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Exactly what it says on the tin. I can imagine constructing such keystreams from:

  • The binary expansions of irrational numbers
  • Chaotic systems like the logistic map or the Lorenz attractor.
  • The bitmap representations of images.
  • Extracting the y-coordinate from equations such as $sin(x)$ or $e^x \mod 1$ every n increments.
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  • $\begingroup$ Those generators may possibly return a endless amount of random-like data, but it remains to be seen if they would be good cryptographic stream ciphers or entropy pools. Bitmap representations of images would be either not be very secure, or not very practical. $\endgroup$
    – Maarten Bodewes
    Feb 1 '15 at 10:20
  • $\begingroup$ The main problem with chaotic keystreams is proving their properties in the strong sense, as required by cryptographic applications, beyond random-looking distributions. $\endgroup$
    – kodlu
    Feb 3 '15 at 0:04
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Well, it's easy to show that any generated with a bound on the amount of internal state must eventually become periodic. It's also easy, once we allow a generator whose internal state grows arbitrary large over time, to design a generator that never repeats (and the state growth required is actually quite reasonable).

That said, I don't know of any proposed stream ciphers that attempts to be completely aperiodic. I suspect that it's mostly because aperiodicity is not a realistic requirement (as opposed to the nicety of being implementable within a bounded memory space). After all, it is easy to have, even with a bounded state generator, such a large period (e.g. a period of $>10^{100}$) that we never have to worry about it. After all, if we never generate that much output, does it really matter if it would repeat if we did?

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    $\begingroup$ There is no kill like overkill. $\endgroup$
    – Melab
    Feb 1 '15 at 22:28

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