Negabent Functions in the Maiorana–McFarland Class

Abstract

Abstract. Boolean functions which are simultaneously bent and negabent are studied. Transformations that leave the bent-negabent property invariant are presented. A construction for infinitely many bentnegabent Boolean functions in 2mn variables (m > 1) and of algebraic degree at most n is described, this being a subclass of the MaioranaMcFarland class of bent functions. Finally it is shown that a bentnegabent function in 2n variables from the Maiorona-McFarland class has algebraic degree at most n − 1.

Nega–Hadamard Transform, Bent and Negabent Functions

“…Our study results simple proof of the main result in the paper [17] that all the symmetric negabent functions must be affine. -A characterization of some bent-negabent functions in Maiorana-McFarland class is obtained in Section 5, thus complementing some results of [19].…”

“…Claim (a) follows from Lemma 1 of [19], since (u) . We now show the first identity of (b) (the second is absolutely similar).…”

“…Conversely, it is known that all affine functions are negabent [19]. Therefore, symmetric functions on even number of variables, if affine, are negabent.…”

“…Towards a nega-periodic analogue of the bent criteria, one can use nega-Hadamard transform and investigate Boolean functions with nega flat spectrum. This motivated several works in the area of Boolean functions [11,13,14,19] in the last few years. In this paper we concentrate on the nega-Hadamard transform in more details.…”

The selfnegadual properties of generalised quadratic Boolean functions

“…their phase representations are eigenvectors of the Walsh-Hadamard transform -such functions are therefore bent. Another direction is to classify and construct Boolean functions that are negabent or bent-negabent (both bent and negabent) [15,16]. We show that a function in ZRM(2, n) can never be both selfdual and negabent.…”