Boolean function

Abstract: Boolean plateaued functions and vectorial functions with plateaued components play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Our knowledge on them is not at a level corresponding to their importance. We introduce new characterizations of plateaued Boolean functions. We give the characterizations of vectorial functions whose components are all plateaued (with possibly different amplitudes), that we simply call plateaued, by means of the value distributions of their derivatives (we characterize similarly those functions whose components are partially bent) and autocorrelation functions, and of the power moments of their Walsh transform. This allows us to derive several characterizations of almost perfect nonlinear (APN) functions in this framework. We prove that all the main results known for quadratic APN functions extend to plateaued functions, allowing the study of their APN-ness to be simplified. We show that if, additionally, the component functions are all unbalanced, this study is still simpler: the APN-ness of such functions depends only on their value distribution. This allows proving, for instance, that any plateaued $(n,n)$ -function, $n$ even, having similar value distribution as the APN power functions, is APN, and has the same extended Walsh spectrum as the APN Gold functions. As by-products, we obtain a few other new results. For instance, any plateaued function in even dimension, which is Carlet-Charpin-Zinoviev (CCZ)-equivalent to a Gold or Kasami APN function, is necessarily extended affine (EA)-equivalent to it.

Abstract: A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC-cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from nondisjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. The programs for both algorithms use advantages of C language to speed up the execution. The comparison of different versions of the algorithm has been carried out. The algorithm and its implementation provide the fastest and most comprehensive program (having many options) known to the authors for the calculation of the Rademacher-Walsh transform. It successfully overcomes all drawbacks in the calculation of the transform from the design automation system based on spectral method-the SPECSYS system from Drexel University, which uses fast Walsh transform. >

Abstract: An algorithm for Boolean matching based on binary decision diagrams using a level-first search strategy is presented. This algorithm is generally not restricted to circuits with just a few inputs and can be used for both technology mapping and logic verification. Unlike depth-first and breadth-first strategies, a level-first strategy permits significant pruning of the search space. A set of filters that further improve the performance of the matching algorithm is described. A method of analyzing the effectiveness of a filter is presented, and the various filters are ranked on the basis of their effect/cost ratio. Experimental results on a number of benchmark circuits are presented, comparing the basic matching algorithm with and without the use of various filters. It is shown how the matching algorithm and the filters can be extended to Boolean functions with don't cares. >

Abstract: We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.