Nega–Hadamard Transform, Bent and Negabent Functions

Abstract

Abstract. In this paper we start developing a detailed theory of negaHadamard transforms. Consequently, we derive several results on negabentness of concatenations, and partially-symmetric functions. We also obtain a characterization of bent-negabent functions in a subclass of Maiorana-McFarland set. As a by-product of our results we obtain simple proofs of several existing facts.

Bent and generalized bent Boolean functions

“…The next two results were shown in a different context in [17]. One can straightforwardly infer, by modifying those proofs that these result hold under the current notions, as well.…”

Further results on constructions of generalized bent Boolean functions

“…In recent years several researchers have proposed generalizations of Boolean functions [6][7][8][9] and studied the effect of the Walsh-Hadamard transform on these classes. In [6], Schmidt presented the connection between words in multicode code-division multiple access (MC-CDMA) systems and generalized bent functions from Z m 2 to Z 4 , and considered functions from Z n 2 to Z q from the viewpoint of cyclic codes over rings.…”

On Negabent Functions and Nega-Hadamard Transform

“…Negabent functions and bent-negabent functions have been extensively studied during the last few years [3][4][5][6][7][8][9][10][11]. Parker and Pott [3] presented several constructions and classifications on bent-negabent.…”

“…Sarkar [5] studied the symmetric negabent functions and obtained that a symmetric function is negabent if and only if it is affine. Stȃnicȃ et al [6,9] gave the detailed study of some of properties of nega-Hadamard transform and derived several results on negabentness of concatenations. They pointed out that the algebraic degree of an -variable negabent function is at most ⌈ /2⌉.…”