A general secondary construction of Boolean functions including the indirect sum and its generalizations

TL;DR

This paper introduces a unified secondary construction for Boolean functions that encompasses classic and generalized methods, providing detailed Walsh transform analysis and new insights into their spectral properties.

Contribution

It presents a general secondary construction of Boolean functions that unifies existing methods and analyzes their Walsh transforms in depth.

Findings

Unifies known secondary constructions of Boolean functions.

Provides detailed Walsh transform analysis of the constructed functions.

Observes new spectral properties related to specific Boolean function combinations.

Abstract

We study a secondary construction of Boolean functions, which generalizes the direct sum and the indirect sum. We detail how these two classic secondary constructions are particular cases of this more general one, as well as two known generalizations of the indirect sum. This unifies the known secondary constructions of Boolean functions. We study very precisely the Walsh transform of the constructed functions. This leads us to an interesting observation on the Walsh transforms $W_g,W_{g'},W_{g''}$, and $W_{g\oplus g'\oplus g''}$ when $g,g',g''$ are Boolean functions such that $(g\oplus g')(g\oplus g'')$ equals the zero function.