Several pseudorandom number generators based on the inversive congruential method have been designed as appealing alternatives to those based on the classical linear congruential method. This paper unveils the first functional graph structure of an inversive pseudorandom number generator over ring \({\mathbb {Z}}_{p^{e}}\) , resolving two foundational gaps: By transforming the generator into a second order linear congruential recurrence relation, we derives a complete and explicit expression of the least period of sequences generated from all initial states in the domain; The graph structure is rigorously analyzed and shown to consist of two distinct components, namely, multiple cycles of varying lengths and a single unilateral connected digraph whose structure is consistent relative to parameter e. Moreover, the graph structure analysis method offers a fresh perspective for analyzing the randomness and periodicity of random number generators. The adopted analysis methodology can be extended to study the graph structure and dynamics of other nonlinear maps.
