Design of new classes of plateaued functions and its 4-decomposition

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.