Period Distribution of Inversive Pseudorandom Number Generators Over Galois Rings

In 2009, Sol\'{e} and Zinoviev (\emph{Eur. J. Combin.}, vol. 30, no. 2, pp. 458-467, 2009) proposed an open problem of arithmetic interest to study the period of the inversive pseudorandom number generators (IPRNGs) and to give conditions bearing on $a, b$ to achieve maximal period, we focus on resolving this open problem. In this paper, the period distribution of the IPRNGs over the Galois ring $({\rm Z}_{p^{e}},+,\times)$ is considered, where $p>3$ is a prime and $e\geq 2$ is an integer. The IPRNGs are transformed to 2-dimensional linear feedback shift registers (LFSRs) so that the analysis of the period distribution of the IPRNGs is transformed to the analysis of the period distribution of the LFSRs. Then, by employing some analytical approaches, the full information on the period distribution of the IPRNGs is obtained, which is to make exact statistics about the period of the IPRNGs then count the number of IPRNGs of a specific period when $a$, $b$ and $x_{0}$ traverse all elements in ${\rm Z}_{p^{e}}$. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods.