Entropy bounds for Glass networks

TL;DR

The paper develops a rigorous method to bound the true entropy of chaotic Glass networks—models built from piecewise-linear ODEs—by enhancing Farcot's transition-graph approach with trapping regions and returning-cone analysis. It constructs progressively refined symbolic representations, TG_r and TG_r(k), that separate realizable dynamics from forbidden ones, yielding upper bounds that converge to the actual entropy $h_{\phi(\mathscr{D})}$ as refinement increases. Through an explicit 4-variable example, the authors demonstrate dramatic improvements over the original TG-based bound, moving from $0.873$ down to about $0.081$ with modest computational effort. The approach provides a practical, principled path to quantify entropy in Glass networks and supports the design of true random number generators based on deterministic chaotic circuits with thermal jitter.

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.

Figures (10)

• Figure 1: TG for example Glass network in Equation \ref{['eq:example']}.
• Figure 1: Blocks of length 120 vs number of transitions up to $10^9$
• Figure 2: Projections of a phase portrait for example Glass network in Equation \ref{['eq:example']}.
• Figure 2: Logarithm (base 2) of number of blocks of length $n$ for $2\le n\le 126$ using a simulation of $10^9$ transitions (dots) at each $n$. The solid line is the least squares best fit, which has slope $\approx 0.0670258$.
• Figure 3: $\mathop{\mathrm{TG}}\nolimits$ for example Glass network in Equation \ref{['eq:example']} with cycle $A$ outlined in Red (a) and cycle $B$ outlined in Blue (b).

Theorems & Definitions (23)

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