Entropy bounds for Glass networks

TL;DR

This paper develops a method to estimate the maximum entropy of chaotic Glass networks, which are promising for true random number generation, by using symbolic dynamics to assess their irregularity.

Contribution

It introduces a procedure to calculate upper bounds on the entropy of Glass networks with improved accuracy and convergence properties compared to previous methods.

Findings

The method provides tight upper bounds close to actual entropy after few refinements.

The approach converges to the true entropy in the limit of refinements.

Demonstrated effectiveness on an example network with numerical validation.

Abstract

We propose that chaotic Glass networks (a class of piecewise-linear Ordinary Differential Equations) are good candidates for the design of true random number generators. A Glass network design has the advantage of involving only standard Boolean logic gates. Furthermore, an already chaotic (deterministic) system combined with random ``jitter'' due to thermal noise can be used to generate random bit sequences in a more robust way than noisy limit-cycle oscillators. Since the goal is to generate bit sequences with as large a positive entropy as possible, it is desirable to have a theoretical method to assess the irregularity of a large class of networks. We develop a procedure here to calculate good upper bounds on the entropy of a Glass network, by means of symbolic representations of the continuous dynamics. Our method improves on a result by Farcot (2006), and allows in principle for an arbitrary level of precision by refinements of the estimate, and we show that in the limiting case, these estimates converge to the true entropy of the symbolic system corresponding to the continuous dynamics. As a check on the method, we demonstrate for an example network that our upper bound after only a few refinement steps is very close to the entropy estimated from a long numerical simulation.