Every cubic Boolean function in 8 variables is the sum of not more than 4 bent functions

Boolean functions with extremal nonlinear properties are called bent functions. They are exactly those functions that have the maximal possible Hamming distance to the class of all affine Boolean functions in n variables. Note that degree of a bent function is not more than n/2. One of the most important problem in bent functions is to find the number of them. In [1] we introduced a new approach to this problem and formulated the following hypothesis: any Boolean function in n variables of degree not more than n/2 can be represented as the sum of two bent functions in n variables (n is even, n > 2). In general, it is interesting to obtain decompositions in constant number of bent functions. In this paper we study bent decompositions for Boolean functions in 8 variables. Recall that Boolean functions f and g in n variables are affine equivalent, if there exist nonsingular binary n × n matrix A, vectors u, v of length n and constant λ ∈ Z2, such that g(x) = = f(Ax+u)+〈v, x〉+λ, where 〈v, x〉 = x1v1+. . .+xnvn is the inner product. We study bent decompositions only for affine nonequivalent Boolean functions due to the following facts: • A Boolean function affine equivalent to a bent function is bent too. • Let a Boolean function f in n variables be represented as the sum of k bent functions. Then every Boolean function affine equivalent to f also can be represented as the sum of k bent functions. In [2] it is proven that every quadratic Boolean function in n variables (n is even) is the sum of two bent functions in n variables. The proof of this fact was based on the known affine classification of all quadratic Boolean functions in n variables (due to the Dickson’s theorem). Thus, let us consider Boolean functions of degree 3.