On the affine classification of cubic bent functions

We consider cubic boolean bent functions, each cubic monomial of which contains the same variable. We investigate canonical forms of these functions under affine transformations of variables. In particular, we refine the affine classification of cubic bent functions of 8 variables. 1 Preliminaries Let Vn be an n-dimensional vector space over the field F2, and Fn be the set of all boolean functions Vn → F2. We identify a function f ∈ Fn of x = (x1, . . . , xn) with its algebraic normal form, that is, a polynomial of the ring F2[x1, . . . , xn] reduced modulo the ideal (x 2 1 − x1, . . . , x 2 n − xn). Denote by deg f the degree of such polynomial. The Walsh–Hadamard transform associates with f ∈ Fn the function