Analysis of (n, n)-Functions Obtained From the Maiorana-McFarland Class

Abstract

Pott et al. (2018) showed that $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ , $n = 2m$ , $r\ge 1$ , is a nontrivial example of a vectorial function with the maximal possible number $2^{n}-2^{m}$ of bent components. Mesnager et al. (2019) generalized this result by showing conditions on $\Lambda (x) = x + \sum _{j=1}^\sigma \alpha _{j}x^{2^{t_{j}}}$ , $\alpha _{j}\in {\mathbb F} _{2^{m}}$ , under which $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions $F(x) = {\rm Tr^{n}_{m}}(\gamma \mathcal {F}(x))$ , $\gamma \in {\mathbb F}_{2^{n}}\setminus {\mathbb F} _{2^{m}}$ , which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations $\mathcal {D}_{a}F(x) = F(x) + F(x+a) + F(a) = 0$ , which forms a spread of ${\mathbb F}_{2^{n}}$ . Analysing these spreads, we can infer neat conditions for functions $H(x) = (F(x),G(x))$ from ${\mathbb F}_{2^{n}}$ to ${\mathbb F}_{2^{m}}\times {\mathbb F} _{2^{m}}$ to exhibit small differential uniformity (for instance for $\Lambda (x) = x$ and $r=0$ this fact is used in the construction of Carlet’s, Pott-Zhou’s, Taniguchi’s APN-function). For some classes of $H(x)$ we determine differential uniformity and with a method based on Bezout’s theorem nonlinearity.