Contents
Impenetrable	Our fractal transformation ciphers cannot be penetrated
Logic Deduction	Binary logic deduction proof
Cannot be Brute-Forced	Fractal Transformation cannot be brute-force cracked
Cipher Non-Determinism	The payload no longer exists within the cipher
Mathematical Proofs	Our fractal transformation is infinitely complex

Impenetrable

We present or logic proof that the cipher created by our fractal transformation technology is impenetrable.

Impenetrable and uncrackable by any means, including Quantum Computers.

This is a world-first as cryptography has not previously had a form of encryption that is impenetrable.

Proof

The foundations of our proof rest on logic deduction and mathematical proofs.

Logic Deduction

The concise deduction is as follows:

As the unknown fractal transform is the same size as the payload,
all random, semi-sensible and sensible cipher bit combinations are payload candidates,
and there is no way to distinguish the payload from other sensible candidates.

The deduction is robust as a binary string of matching length can define all possible text, images, audio and any data whatsoever.

When a cipher is created by transforming a binary payload with an unknown fractal transform stream of the same length, all cipher bit-combinations become payload candidates.

Let's use an example 24 character email payload, allowing 36 characters (26 letters a - z and 10 numbers 0 - 9) and ignoring @ and . characters. We examine 22 binary character combinations where each character uses 8 bits (2 ^ 8 = 256 combinations). To keep things simple we calculate sensible candidates using 2 ^ 5 = 32 combinations (4 less than 36):

Name	Formula	Total Combinations
Total Combinations	2^8^22	95,780,971,304,118,100,000,000,000,000,000,000,000,000,000,000,000,000
Sensible Candidates	2^5^22	1,298,074,214,633,710,000,000,000,000,000,000

In this case sensible candidates make up only 0.000000000000000001355252715607% of all possible candidates, yet they still amount to 1.298 Undecillion sensible email candidates (for a 24 character email), proving that the email payload (which is one of those) cannot be distinguished from all the other sensible candidates.

Similarly, a transformed graphic payload will have equally large numbers of sensible graphic candidates, and the same goes for audio, html, xml, word, excel and pdf payloads.

There is no way to distinguish the payload from all possible sensible bit-combinations, and there are always a great many perfectly sensible bit-combinations.

Cannot be Brute-Forced

Classic encryption such as AES can be brute-force cracked by trying every possible key as:

1. Keys have a fixed size
2. There is a limited number of possible keys to try
3. Keys act directly on the payload
4. Only the correct key returns a sensible result
5. Processing time required has been dramatically reduced by quantum computers

Fractal Transformation cannot be brute-force cracked as:

1. Keys can be any size
2. There is an infinite number of possible keys
3. Keys do not act on the payload
4. There is an infinite number of possible fractal transforms
5. You cannot distinguish between the very large number of possible sensible candidates (see above)
6. The precise fractal mapping and configuration used to transform the payload is also required

Each key initialises a non-deterministic fractal stream that must be generated according to its original fractal mapping and configuration. While it is theoretically possible to try an infinite number of keys, you will have no way of distinguishing between all the possible sensible payload candidates.

Cipher Non-Determinism

Ciphers created with Classic block encryption such as AES are deterministic as:

1. The key shape acts directly on payload blocks
2. The algorithm creates a linear relationship between key, payload and cipher
3. The cipher deterministically contains the payload relative to the key

Ciphers created with Fractal Transformation are non-deterministic as:

1. The key shape does not act directly on the payload
2. The key maps to one of an infinite number of possible fractal portals
3. Each portal is a gateway to a unique and infinitely complex fractal stream
4. Each fractal stream iteration is unpredictable and non-deterministic
5. The payload is overwritten by the fractal stream
6. The resulting cipher is a whole-of-payload fractal transformation remnant
7. The resulting cipher no longer contains the payload

The fractal stream acting on the payload has zero determinism related to the key, which is discarded at the time the fractal portal is identified. Each fractal stream iteration has zero determinism related to previous iterations. Fractal transformation is a unique infinitely complex whole-of-payload overwrite. Resulting fractal ciphers are absent the payload without the exact whole-of-payload fractal stream reversal.

Mathematical Proofs

The logic proof relies on a high quality, complex, unique and unpredictable unknown fractal stream.

Our unknown fractal stream is derived from the Mandelbrot Fractal.

Our Fractal Transformation technology maps unlimited size keys to an infinite number of possible Fractal Portals, each initialising highly complex, unique and unpredictable Fractal Streams.

Mitsuhiro Shishikura has created the formal mathematical proof that the Mandelbrot Fractal is infinitely complex. His proof can be examined here: Mitsuhiro Shishikuras Proof.

It is common knowledge that there is an infinite number of possible x,y locations on a limited plain. The Mandelbrot Fractal responds with a unique and unpredictable value for each of the infinite possible x,y locations.

Our Fractal Transformation selects new x,y locations based on the unpredictable value of previous x,y locations.

As the values of all possible Mandelbrot x,y locations are proven to be infinitely complex, the quality of our Fractal Stream is also infinitely complex.