Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals

TL;DR

This paper introduces four systematic methods for constructing bent-negabent functions with various variable counts, determines their algebraic properties, and explores their duals, including a novel construction within the rotation symmetric class.

Contribution

It provides new systematic constructions of bent-negabent functions for different variable sizes and introduces the first such functions in the generalized rotation symmetric class.

Findings

Constructed bent-negabent functions on 4k, 8k, 4k+2, and 8k+2 variables.

Determined algebraic normal forms and duals of these functions.

Presented a method to construct 2-rotation symmetric bent-negabent functions with any algebraic degree.

Abstract

Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those bent-negabent functions which have the maximum algebraic degree. At last, by modifying the truth tables of a class of quadratic 2-rotation symmetric bent-negabent functions, we present a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees. Considering that there are probably no bent-negabent functions in the rotation symmetric class, it is the first significant attempt to construct bent-negabent functions in the generalized rotation symmetric class.