Computing Walsh coefficients from the algebraic normal form of a Boolean function

Abstract

We study the relationship between the Walsh Transform of a Boolean function and its Algebraic Normal Form(ANF), and present algorithms that compute the Walsh coefficients at a small set of points in terms of certain parameters derived from the ANF of a Boolean function. In the first part of this paper, based on the previous result by Gupta and Sarkar, we investigate the formula in Gupta-Sarkar's algorithm in a novel iterative method and obtain a recurrence relation for the Walsh Transform of a Boolean function. The second part is devoted to applying this formula to algorithms to evaluate it. Experimental result shows that for the specified classes of Boolean functions, our algorithms can perform better than Gupta-Sarkar's algorithm. For example, the proposed algorithm "ComputeWalsh" is able to compute the Walsh coefficients of the functions for which the complexity of Gupta-Sarkar's algorithm is impractical. Besides, for functions acting on high number of variables (m>30) and having low number of monomials, the proposed algorithms are advantageous over the Fast Walsh Transform which is a standard method of computing the Walsh Transform with a complexity of O(m2 m ) operations.