A conjecture about Gauss sums and bentness of binomial Boolean functions

TL;DR

This paper explores the relationship between Gauss sums and the bentness of binomial Boolean functions, proposing a conjecture and providing partial proofs supported by experimental data.

Contribution

It introduces a conjecture linking Gauss sums to bentness and offers an explicit formula involving Kloosterman sums for certain extension degrees, supported by experimental evidence.

Findings

Conjectured explicit formula for Walsh transform involving Kloosterman sums.

Supported the conjecture with extensive experimental data.

Established equivalence between the formula's validity and a known bentness characterization.

Abstract

In this note, the polar decomposition of binary fields of even extension degree is used to reduce the evaluation of the Walsh transform of binomial Boolean functions to that of Gauss sums. In the case of extensions of degree four times an odd number, an explicit formula involving a Kloosterman sum is conjectured, proved with further restrictions, and supported by extensive experimental data in the general case. In particular, the validity of this formula is shown to be equivalent to a simple and efficient characterization for bentness previously conjectured by Mesnager.