Abstract
The Möbius 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 Möbius computation variable by variable and new computation properties. As a consequence, we propose new algorithms which can produce a huge speed up of the Möbius computation for sub-families of Boolean function. Furthermore we compute directly the Möbius 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.
Highlights
Numerous studies of Boolean functions have been conducted in various fields like cryptography and error correcting codes [7], Boolean circuits and Boolean Decision Diagram (BDD) [3], Boolean logic [1] or constraint satisfaction problems [11]
The major contribution of our work is to introduce a polynomial form without reference of a specific Boolean function; since the indeterminates indicate the variables which occurs in the Algebraic Normal Form (ANF) and not the number of variables
Which allow us to give a new point of view of the Mobius transform and to manipulate Boolean functions of various number of variables via different Mobius transform operators
Summary
Numerous studies of Boolean functions have been conducted in various fields like cryptography and error correcting codes [7], Boolean circuits and Boolean Decision Diagram (BDD) [3], Boolean logic [1] or constraint satisfaction problems [11]. It occurs in various other fields like Ordered Binary Decision Diagrams (OBDD) [21] or Modal Logics [23] These decompositions allow us to rewrite a Boolean function with n variables into two Boolean functions with n − 1 variables, while the expansions perform the same acts in reverse, they allows us to build a Boolean function with n variables with two Boolean functions with n − 1 variables. It is possible to go from one world to another by fixing the number of variables of Boolean functions We prove that this new approach provides better algorithms to perform Mobius transform (from ANF to truth table) when we have few monomials in the ANF and monomials of high degree.
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