Recent Results on Balanced Symmetric Boolean Functions

Abstract

This paper focuses on the balancedness of symmetric Boolean functions. We prove a conjecture presented by Canteaut and Videau, which states that the balanced symmetric Boolean functions of fixed algebraic degree are trivially balanced when the number of variables is large enough. Denoted by σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,d</sub> , the n-variable elementary symmetric Boolean function of degree d. As an application of this result to elementary symmetric Boolean functions, we show that all the trivially balanced elementary symmetric Boolean functions are of the form σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2t+1l-1,2t</sub> , where t and l are any positive integers. It implies that Cusick et al.'s conjecture, which claims that σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2t+1l-1,2t</sub> is the only nonlinear balanced elementary symmetric Boolean functions, is equivalent to the conjecture that all the balanced elementary symmetric Boolean functions are trivially balanced.