Atomic Formulation of the Boolean Curve Fitting Problem

Abstract

Boolean curve fitting is the process of finding a Boolean function that takes given values at certain points in its Boolean domain. The problem boils down to solving a set of ‘big’ Boolean equations that may or may not be consistent. The usual formulation of the Boolean curve fitting problem is quite complicated, indeed. In this paper, we formulate the Boolean curve fitting problem using the technique of atomic decomposition of Boolean equations. This converts the problem into a set of independent switching equations. We present the solution of these switching equations and express the solution in very simple and compact forms. We also present the consistency and uniqueness conditions for this problem again in very compact forms. A few illustrative examples are given. These examples clearly pinpoint the simplicity gained by the Boolean-equation solving step within the overall Boolean-fitting procedure. The method presented here can be applied to the design of Boolean functions for cryptographic systems.