The Haar Analysis of Nonlinearity Cryptographic Security Criterion

One of the main security criteria that mostly influence the strength of a conventional cipher system is the Nonlinearity criterion. A classical cipher system is considered strong if it is characterized by high nonlinearity. That is, the underlying Boolean function employed within the system is required to be far from being linear. The measure of a Boolean function's nonlinearity can be done with the help of the Walsh transform as a tool. This paper presents an analysis of the nonlinearity criterion from the Haar transform domain perspective. The paper proceeds by utilizing the connection between the Haar and the Walsh transforms. The work provides two main contributions in the form of the definition of nonlinearity given the Haar spectrum of an arbitrary Boolean function and a Haar-Walsh hybrid algorithm for measuring the nonlinearity given the Haar spectrum. The derived algorithm is then simulated along with the Walsh benchmark and the results are presented in a comparison study. The paper then follows up by presenting a discussion on the computational advantages of employing the Haar-Walsh hybrid method in terms of time and space complexities. It is shown that, the Haar-Walsh hybrid method performs better when the transformed function is characterized by high number of input variables (n > 11). The paper concludes with a summary of the presented work and suggestions for future work.