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I found this paper of Cuomo and Oppenheim, where they use a Lorenz system to define an encryption scheme for signals. There is also this blog post describing and implementing the technique. The technique uses an interesting property of the Lorenz system known as "synchronization," which as far as I know occurs in other dynamical systems as well. Moreover, the chaotic nature of these systems intuitively makes guessing the parameters of the system infeasible, so a natural shared secret key is the set of constants used to define the system.

I was wondering if there has been any theoretical work in cryptography that deals with dynamical systems. Specifically, are there established cryptographic hardness assumptions for specific dynamical systems? It seems like it would be a different flavor of hardness assumption from the usual ones, because the methods naturally introduce some noise into the decrypted message.

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Personally, I wouldn't bother with chaos-based encryption. I have yet to see any scheme based upon chaos that strikes me as very promising from either a practical or a theoretical perspective. The work on using chaos for encryption that I've seen ranks pretty high on my "bogometer". I never see any sort of security proof or plausible evidence of security, and many such systems turn out to be broken when you take a close look. And I find it hard to see any benefit over standard well-known approaches (e.g., CBC mode encryption; information-theoretic work based upon Wyner's wiretap channel; etc.).

But if you want to find something in this space, you might look at the following paper:

It has plenty of theoretical/foundational content, and is in the vicinity of the sort of thing you mention.

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    $\begingroup$ Do you think that the intuitive premise of chaos is unsuitable for encryption, or that if it were viable someone would have invented a solid example by now? I can imagine one reason for a lack of security proofs: the communities are disjoint. For physicists experiments lining up with intuition is often proof enough. It's no excuse, but just the reason for my question :) $\endgroup$
    – Jeremy Kun
    May 24, 2014 at 2:31
  • $\begingroup$ @JeremyKun, Yup, that's probably part of it. My sense is that the schemes I've seen were invented by people who don't know the field of crypto, and cryptosystems designed by people who aren't familar with the field often fare poorly. That said, I don't see chaos as a promising approach to symmetric-key crypto. It's a cute buzzword that sounds good, but I see no reason why chaos would lead to symmetric-key cryptosystems that are competitive with state-of-the-art systems on security and performance (and many reasons why it wouldn't: e.g., floating-point round-off). $\endgroup$
    – D.W.
    May 24, 2014 at 2:35
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    $\begingroup$ I'm willing to forget about competitiveness. But I would find it theoretically interesting if a genuinely new hardness assumption arose from this field, something closer to learning parity with noise than to factoring/discrete log. $\endgroup$
    – Jeremy Kun
    May 24, 2014 at 2:46

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