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

TL;DR

This paper characterizes negabent functions, explores their spectral properties, and introduces a construction method for bent-negabent functions with maximum algebraic degree, solving two open problems in the field.

Contribution

It provides necessary and sufficient conditions for negabent functions, analyzes their spectra, and constructs bent-negabent functions with maximum algebraic degree, advancing understanding of their structure.

Findings

Negabent functions have at most 4 nega spectrum values.

The nega spectrum distribution of negabent functions is determined.

Maximum algebraic degree of bent-negabent functions is n/2 for even n.

Abstract

We present necessary and sufficient conditions for a Boolean function to be a negabent function for both even and odd number of variables, which demonstrate 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 4 values. We determine the nega spectrum distribution of negabent functions. Further, we provide a method to construct bent-negabent functions in $n$ variables ($n$ even) of algebraic degree ranging from 2 to $\frac{n}{2}$, which implies that the maximum algebraic degree of an $n$-variable bent-negabent function is equal to $\frac{n}{2}$. Thus, we answer two open problems proposed by Parker and Pott and by St\v{a}nic\v{a} \textit{et al.} respectively.