Equivalence Classes of Boolean Functions for First-Order Correlation

This paper presents a complete characterization of the first order correlation immune Boolean functions that includes the functions that are 1-resilient. The approach consists in defining an equivalence relation on the full set of Boolean functions with a fixed number of variables. An equivalence class in this relation, called a first-order correlation class, provides a measure of the distance between the Boolean functions it contains and the correlation-immune Boolean functions. The key idea consists on manipulating only the equivalence classes instead of the set of Boolean functions. To achieve this goal, a class operator is introduced to construct a class with n variables from two classes of n - 1 variables. In particular, the class of 1-resilient functions on n variables is considered. An original and efficient method to enumerate all the Boolean functions in this class is proposed by performing a recursive decomposition of classes with less variables. A bottom up algorithm provides the exact number of 1-resilient Boolean functions with seven variables which is 23478015754788854439497622689296. A tight estimation of the number of 1-resilient functions with eight variables is obtained by performing a partial enumeration. It is conjectured that the exact complete enumeration for general n is intractable.