Message-Id: <199712290558.VAA01669@mail.san.rr.com>
From: "Jay Hill" <ehill1@san.rr.com>
To: <fractint@lists.xmission.com>
Subject: Re: (fractint) Fractal encryption
Date: Sun, 28 Dec 1997 21:57:42 -0800
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Hi Martin,

----------
> From: Martin Hecko <martineno@hotmail.com>

> I don't know if this topic has been discussed on this list before, but 
> here is my question.
> Could it be possible to use fractals as a way to encrypt information? 
> How strong the encryption would be?

It has not appeared here, but has been discussed on sci.crypt. Usually 
getting a general brush off.
 
> I have thought of very simple method. For simplicity's sake lets use the 
> regular mandelbrot formula. The password (or the passphrase) would be 
> the initial parameters for an iteration.
> Then take the first part of plain text, make first iteration of the 
> formula, run the result through some convenient hash algorithm and xor 
> the result with the plain text. Repeat the above steps for the rest of 
> the plaintext.

I have thought about this at length. Here are some specific ideas.
You are suggesting a stream cipher using the MSet as a pseudo-random 
number generator.  The key you use is, as you say, the coordinates, 
c, in the equation z[n+1]=z[n]*z[n]+c, z[0]=0.

> I don't know if it would work (due to the fact that  the floating point 
> calculations would have to be converted to integers and this could cause 
> inaccuracies in results).

Floating point must be done with plenty of precision, double (64 bit) 
is a minimum. 

The main weakness is that the values z[n] are not uniformly distributed.
Although this may be offset by a hash function, it can limit the search 
space of the attacker.  Even worse, key values not carefully chosen
lead quickly to divergence to infinity or to a periodic attractor. Only
chaotic c are acceptable. While marginally chaotic c are known
on the boundaries of the components, the only really chaotic values
can be found accurately on the real line.  This greatly limits your key 
space - to one dimension.  

A possibility of a two dimensional key space might be to include the 
initial z[0] as part of the key rather than using z[0]=0.  Care must be 
taken that c is in fact not in a bud or cardioid (a component).

For every point that is truly chaotic, a periodic point (in a component) 
is only an infinitesimal distance away.  If c is in a component, then 
the z[n] sequence is doomed to periodicity.

The Lyapunov exponent is a measure the chaos. But it can only be 
estimated by following the sequence to great lengths.  The Lyapunov 
exponent will indicate chaotic behavior right up to the point where the 
first period is reached. It will then drop in one iteration into non-chaos 
values - a disaster if it occurs in your message! Since, at that point 
your sequence may start repeating.

A large Lyapunov exponent indicates chaos, which is desirable in the 
sequence. However, it also indicates divergence when a very small 
perturbation (one bit) occurs incorrectly in the arithmetic. Therefore, the 
message sender and receiver must use exactly identical arithmetic 
processing. This can be a problem across platforms.

> I realize that this is not exactly a topic about Fractint itself, but I 
> think there are many knowledgable people on this list that have 
> potential to contribute to this discourse.

> Martin


Jay

PS Taking an off topic penalty.....

brain              { ; A brain after too much crypto thinking
                     ; by Jay Hill, 1997
  reset=1960 type=phoenixcplx
  center-mag=1.1175e-005/-8.65e-007/0.6666663
  params=0.27/0.02/0.25/0.1/0 float=y maxiter=256 outside=atan
  colors=000<62>wwwwwwvvv<61>KKKKKKKKL<76>bbvccwccv<43>112000000000
  }



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