Abstract
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.
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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)
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)
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)
Barg, S.: Some new NP-complete coding problems. Probl. Peredachi Inf. 30, 23–28 (1994)
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)
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)
Bouyukliev, I.: About the code equivalence. Ser. Coding Theory Cryptol. 3, 126–151 (2007)
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)
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)
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)
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)
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
Feulner, T.: The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Adv. Math. Commun. 3, 363–383 (2009)
Fripertinger, H.: Enumeration of linear codes by applying methods from algebraic combinatorics. Grazer Math. Ber. 328, 31–42 (1996)
Fripertinger, H.: Enumeration of the semilinear isometry classes of linear codes. Bayrether Mathematische Schriften 74, 100–122 (2005)
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)
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)
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)
Harari, S.: A new authentication algorithm. In: Wolfmann, J., Cohen, G. (eds.) Coding Theory 1988. LNCS, vol. 388, pp. 91–105. Springer, Heidelberg (1989)
Human, W.C.: Codes and groups. In: Pless, V., Human, W.C. (eds.) Handbook of Coding Theory, pp. 1345–1440. Elsevier, North-Holland (1998)
Kaski, P., Östergård, P.R.J.: Classification Algorithms for Codes and Designs. Algorithms and Computation in Mathematics, vol. 15. Springer, Heidelberg (2006)
MacWilliams, F.J.: Error-correcting codes for multiple-level transmission. Bell. Syst. Tech. J. 40, 281–308 (1961)
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)
Overbeck, R., Sendrier, N.: Code-based cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 95–145. Springer (2009)
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)
Petrank, E., Roth, R.M.: Is code equivalence easy to decide? IEEE Trans. Inform. Theory 43, 1602–1604 (1997)
Sendrier, N.: On the dimension of the hull. SIAM J. Discrete Math. 10(2), 282–293 (1997)
Sendrier, N.: Finding the permutation between equivalent linear codes: The support splitting algorithm. IEEE Trans. Inform. Theory 26, 1193–1203 (2000)
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), https://www.rocq.inria.fr/secret/PUBLICATIONS/codeq3.pdf
Skersys, G.: Calcul du groupe d’automorphisme des codes. Détermination de l’ equivalence des codes. Thèse de doctorat, Université de Limoges (October 1999)
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)
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)
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)
Vertigan, D.: Bicycle dimension and special points of the Tutte polynomial. Journal of Combinatorial Theory, Series B 74, 378–396 (1998)
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Sendrier, N., Simos, D.E. (2013). The Hardness of Code Equivalence over \(\mathbb{F}_q\) and Its Application to Code-Based Cryptography. In: Gaborit, P. (eds) Post-Quantum Cryptography. PQCrypto 2013. Lecture Notes in Computer Science, vol 7932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38616-9_14
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DOI: https://doi.org/10.1007/978-3-642-38616-9_14
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