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.

翻译：通过确定性系统生成随机数和伪随机数是理论研究和工程应用中长期存在的挑战。基于逆同余法设计的多种伪随机数生成器，已成为经典线性同余法生成器的有吸引力的替代方案。本文通过将环 $\mathbb{Z}_e$ 上的逆型伪随机数生成器迭代生成的序列转化为二阶线性同余递推关系，揭示了该序列的最小周期。根据序列是周期性的还是最终周期性的，定义域中的所有状态可归为两类对象：若干不同长度的循环以及一个单向连通有向图，该有向图的结构相对于参数 $e$ 保持不变。通过严格的理论分析精确揭示了环 $\mathbb{Z}_e$ 上生成器的图结构，并通过实验进行了验证。所采用的分析方法可推广至研究其他非线性映射的图结构。