On derivatives of Boolean bent functions

Bent function can be defined as a Boolean function $f(x)$ in $n$ variables ($n$ is even) such that for any nonzero vector $y$ its derivative $D_yf(x)=f(x)\oplus f(x\oplus y)$ is balanced, that is, it takes values $0$ and $1$ equally often. Whether every balanced function is a derivative of some bent function or not is an open problem. In this paper, special case of this problem is studied. It is proven that every non-constant affine function in $n$ variables, $n\geqslant4$, $n$ is even, is a derivative of $(2^{n-1}-1)|\mathcal{B}_{n-2}|^2$ bent functions, where $|\mathcal{B}_{n-2}|$ is the number of bent functions in $n-2$ variables. New iterative lower bounds for the number of bent functions are presented.