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

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