On the computation of the M{\"o}bius transform
Abstract
The M{\"o}bius transform is a crucial transformation into the Boolean world; it allows to change the Boolean representation between the True Table and Algebraic Normal Form. In this work, we introduce a new algebraic point of view of this transformation based on the polynomial form of Boolean functions. It appears that we can perform a new notion: the M{\"o}bius computation variable by variable and new computation properties. As a consequence, we propose new algorithms which can produce a huge speed up of the M{\"o}bius computation for sub-families of Boolean function. Furthermore we compute directly the M{\"o}bius transformation of some particular Boolean functions. Finally, we show that for some of them the Hamming weight is directly related to the algebraic degree of specific factors.
Keywords
Cite
@article{arxiv.2004.11146,
title = {On the computation of the M{\"o}bius transform},
author = {Morgan Barbier and Hayat Cheballah and Jean-Marie Le Bars},
journal= {arXiv preprint arXiv:2004.11146},
year = {2020}
}