On Walsh Spectrum of Cryptographic Boolean Function

<p>Walsh transformation of a Boolean function ascertains a number of cryptographic properties of the Boolean function viz, non-linearity, bentness, regularity, correlation immunity and many more. The functions, for which the numerical value of Walsh spectrum is fixed, constitute a class of Boolean functions known as bent functions. Bent functions possess maximum possible non-linearity and therefore have a significant role in design of cryptographic systems. A number of generalisations of bent function in different domains have been proposed in the literature. General expression for Walsh transformation of generalised bent function (GBF) is derived. Using this condition, a set of Diophantine equations whose solvability is a necessary condition for the existence of GBF is also derived. Examples to demonstrate how these equations can be utilised to establish non-existence and regularity of GBFs is presented.</p>

A Set of Split-Matrix Ordered Walsh Functions

A new set of Walsh functions is defined in terms of the split-matrix ordering. The intrinsic properties of the functions are analyzed. The relationship and the associated conversion rules between the newly defined functions and other typical Walsh functions are discussed. It is shown that the proposed Walsh functions provide a straighforward and efficient algorithm for fast Discrete Walsh Transformation.