. Berger T.P.: Isometries for rank distance and permutation group of Gabidulin codes. IEEE Trans. Inf. Theory 49(11), 3016–3019 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  2. Chabaud F., Stern J.: The cryptographic security of the syndrome decoding problem for rank distance codes. In: Advances in Cryptology—ASIACRYPT ’96, International Conference on the Theory and Applications of Cryptology and Information Security, Kyongju, Korea, November 3–7, 1996, Proceedings, pp. 368–381 (1996).

  3. Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Pereda. Inform. 21(1), 3–16 (1985).

    MathSciNet  MATH  Google Scholar 

  4. Gabidulin E.M.: Attacks and counter-attacks on the GPT public key cryptosystem. Des. Codes Cryptogr. 48(2), 171–177 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  5. Gabidulin E.M., Paramonov A.V., Tretjakov O.V.: Ideals over a non-commutative ring and their application in cryptology. In: Proceedings of the 10th Annual International Conference on Theory and Application of Cryptographic Techniques, EUROCRYPT’91, pp. 482–489. Springer, Berlin (1991).

  6. Gabidulin E.M., Rashwan H., Honary B.: On improving security of GPT cryptosystems. In: IEEE International Symposium on Information Theory, 2009 (ISIT 2009), pp. 1110–1114 (2009).

  7. Gaborit P., Ruatta O., Schrek J., Zémor G.: New results for rank-based cryptography. In: Progress in Cryptology—AFRICACRYPT 2014—7th International Conference on Cryptology in Africa, Marrakesh, Morocco, May 28–30, 2014, Proceedings, pp. 1–12 (2014).

  8. Gaborit P., Ruatta O., Schrek J.: On the complexity of the rank syndrome decoding problem. IEEE Trans. Inf. Theory 62(2), 1006–1019 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  9. Gibson J.K.: Severely denting the Gabidulin version of the McEliece public key cryptosystem. Des. Codes Cryptogr. 6(1), 37–45 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  10. Giorgetti M., Previtali A.: Galois invariance, trace codes and subfield subcodes. Finite Fields Their Appl. 16(2), 96–99 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  11. Horlemann-Trautmann A.-L., Marshall K.: New criteria for MRD and Gabidulin codes and some rank-metric code constructions. arXiv:1507.08641 (2015).

  12. Horlemann-Trautmann A.-L., Marshall K., Rosenthal J.: Considerations for rank-based cryptosystems. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT 2016), Barcelona, pp. 2544–2548 (2016).

  13. Kshevetskiy A.: Security of GPT-like public-key cryptosystems based on linear rank codes. In: 3rd International Workshop on Signal Design and Its Applications in Communications, 2007 (IWSDA 2007), pp. 143–147 (2007).

  14. Loidreau P.: Designing a rank metric based McEliece cryptosystem. In: Proceedings of the Third International Conference on Post-Quantum Cryptography (PQCrypto’10), pp. 142–152. Springer, Berlin (2010).

  15. McEliece R.J.: A public-key cryptosystem based on algebraic coding theory. Deep Space Netw. Progr. Rep. 44, 114–116 (1978).

    Google Scholar 

  16. Morrison K.: Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes. IEEE Trans. Inf. Theory 60(11), 1–12 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  17. Ourivski A.V., Johansson T.: New technique for decoding codes in the rank metric and its cryptography applications. Probl. Inf. Transm. 38(3), 237–246 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. Overbeck R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  19. Rashwan H., Gabidulin E.M., Honary B.: A smart approach for GPT cryptosystem based on rank codes. In: IEEE International Symposium on Information Theory, ISIT 2010, June 13–18, 2010, Austin, Texas, USA, Proceedings, pp. 2463–2467 (2010).

  20. Sidelnikov V.M., Shestakov S.O.: On insecurity of cryptosystems based on generalized Reed-Solomon codes. Discrete Math. Appl. 2, 439–444 (1992).

    MathSciNet  Google Scholar 

  21. Silva D., Kschischang F.R.: Fast encoding and decoding of Gabidulin codes. In: IEEE International Symposium on Information Theory, 2009 (ISIT 2009), pp. 2858–2862 (2009).

  22. Wachter-Zeh A.: Bounds on list decoding of rank-metric codes. IEEE Trans. Inf. Theory 59(11), 7268–7277 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  23. Wachter-Zeh A., Afanassiev V., Sidorenko V.: Fast decoding of Gabidulin codes. Des. Codes Cryptogr. 66(1), 57–73 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. Wan Z.-X.: Geometry of matrices. World Scientific, Singapore (1996). In memory of Professor L.K. Hua (1910–1985).

Download references

Acknowledgements

This work was supported by SNF Grant No. 149716.

Author information

Authors and Affiliations

  1. University of St. Gallen, St. Gallen, Switzerland

    Anna-Lena Horlemann-Trautmann

  2. Universität Zürich, Zurich, Switzerland

    Kyle Marshall & Joachim Rosenthal

Authors
  1. Anna-Lena Horlemann-Trautmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kyle Marshall
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joachim Rosenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kyle Marshall.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the Special Issue on Network Coding and Designs.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Horlemann-Trautmann, AL., Marshall, K. & Rosenthal, J. Extension of Overbeck’s attack for Gabidulin-based cryptosystems. Des. Codes Cryptogr. 86, 319–340 (2018). https://doi.org/10.1007/s10623-017-0343-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-017-0343-7

Keywords

Mathematics Subject Classification

Search

Navigation