The degree of proximity of the Boolean function reduced representation to the class of monomial functions according to basis selection

It is known that a definition of Boolean bent functions is invariant under any linear nonsingular transformation of the variables. This paper investigates the effect of the basis selection of the Boolean function reduced representation on it's property “to be a hyper-bent function”. The following results are obtained: 1) for any bent function of 4 variables there exist two bases of the vector space $\left({\mathbb{F}_{2^4}}\right)_{\mathbb{F}_2}$ such that the reduced representation of this function in the first basis is a hyper-bent function, and in the second basis is not. 2) For any even $n>4$ there exist two bases of the vector space $\left({\mathbb{F}_{2^n}}\right)_{\mathbb{F}_2}$ and the function of $n$ variables such that the reduced representation of this function in the first basis is a hyper-bent function, and in the second basis is not.