Using Wiedemann’s Algorithm to Compute the Immunity Against Algebraic and Fast Algebraic Attacks

  • Frédéric Didier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4329)


We show in this paper how to apply well known methods from sparse linear algebra to the problem of computing the immunity of a Boolean function against algebraic or fast algebraic attacks. For an n-variable Boolean function, this approach gives an algorithm that works for both attacks in O(n2 n D) complexity and O(n2 n ) memory. Here \(D = \binom{n}{d}\) and d corresponds to the degree of the algebraic system to be solved in the last step of the attacks. For algebraic attacks, our algorithm needs significantly less memory than the algorithm in [ACG + 06] with roughly the same time complexity (and it is precisely the memory usage which is the real bottleneck of the last algorithm). For fast algebraic attacks, it does not only improve the memory complexity, it is also the algorithm with the best time complexity known so far for most values of the degree constraints.


algebraic attacks algebraic immunity fast algebraic attacks Wiedemann’s algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ACG+06]
    Armknetcht, F., Carlet, C., Gaborit, P., Künzli, S., Meier, W., Ruatta, O.: Efficient computation of algebraic immunity for algebraic and fast algebraic attacks. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 147–164. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. [Arm04]
    Armknetch, 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. [BLP06]
    Braeken, A., Lano, J., Preneel, B.: Evaluating the resistance of stream ciphers with linear feedback against fast algebraic attacks. In: Batten, L.M., Safavi-Naini, R. (eds.) ACISP 2006. LNCS, vol. 4058, pp. 40–51. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. [BP05]
    Braeken, A., Preneel, B.: On the algebraic immunity of symmetric Boolean functions (2005),
  5. [Car04]
    Carlet, C.: Improving the algebraic immunity of resilient and nonlinear functions and constructing bent functions (2004),
  6. [CM03]
    Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 346–359. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. [Cop94]
    Coppersmith, D.: Solving linear equations over GF(2) via block Wiedemann algorithm. Math. Comp. 62(205), 333–350 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [COS86]
    Coppersmith, D., Odlyzko, A., Schroeppel, R.: Discrete logarithms in GF(p). Algorithmitica 1, 1–15 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Cou03]
    Courtois, N.: 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
  10. [DGM04]
    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
  11. [DM06]
    Dalai, D.K., Maitra, S.: Reducing the number of homogeneous linear equations in finding annihilators. In: Gong, G., Helleseth, T., Song, H.-Y., Yang, K. (eds.) SETA 2006. LNCS, vol. 4086, pp. 376–390. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. [DMS05]
    Dalai, D.K., Maitra, S., Sarkar, S.: Basic theory in construction of Boolean functions with maximum possible annihilator immunity (2005),
  13. [DT06]
    Didier, F., Tillich, J.-P.: Computing the algebraic immunity efficiently. In: Robshaw, M.J.B. (ed.) FSE 2006. LNCS, vol. 4047, pp. 359–374. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. [FA03]
    Faugère, J.-C., Ars, G.: An algebraic cryptanalysis of nonlinear filter generator using Gröbner bases. Rapport de Recherche INRIA, 4739 (2003)Google Scholar
  15. [FJ03]
    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
  16. [HR04]
    Hawkes, P., Rose, G.C.: Rewriting variables: The complexity of fast algebraic attacks on stream ciphers. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 390–406. Springer, Heidelberg (2004)Google Scholar
  17. [Mas69]
    Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory IT-15, 122–127 (1969)CrossRefMathSciNetGoogle Scholar
  18. [MPC04]
    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
  19. [Odl84]
    Odlyzko, A.M.: Discrete logarithms in finite fields and their cryptographic significance. In: Theory and Application of Cryptographic Techniques, pp. 224–314 (1984)Google Scholar
  20. [Wie86]
    Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory IT-32, 54–62 (1986)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Frédéric Didier
    • 1
  1. 1.Projet CODES, INRIA Rocquencourt, Domaine de VoluceauLe Chesnay cedex

Personalised recommendations