Properties and constructions of coincident functions
This work addresses a theoretical problem in Boolean function analysis, likely for researchers in cryptography or coding theory, but appears incremental as it builds on a recently introduced class.
The paper tackled the problem of understanding coincident functions, a class of Boolean functions invariant under the Möbius transform, by proposing an innovative method to handle the transform, enabling composition and decompositions, and experimentally showed that for many features, coincident functions resemble any Boolean functions.
Extensive studies of Boolean functions are carried in many fields. The Mobius transform is often involved for these studies. In particular, it plays a central role in coincident functions, the class of Boolean functions invariant by this transformation. This class -- which has been recently introduced -- has interesting properties, in particular if we want to control both the Hamming weight and the degree. We propose an innovative way to handle the Mobius transform which allows the composition between several Boolean functions and the use of Shannon or Reed-Muller decompositions. Thus we benefit from a better knowledge of coin-cident functions and introduce new properties. We show experimentally that for many features, coincident functions look like any Boolean functions.