On the Nonlinearity of Discrete Logarithm in \(\mathbb F_{2^n}\)

  • Risto M. Hakala
  • Kaisa Nyberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)


In this paper, we derive a lower bound to the nonlinearity of the discrete logarithm function in \(\mathbb F_{2^n}\) extended to a bijection in \(\mathbb F_2^n\). This function is closely related to a family of S-boxes from \(\mathbb F_2^n\) to \(\mathbb F_2^m\) proposed recently by Feng, Liao, and Yang, for which a lower bound on the nonlinearity was given by Carlet and Feng. This bound decreases exponentially with m and is therefore meaningful and proves good nonlinearity only for S-boxes with output dimension m logarithmic to n. By extending the methods of Brandstätter, Lange, and Winterhof we derive a bound that is of the same magnitude. We computed the true nonlinearities of the discrete logarithm function up to dimension n = 11 to see that, in reality, the reduction seems to be essentially smaller. We suggest that the closing of this gap is an important problem and discuss prospects for its solution.


Symmetric cryptography Boolean functions S-boxes nonlinearity discrete logarithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brandstätter, N., Lange, T., Winterhof, A.: On the non-linearity and sparsity of Boolean functions related to the discrete logarithm in finite fields of characteristic two. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 135–143. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Carlet, C., Feng, K.: An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 425–440. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Carlet, C., Feng, K.: An infinite class of balanced vectorial Boolean functions with optimum algebraic immunity and good nonlinearity. In: Chee, Y.M., Li, C., Ling, S., Wang, H., Xing, C. (eds.) IWCC 2009. LNCS, vol. 5557, pp. 1–11. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Cochrane, T.: On a trigonometric inequality of Vinogradov. Journal of Number Theory 27(1), 9–16 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Feng, K., Liao, Q., Yang, J.: Maximal values of generalized algebraic immunity. Designs, Codes and Cryptography 50(2), 243–252 (2009)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Konyagin, S., Lange, T., Shparlinski, I.: Linear complexity of the discrete logarithm. Designs, Codes and Cryptography 28(2), 135–146 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lidl, R., Niederreiter, H.: Finite fields. In: Encyclopedia of Mathematics and its Applications, 2nd edn., vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  8. 8.
    Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Risto M. Hakala
    • 1
  • Kaisa Nyberg
    • 1
    • 2
  1. 1.Department of Information and Computer ScienceAalto University School of Science and TechnologyAaltoFinland
  2. 2.Nokia Research CenterFinland

Personalised recommendations