In this paper, we consider the spectra of Boolean functions with respect to the nega-Hadamard transform. Based on the properties of the nega-Hadamard transform and the solutions of the Diophantine ...equations, we investigate all possibilities of the nega-Hadamard transform of Boolean functions with exactly two distinct nega-Hadamard coefficients.
Aaronson defined Forrelation (2010) as a measure of correlation between a Boolean function
f
and the Walsh–Hadamard transform of another function
g
. In a recent work, we have studied different ...cryptographically important spectra of Boolean functions through the lens of Forrelation. In this paper, we explore a similar kind of correlation in terms of nega-Hadamard transform. We call it nega-Forrelation and obtain a more efficient sampling strategy for nega-Hadamard transform compared to the existing results. Moreover, we present an efficient sampling strategy for nega-crosscorrelation (and consequently nega-autocorrelation) spectra too, by tweaking the nega-Forrelation technique. Finally, we connect the hidden shift finding algorithm for bent functions (Rötteler, 2010) with the Forrelation algorithm and extend it for the negabent functions.
In design of secure cryptosystems and CDMA communications, the negabent functions play a significant role. The generalized Boolean functions have been extensively studied by Schmidt and established ...several important results in this setup. In this paper, several characteristics of the generalized nega-Hadamard transform (GNT) of generalized Boolean functions like inverse of GNT, generalized nega-cross correlation, generalized nega-Parseval’s identity, relationship between GNT and generalized nega-cross correlation have analyzed. We studied the GNT for the derivative of this setup of functions and established the connection of generalized Walsh-Hadamard transform and GNT of derivatives of these functions. Also, the GNT of composition of vectorial Boolean function and generalized Boolean function is presented. Further, the generalized nega-convolution theorem for generalized Boolean function is obtained.
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