Contents
Impenetrable	Our fractal transformation ciphers cannot be penetrated
Logic Deduction	Whole-of-Payload Transformation
Mathematical Proof	Our fractal transformation is infinitely complex

Impenetrable

We present our logic proofs 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.

FES Impenetrability via Whole-of-Payload Transformation

Logic Statement:

If a payload is offset or XORed by an unpredictable, unknown byte stream of the same size, all possible bit combinations of the cipher are equally valid payload candidates.

Resulting in:

1. Cipher-to-Payload Equivalence:
   • Since the fractal stream matches the payload size, every possible cipher bit combination is a valid payload candidate.
2. Sensible Candidates Are Uniquely Masked:
   • Among these combinations, a subset represents sensible outputs (e.g., plaintext email addresses, documents, or images).
   • FES ensures these sensible candidates are indistinguishable from one another and from random noise.
3. Impossibility of Distinction:
   • Without knowledge of the exact fractal stream (which is only reproducible with the exact FES key), there is no cryptanalytic method to isolate the original payload from other sensible candidates.
   • Every bit combination, whether meaningful or random, is equally probable.
4. Sensible Candidate Explosion:
   • For a typical 24-byte payload (e.g., an email address), the space of sensible candidates is exponentially smaller than the total bitspace:
     • Total combinations: \( 2^{8 \cdot 22} = 2^{176} \approx 9.58 \times 10^{52} \)
     • Sensible candidates: \( \sim 2^{5 \cdot 22} = 2^{110} \approx 1.3 \times 10^{33} \)
   • Even this drastically reduced subset remains astronomically large, and distinguishing one sensible candidate from another is mathematically impossible.
5. Conclusion:
   • FES guarantees that:
     • All random, semi-sensible, and sensible bit combinations are valid payload candidates.
     • No cryptanalytic method can identify the original payload among other sensible candidates.
   • This ensures that breaking the cipher is as hard as enumerating and verifying all possible payloads, an intractable problem even for quantum computers.

Why the Fractal Stream Matters

The "Whole-of-Payload Transformation" proof demonstrates that every bit combination of an FES cipher is a plausible payload, making decryption without the fractal stream impossible.

But what makes the fractal stream so special? The answer lies in fractal navigation—a mathematically infinite, unpredictable process that ensures no two streams are alike.

Learn more about FES’s fractal navigation and why it stands as a separate, complementary proof of impenetrability.

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.