Gram matrices of bent functions and properties of subfunctions of quadratic self-dual bent functions

A Boolean function in even number of variables $n$ is called a bent function if it has flat Walsh — Hadamard spectrum consisting of numbers $\pm2^{n/2}$. A bent function is called self-dual if it coincides with its dual bent function. Previously the author obtained a sufficient condition for subfunctions in $n-2$ variables of a self-dual bent function in $n$ variables, obtained by fixing the first two variables, to be bent. In this paper, we prove that for quadratic self-dual bent functions this condition is not necessary for $n\geqslant6$. The concept of the Gram matrices of Boolean functions is introduced, the general form of the Gram matrix of a bent function and its dual function are obtained. It is proved that if the Gram matrix of a bent function in $n$ variables is non-invertible, then its subfunctions in $n-2$ variables, obtained by fixing the first two variables, are bent functions. It is also proved that the subfunctions of its dual bent function are also bent functions.