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 (x1− x1, . . . , x 2 n − xn). Denote by deg f the degree of such polynomial. The Walsh–Hadamard transform associates with f ∈ Fn the function ∧ f (u) = ∑ x∈Vn χ(f(x) + x · u), u ∈ Vn, where χ(a) = (−1) is the additive character of F2 and x ·u is the dot product of two vectors. Denote by ∧ F n the image of Fn under the mapping f 7→ ∧ f . Let Bn be the set of all boolean bent functions of n variables: f ∈ Bn if | ∧ f (u)| = 2 for all u ∈ Vn. Bent functions were introduced by Rothaus in 1976 [7], and since then have been widely studied. It is clear that Bn 6= ∅ only for even n. Therefore, when we write Bn, we mean that n is even. Let AGLn be the general affine group of transformations of Vn. An element σ ∈ AGLn acts as follows: σ(x) = xA + b, where A is an invertible n × n matrix over F2, b ∈ Vn. Extend the action of AGLn on Fn in a natural way: σ(f)(x) = f(xA+ b),