On the nonlinearity of discrete logarithm in F2n

Authors:
Risto M. Hakala;Kaisa Nyberg
Affiliations:
Department of Information and Computer Science, Aalto University School of Science and Technology, Aalto, Finland;Department of Information and Computer Science, Aalto University School of Science and Technology, Aalto, Finland and Nokia Research Center, Finland
Venue:
SETA'10 Proceedings of the 6th international conference on Sequences and their applications
Year:
2010

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Abstract

In this paper, we derive a lower bound to the nonlinearity of the discrete logarithm function in F2n extended to a bijection in F2n. This function is closely related to a family of S-boxes from F2n to F2m 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.