# Chaotic Henon-Logistic Map Integration: A Powerful Approach for Safeguarding Digital Images
## 2 Related Works  
The Collatz Conjecture, introduced by German mathematician [PERSON] in 1937, presents a fascinating mathematical puzzle that remains unsolved [17]. This conjecture revolves around a straightforward iterative procedure applied to any positive integer. Beginning with a given number, \(t_{0}\), \(t_{i}\) is obtained using the formula  
\[t_{i}=\left\{ \begin{array}{ll}\frac{t_{i-1}}{2}&\text{if }\bmod(t_{i-1},2)=0\\ 3t_{i-1}+1&\text{otherwise}\end{array} \right.\]  
This process is repeated with the resulting number, generating a sequence. It is widely believed that the sequence will inevitably reach the repeating cycle 4-2-1 regardless of the starting integer. Despite extensive efforts, no definitive proof or disproof has been found, but its simplicity and enduring mystery have made it a compelling focus of mathematical inquiry [17, 18]. Researchers have found a creative application for the Collatz Conjecture in generating cryptographic keys in image encryption. The deterministic nature of the Collatz sequence, coupled with its chaotic behavior for certain starting values, forms the foundation of a unique key generation algorithm. In this approach, the initial conditions of the Collatz Conjecture become a seed to generate a numerical sequence used as a cryptographic key. By mapping the pixel values of an image through this key-generated sequence, a complex and pseudo-random pattern is introduced, enhancing the security of the encryption process. The unpredictability and inherent complexity of the Collatz Conjecture offer a fresh perspective on key generation for image encryption, providing a potential avenue for developing robust and unconventional cryptographic techniques [17; 18]. Chaotic maps are mathematical functions that exhibit chaos, characterized by sensitivity to initial conditions, topological mixing, and dense periodic orbits. These properties make chaotic maps useful in various fields, especially in cryptography for encryption and secure communications. There are several types of chaotic maps, each with distinct features and applications [19; 20; 21; 22; 23; 24]:  
1. Logistic Map: The logistic map, a polynomial mapping of degree 2, is one of the most well-known and basic chaotic maps. It is frequently used as a prime illustration of how extremely basic nonlinear dynamical equations can give rise to complex and chaotic behavior. 2. Tent Map: The tent map is piecewise linear and exhibits chaotic behavior for certain parameter values. It is characterized by its tent-like shape, and it is commonly used in studies of dynamical systems due to its simplicity and the ease with which one can analyze its properties. 3. Chaotic Henon Map: A two-dimensional map, the chaotic Henon map is one of the most studied examples of dynamical systems that exhibit chaotic behavior. Introduced by [PERSON], it is renowned for its strange attractor, a complicated set of fractal structures in its phase space. 4. Chebyshev Map: This map uses Chebyshev polynomials and is known for its use in pseudo-random number generation and cryptographic algorithms. The Chebyshev map exhibits chaotic behavior for certain parameters and is appreciated for its complex dynamical properties. Each of these maps has its own set of characteristics and applications, and the choice of a specific map often depends on the requirements of the system being designed, particularly in terms of the required cryptographic strength and computational efficiency. When delving into the intricacies of throughput in image encryption methods, as delineated in Table 1, a discernible pattern emerges, particularly concerning the performance of various chaotic approaches. Notably, hyper-chaotic approaches tend to outshine their counterparts. This distinction can be attributed to their inherent complexity and dynamic characteristics. Unlike standard chaotic systems, which rely on a single positive Lyapunov exponent, hyper-chaotic systems boast multiple such exponents, enriching the encryption process with an additional layer of unpredictability and security.