Algebraic Cryptanalysis of McEliece Variants with Compact Keys

  • Jean-Charles Faugère
  • Ayoub Otmani
  • Ludovic Perret
  • Jean-Pierre Tillich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6110)

Abstract

In this paper we propose a new approach to investigate the security of the McEliece cryptosystem. We recall that this cryptosystem relies on the use of error-correcting codes. Since its invention thirty years ago, no efficient attack had been devised that managed to recover the private key. We prove that the private key of the cryptosystem satisfies a system of bi-homogeneous polynomial equations. This property is due to the particular class of codes considered which are alternant codes. We have used these highly structured algebraic equations to mount an efficient key-recovery attack against two recent variants of the McEliece cryptosystems that aim at reducing public key sizes. These two compact variants of McEliece managed to propose keys with less than 20,000 bits. To do so, they proposed to use quasi-cyclic or dyadic structures. An implementation of our algebraic attack in the computer algebra system Magma allows to find the secret-key in a negligible time (less than one second) for almost all the proposed challenges. For instance, a private key designed for a 256-bit security has been found in 0.06 seconds with about 217.8 operations.

References

  1. 1.
    Adams, W., Loustaunau, P.: An Introduction to Gröbner Bases. American Mathematical Society, Providence (July 1994)MATHGoogle Scholar
  2. 2.
    Avanzi, R.: Lightweight asymmetric cryptography and alternatives to rsa, ecrypt european network of excellence in cryptology (2005), http://www.ecrypt.eu.org/ecrypt1/documents/D.AZTEC.2-1.2.pdf
  3. 3.
    Baldi, M., Bodrato, M., Chiaraluce, G.F.: A new analysis of the McEliece cryptosystem based on QC-LDPC codes. In: Ostrovsky, R., De Prisco, R., Visconti, I. (eds.) SCN 2008. LNCS, vol. 5229, pp. 246–262. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Baldi, M., Chiaraluce, G.F.: Cryptanalysis of a new instance of McEliece cryptosystem based on QC-LDPC codes. In: IEEE International Symposium on Information Theory, Nice, France, March 2007, pp. 2591–2595 (2007)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., Loidreau, P.: How to mask the structure of codes for a cryptographic use. Designs Codes and Cryptography 35(1), 63–79 (2005)MATHCrossRefMathSciNetGoogle 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.
    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
  9. 9.
    Bernstein, D.J., Lange, T., Peters, C., van Tilborg, H.: Explicit bounds for generic decoding algorithms for code-based cryptography. In: Pre-proceedings of WCC 2009, pp. 168–180 (2009)Google Scholar
  10. 10.
    Biswas, B., Sendrier, N.: McEliece cryptosystem implementation: Theory and practice. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 47–62. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck (1965)Google Scholar
  12. 12.
    Cox, D.A., Little, J.B., O’Shea, D.: Ideals, Varieties, and algorithms: an Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer, New York (2001)Google Scholar
  13. 13.
    Faugère, J.-C.: A new efficient algorithm for computing gröbner bases (f4). Journal of Pure and Applied Algebra 139(1-3), 61–88 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Faugère, J.-C.: A new efficient algorithm for computing gröbner bases without reduction to zero: F5. In: ISSAC 2002, pp. 75–83. ACM press, New York (2002)CrossRefGoogle Scholar
  15. 15.
    Faugère, J.-C., Levy-dit Vehel, F., Perret, L.: Cryptanalysis of minrank. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 280–296. Springer, Heidelberg (2008)Google Scholar
  16. 16.
    Faugère, J.-C., El Din, M.S., Spaenlehauer, P.-J.: Gröbner bases of bihomogeneous ideals generated by polynomials of bidegree (1,1): Algorithms and complexity. CoRR, abs/1001.4004 (2010)Google Scholar
  17. 17.
    Faugère, J.-C., Gianni, P.M., Lazard, D., Mora, T.: Efficient computation of zero-dimensional gröbner bases by change of ordering. J. Symb. Comput. 16(4), 329–344 (1993)MATHCrossRefGoogle Scholar
  18. 18.
    Faure, C., Minder, L.: Cryptanalysis of the McEliece cryptosystem over hyperelliptic curves. In: Proceedings of the eleventh International Workshop on Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, June 2008, pp. 99–107 (2008)Google Scholar
  19. 19.
    Finiasz, M., Sendrier, N.: Security bounds for the design of code-based crypto systems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 88–105. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Gabidulin, E., Paramonov, A.V., Tretjakov, O.V.: Ideals over a non-commutative ring and their applications to cryptography. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 482–489. Springer, Heidelberg (1991)Google Scholar
  21. 21.
    Gaborit, P.: Shorter keys for code based cryptography. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 81–91. Springer, Heidelberg (2006)Google Scholar
  22. 22.
    Gibson, J.K.: Severely denting the Gabidulin version of the McEliece public key cryptosystem. Design Codes and Cryptography 6(1), 37–45 (1995)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Janwa, H., Moreno, O.: McEliece public key cryptosystems using algebraic-geometric codes. Designs Codes and Cryptography 8(3), 293–307 (1996)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes, 5th edn. North-Holland, Amsterdam (1986)Google Scholar
  25. 25.
    McEliece, R.J.: A Public-Key System Based on Algebraic Coding Theory, pp. 114–116. Jet Propulsion Lab. (1978); DSN Progress Report 44Google Scholar
  26. 26.
    Minder, L., Shokrollahi, A.: Cryptanalysis of the Sidelnikov cryptosystem. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 347–360. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  27. 27.
    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)Google Scholar
  28. 28.
    Niederreiter, H.: A public-key cryptosystem based on shift register sequences. In: Pichler, F. (ed.) EUROCRYPT 1985. LNCS, vol. 219, pp. 35–39. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  29. 29.
    Otmani, A., Tillich, J.P., Dallot, L.: Cryptanalysis of McEliece cryptosystem based on quasi-cyclic ldpc codes. In: Proceedings of First International Conference on Symbolic Computation and Cryptography, Beijing, China, April 28-30, pp. 69–81. LMIB Beihang University (2008)Google Scholar
  30. 30.
    Overbeck, R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptology 21(2), 280–301 (2008)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sidelnikov, V.M.: A public-key cryptosytem based on Reed-Muller codes. Discrete Mathematics and Applications 4(3), 191–207 (1994)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Sidelnikov, V.M., Shestakov, S.O.: On the insecurity of cryptosystems based on generalized Reed-Solomon codes. Discrete Mathematics and Applications 1(4), 439–444 (1992)CrossRefMathSciNetGoogle 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 (1988)CrossRefGoogle Scholar
  34. 34.
    Gauthier Umana, V., Leander, G.: Practical key recovery attacks on two McEliece variants (2009), http://eprint.iacr.org/2009/509
  35. 35.
    Wieschebrink, C.: Cryptanalysis of the Niederreiter public key scheme based on GRS subcodes. eprint 452 (2009), http://eprint.iacr.org/2009/452.pdf

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jean-Charles Faugère
    • 1
  • Ayoub Otmani
    • 2
    • 3
  • Ludovic Perret
    • 1
  • Jean-Pierre Tillich
    • 2
  1. 1.SALSA Project - INRIA (Centre Paris-Rocquencourt), UPMC, Univ Paris 06 - CNRS, UMR 7606, LIP6ParisFrance
  2. 2.SECRET Project - INRIA Rocquencourt, Domaine de Voluceau, B.P. 105Le Chesnay CedexFrance
  3. 3.GREYC - Université de Caen - Ensicaen, Boulevard Maréchal JuinCaen CedexFrance

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