Control-Network-Systems/docs/Chapters/11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md

Chryptography with Chaotic Systems

Deterministic Chaotic Systems

These are non-linear differential equations that are very sensitive to initial conditions. One of the most famous is the Lorentz-Attractor.

Since we cannot get back to these initial conditions, these are unpredictable systems.

Note

s(x) is a sync message added to the other system to get it produce the same output as the first one.

Here it is represented its effect in red.

In yellow you see a simple gain factor coming from the same s(x)

Let's say we have 2 of these systems:

\begin{align*}
    x(t); z(t) &\triangleq \text{chaotic systems}\\
    e(t) = x(t) - z(t) \rightarrow 0 
    &\triangleq \text{synchronization} \\
    s(x) = f(x) + Kx &\triangleq \text{synchronization signal}

   
\end{align*}\\

\begin{cases}
        \dot{x}(t) = Ax(t) + Bf(x) + c\\
        \dot{z}(t) = Az(t) + Bf(z) + c 
        \textcolor{red}{+ Bf(x) - Bf(z)}\\
        \dot{e}(t) = \dot{x}(t) - \dot{z}(t)
\end{cases}\\

\begin{align*}
    \dot{e}(t) &= \dot{x}(t) - \dot{z}(t) = \\
   &=  Ax(t) + Bf(x) + c -  Az(t) - Bf(z) - c = \\
   &=  A(x(t) - z(t))+ B(f(x) - f(z)) = \\
   &=  Ae(t)+ B(f(x) - f(z)) = \\
   &=  Ae(t)+ B(f(x) - f(z)\textcolor{red}{-f(x) + f(z)}) = \\
   &=  Ae(t) \longrightarrow e(t) \rightarrow 0 
   \text{ if } eig(A)\in \R^- \\

   &= Ae(t) \textcolor{yellow}{-BKx + BKz} = (A - BK)e
\end{align*}

At the end of all, you can basically add a message into \hat{s}(x) = s(x) + m(t) and after getting the new value z(t) I can extract from here the message:

\hat{s}(x) - f(z) -Kz = f(x) + Kx + m(t) - f(x) - Kx = m(t)