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Stream Ciphers pp 131-154 | Cite as

Algebraic Attacks

  • Andreas Klein

Abstract

As the name suggests algebraic attacks need a lot of algebra. So we take a crash course in solving systems of nonlinear equations. At the end of the chapter we will look on two real word examples to see how the theory pays off. We will see how algebraic attacks breaks the eStream candidate LILI-128 and we will revisit E 0 to see how algebraic attacks put pressure on it.

Keywords

Gaussian Elimination Monomial Ideal Multivariate Polynomial Division Algorithm Elementary Symmetric Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 6.
    Amrhein, B., Gloor, O., Küchlin, W.: On the walk. Theor. Comput. Sci. 187, 179–202 (1997) MATHCrossRefGoogle Scholar
  2. 8.
    Armknecht, F., Krause, M.: Algebraic attacks on combiners with memory. In: Proceedings of Crypto 2003. LNCS, vol. 2729, pp. 162–176. Springer, Berlin (2003) CrossRefGoogle Scholar
  3. 9.
    Ars, G., Faugère, J.C., Imai, H., Kawazoe, M., Sugita, M.: Comparison between XL and Gröbner basis algorithms. In: Advances in Cryptology—ASIACRYPT 2004. LNCS, vol. 3329, pp. 338–353. Springer, Berlin (2004) CrossRefGoogle Scholar
  4. 39.
    Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K., Reidel, D. (eds.) Multidimensional Systems Theory, pp. 184–232. Reidel, Dordrecht (1985) CrossRefGoogle Scholar
  5. 55.
    Collart, S., Kalkbrener, M., Mall, D.: Converting bases with the Gröbner walk. J. Symb. Comput. 24, 465–469 (1997) MathSciNetMATHCrossRefGoogle Scholar
  6. 62.
    Courtois, N.: Fast algebraic attacks on stream ciphers with linear feedback. In: Proceedings of Crypto 2003. LNCS, vol. 2729, pp. 177–194. Springer, Berlin (2003) Google Scholar
  7. 63.
    Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Advances in Cryptology—EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Berlin (2000) CrossRefGoogle Scholar
  8. 64.
    Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Proceedings of Eurocrypt 2003. LNCS, vol. 2656, pp. 345–359. Springer, Berlin (2003). An extended version is available at http://www.cryptosystem.net/stream/ CrossRefGoogle Scholar
  9. 72.
    Dawson, E., Clark, A., Golić, J., Millan, W., Penna, L., Simpson, L.: The LILI-128 keystream generator. In: Proc. of First NESSIE Workshop (2001) Google Scholar
  10. 87.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139(1), 61–88 (1999). Available online http://fgbrs.lip6.fr/@papers/F99a.pdf MathSciNetMATHCrossRefGoogle Scholar
  11. 88.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 75–83. ACM, New York (2002). Available online http://fgbrs.lip6.fr/@papers/F02a.pdf CrossRefGoogle Scholar
  12. 89.
    Faugère, J.C., Gianni, P., 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
  13. 100.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  14. 103.
    Gebauer, R., Möller, H.M.: On an installation of Buchberger’s algorithm. In: Robbiano, L. (ed.) Computational Aspects of Communicative Algebra, pp. 141–152. Academic Press, New York (1988) Google Scholar
  15. 124.
    Hawkes, P., Rose, G.G.: Rewriting variables: the complexity of fast algebraic attacks on stream ciphers. In: Advances in Cryptology—CRYPTO 2004. Lecture Notes in Comput. Sci., vol. 3152, pp. 390–406. Springer, Berlin (2004) CrossRefGoogle Scholar
  16. 132.
    Huang, X., Huang, W., Liu, X., Wang, C., Wang, Z.J., Wang, T.: Reconstructing the nonlinear filter function of LILI-128 stream cipher based on complexity (2007). http://arxiv.org/abs/cs.CR/0702128
  17. 147.
    Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem. In: Proceedings of CRYPTO ’99. Springer, Berlin (1999) Google Scholar
  18. 224.
    Robbiano, L.: Term orderings on the polynomial ring. In: EUROCAL’85. LNCS, vol. 204, 513–517 (1985) CrossRefGoogle Scholar
  19. 232.
    Sarkar, P., Maitra, S.: Nonlinearity bounds and construction of resilient Boolean functions. In: Advances in Cryptology—CRYPTO 2000. LNCS, vol. 1880, pp. 515–532. Springer, Berlin (2000) CrossRefGoogle Scholar
  20. 253.
    The SINGULAR computer algebra system. http://www.singular.uni-kl.de/
  21. 266.
    Traverso, C.: Hilbert functions and Buchberger’s algorithm. J. Symb. Comput. 22, 355–376 (1997) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Andreas Klein
    • 1
  1. 1.Dept. of Pure Mathem. & Computer AlgebraState University of GhentGhentBelgium

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