On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators

Abstract

<p style='text-indent:20px;'>Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period <inline-formula><tex-math id="M2">\begin{document}$ 2^n $\end{document}</tex-math></inline-formula> and a filtering function in <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called <i>invariant under the WG transform</i>, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple uniformity in output sequences, up to <inline-formula><tex-math id="M5">\begin{document}$ t = (n-m+1) $\end{document}</tex-math></inline-formula>. The conjecture of Mandal <i>et al.</i> in [<xref ref-type="bibr" rid="b29">29</xref>] about the ideal tuple distribution on the WG transformation is proved. It is also shown that the Gold functions and quadratic functions can guarantee <inline-formula><tex-math id="M6">\begin{document}$ (n-m+1) $\end{document}</tex-math></inline-formula>-tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.</p>