Characterization of Negabent Functions and Construction of Bent-Negabent Functions With Maximum Algebraic Degree

Abstract

We present necessary and sufficient conditions for a Boolean function to be a negabent function for both an even and an odd number of variables, which demonstrates the relationship between negabent functions and bent functions. By using these necessary and sufficient conditions for Boolean functions to be negabent, we obtain that the nega spectrum of a negabent function has at most four values. We determine the nega spectrum distribution of negabent functions. Further, we provide a method to construct bent-negabent functions in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> variables ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> even) of algebraic degree ranging from 2 to [( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> )/2], which implies that the maximum algebraic degree of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -variable bent-negabent function is equal to [( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> )/2]. Thus, we answer two open problems proposed by Parker and Pott and by Stănică et al.