Correlation Immunity and Resiliency of Boolean Functions from Haar Domain Perspective

The strength of any conventional cipher system relies on the underlying cryptographic Boolean functions employed in the system. The design of such systems requires that the employed Boolean functions meet specific security criteria. Two of such criteria are the correlation immunity and resiliency of a given Boolean function. To determine whether such criteria are met, a designer needs the help of spectral transform tool and in this case the Walsh spectral transform. Most of the cryptographic criteria have been generalized in terms of the Walsh transform. In this paper, we present an alternative view of such criteria from the Haar spectral transform point of view. The Haar along with the Walsh are the two methods considered suitable for representing Boolean functions. The paper exploits the analogy between the two transforms to derive the Haar general representation of the correlation-immunity and the resiliency security criteria. The paper presents the Haar-based conditions on which a given Boolean function should meet to be considered correlationimmune of order k (CI(k)) or resilient of order k (R(k)). In addition, the paper presents a Haar-based algorithm for testing correlation-immunity of an arbitrary Boolean function including experimental results related to the algorithm. The results in this presentation are based on a simulation study of the Haar-based algorithm in comparison to its Walsh-counterpart. The results portray the computational advantage of the Haar method over the Walsh approach for the correlation-immunity measure. The paper includes as well, a discussion on the worst-case scenario with advantages and flexibility of the Haar method in conjunction with the lower order Walsh transform. A summary of the work is then presented in the conclusion of the paper.