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<p>Functional graph analysis offers a powerful framework for characterizing the dynamics and structural properties of nonlinear dynamical systems, particularly pseudorandom number generators. As one of the most widely used pseudorandom number generators, the inversive pseudorandom number generator (<i>IPRNG</i>) was introduced four decades ago and remains poorly understood in terms of its internal behavior. This paper presents a structure-oriented approach to dynamically enhancing the generator over the ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathbb {Z}}_{2^e}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <msup> <mn>2</mn> <mi>e</mi> </msup> </msub> </math></EquationSource> </InlineEquation> by analyzing its functional graph, treating the iterative process as state transitions in a discrete dynamical system, and examining the distributions of fixed points and periodic orbits. By leveraging the underlying graph topology, we identify a coupling mechanism between the <i>IPRNG</i> and the Logistic map, thereby enabling the construction of sequences attaining the maximum period. Consequently, the coupled system passes the NIST statistical test suite. This work not only elucidates the structural dynamics of the <i>IPRNG</i> but also introduces a general framework for enhancing pseudorandom number generators via directed graph analysis.</p>

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Graph structure of an inversive pseudorandom number generator over ring \({\mathbb {Z}}_{2^e}\)

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Yuling Dai,
  • Xiaoxiong Lu,
  • Chengqing Li

摘要

Functional graph analysis offers a powerful framework for characterizing the dynamics and structural properties of nonlinear dynamical systems, particularly pseudorandom number generators. As one of the most widely used pseudorandom number generators, the inversive pseudorandom number generator (IPRNG) was introduced four decades ago and remains poorly understood in terms of its internal behavior. This paper presents a structure-oriented approach to dynamically enhancing the generator over the ring \({\mathbb {Z}}_{2^e}\) Z 2 e by analyzing its functional graph, treating the iterative process as state transitions in a discrete dynamical system, and examining the distributions of fixed points and periodic orbits. By leveraging the underlying graph topology, we identify a coupling mechanism between the IPRNG and the Logistic map, thereby enabling the construction of sequences attaining the maximum period. Consequently, the coupled system passes the NIST statistical test suite. This work not only elucidates the structural dynamics of the IPRNG but also introduces a general framework for enhancing pseudorandom number generators via directed graph analysis.