An upper bound on the number of iterations for transforming a Boolean function of degree greater or equal than 4 to a function of degree 3

The difficulty involved in characterizing the weight distribution of all Bo olean functions of degree greater than or equal to 3 is well-known [2, p. 4 46]. In [1] the author introduces a transformation on Boolean functions whi ch changes their weights in a way that is easy to follow, and which, when i terated, reduces the degree of the function to 2 or 3. He concludes that it is just as difficult to characterize the weight of any function of degree 3 as it is for any other degree. The application of this transformation on a Boolean function defined on V-m = GF(2)(m), m is an element of N, increas es the number of its variables by two. On the other hand, in order to reduc e the degree of a function to 2 or 3 it is necessary to apply the tranforma tion a number of times that grows exponentially with respect to m. In this paper, a factorization method on Boolean functions that allows the establis hment of an upper bound for the number of applications of the transformatio n is presented. It shows that, in general, it is possible to significantly decrease the number of iterations in this process of degree reduction.