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Chaos in a ring circuit

Farcot, Etienne; Best, Scott; Edwards, Roderick; Belgacem, Ismail; Xu, Xiaolin; Gill, Patrick

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Scott Best

Roderick Edwards

Ismail Belgacem

Xiaolin Xu

Patrick Gill


A ring-shaped logic circuit is proposed here as a robust design for a True Random Number Generator (TRNG). Most existing TRNGs rely on physical noise as a source of randomness, where the underlying idealized deterministic system is simply oscillatory. The design proposed here is based on chaotic dynamics and therefore intrinsically displays random behavior, even in the ideal noise-free situation. The paper presents several mathematical models for the circuit having different levels of detail. They take the form of differential equations using steep sigmoid terms for the transfer functions of logic gates. A large part of the analysis is concerned with the hard step-function limit, leading to a model known in mathematical biology as a Glass network. In this framework, an underlying discrete structure (a state space diagram) is used to describe the likely structure of the global attractor for this system. The latter takes the form of intertwined periodic paths, along which trajectories alternate unpredictably. It is also invariant under the action of the cyclic group. A combination of analytical results and numerical investigations confirms the occurrence of symmetric chaos in this system, which when implemented in (noisy) hardware, should therefore serve as a robust TRNG.


Farcot, E., Best, S., Edwards, R., Belgacem, I., Xu, X., & Gill, P. (2019). Chaos in a ring circuit. Chaos, 29(4), Article 043103.

Journal Article Type Article
Acceptance Date Mar 5, 2019
Online Publication Date Apr 5, 2019
Publication Date Apr 5, 2019
Deposit Date Apr 10, 2019
Publicly Available Date Apr 6, 2020
Journal Chaos: An Interdisciplinary Journal of Nonlinear Science
Print ISSN 1054-1500
Electronic ISSN 1089-7682
Publisher American Institute of Physics
Peer Reviewed Peer Reviewed
Volume 29
Issue 4
Article Number 043103
Keywords Mathematical Physics; General Physics and Astronomy; Applied Mathematics; Statistical and Nonlinear Physics
Public URL
Publisher URL
Additional Information This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article may be found at


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