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Author Topic: makes it sense to assosicate chaos<->encryption !?  (Read 1590 times)
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cKleinhuis
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« on: January 02, 2014, 01:01:50 PM »

hi all, i am struggling finding a right keynote for my new year issue show, i want to include some words about security and
chaos theory, i think it just feels right to use chaos theory for such things as well, what do you think or is it just ridiculous ?!

there seems to be not much going on in this field, i think we are going to open up a "security" sub board soon, this yielded
a quick search on google :

http://www.codeproject.com/Articles/406389/Fractal-encryption-algorithm

and

http://www.codeproject.com/Articles/311809/Chaos-Based-Encryption
« Last Edit: January 02, 2014, 01:08:34 PM by cKleinhuis » Logged

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divide and conquer - iterate and rule - chaos is No random!
hobold
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« Reply #1 on: January 02, 2014, 06:28:13 PM »

Beware of your intuitive understanding of the word "chaos". In mathematics, namely in chaos theory, that word is used as an abstract label to denote a defined set of qualities / constraints. Good encryption might require another kind of chaos than what fractal geometry can provide.

Consider, for example, the self-similarity that is inherent in all fractals. This kind of strong correlation would probably weaken an encryption algorithm. Fractals are, in a manner of speaking, not random enough for strong encryption. The essence of fractal geometry is that there is structure in chaos. The essence of cryptography is that there is as little discernible structure as possible.
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Pauldelbrot
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« Reply #2 on: January 02, 2014, 07:52:53 PM »

The technical definition of "chaos" in these systems is "sensitive dependence on initial conditions". All strong ciphers already exhibit chaos, inasmuch as that small perturbations to the plaintext cause enormous changes in the ciphertext. Ditto secure hashes -- small changes to the thing being hashed, or to the salt, will scramble the hash output completely. This both makes collisions rare and makes computing an inverse function extremely difficult, limiting codebreakers to brute-force approaches (then defeated with big enough key-sizes) or wholly non-cryptanalytic attacks (steal the plaintext before encryption, MITM attacks, etc.) -- contrast this with a simple substitution cipher, a Vigenere cipher, or anything similar where changing one letter of the input changes one letter or maybe a bit more of the output, and changing the key a tiny bit likewise changes the output a tiny bit, and how these facts are exploited to do cryptanalysis on these ciphers.

However, the chaos that appears in modern strong crypto systems isn't really useful for generating fractals. As hobold suggested, the chaos in good ciphers so thoroughly mixes space that any fractals you tried to plot from these would be black boxes, white noise, or other not very interesting shapes. (Consider, for example, the limit of a Peano spacefilling curve -- a black box. Good ciphers mix and fold at least as much and as thoroughly.)
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cKleinhuis
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« Reply #3 on: January 02, 2014, 10:18:40 PM »

we dont want nice pictures from security algorithms wink
the severe dependency of starting conditions is a good point related to chaos theory wink
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divide and conquer - iterate and rule - chaos is No random!
hobold
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« Reply #4 on: January 03, 2014, 02:43:17 AM »

One definition of mathematical chaos I remember from a few decades ago is based on three qualities:

1. sensitive
If you iterate two nearby, but not identical, non-periodic points X and Y, regardless of how close together they start out, there exists a finite number of iterations N such that the iterated points X_N and Y_N are arbitrarily far apart (limited only by the diameter of the resulting fractal). Intuitively, all points tend to drift apart under continued iteration.

2. mixing
If you iterate two disjoint open subsets A and B (think two small intervals excluding their borders), then there exists a finite number of iterations of N such that the iterated images A_N and B_N overlap in at least one point. Intuitively, the iteration not only stretches intervals (in condition 1), but also folds them back on themselves.

3. periodic points are dense
Here "dense" is another mathematically defined term meaning that any open subset of the resulting fractal contains infinitely many distinct periodic points.


I don't think this definition is an established standard, but it was a useful starting point back when fractals were a new and hot topic. The above conditions might not catch all kinds of chaos, but suffice for one-dimensional cases like the logistical map (a.k.a. "Feigenbaum Scenario").


Anyway, the mappings used in encryption are usually sensitive in a loose sense (not in strict sense because encryptions typically operate on finite sets, not on a continuum of real or complex (or triplex) numbers). Periodic points would be a cryptographical weakness (at least for small periods; maximal periods are unavoidable in a finite number space), and they are certainly not dense in the number space that encryption is working on (again dismissing the problem of a proper definition for finite number spaces). The mixing property seems to be intuitively fulfilled, but does not at all carry over from continuous sets to finite sets, so intuition is misleading here.

So, summing up, I'll say I would be very surprised if somebody were able to base a provably strong encryption on fractal chaos.
« Last Edit: January 03, 2014, 02:51:31 AM by hobold » Logged
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