# Secure Image Transmission Using Multilevel Chaotic Encryption and Video Steganography
## 3 Methods and Materials
### Henon Map  
The Henon map is a two-dimensional quadratic map introduced by [PERSON] in 1976. It is a simplified model used to study chaotic systems, particularly the Lorenz system, and features a \"strange attractor\" [29].  
\[x_{n}+1=1-ax_{n}^{2}+y_{n}\]  
\[y_{n}+1=bx_{n}\]  
\(x_{n}\) represents the position in the 2D plane at the iteration and is updated using a nonlinear relationship, making it very sensitive to initial conditions.  
\(y_{n}\) represents momentum or the control variable associated with motion in the system.  
It is updated linearly, where \(x_{n}\) is multiplied by \(a\) constant \(b\).  
\(a\): The parameter '\(a\) in the Henon map is a scaling factor that nonlinearly influences the transformation of \(x_{n}\).  
Where \(a=1.4\), \(b=0.3\), \(x_{n}=-0.3\), and \(y_{n}=0.34\). This map has the following features:  
Chaotic behavior exhibits sensitivity to initial conditions and parameter values;  
Attractor, which has a fractal structure, resembling a Cantor set;  
Stability of fixed points that depends on the values of \(a\) and \(b\)[29].  
The tent map is a one-dimensional piecewise chaotic map used in image encryption and other chaos-based applications. Its name comes from the tent-like shape of its graph [30].  
\[F(x)=\begin{array}{c}rx,\ \ \ \ \ \ \ \ \ \ \ \ 0\leq x<0.5\\ r(1-x),\ \ 0.5\leq x\leq 1\end{array},\]  
\(x\) represents the position in the system at iteration \(n\) and is updated according to the previous value and the limits set in the equation.  
\(r\) is the control parameter, which determines the degree of growth or decline in the system.  
Where \(r=1.98\), and \(x=0.499\). This map has the following features:  
Chaos generation, producing chaotic sequences for \(\mu>1\);  
Lyapunov exponent, which is positive for chaotic behavior, indicating sensitivity to initial conditions;Bifurcation as \(\upmu\) varies.  
Each chaotic map employed in our proposed system has been rigorously tested under various initial conditions and parameter values. The ultimate output value ranges were not arbitrary but were based on extensive experimentation with a broad range of values. These ranges (\([0,1]\) for Arnold, \([-1,1]\) for Ikeda, \([-10,10]\) for ginger bread, etc.) have been chosen specifically to yield output distributions that when binarized simply with thresholding, produce bitstreams with balance, i.e., approximately 50% 0 s and 50% 1 s, as this is a key requirement for achieving high entropy, passing NIST randomness tests, and maintaining cryptographic key strength. Inappropriate range selections (narrow or wide) resulted in biased binary outputs and failed randomness tests.  
Moreover, each chaotic system makes its stability unpredictable. For instance, the Ikeda map's stable chaos oscillations lie in \([-1,1]\), and the tent map's working range is in the region where \(r\approx 1.98\), and \(x\in[0,1]\). Our choice covers all chaotic outputs in the most dynamic areas of each map, so that maximum entropy and diffusion are achieved without losing the system's determinism. Therefore, the selected output intervals provide theoretical foundations and optimized advantages for secure key provisioning.