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

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.