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Information-Set Decoding for Linear Codes over Fq

  • Christiane Peters
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6061)

Abstract

The best known non-structural attacks against code-based cryptosystems are based on information-set decoding. Stern’s algorithm and its improvements are well optimized and the complexity is reasonably well understood. However, these algorithms only handle codes over F 2.

This paper presents a generalization of Stern’s information-set- decoding algorithm for decoding linear codes over arbitrary finite fields F q and analyzes the complexity. This result makes it possible to compute the security of recently proposed code-based systems over non-binary fields.

As an illustration, ranges of parameters for generalized McEliece cryptosystems using classical Goppa codes over F 31 are suggested for which the new information-set-decoding algorithm needs 2128 bit operations.

Keywords

Generalized McEliece cryptosystem security analysis Stern attack linear codes over Fq information-set decoding 

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References

  1. 1.
    Berger, T.P., Cayrel, P.-L., Gaborit, P., Otmani, A.: Reducing key length of the McEliece cryptosystem. In: Preneel, B. (ed.) Progress in Cryptology – AFRICACRYPT 2009. LNCS, vol. 5580, pp. 77–97. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Berger, T.P., Loidreau, P.: How to mask the structure of codes for a cryptographic use. Designs, Codes and Cryptography 35(1), 63–79 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.A.: On the inherent intractability of certain coding problems. IEEE Transactions on Information Theory 24, 384–386 (1978)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bernstein, D.J., Lange, T., Peters, C.: Attacking and defending the McEliece cryptosystem. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 31–46. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bernstein, D.J., Lange, T., Peters, C., van Tilborg, H.C.A.: Explicit bounds for generic decoding algorithms for code-based cryptography. In: Pre-Proceedings of WCC 2009, pp. 168–180 (2009)Google Scholar
  6. 6.
    Canteaut, A., Chabaud, F.: A new algorithm for finding minimum-weight words in a linear code: application to McEliece’s cryptosystem and to narrow-sense BCH codes of length 511. IEEE Transactions on Information Theory 44(1), 367–378 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Finiasz, M., Sendrier, N.: Security bounds for the design of code-based cryptosystems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 88–105. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Hallgren, S., Vollmer, U.: Quantum computing. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 15–34. Springer, Berlin (2009)CrossRefGoogle Scholar
  9. 9.
    Janwa, H., Moreno, O.: McEliece public key cryptosystems using algebraic-geometric codes. Designs, Codes and Cryptography 8(3), 293–307 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lee, P.J., Brickell, E.F.: An observation on the security of McEliece’s public-key cryptosystem. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 275–280. Springer, Heidelberg (1988)Google Scholar
  11. 11.
    Leon, J.S.: A probabilistic algorithm for computing minimum weights of large error-correcting codes. IEEE Transactions on Information Theory 34(5), 1354–1359 (1988)CrossRefMathSciNetGoogle Scholar
  12. 12.
    McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Jet Propulsion Laboratory DSN Progress Report 42–44 (1978), http://ipnpr.jpl.nasa.gov/progress_report2/42-44/44N.PDF
  13. 13.
    Misoczki, R., Barreto, P.S.L.M.: Compact McEliece keys from Goppa codes. In: Jacobson Jr., M.J., Rijmen, V., Safavi-Naini, R. (eds.) SAC 2009. LNCS, vol. 5867, pp. 376–392. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Overbeck, R., Sendrier, N.: Code-based cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 95–145. Springer, Berlin (2009)CrossRefGoogle Scholar
  15. 15.
    Prange, E.: The use of information sets in decoding cyclic codes. IRE Transactions on Information Theory 8(5), 5–9 (1962)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Stern, J.: A method for finding codewords of small weight. In: Wolfmann, J., Cohen, G. (eds.) Coding Theory 1988. LNCS, vol. 388, pp. 106–113. Springer, Heidelberg (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christiane Peters
    • 1
  1. 1.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenNetherlands

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