Using a reservoir computer to learn chaotic attractors with chaos and cryptography applications

What is it about?

Using the machine learning approach known as reservoir computing, it is possible to train one dynamical system to emulate another. We show that such trained reservoir computers reproduce the properties of the attractor of the chaotic system sufficiently well to exhibit chaos synchronisation. That is, the trained reservoir computer, weakly driven by the chaotic system, will synchronise with the chaotic system. Conversely, the chaotic system, weakly driven by a trained reservoir computer, will synchronise with the reservoir computer. We illustrate this behavior on the Mackey-Glass and Lorenz systems. We then show that trained reservoir computers can be used to crack chaos-based cryptography and illustrate this on a chaos cryptosystem based on the Mackey-Glass system. We conclude by discussing why reservoir computers are so good at emulating chaotic systems.

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Photo by Darius Bashar on Unsplash

Photo by Darius Bashar on Unsplash

Why is it important?

Time series forecasting has been investigated with several different machine learning techniques, in addition to reservoir computing. These include support vector machines, and auto regressive models and neural networks. It would of course be very interesting to compare reservoir computing with other machine learning approaches for the above tasks of emulating chaotic systems, learning their parameters, chaos synchronisation, cracking chaos cryptography, etc... Such a comparison goes however beyond the present work. We expectthat reservoir computing will probably report favourably in such a comparison. Indeed as noted above to the best of our knowledge reservoir computers hold the record for predicting the future trajectory of chaotic systems. (Most likely this is because reservoir computers, being recurrent dynamical systems themselves, already encode much of the structure which needed for such a task). An advantage of reservoir computers is that they are particularly easy to train, using only a linear regression. The present work builds on previous works which showed that reservoir computers with output feedback can emulate chaotic dynamical systems. Previous works focused on forecasting trajectories and predicting spatiotemporal chaos, inferring hidden degrees of freedom, estimating Lyapunov exponents. Here we show that trained reservoir computers can synchronise with another chaotic system, thereby demonstrating that the trained reservoir computer has an attractor with similar geometry and stability properties as the original system. We then show how a reservoir computer can be used to crack chaos based cryptography. It is interesting to note that cracking chaos based cryptography seems comparatively easy for the reservoir computer. Indeed, while for the time series prediction task we used reservoirs with N = 1500 neurons, for the cryptography application we only used N = 250 neurons. In addition in the presence of noise in the transmission line the reservoir computer in fact performed better than the system used by Bob. This is in part because the reservoir computer we used comprises a low pass filter. As noted above, reservoir computers can be implemented in hardware implementations, with good performance and high speed.

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