Imagine that you're zooming in somewhere in the mandelbrot set. Where in the image are you, and what are the chances of stumbling upon this exact spot by accident?

Well, the depth and uniqueness of fractals could be used for generating encryption keys. Instead of really long sequences of numbers,
you could use a zoomed-in section of the Mandelbrot set. Then, instead of having a long number, you just specify the fractal formula, and a zoom ratio.

NASA Tech Briefs, Jul 2001http://findarticles..._200107/ai_n8971625 Nice idea, but fairly well rehearsed - here's one example (they use Lorenz shapes, rather than the M-set here) [zen_tom, Nov 26 2008]

Doesn't everybody have the key to the master-fractal though?

...although, this is useful for many applications, of course. I'm no master programmer, and I probably couldn't crack the code even if given the fractal set. You could just as well pick every other number of Pi or the lucky primes for a code though, and I'm not sure how a fractal code would work either. However, it is still very neat if it is some sort of digital fingerprinting of fractal patterns. I suppose that the big problem would boil down to hackers guessing the coordinants of a square or other polynomial on the imaginary plane that the computer programmers decided on. [+]

I guess one cool think about this is that it gives an instant visual presentation to an end-user. SSH started doing a little ascii picture to help humans see problems with patterns in ssh keys, but this takes it one step further, into the 90s, with total Fractal radicalness.

I seem to remember a Mandelbrot Set can be created with just 3 variables and your zoom-in ratio would be a fourth. And a 5th is the specific set. So much easier to remember 5 digits than a password. The login shows a beautiful moving fractal pattern. When you enter the correct digits, it zooms in to the spot entered and lets you in. I'm no mathematician, but I think the variations are infinite. Maybe not, but do try it.

There's a few issues here - first, position. Zoom alone is not useful since the point 0,0 of most fractals is likely to be a known quantity that's independent of zoom - so you need to add x and y coordinates. That's now 3 values to remember, x, y and z. Then there's the question not only of the formula used, but the interpretation. Fractal formulae are iterative and measure the approach of a field of values towards infinity over a number of iterations - if you don't intend to wait forever for the results (essentially a binary yes, or no for each point) you instead choose an arbitrary number of iterations, and a cutoff value beyond which you assume infinity yawns inexorably, so that's two more numbers - meaning that you now have to remember all 5 variables x,y,z,i,c and the fractal formula in order to reliably recreate a particular 'bit' of fractal. You could probably standardise i and c, but that still leaves someone to choose three numbers, x and y and z - x and y for the Mandelbrot set must both be in the range -2 to +1.5 After a few zooms, you're looking at numbers like (-0.63245,-0.55467,1200) which is almost like choosing a great-big number in the first place (i.e. what [kamathln] said)

If you keep people to a fixed number of significant digits, and standardise your i and c values, that means that the number of real alternatives drops to a number much lower than all the clever maths warrants - i.e. it becomes vulnerable to a guided brute-force cracking attempt - and you might as well have used a 4-6 digit pin.

You have a set (the message set) of binary digits (M1 thru Mi) that you would like to communicate. You would like to convert the set M to a set C(the cipher set) C1 thru Ci. You want to do this because the message set (M) is "visible" to others, and the cipher set (C) is "invisible" to others. This requires you to have a set K (the key set) and an operation of that set on the other set. The most used operation is the XOR. So Mi XOR Ki = Ci. You can successfully hide anything in this operation. But you want the other, and only the other, party to successfully get the set M.
This means you must agree on a set K.

The set K must be previously, and securely, communicated (OTP). Or, and this is where your idea has some use, be securely and correctly derived.
Securely and correctly derived, ( by Diffie, Hellman, (Merkle) key exchange) means a one-way function. It is not possible for a fractal set to be computationally as complex as the factorisation problem. It is just is... Prime decompostion does not follow some algorithm, whereas fractals do. So even if you could magigraphically transmute your location on the specific fractal set to your friend, the oppurtunity for someone else to derive meaning given the ENTIRE set is not computationally insignificant enough. Sure, it is hard, just not hard enough.

If you boil this idea down enough what are describing is, in essence, a table. If you boil it down further you are describing the ASCII table. Where parties agree on a meaning that will be transmitted by a "compressed" format, each with knowledge of the original format. There is no encryption here, move on folks...