On Walsh Spectrum of Cryptographic Boolean Function

Abstract

<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>