The Hardness of Code Equivalence over \(\mathbb{F}_q\) and Its Application to Code-Based Cryptography

  • Nicolas Sendrier
  • Dimitris E. Simos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7932)


The code equivalence problem is to decide whether two linear codes over \(\mathbb{F}_{q}\) are identical up to a linear isometry of the Hamming space. In this paper, we review the hardness of code equivalence over \(\mathbb{F}_q\) due to some recent negative results and argue on the possible implications in code-based cryptography. In particular, we present an improved version of the three-pass identification scheme of Girault and discuss on a connection between code equivalence and the hidden subgroup problem.


Code Equivalence Isometry Hardness Zero-Knowledge Protocols Quantum Fourier Sampling Linear Codes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aguilar, C., Gaborit, P., Schrek, J.: A new zero-knowledge code based identification scheme with reduced communication. In: 2011 IEEE Information Theory Workshop (ITW), pp. 648–652 (2011)Google Scholar
  2. 2.
    Assmus, E.F.J., Key, J.D.: Designs and their Codes. Cambridge Tracts in Mathematics, vol. 103. Cambridge University Press (1992), second printing with corrections (1993)Google Scholar
  3. 3.
    Babai, L., Codenotti, P., Grochow, J.A., Qiao, Y.: Code equivalence and group isomorphism. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 1395–1408. SIAM (2011)Google Scholar
  4. 4.
    Barg, S.: Some new NP-complete coding problems. Probl. Peredachi Inf. 30, 23–28 (1994)MathSciNetGoogle Scholar
  5. 5.
    Berlekamp, E., McEliece, R., van Tilborg, H.: On the inherent intractability of certain coding problems (corresp.). IEEE Transactions on Information Theory 24, 384–386 (1978)zbMATHCrossRefGoogle Scholar
  6. 6.
    Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-Correcting Linear Codes: Classification by Isometry and Applications. Algorithms and Computation in Mathematics, vol. 18. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  7. 7.
    Bouyukliev, I.: About the code equivalence. Ser. Coding Theory Cryptol. 3, 126–151 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cayrel, P.L., Gaborit, P., Girault, M.: Identity-based identification and signature schemes using correcting codes. In: Augot, D., Sendrier, N., Tillich, J.P. (eds.) Workshop on Coding and Cryptography - WCC 2007, pp. 69–78. INRIA (2007)Google Scholar
  9. 9.
    Cayrel, P.-L., Véron, P., El Yousfi Alaoui, S.M.: A zero-knowledge identification scheme based on the q-ary syndrome decoding problem. In: Biryukov, A., Gong, G., Stinson, D.R. (eds.) SAC 2010. LNCS, vol. 6544, pp. 171–186. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Courtois, N.T., Finiasz, M., Sendrier, N.: How to achieve a McEliece-based digital signature scheme. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 157–174. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Dinh, H., Moore, C., Russell, A.: McEliece and niederreiter cryptosystems that resist quantum fourier sampling attacks. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 761–779. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Dinh, H., Moore, C., Russell, A.: Quantum fourier sampling, code equivalence, and the quantum security of the mceliece and sidelnikov cryptosystems. Tech. rep. (2011), also available as arXiv:1111.4382v1 Google Scholar
  13. 13.
    Feulner, T.: The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Adv. Math. Commun. 3, 363–383 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fripertinger, H.: Enumeration of linear codes by applying methods from algebraic combinatorics. Grazer Math. Ber. 328, 31–42 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Fripertinger, H.: Enumeration of the semilinear isometry classes of linear codes. Bayrether Mathematische Schriften 74, 100–122 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fripertinger, H., Kerber, A.: Isometry classes of indecomposable linear codes. In: Giusti, M., Cohen, G., Mora, T. (eds.) AAECC 1995. LNCS, vol. 948, pp. 194–204. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  17. 17.
    Girault, M.: A (non-practical) three-pass identification protocol using coding theory. In: Seberry, J., Pieprzyk, J. (eds.) AUSCRYPT 1990. LNCS, vol. 453, pp. 265–272. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  18. 18.
    Grochow, J.A.: Matrix lie algebra isomorphism. Tech. Rep. TR11-168, Electronic Colloquium on Computational Complexity (2011), also available as arXiv:1112.2012, IEEE Conference on Computational Complexity (2012) (to appear)Google Scholar
  19. 19.
    Harari, S.: A new authentication algorithm. In: Wolfmann, J., Cohen, G. (eds.) Coding Theory 1988. LNCS, vol. 388, pp. 91–105. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  20. 20.
    Human, W.C.: Codes and groups. In: Pless, V., Human, W.C. (eds.) Handbook of Coding Theory, pp. 1345–1440. Elsevier, North-Holland (1998)Google Scholar
  21. 21.
    Kaski, P., Östergård, P.R.J.: Classification Algorithms for Codes and Designs. Algorithms and Computation in Mathematics, vol. 15. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  22. 22.
    MacWilliams, F.J.: Error-correcting codes for multiple-level transmission. Bell. Syst. Tech. J. 40, 281–308 (1961)MathSciNetGoogle Scholar
  23. 23.
    McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Tech. Rep. DSN Progress Report 42-44, California Institute of Technology, Jet Propulsion Laboratory, Pasadena, CA (1978)Google Scholar
  24. 24.
    Overbeck, R., Sendrier, N.: Code-based cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 95–145. Springer (2009)Google Scholar
  25. 25.
    Peters, C.: Information-set decoding for linear codes over \(\mathbb{F}_q\). In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 81–94. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  26. 26.
    Petrank, E., Roth, R.M.: Is code equivalence easy to decide? IEEE Trans. Inform. Theory 43, 1602–1604 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sendrier, N.: On the dimension of the hull. SIAM J. Discrete Math. 10(2), 282–293 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Sendrier, N.: Finding the permutation between equivalent linear codes: The support splitting algorithm. IEEE Trans. Inform. Theory 26, 1193–1203 (2000)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sendrier, N., Simos, D.E.: How easy is code equivalence over \(\mathbb{F}_q\)? In: WCC 2013: Proceedings of the 8th International Workshop on Coding and Cryptography (preprint 2012) (to appear, 2013),
  30. 30.
    Skersys, G.: Calcul du groupe d’automorphisme des codes. Détermination de l’ equivalence des codes. Thèse de doctorat, Université de Limoges (October 1999)Google Scholar
  31. 31.
    Stern, J.: An alternative to the fiat-shamir protocol. In: Quisquater, J.J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 173–180. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  32. 32.
    Stern, J.: A new identification scheme based on syndrome decoding. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 13–21. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  33. 33.
    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
  34. 34.
    Vertigan, D.: Bicycle dimension and special points of the Tutte polynomial. Journal of Combinatorial Theory, Series B 74, 378–396 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nicolas Sendrier
    • 1
  • Dimitris E. Simos
    • 1
    • 2
  1. 1.Project-Team SECRETINRIA Paris-RocquencourtLe Chesnay CedexFrance
  2. 2.SBA ResearchViennaAustria

Personalised recommendations