On the computation of the M"obius transform

The M"obius 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"obius 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"obius computation for sub-families of Boolean function. Furthermore we compute directly the M"obius 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.