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.
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Hakala, R.M., Nyberg, K. (2010). On the Nonlinearity of Discrete Logarithm in \(\mathbb F_{2^n}\) . In: Carlet, C., Pott, A. (eds) Sequences and Their Applications – SETA 2010. SETA 2010. Lecture Notes in Computer Science, vol 6338. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15874-2_29
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DOI: https://doi.org/10.1007/978-3-642-15874-2_29
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