On the $q$-Bentness of Boolean Functions

TL;DR

This paper proves Klapper's conjecture that no Boolean function can have $q$-transform coefficients of magnitude $2^{n/2}$, by analyzing partial difference sets, and introduces almost $q$-bent functions as a near approximation.

Contribution

The paper confirms Klapper's conjecture for all $n$ by proving the nonexistence of $q$-bent functions through group-theoretic analysis and introduces the concept of almost $q$-bent functions.

Findings

No $q$-bent functions exist for any non-constant $q$.

The proof involves the nonexistence of certain partial difference sets.

A new class of functions called almost $q$-bent functions is introduced.

Abstract

For each non-constant $q$ in the set of $n$-variable Boolean functions, the {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. Klapper conjectured that no Boolean function exists with its $q$-transform coefficients equal to $\pm 2^{n/2}$ (such function is called $q$-bent). In our early work, we only gave partial results to confirm this conjecture for small $n$. Here we prove thoroughly that the conjecture is true by investigating the nonexistence of the partial difference sets in Abelian groups with special parameters. We also introduce a new family of functions called almost $q$-bent functions, which are close to $q$-bentness.