Haar spectra of binary bent functions

Abstract

The paper presents a discussion of Haar spectra for binary bent functions. The presentation is restricted to bent functions of n = 4 variables, since in this case the total of 896 bent functions allows an exhaustive search and discussion of their features. Possibilities for a straightforward extension to functions of a larger number of variables are illustrated by examples for n = 6. The definition and multiresolution feature of Haar coefficients and their organization into packets provides a deeper insight into bent functions, and in particular the relationships between values in their function vectors. In the spectral domain, these relationships are expressed as the appearance of Walsh spectra for smaller number of variables in the Haar spectra of bent functions for a given n. Reconstruction of function values from Haar coefficients shows that for bent functions, function values appear in pairs of values in certain orders. Destroying these pairs results in non-bent functions.