Crosscorrelation Spectra of Dillon and Patterson-Wiedemann type Boolean Functions

In this paper we study the additive crosscorrelation spectra between two Boolean functions whose supports are union of certain cosets. These functions on even number of input variables have been introduced by Dillon and we refer to them as Dillon type functions. Our general result shows that the crosscorrelation spectra between any two Dillon type functions are at most 5-valued. As a consequence we find that the crosscorrelation spectra between two Dillon type bent functions on n-variables are at most 3-valued with maximum possible absolute value at the nonzero points being ≤ 2 n 2 . Moreover, in the same line, the autocorrelation spectra of Dillon type bent functions at different decimations is studied. Further we demonstrate that these results can be used to show the existence of a class of polynomials for which the absolute value of the Weil sum has a sharper upper bound than the Weil bound. Patterson and Wiedemann extended the idea of Dillon for functions on odd number of variables. We study the crosscorrelation spectra between two such functions and then use the results for calculating the autocorrelation spectra too.