On the pseudorandom properties of $ k $-ary Sidel'nikov sequences

Abstract

<p style='text-indent:20px;'>In 2002 Mauduit and Sárközy started to study finite sequences of <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $\end{document}</tex-math></inline-formula> is a finite set of <inline-formula><tex-math id="M4">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols. Later many pseudorandom sequences of <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences with length <inline-formula><tex-math id="M7">\begin{document}$ q-1 $\end{document}</tex-math></inline-formula> by using the estimates for certain character sums with exponential function, where <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size <inline-formula><tex-math id="M9">\begin{document}$ \phi(q-1) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M10">\begin{document}$ k $\end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that <inline-formula><tex-math id="M11">\begin{document}$ k = o(q^{\frac{1}{4}}) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> denotes Euler's totient function.</p>