Open Problems Related to Algebraic Attacks on Stream Ciphers

  • Anne Canteaut
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3969)


The recently developed algebraic attacks apply to all keystream generators whose internal state is updated by a linear transition function, including LFSR-based generators. Here, we describe this type of attacks and we present some open problems related to their complexity. We also investigate the design criteria which may guarantee a high resistance to algebraic attacks for keystream generators based on a linear transition function.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, R.J.: Searching for the optimum correlation attack. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 137–143. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  2. 2.
    Armknecht, F.: Improving fast algebraic attacks. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 65–82. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Armknecht, F.: Algebraic attacks and annihilators. In: Proceedings of the Western European Workshop on Research in Cryptology (WEWoRC 2005). Lecture Notes in Informatics. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Armknecht, F., Krause, M.: Algebraic attacks on combiners with memory. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 162–175. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Ars, G., Faugère, J.-C., Imai, H., Kawazoe, M., Sugita, M.: Comparison between XL and gröbner basis algorithms. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 338–353. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Bardet, M., Faugère, J.-C., Salvy, B., Yang, B.-Y.: On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proc. International Conference on Polynomial System Solving (ICPSS 2004) (2004)Google Scholar
  7. 7.
    Bardet, M., Faugère, J.-C., Salvy, B., Yang, B.-Y.: Asymptotic behaviour of the degree of regularity of semi-regular polynomial systems. In: MEGA 2005, Porto Conte, Italy (May 2005)Google Scholar
  8. 8.
    Carlet, C., Prouff, E.: On a new notion of nonlinearity relevant to multi-output pseudo-random generators. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 291–305. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic programming. Journal of Symbolic Computation (9), 251–280 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Courtois, N.T.: Fast algebraic attacks on stream ciphers with linear feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 176–194. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 345–359. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Courtois, N.T., Klimov, A.B., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Courtois, N.T.: Algebraic attacks on combiners with memory and several outputs. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 3–20. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Courtois, N.T., Pieprzyk, J.: Cryptanalysis of block ciphers with overdefined systems of equations. In: ESORICS 2002. LNCS, pp. 267–287. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Dalai, D.K., Gupta, K.C., Maitra, S.: Results on algebraic immunity for cryptographically significant boolean functions. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 92–106. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Dalai, D.K., Gupta, K.C., Maitra, S.: Cryptographically significant boolean functions: Construction and analysis in terms of algebraic immunity. In: Gilbert, H., Handschuh, H. (eds.) FSE 2005. LNCS, vol. 3557, pp. 98–111. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Dalai, D.K., Sarkar, S., Maitra, S.: Balanced Boolean functions with maximum possible algebraic immunity, April 2005 (preprint)Google Scholar
  18. 18.
    Diem, C.: The XL-algorithm and a conjecture from commutative algebra. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 323–337. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F 4). Journal of Pure and Applied Algebra 139(1-3), 61–88 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5). In: Proceedings of the 2002 international symposium on Symbolic and algebraic computation. ACM, New York (2002)Google Scholar
  21. 21.
    Faugère, J.-C., Ars, G.: An algebraic cryptanalysis of nonlinear filter generators using Gröbner bases. Technical Report 4739, INRIA (2003), Available at:
  22. 22.
    Faugère, J.-C., Joux, A.: Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. 23.
    Key, J.D., McDonough, T.P., Mavron, V.C.: Information sets and partial permutation decoding for codes from finite geometries. Finite Fields and Their Applications (to appear, 2005)Google Scholar
  24. 24.
    Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of boolean functions. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 474–491. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Pasalic, E.: On algebraic immunity of Maiorana-McFarland like functions and applications of algebraic attack. In: Proceedings of the ECRYPT Symmetric Key Encryption Workshop (SKEW), Aarhus, Danemark (May 2005)Google Scholar
  26. 26.
    Shannon, C.E.: Communication theory of secrecy systems. Bell system technical journal 28, 656–715 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Steel, A.: Allan Steel’s Gröbner basis timings page (2004),
  28. 28.
    Zhang, M., Chan, A.H.: Maximum correlation analysis of nonlinear S-boxes in stream ciphers. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 501–514. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Anne Canteaut
    • 1
  1. 1.INRIA – projet CODESLe ChesnayFrance

Personalised recommendations