Paper 2019/430

Composition of Boolean Functions: An Application to the Secondary Constructions of Bent Functions

Guangpu Gao, Dongdai Lin, Wenfen Liu, and Yongjuan Wang

Abstract

Bent functions are optimal combinatorial objects and have been attracted their research for four decades. Secondary constructions play a central role in constructing bent functions since a complete classification of this class of functions is elusive. This paper is devoted to establish a relationship between the secondary constructions and the composition of Boolean functions. We firstly prove that some well-known secondary constructions of bent functions, can be described by the composition of a plateaued Boolean function and some bent functions. Then their dual functions can be calculated by the Lagrange interpolation formula. By following this observation, two secondary constructions of bent functions are presented. We show that they are inequivalent to the known ones, and may generate bent functions outside the primary classes $\mathcal{M}$ and $% \mathcal{PS}$. These results show that the method we present in this paper is genetic and unified and therefore can be applied to the constructions of Boolean functions with other cryptographical criteria.