Nonlinearity Computation for Sparse Boolean Functions

An algorithm for computing the nonlinearity of a Boolean function from its algebraic normal form (ANF) is proposed. By generalizing the expression of the weight of a Boolean function in terms of its ANF coefficients, a formulation of the distances to linear functions is obtained. The special structure of these distances can be exploited to reduce the task of nonlinearity computation to solving an associated binary integer programming problem. The proposed algorithm can be used in cases where applying the Fast Walsh transform is infeasible, typically when the number of input variables exceeds 40

On the computation of the M{\"o}bius transform

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