Some Spectral Invariant Operations for Multiple-Valued Functions with Homogeneous Disjoint Products in the Polynomial Form

Abstract

It has long been known that some transformationsof the Boolean functions affect only the permutation of somesubset of coefficients in the Walsh-Hadamard spectrum or justchange the sign of some coefficients. These operations are knownas spectral invariant operations. It exists a generalization of theseinvariant operations for multi-valued functions and Vilenkin-Chrestenson spectrum. Here some new spectral invariant operationswill be defined for functions with p = 3 and with n ≥ 5 variables, which have disjoint products of two variables in theirpolynomial forms. As a result of these new operations only thevalues of some subsets of spectral coefficients will by permuted, like in the case of invariant operations which are known untilnow. This property of spectral invariant operations has important consequences on multiple-valued bent functions. Any functionobtained by the application of one or more spectral invariantoperations to a bent function will be also a bent function. It willbe shown that the defined new invariant operations are usefulfor characterization of multi-valued bent functions.