Raziskovalni matematični seminar

This talk focuses on the construction of two new classes of plateaued functions, denoted by $\mathcal{C}$ and $\mathcal{D}_0$, both of which are derived from the $\mathcal{GMM}$ class.  We introduce the class $\mathcal{C}$ of $s$-plateaued functions defined as 

$$f(x, y) = x \cdot \phi(y) + 1_{L^{\perp}}(y),$$

where $0 < s < n$ and $L^{\perp}$ is a linear subspace of $\mathbb{F}_2^{\frac{n+s}{2}}$. This class is presented as a special subclass of the broader $\mathcal{D}$ class. 

Furthermore, we demonstrate that the subclass $\mathcal{D}_0$ can still be constructed even when the mapping $\phi$ is not injective. However, in this case, the sufficient conditions are more complicated than in the bent function.

In the final part of the talk, I present an initial approach for decomposing a plateaued function $f \in \mathcal{B}_n$ into four component functions $f_i \in \mathcal{B}_{n-2}$, for $i \in [1, 4]$. The 4-decomposition of a plateaued function $f = f1||f2||f3||f4 \in \cB_n$ is explained by describing three atmost cases:

1.     All component functions $f_i \in \mathcal{B}_{n-2}$ $(i \in [1, 4])$ are plateaued.
2.     All $f_i$ are functions with $5$-valued Walsh spectra.
3.     A mixed 4-decomposition.

This is a joint work with Enes Pasalic and Sadmir Kudin.