Enhanced Image Encryption Using Six-Dimensional Hyperchaotic Systems

Chaotic-based image encryption algorithms have found widespread applications across various industries. However, many encryption schemes utilizing low-dimensional chaotic systems suffer from insufficient security measures. To address this limitation, this paper introduces an innovative image encryption algorithm that combines a 6D high-dimensional chaotic system with DNA encoding techniques.

The proposed methodology first applies multiple random chaotic sequences to perform diffusion and permutation operations on the original image sequence. Subsequently, at the DNA level, different chaotic sequences further diffuse and permute the generated sequences. Finally, these diverse encoding sequences are combined to form the encrypted image.

Introduction to High-Dimensional Chaotic Encryption

With the rapid advancement of digital image processing technology, image encryption has become increasingly crucial in information security. Chaotic encryption algorithms have emerged as a prominent research area due to their excellent diffusion and confusion properties. Nevertheless, conventional encryption methods based on low-dimensional chaotic systems face challenges such as limited key space and inadequate security.

Proposed 6D Hyperchaotic System Approach

To overcome the shortcomings of low-dimensional chaotic encryption, this paper presents an algorithm based on a 6D high-dimensional chaotic system. The implementation utilizes this 6D system as a pseudo-random sequence generator, enhanced with DNA encoding technology to significantly improve security.

Algorithm Implementation Procedure

1. Key Generation: Six initial values and control parameters are generated to construct the encryption key for the 6D hyperchaotic system.
2. Image Preprocessing: The original image is converted to a binary representation and divided into 8×8 pixel blocks.
3. Diffusion Process: Six chaotic sequences are employed to diffuse each pixel block, modifying pixel values.
4. Permutation Process: Different chaotic sequences permute the diffused pixel blocks, altering pixel positions.
5. DNA Encoding: The permuted pixel blocks are transformed into DNA sequences, which undergo additional diffusion and permutation using distinct chaotic sequences.
6. Image Reconstruction: The various encoded sequences are combined to form the final encrypted image.

Performance Evaluation

To assess the algorithm's effectiveness, comprehensive experiments were conducted with the following results:

• Image Entropy: The encrypted image entropy approaches 8, indicating superior randomness.
• Pixel Correlation: Extremely low pixel correlation demonstrates excellent confusion properties.
• Image Complexity: The key space exceeds 2³⁰⁰, providing high security against brute-force attacks.
• Robustness: The encrypted image exhibits strong resistance against geometric and truncation attacks.

Entropy Analysis Implementation

The following MATLAB function implements entropy calculation for image analysis:

function entropy_values = calculate_image_entropy(image_data)
%{
================================================================================
                            calculate_image_entropy
================================================================================
Definition:
Entropy in information theory is directly analogous to the entropy 
in statistical thermodynamics. The analogy results when the values of 
the random variable designate energies of microstates, 
so Gibbs formula for the entropy is formally identical to Shannon's formula.

Application:
The function calculate_image_entropy computes the entropy of an image,
indicating the encryption algorithm's performance.
%}
    entropy_values = zeros(1,3);
    
    function entropy = compute_single_entropy(image_channel)
        image_channel = double(image_channel);
        [height, width] = size(image_channel);
        image_vector = transpose(image_channel(:));
        histogram = zeros(1, 256);
    
        for intensity = 1:256
            histogram(intensity) = sum(image_vector == (intensity-1));
            histogram(intensity) = histogram(intensity)/(height*width);
        end
    
        entropy = -histogram(histogram>0)*transpose(log2(histogram(histogram>0)));
    end
    
    entropy_values(1) = compute_single_entropy(image_data(:,:,1));
    entropy_values(2) = compute_single_entropy(image_data(:,:,2));
    entropy_values(3) = compute_single_entropy(image_data(:,:,3));
end

Experimental Findings

The experimental results demonstrate that the proposed algorithm significantly outperforms reference methods in key security metrics. The entropy values approaching the theoretical maximum of 8 confirm the encryption's effectiveness in producing uniformly distributed pixel values. The minimal pixel correlation across horizontal, vertical, and diagonal directions validates the algorithm's confusion capabilities.

Furthermore, the substantial key space of 2³⁰⁰ ensures protection against exhaustive search attacks. The algorithm's resilience against geometric transformations and cropping attacks makes it suitable for applications where encrypted images might undergo partial alterations.

References

Li, Q., Chen, L. An image encryption algorithm based on 6-dimensional hyper chaotic system and DNA encoding. *Multimed Tools Appl* **83**, 5351–5368 (2024). https://doi.org/10.1007/s11042-023-15550-3