An encryption/decryption algorithm based on Lorenz's Strange Attractor, and WebSockets for communication.

Both the server and the client have a running Lorenz Strange Attractor.
The Attractors agree on the same pre-conditions, but each starts at different positions in space, resulting in vastly different trajectories.
The basic idea is:

Create public and private keys (X25519 eliptic curve Diffie-Hellman) 
  -> reach consensus on the system parameters (σ, ρ) using the shared secret
  -> start Attractors with different coordinates/trajectories 
  -> sync them 
  -> create a stream cipher on the server and client 
  -> encrypt with client cipher 
  -> send encrypted message 
  -> decrypt with server cipher
  -> desync the Attractors

The stream cipher is constructed with the current y coordinate of the Attractor at each frame.
Because the attractors are synced, the y coordinates should be the same, and so the server can decrypt the message.

The whole process takes about 30-55ms.

But how can we sync these seemingly chaotic systems?

Pecora and Carroll [1], and later Steven Strogatz [2], described an easy way of syncing two or more chaotic systems:

• Take one of the Systems, the Driver, that will transmit its current state in a one-way communication tunnel.
• The other system becomes the Reciever.
• If we force the Reciever's x coordinate to be equal to the x coordinate from the Driver system, we observe that, after a small number of iterations, the systems become synced

In this example, the bottom Attractor, the Reciever, struggles to display the normal Butterfly-like behavior at first, but then, after a few seconds, for each new point the other coordinates start coming closer and closer to the Driver Attractor, until they are, in Steven Strogatz’s words, dancing in perfect sync with their doppelgänger.

This implementation defines the Attractor on the client side as the Driver and the server side as the Reciever.

• They are Deterministic, meaning that, given the same pre-conditions, the outcome will always be the same.
• They are also very sensitive to those pre-conditions. The smallest of changes means a huge difference in the outcome, which is one of reasons why they are called chaotic (the other is that it is hard to predict what will happen next). This quality makes it worthwhile, because it means we can create secure ciphers.

Run the server:

cargo run --bin server

And the client:

cargo run --bin client

in separate terminal windows, write something on the client, and watch it get encoded on the client and decoded on the server.

Run the tests with the command:

cargo test

The testing suite is made up of:

• Unit Tests

  • Encryption function
  • Decryption function
  • Lorenz Attractor Syncing

• Integration Tests

  • 100 Non-Concurrent Clients
  • 50 Concurrent Clients

• Client Verification with Keys
• Server and Client Agreement on Different Pre-Conditions
• Two-way Encryption/Decryption
• Add more Attractors and a way for the Server and Client to reach a consensus on which one to use
• Add more capacity for concurrent clients

Please note that I did not formally prove this algorithm.
It may not be suitable for real-world applications, as it may contain security concerns and/or not be 100% accurate all of the time.

References:

• [1] Synchronization in chaotic systems (Link)

  • Authors: Pecora, Louis M., and Thomas L. Carroll.
  • Journal: Physical review letters 64.8 (1990): 821.

• [2] Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering (Link)

  • Author: Strogatz, Steven H
  • Publisher: CRC press
  • Edition: 2018

Inspiration for this project:

• A Chaos Based Encryption Method Using Dynamical Systems with Strange Attractors (Link)

  • Authors: Sheikholeslam, S. Arash
  • Journal: SECRYPT. 2009

• Fast, parallel and secure cryptography algorithm using Lorenz's attractor (Link)

  • Authors: Marco, Anderson Gonçalves, Alexandre Souto Martinez, and Odemir Martinez Bruno
  • Journal: International Journal of Modern Physics C, Volume: 21, Issue: 3(2010) pp. 365-382

• Syncing GIF
  • Author: Iacopo Garizio
  • Website: Synchronizing Lorenz attractors I