A conjecture about Gauss sums and bentness of binomial Boolean functions
This work addresses a theoretical problem in cryptography and coding theory for researchers studying Boolean functions, but it appears incremental as it builds on and supports a previous conjecture.
The paper tackles the problem of evaluating the Walsh transform of binomial Boolean functions by reducing it to Gauss sums, and conjectures an explicit formula involving Kloosterman sums for extensions of degree four times an odd number, supported by experimental data and shown to be equivalent to a prior bentness characterization.
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.