Derivatives of bent functions in connection with the bent sum decomposition problem

In this paper, we investigate when a balanced function can be a derivative of a bent function. We prove that every nonconstant affine function in an even number of variables n is a derivative of \((2^{n-1}-~1)\) \(\mid {\mathcal {B}}_{n-2}\mid ^2\) bent functions, where \({\mathcal {B}}_n\) is the set of all bent functions in n variables. Based on this result, we propose a new iterative lower bound for the number of bent functions. We study the property of balanced functions that depend linearly on at least one of their variables to be derivatives of bent functions. We show the connection between this property and the “bent sum decomposition problem”. We use this connection to prove that if a balanced quadratic Boolean function is a derivative of a Boolean function, then this function is a derivative of a bent function.