# Chaotic Henon-Logistic Map Integration: A Powerful Approach for Safeguarding Digital Images
## 3 The Proposed Scheme  
This paper presents an innovative integrated chaos-based algorithm for image cryptography, combining the chaotic Henon map and the chaotic logistic map (CLM) to enhance the transmission security of images. The proposed method effectively meets the key requirements of data cryptography, as follows:  
* Robust encryption key: The algorithm generates a strong encryption key that is highly resistant to brute-force attacks, providing a robust security framework that reassures readers about the system's security. * Sensitivity to minor changes: Leveraging the unique characteristics of chaotic systems, the method ensures that even the slightest alterations in the input image or encryption key result in a completely different encrypted output. This high sensitivity instills confidence in the system's ability to protect data from unauthorized parties. * Optimal quality parameters: The approach maintains high-quality parameters throughout decryption and encryption, guaranteeing the similarity between the decrypted and original images, without compromising on quality or fidelity. * High MSE during encryption: A high Mean Square Error (MSE) between the original and encrypted images ensures that the encrypted image appears highly distorted, preventing attackers from interpreting the content. This distortion enhances the algorithm's security. * Decryption with zero MSE: Following decryption, the MSE should return to zero, confirming that the original image has been perfectly restored without losing data or integrity. * Efficient encryption and decryption: The method prioritizes speed, minimizing the time required for encryption and decryption while maximizing processing efficiency. This emphasis on efficiency reassures readers that the system is practical and responsive for real-world applications. * Broad image compatibility: The algorithm is designed to work effectively with various image formats, resolutions, and color spaces. Its adaptability ensures reliable performance across diverse image types, making it suitable for real-world use cases. A graphical abstract of the proposed method is shown in Figure 1, illustrating the generation and application of cryptographic keys using chaos theory for image encryption. Initially, a 1D key is generated using the logistic map generator, while a 2D key is created using the chaotic Henon map generator. These chaotic maps produce highly unpredictable and complex keys, which are essential for robust encryption. The abstract visually depicts how these keys are used in the encryption algorithm, transforming plain images into encrypted ones. This dual-key approach leverages the inherent randomness of chaotic maps, ensuring enhanced security and resistance against cryptographic attacks. Figure 1: Graphical abstract of the proposed method. The chaotic logistic map (CLM) is a mathematical framework that exhibits chaotic dynamics. It operates as a discrete-time dynamical system and is described by the following recursive Equation (1):  
\[X_{n+1}=rX_{n}(X_{n}-1) \tag{1}\]  
In this context, \(X_{n}\) represents the state of the system at the time step \(n\), \(r\) is a parameter that influences the system's dynamics, and \(x\) denotes the system's initial state. The state variable \(x\) typically lies in the interval \([0,1]\) during iteration. The value of \(r\) dictates the behavior of the system, as follows:  
* For \(r<1\), the system stabilizes at a single point. * For \(1\leq r\leq 3\), the system follows a periodic pattern. * For \(3\leq r\leq 3.57\), the system experiences chaotic behavior. * For \(r>3.57\), the system becomes divergent. Two one-dimensional (1D) CLM keys are randomly generated using the CLM method to shuffle pixel positions both row-wise and column-wise. By utilizing the CLM model with the parameters \(r\) and \(x\), the dimensions of the 1D keys are determined by the parameters \(R\) and \(C\).