Generic Constructions of Five-Valued Spectra Boolean Functions

Explicit infinite families of bent functions outside the completed Maiorana–McFarland class

“…The above mentioned fact regarding a bent function and its dual (with respect to the inclusion in MM # ) is then useful when the so-called 4-decomposition of bent functions (say on F n 2 ) is considered, which regards the decomposition into the cosets of an (n − 2)-dimensional subspace V of F n 2 . It was originally investigated by Canteaut and Charpin [3] in terms of the second-order derivatives of the dual function, whereas the similar properties were recently stated using duals of the cosets of V [14]. The main conclusion in [3] is that there are exactly three possible cases of this 4-decomposition of a bent function, namely, all four restrictions being bent, semi-bent, or 5valued spectra functions.…”

“…We first recall certain notations, introduced in [14] and also used in [17], useful in handling a 5-valued spectra Boolean function which has two different non-zero absolute values.…”

“…Let the WHT spectrum of a function f : F n 2 → F 2 contain the values 0, ±c 1 , ±c 2 (c 1 = c 2 ), where c 1 , c 2 ∈ N. Some of the results in [14] are stated in a more general context, but since the 4-decomposition of bent functions is our main objective we only consider the cases c 1 = 2 n/2 and c 2 = 2 (n+2)/2 above. For i = 1, 2, by S [i] f ⊂ F n 2 we denote the set…”

“…, x 2 λ i −1 } is ordered lexicographically. A more specific method for designing 5-valued spectra functions on F n 2 (thus W f (u) ∈ {0, ±2 n/2 , ±2 n+2 2 }), originally considered in [14], will be used in Sect. 3.4 for specifying suitable quadruples of such functions whose concatenation will give bent functions outside MM # .…”

“…The main conclusion in [3] is that there are exactly three possible cases of this 4-decomposition of a bent function, namely, all four restrictions being bent, semi-bent, or 5valued spectra functions. For each of the cases, using the necessary and sufficient conditions in [14] (see Theorem 2.2), we provide generic methods (at least one) for designing bent functions provably outside MM # . For instance, in the elementary case of defining a bent function h(x, y 1 , y 2 ) = f (x)⊕ y 1 y 2 on F n+2 2 using any bent function f on F n 2 (corresponding to a bent 4-decomposition since h = f || f || f ||(1 ⊕ f )), we show that h is outside MM # if and only if f is outside MM # .…”