Chaos in a ring circuit

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.