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

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

Abstract

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.

Keywords

Symmetric cryptography Boolean functions S-boxes nonlinearity discrete logarithm 

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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

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