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A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where $\tilde{F}(x)=(f_1(x),\ldots,f_m(x))$ is a vectorial bent function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, and $a_i$, $1\le i\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $F(x) = x^{2^r}{\rm Tr^n_m}(\Lambda(x))$, where $\Lambda$ is a linearized permutation of $\mathbb{F}_{2^m}$. In the first part of this article it is shown that plateaued functions with $2^n-2^m$ bent components can have nonlinearity at most $2^{n-1}-2^{\lfloor\frac{n+m}{2}\rfloor}$, a bound which is attained by the example $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r

We introduce a method based on Bezout's theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on $\mathbb{F}_{2^{2m}}$. Among those are the functions of Taniguchi 2019, Carlet 2011, and Zhou and Pott 2013, all of which are APN under certain conditions. This approach helps to understand why the majority of the functions in those classes have solely bent and semibent components, which in the case of APN functions is called the classical spectrum. More precisely, we show that all Taniguchi functions have the classical spectrum independent from being APN. We determine the nonlinearity of all functions belonging to Carlet's class and to the class of Zhou and Pott, which also confirms with comparatively simple proofs earlier results on the Walsh spectrum of APN-functions in these classes. Using the Hasse-Weil bound, we show that some simple sufficient conditions for the APN-ness of the Zhou-Pott functions, which are given in the original paper, are also necessary.

Zhou 2013 introduced modified planar functions to describe $(2^n,2^n,2^n,1)$ relative difference sets $R$ as a graph of a function on the finite field $\F_{2^n}$, and pointed out that projections of $R$ are difference sets that can be described by negabent or bent$_4$ functions, which are Boolean functions given in multivariate form. Objective of this paper is to contribute to the understanding of these component functions of modified planar functions. We first completely describe a multivariate version of modified planar functions in terms of their bent$_4$ components. In the second part we characterize the component functions of (univariate) modified planar functions in terms of appropriate generalizations of the Walsh-Hadamard transform, with respect to which they have a flat spectrum. We hereby obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function.

In this article we give a Drinfeld modular interpretation for various towers of function fields meeting Zink's bound.

We give a complete characterization of all Galois subfields of the generalized Giulietti--Korchm\'aros function fields $\mathcal C_n / \fqn$ for $n\ge 5$. Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields.

The Walsh transform $\widehat{Q}$ of a quadratic function $Q:F_{p^n}\rightarrow F_p$ satisfies $|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$ for all $b\in F_{p^n}$, where $0\le s\le n-1$ is an integer depending on $Q$. In this article, we study the following three classes of quadratic functions of wide interest. The class $\mathcal{C}_1$ is defined for arbitrary $n$ as $\mathcal{C}_1 = \{Q(x) = Tr(\sum_{i=1}^{\lfloor (n-1)/2\rfloor}a_ix^{2^i+1})\;:\; a_i \in F_2\}$, and the larger class $\mathcal{C}_2$ is defined for even $n$ as $\mathcal{C}_2 = \{Q(x) = Tr(\sum_{i=1}^{(n/2)-1}a_ix^{2^i+1}) + {\rm Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in F_2\}$. For an odd prime $p$, the subclass $\mathcal{D}$ of all $p$-ary quadratic functions is defined as $\mathcal{D} = \{Q(x) = Tr(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1})\;:\; a_i \in F_p\}$. We determine the distribution of the parameter $s$ for $\mathcal{C}_1, \mathcal{C}_2$ and $\mathcal{D}$. As a consequence we obtain the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $p > 2$, our results yield the distribution of the co-dimensions for the rotation symmetric quadratic $p$-ary functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the subcodes of the second order Reed-Muller codes corresponding to $\mathcal{C}_1$ and $\mathcal{C}_2$.