Analysis of Autocorrelation Function of Boolean Functions in Haar Domain

Abstract

Design of strong symmetric cipher systems requires that the underlying cryptographic Boolean function meet specific security requirements. Some of the required security criteria can be measured with the help of the Autocorrelation function as a tool, while other criteria can be measured using the Walsh transform as a tool. The connection between the Walsh transform and the Autocorrelation function is given by the well known Wiener-Khintchine theorem. In this paper, we present an analysis of the Autocorrelation function from the Haar spectral domain. We start by presenting a brief review on Boolean functions and the Autocorrelation function. Then we exploit the analogy between the Haar and Walsh in deriving the Haar general representation of the Autocorrelation function. The derivations are carried out in two ways namely, in terms of individual spectral coefficients, and based on zones within the spectrum. The main contribution of the paper is the establishment of the link between the Haar transform and the Wiener-Khintchine theorem. This is done by deducing the connection between the Haar transform, the Autocorrelation, and the Walsh power spectrum for an arbitrary Boolean function. In the process we show that, the same characteristics of the Wiener-Khintchine theorem holds locally within the Haar spectral zones, instead of globally as with the Walsh domain. The Haar general representations of Autocorrelation function are given for arbitrary Boolean functions in general and Bent Boolean functions in particular. Finally, we present a conclusion of the work with a summary of findings and future work.