Skip to main content
Springer Nature Link
Log in

Distinguishing and Recovering Generalized Linearized Reed–Solomon Codes

  • Conference paper
  • First Online:
Code-Based Cryptography (CBCrypto 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13839))

Included in the following conference series:

Abstract

We study the distinguishability of linearized Reed–Solomon (LRS) codes by defining and analyzing analogs of the square-code and the Overbeck distinguisher for classical Reed–Solomon and Gabidulin codes, respectively. Our main results show that the square-code distinguisher works for generalized linearized Reed–Solomon (GLRS) codes defined with the trivial automorphism, whereas the Overbeck-type distinguisher can handle LRS codes in the general setting. We further show how to recover defining code parameters from any generator matrix of such codes in the zero-derivation case. For other choices of automorphisms and derivations simulations indicate that these distinguishers and recovery algorithms do not work. The corresponding LRS and GLRS codes might hence be of interest for code-based cryptography.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    In fact, the distinguisher recognizes a GLRS code with probability one. But, with a small probability, it might wrongly declare a non-GLRS code to be a GLRS code.

  2. 2.

    In fact, the distinguisher recognizes a GLRS code with probability one. But, with a small probability, it might wrongly declare a non-GLRS code to be a GLRS code.

References

  1. Alagic, G., et al.: Status report on the third round of the NIST post-quantum cryptography standardization process (2022)

    Google Scholar 

  2. Alfarano, G.N., Lobillo, F.J., Neri, A., Wachter-Zeh, A.: Sum-rank product codes and bounds on the minimum distance. Finite Fields Appl. 80, 102013 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barra, A., Gluesing-Luerssen, H.: MacWilliams extension theorems and the local-global property for codes over Frobenius rings. J. Pure Appl. Algebra 219(4), 703–728 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger, T.P., Loidreau, P.: How to mask the structure of codes for a cryptographic use. Des. Codes Crypt. 35(1), 63–79 (2005). https://doi.org/10.1007/s10623-003-6151-2

    Article  MathSciNet  MATH  Google Scholar 

  5. Caruso, X.: Residues of skew rational functions and linearized Goppa codes. arXiv preprint arXiv:1908.08430v1 (2019)

  6. Caruso, X., Durand, A.: Duals of linearized Reed-Solomon codes. Des. Codes Crypt. 91, 241–271 (2022). https://doi.org/10.1007/s10623-022-01102-7

    Article  MathSciNet  MATH  Google Scholar 

  7. Castryck, W., Decru, T.: An efficient key recovery attack on SIDH (preliminary version). Cryptology ePrint Archive ia.cr/2022/975 (2022)

    Google Scholar 

  8. Gabidulin, E.M., Paramonov, A.V., Tretjakov, O.V.: Ideals over a non-commutative ring and their application in cryptology. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 482–489. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-46416-6_41

    Chapter  MATH  Google Scholar 

  9. Gabidulin, E.M.: Attacks and counter-attacks on the GPT public key cryptosystem. Des. Codes Crypt. 48(2), 171–177 (2008). https://doi.org/10.1007/s10623-007-9160-8

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  11. Horlemann-Trautmann, A.L., Marshall, K., Rosenthal, J.: Extension of Overbeck’s attack for Gabidulin-based cryptosystems. Des. Codes Crypt. 86(2), 319–340 (2018). https://doi.org/10.1007/s10623-017-0343-7

    Article  MathSciNet  MATH  Google Scholar 

  12. Lam, T.Y., Leroy, A.: Vandermonde and Wronskian matrices over division rings. J. Algebra 119(2), 308–336 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu, H.f., Kumar, P.V.: A unified construction of space-time codes with optimal rate-diversity tradeoff. IEEE Trans. Inf. Theor. 51(5), 1709–1730 (2005)

    Google Scholar 

  14. Martínez-Peñas, U.: Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring. J. Algebra 504, 587–612 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Martínez-Peñas, U.: Hamming and simplex codes for the sum-rank metric. Des. Codes Crypt. 88(8), 1521–1539 (2020). https://doi.org/10.1007/s10623-020-00772-5

    Article  MathSciNet  MATH  Google Scholar 

  16. Martínez-Peñas, U., Kschischang, F.R.: Reliable and secure multishot network coding using linearized Reed-Solomon codes. IEEE Trans. Inf. Theor. 65(8), 4785–4803 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. The Deep Space Network Progress Report, vol. 42–44, pp. 114–116 (1978)

    Google Scholar 

  18. Neri, A.: Twisted linearized Reed-Solomon codes: a skew polynomial framework. J. Algebra 609, 792–839 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Overbeck, R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2007). https://doi.org/10.1007/s00145-007-9003-9

    Article  MathSciNet  MATH  Google Scholar 

  20. Overbeck, R.: A new structural attack for GPT and variants. In: Dawson, E., Vaudenay, S. (eds.) Mycrypt 2005. LNCS, vol. 3715, pp. 50–63. Springer, Heidelberg (2005). https://doi.org/10.1007/11554868_5

    Chapter  Google Scholar 

  21. Overbeck, R.: Public key cryptography based on coding theory. Ph.D. thesis, Technical University of Darmstadt (2007)

    Google Scholar 

  22. Puchinger, S., Renner, J., Rosenkilde, J.: Generic decoding in the sum-rank metric. In: 2020 IEEE International Symposium on Information Theory, pp. 54–59 (2020)

    Google Scholar 

  23. Rashwan, H., Gabidulin, E.M., Honary, B.: A smart approach for GPT cryptosystem based on rank codes. In: 2010 IEEE International Symposium on Information Theory, pp. 2463–2467 (2010)

    Google Scholar 

  24. Schmidt, G., Sidorenko, V., Bossert, M.: Decoding Reed-Solomon codes beyond half the minimum distance using shift-register synthesis. In: 2006 IEEE International Symposium on Information Theory, pp. 459–463 (2006)

    Google Scholar 

  25. Sidelnikov, V.M., Shestakov, S.O.: On insecurity of cryptosystems based on generalized Reed-Solomon codes. Discrete Math. Appl. 2(4), 439–444 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stein, W.A., et al.: Sage mathematics software (version 9.7). The Sage Development Team (2022). http://www.sagemath.org

  27. Wieschebrink, C.: An attack on a modified Niederreiter encryption scheme. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 14–26. Springer, Heidelberg (2006). https://doi.org/10.1007/11745853_2

    Chapter  Google Scholar 

  28. Wieschebrink, C.: Cryptanalysis of the Niederreiter public-key scheme based on GRS subcodes. In: International Workshop on Post-quantum Cryptography, pp. 61–72 (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Communications and Navigation, German Aerospace Center (DLR), Oberpfaffenhofen-Wessling, Germany

    Felicitas Hörmann & Hannes Bartz

  2. School of Computer Science, University of St. Gallen, St. Gallen, Switzerland

    Felicitas Hörmann & Anna-Lena Horlemann

Authors
  1. Felicitas Hörmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hannes Bartz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna-Lena Horlemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Felicitas Hörmann .

Editor information

Editors and Affiliations

  1. ENAC - University of Toulouse, Toulouse, France

    Jean-Christophe Deneuville

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hörmann, F., Bartz, H., Horlemann, AL. (2023). Distinguishing and Recovering Generalized Linearized Reed–Solomon Codes. In: Deneuville, JC. (eds) Code-Based Cryptography. CBCrypto 2022. Lecture Notes in Computer Science, vol 13839. Springer, Cham. https://doi.org/10.1007/978-3-031-29689-5_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-29689-5_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-29688-8

  • Online ISBN: 978-3-031-29689-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics