Balanced Boolean Functions with Nonlinearity > 2^{n-1} - 2^{(n-1)/2}

Recently, balanced 15-variable Boolean functions with nonlinearity 16266 were obtained by suitably modifying unbalanced Patterson-Wiedemann (PW) functions, which possess nonlinearity 2^{n-1}-2^{(n-1)/2}+20 = 16276. In this short paper, we present an idempotent interpreted as rotation symmetric Boolean function) with nonlinearity 16268 having 15 many zeroes in the Walsh spectrum, within the neighborhood of PW functions. Clearly this function can be transformed to balanced functions keeping the nonlinearity and autocorrelation distribution unchanged. The nonlinearity value of 16268 is currently the best known for balanced 15-variable Boolean functions. Furthermore, we have attained several balanced 13-variable Boolean functions with nonlinearity 4036, which improves the recent result of 4034

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 nd 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 . Moreover, in the same line, the autocorrelation spectra of Dillon type bent functions at dierent 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

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 +1. 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