Re: Lyapunov Exponents

From: Dr Chaos (mbNOSPAMkennel_at_yahoo.dot.com)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 13:16:59 -0700

Lou Pecora wrote:
> In article <412F3B96.8080808@netscape.net>,
> "Roger L. Bagula" <rlbtftn@netscape.net> wrote:
>
>
>>Dear Dr. Chaos,
>>As I understand it chaotic bidirectional synchronization
>>has been rejected by cryptographers as being to
>>"easy" to decrypt and it is only that
>>chaotic system resembles "noise" and is secret
>>that make such systems useful.
>>I had a cytologist on my case about it at the start of the year.
>>Chaotic systems ( 3d types) are usually too close to "smooth"
>>compared to noise like that of a Mandelbrot cartoon/
>>fractional Brownian type.
>
>
> Bi- or uni-directional flows (ODEs) are probably not a good choice for
> encryption. As you say they are too smooth. However, chaotic maps are
> better choices.

Why does smoothness matter? Is there something peculiar about
having localized high derivatives (high local lyapunov exponents)?
Are discontinuous derivatives, or non-invertibility advantageous?

> But as you get closer to maps and then digital forms
> you converge toward some of what's being done now in cryptology. For
> example, a standard approach to creating pseudo-random series is the
> shift register. That is just a digital form of the chaotic map called
> the Bernoulli shift map.

What about flows with equivalent symbolic dynamics?

> I suspect there are still some things to be
> mined from the merging of spread-spectrum and encryption fields with
> nonlinear dynamics, but the fields remain very far apart. I'm not an
> expert in communications or cryptology, but this is my sense of the
> situation right now. We nonlinear people have been too naive about
> cryptology and communication issues (speaking broadly here) and the
> other side has been too dismissive of nonlinear approaches.
>
> IMHO.

matt