Classification of Boolean Functions Where Affine Functions Are Uniformly Distributed

The present paper on classification of -variable Boolean functions highlights the process of classification in a coherent way such that each class contains a single affine Boolean function. Two unique and different methods have been devised for this classification. The first one is a recursive procedure that uses the Cartesian product of sets starting from the set of one variable Boolean functions. In the second method, the classification is done by changing some predefined bit positions with respect to the affine function belonging to that class. The bit positions which are changing also provide us information concerning the size and symmetry properties of the classes/subclasses in such a way that the members of classes/subclasses satisfy certain similar properties. 1. Introduction Classification of non-linear Boolean functions has been a long standing problem in the field of theoretical computer science. A systematic classification of Boolean functions with -variable having a representative in each class is a welcomed step in this area of study. It has been very accurately considered as vital and meaningful because of two important well-defined reasons: (a) equivalent functions in each class possess similar properties and (b) the number of representatives in each class is much less than that of Boolean functions. Earlier, when two Boolean functions of -variable differ only by permutation or complementation of their variables, they fall into equivalence classes. The formula for counting the number of such equivalence classes is given in [1]. Further, it has also been elaborated in [2] about the procedures of selection of a representative assembly, with one member from each equivalence class. In [3], the linear group and the affine Boolean function group of transformations have been defined and an algorithm has been proposed for counting the number of classes under both groups. The classification of the set of -input functions is specifically based on three criteria: the number of functions, the number of classes, and the number of NPN classes, which are first introduced in [4]. Classification of the affine equivalence classes of cosets of the first order Reed-Muller code with respect to cryptographic properties such as correlation immunity, resiliency, and propagation characteristics has been discussed in [5–8]. Heuristic design of cryptographically strong balanced Boolean function was envisaged in [9]. In [10], three variable Boolean functions in the name of 3-neighborhood cellular automata rules have been classified on the basis of hamming distance