Matrix representations of vectorial Boolean functions and eigenanalysis

Abstract

This paper aims at giving a unified overview on the various representations of vectorial Boolean functions, namely the Walsh matrix, the correlation matrix and the adjacency matrix. A new representation called polynomial matrix is introduced. It is shown that those different representations are similar. For a vectorial Boolean function with the same number of inputs and outputs, an eigenanalysis of those representations is performed. It is shown how eigenvalues and eigenvectors are related to the structure of the graph associated to this function.