Graph Structure of an Inversive Pseudorandom Number Generator over Ring $\mathbb{Z}_{p^{e}}$

Generating random and pseudorandom numbers with a deterministic system is a long-standing challenge in theoretical research and engineering applications. Several pseudorandom number generators based on the inversive congruential method have been designed as attractive alternatives to those based on the classical linear congruential method. This paper discloses the least period of sequences generated by iterating an inversive pseudorandom number generator over the ring $\mathbb{Z}_e$ by transforming it into a two-order linear congruential recurrence relation. Depending on whether the sequence is periodic or ultimately periodic, all states in the domain can be attributed to two types of objects: some cycles of different lengths and one unilateral connected digraph whose structure remains unchanged concerning parameter $e$. The graph structure of the generator over the ring $\mathbb{Z}_e$ is precisely disclosed with rigorous theoretical analysis and verified experimentally. The adopted analysis methodology can be extended to study the graph structure of other nonlinear maps.