Gabidulin Matrix Codes and Their Application to Small Ciphertext Size Cryptosystems

  • Thierry P. Berger
  • Philippe Gaborit
  • Olivier Ruatta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10698)


In this paper we propose a new method to hide the structure of Gabidulin codes for cryptographic applications. At the difference of previous cryptosystems based on Gabidulin codes, we do not try to mask the structure of Gabidulin codes by the use of some distortion methods, but we consider matrix codes obtained from subcodes of binary images of Gabidulin codes. This allows us to remove the properties related to multiplication in the extension field. In particular, this prevents the use of Frobenius for cryptanalysis. Thus, Overbeck’s attack can no longer be applied. In practice we obtain public key with a gain of a factor of order ten compared to the classical Goppa-McEliece scheme with still a small cipher text of order only 1 kbits, better than recent cryptosystems for which the cipher text size is of order 10 kbits. Several results used and proved in the paper are of independent interest: results on structural properties of Gabidulin matrix codes and hardness of deciding whether a code is equivalent to a subcode of a matrix code.


McEliece public key cryptosystem Rank metric Gabidulin codes 


  1. 1.
    Aguilar Melchor, C., Blazy, O., Deneuville, J.-C., Gaborit, P., Zémor, G.: Efficient encryption from random quasi-cyclic codes. CoRR abs/1612.05572 (2016).
  2. 2.
    Alekhnovich, M.: More on average case vs approximation complexity. In: Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS 2003), Cambridge, MA, USA, 11–14 October 2003, pp. 298–307 (2003)Google Scholar
  3. 3.
    Berger, T.P.: Isometries for rank distance and permutation group of Gabidulin codes. IEEE Trans. Inf. Theory 49(11), 3016–3019 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berger, T.P., El Amrani, N.: Codes over \(\cal{L}(GF(2)^m,GF(2)^m)\), MDS diffusion matrices and cryptographic applications. In: El Hajji, S., Nitaj, A., Carlet, C., Souidi, E.M. (eds.) C2SI 2015. LNCS, vol. 9084, pp. 197–214. Springer, Cham (2015). Google Scholar
  5. 5.
    Berger, T.P., Cayrel, P.-L., Gaborit, P., Otmani, A.: Reducing key length of the McEliece cryptosystem. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 77–97. Springer, Heidelberg (2009). CrossRefGoogle Scholar
  6. 6.
    Berger, T.P., Gueye, C.T., Klamti, J.B.: A NP-complete problem in coding theory with application to code based cryptography. In: El Hajji, S., Nitaj, A., Souidi, E.M. (eds.) C2SI 2017. LNCS, vol. 10194, pp. 230–237. Springer, Cham (2017). CrossRefGoogle Scholar
  7. 7.
    Berger, T.P., Loidreau, P.: Designing an efficient and secure public-key cryptosystem based on reducible rank codes. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 218–229. Springer, Heidelberg (2004). CrossRefGoogle Scholar
  8. 8.
    Bogart, K.P., Goldberg, D., Gordon, J.: An elementary proof of the MacWilliams theorem on equivalence of codes. Inf. Control 37, 19–22 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Couvreur, A., Gaborit, P., Gauthier-Umaña, V., Otmani, A., Tillich, J.-P.: Distinguisher-based attacks on public-key cryptosystems using Reed-Solomon codes. Des. Codes Cryptogr. 73(2), 641–666 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Delsarte, P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Faugère, J.-C., Safey El Din, M., Spaenlehauer, P.-J.: Gröbner bases of Bihomogeneous ideals generated by polynomials of bidegree (1,1): algorithms and complexity. J. Symb. Comput. 46(4), 406–437 (2011)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gabidulin, E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. (English translation of Problemy Peredachi Informatsii) 21(1), 1–71 (1985)MathSciNetzbMATHGoogle Scholar
  13. 13.
    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). Google Scholar
  14. 14.
    Gabidulin, E., Rashwan, H., Honary, B.: On improving security of GPT cryptosystems. In: 2009 IEEE International Symposium on Information Theory Proceedings (ISIT), ISIT 2009, pp. 1110–1114 (2009)Google Scholar
  15. 15.
    Gaborit, P., Murat, G., Ruatta, O., Zémor, G.: Low rank parity check codes and their application to cryptography. In: Proceedings of the Workshop on Coding and Cryptography, WCC 2013, Bergen, Norway (2013).
  16. 16.
    Gaborit, P., Ruatta, O., Schrek, J.: On the complexity of the rank syndrome decoding problem. IEEE Trans. Inf. Theory 62(2), 1006–1019 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Huffman, W.C.: Groups and codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory. Elsevier, Amsterdam (1998). Chap. 17Google Scholar
  18. 18.
    Loidreau, P.: A new rank metric codes based encryption scheme. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 3–17. Springer, Cham (2017). CrossRefGoogle Scholar
  19. 19.
    MacWilliams F.J.: Combinatorial properties of elementary Abelian groups Ph.D. thesis, Radcliffe College (1962)Google Scholar
  20. 20.
    McEliece, R.: A public-key cryptosystem based on algebraic coding theory. In: DSN Program Report, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, pp. 114–116, January 1978Google Scholar
  21. 21.
    Misoczki, R., Barreto, P.S.L.M.: Compact McEliece keys from Goppa codes. In: Jacobson, M.J., Rijmen, V., Safavi-Naini, R. (eds.) SAC 2009. LNCS, vol. 5867, pp. 376–392. Springer, Heidelberg (2009). CrossRefGoogle Scholar
  22. 22.
    Misoczki, R., Tillich, J.P., Sendrier, N., Barreto, P.S.: MDPC-McEliece: new McEliece variants from moderate density parity-check codes. In: IEEE International Symposium on Information Theory, ISIT 2013, pp. 2069–2073. IEEE (2013)Google Scholar
  23. 23.
    Otmani, A., Kalachi, H.T., Ndjeya, S.: Improved cryptanalysis of rank metric schemes based on Gabidulin codes. CoRR abs/1602.08549 (2016)Google Scholar
  24. 24.
    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). CrossRefGoogle Scholar
  25. 25.
    Overbeck, R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rashwan, H., Honary, B., Gabidulin, E.M.: On improving security of GPT cryptosystems. In: IEEE International Symposium on Information Theory, ISIT 2009, pp. 1110–1114. IEEE (2009)Google Scholar
  27. 27.
    Roth, R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inf. Theory 37(2), 328–336 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Thierry P. Berger
    • 1
  • Philippe Gaborit
    • 1
  • Olivier Ruatta
    • 1
  1. 1.Univ. Limoges, CNRS, XLIM, UMR 7252LimogesFrance

Personalised recommendations