On the number of factorizations of $ t $ mod $ N $ and the probability distribution of Diffie-Hellman secret keys for many users

Abstract

<p style='text-indent:20px;'>We study the number <inline-formula><tex-math id="M3">\begin{document}$ R_n(t,N) $\end{document}</tex-math></inline-formula> of tuplets <inline-formula><tex-math id="M4">\begin{document}$ (x_1,\ldots, x_n) $\end{document}</tex-math></inline-formula> of congruence classes modulo <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} x_1\cdots x_n \equiv t \pmod{N}. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>As a result, we derive a recurrence for <inline-formula><tex-math id="M6">\begin{document}$ R_n(t,N) $\end{document}</tex-math></inline-formula> and prove some multiplicative properties of <inline-formula><tex-math id="M7">\begin{document}$ R_n(t,N) $\end{document}</tex-math></inline-formula>. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.</p>