Image Encryption Algorithm Based on a New Four-Dimensional Hyper Chaotic System and Second Order Fuzzy Differential Equations

The widespread use of multimedia, particularly images, across fields such as geology, biology, medicine, and the military has enabled technological progress that has also enabled attacks, resulting in the theft or falsification of information [1, 2]. The possibility of hiding information within them or preventing the leakage of essential images exists, as does the presence of open networks for communication between people, which expose data to danger. Therefore, it is crucial to maintain the security of information and the safety of images sent over the internet [3-6]. From this standpoint, the focus has shifted to encrypting images and developing algorithms that are highly secure while also restoring them to full resolution without data loss. Scientists and researchers are compelled to build and innovate algorithms to enhance security [7-10]. One of the techniques used is the substitution box (S-box) generated from chaotic systems, which disperses pixel correlations as a preliminary step before encryption [11, 12].

These chaotic systems have initial values that generate non-periodic, sensitive-to-change random numbers, making them a type of complex, non-linear system. Also, multiple steps or rounds can be used for encryption, enhancing security. Also, two stages are used: configuration and diffusion. Encryption is of two types: block encryption, which treats data and images as blocks of bits or bytes, and stream encryption, which treats data bit by bit or byte by byte [13-16].

Create the new 4D hyper chaotic system, a mathematical model containing thirteen parameters, non-linear terms are obtained as:

$\begin{gathered}\frac{d x}{d t}=a y z-b x-c w+b \sin (y) \\ \frac{d y}{d t}=d x z+e x-y-b x \operatorname{Exp}(z) \\ \frac{d z}{d t}=f x y-g z+h y^2-i \operatorname{Sin}(w) \\ \frac{d w}{d t}=j y z+k x z-l w+m z \operatorname{Exp}(x)\end{gathered}$           (1)

where, value of initial conditions are  x(0) = 0.5, y(0) = 0.4, z(0) = 1.5, w(0) = 0.6, and thirteen initial parameters are a = 5.8, b = 3, c = 0.5, d = 0.8, e = 17, f = 29, g = 2.4, h = 9, i = 5, j = 15, k = 4, l = 2.1 and m = 2.

Lyapunov Exponent and fixed points

The Lyapunov Exponent values are L.E₁ = 2.88108, L.E₂ = 0.0828833, L.E₃ = -2.48959, and L.E₄ = -25.0176, There are two positive values this mean hyper chaotic system The new hyper chaotic system have two fixed points are E₀ {x = 0, y = 0, z = 0, w = 0} and E₁ {x = 0.178759, y = 0.266214, z = 1.72831, w = 5.84311}, and the new hyper chaotic system instable in fixed point.

Phase portraits

The system exhibits complex, chaotic behavior, as shown in the phase diagram. Figures 1 and 2 show the three and two-dimensional strange attractors. Numerical simulations were performed using Mathematica. The generated shape in the phase resembles the flapping of a butterfly's wings, hence the term "butterfly effect," and because even a small change can cause a significant change in the results.

Figure 1. Three-dimensional view of chaotic attractors

Figure 2. Chaotic attractors of 2D x-y phase

Bifurcation diagram

To understand what happens to a small number of $b$ values within a large range $\mathrm{b} \in[4.9,5]$. The bifurcation diagram in Figure 3 shows that as $b$ values increase, a cyclic doubling occurs in the region $4.9 \leq b \leq 5$.

0003.png

Figure 3. The bifurcation diagram of $b \in[4.9,5]$

Sensitivity to initial conditions

One of the most important characteristics of a system is its extreme sensitivity to even the slightest change in values. No matter how close the values of two different initial conditions may seem, they will eventually diverge. Therefore, there will come a time when the system's state becomes unpredictable. Figure 4 illustrates that the evolution of the chaotic path is highly sensitive to the initial conditions. The solid line is (x(0) = 0.5, y(0) = 0.4, z(0) = 1.5, w(0) = 0.6) and the value change is (x(0) = 0.5, y(0) = 0.4, z(0) = 1.5, w(0) = 0.60000000000001) for the dashed line.

Figure 4. Sensitivity tests of the of the trajectory w(t)

Using different color image diminution with varying types of extensions, the Lake image (1024 × 1024 .jpg file), boat image (512 × 512 .png file), and Peppers image (256 × 256 .tiff file) were tested. The algorithm was simulated using Python 3.9.7-amd64 and Visual Studio 2022. The numerous tests conducted yielded excellent results, including a histogram analysis. The results demonstrated that this method is robust, as evidenced by its appearance in various attack analyses, including Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI). It is difficult to hack or break, making it highly suitable for encrypting color images. Also, when decrypting the generated image, it matches the original one exactly, with no errors or missing values, as shown in the Peak Signal-to- Noise Ratio (PSNR) and Mean Squared Error (MSE).

5.1 Histogram analysis

Table 3 illustrates histogram analysis, a crucial statistic for measuring encryption quality. We observe that images in section b show increases and decreases in frequency, representing the repetitions within the image that reveal its content. In contrast, images in section d exhibit a stable distribution, lacking statistical similarity in visual appearance.

Table 3. Encryption results (a) org-imgs, (b) enc-imgs, (c) enc-histogram, (d) dec-imgs, (e) dec-histogram

5.2 Information entropy analysis 

In Table 4, a higher entropy value, close to 8, is obtained using the Shannon entropy measure, indicating higher security when calculating the frequency of image pixels. It is a perfect result that repels statistical attacks. The entropy image value is shown in the table below.

Table 4. Entropy of encrypted and original images

5.3 Correlation coefficient analysis

A coefficient value approaching zero indicates stronger encryption performance, as it implies minimal statistical similarity between the two images. In such cases, the encrypted image lacks discernible visual patterns, making it extremely difficult to interpret or reconstruct without possession of the correct decryption key [20]. Table 5 presents the results evaluating the robustness of the proposed image encryption method, explicitly using the correlation coefficient between the encrypted and original images.

Table 5. Correlation coefficient test for the three directions: (V, H, and D) for the encrypted and original images of the three images 'Peppers', 'Sailboat', and 'Lake'

5.4 Security differential attack test

One important test for evaluating the algorithm's efficiency is differential attack testing. The algorithm must prevent attacks on the encryption. The algorithm must ensure that any slight change in the original image results in a change in the encrypted image. This is measured using NPCR, where accuracy is above 99%, and UACI, where it is around 33%. Table 6 shows the results of the differential tests.

Table 6. NPCR and UACI results

5.5 Execution and complexity analysis of time

Table 7 contains the execution times for encrypting and decrypting the image, and the values are off by 1 second or a few parts. When using a specific encryption algorithm or designing a new one, time is one of the most important factors to consider. Therefore, based on previous tests that demonstrated its strength and the inability to decode it, the encryption time values clearly support the findings of the research. used Python 3.9, Visual Basic 2022, 16GB of RAM, and Windows 11. Table 8 illustrates the time allocation for each operation in the algorithm. The execution time increases with increasing input size and the number of pixels in the image. The time complexity depends entirely on the image size and the operations performed on the image during encryption to generate keys and perform the encryption.

Table 7. Time speed of three different image sizes

Table 8. The execution time of the algorithm

5.6 Analysis of MSE and PSNR ratio

Two essential metrics for evaluating decryption quality are PSNR and MSE. The current algorithm returns zero, indicating a perfect match between the decrypted image and the original, as shown in Table 9.

5.7 Key space analysis

One of the important properties of algorithms is the size of the key space. The key space represents the number of possible keys generated in the system. The larger the space, the greater the ability to repel attacks, as the key used cannot be predicted. Based on the system's equations, the space is $(10^{14})^{17} =10^{238} \simeq 2^{757}$, which is larger than (2¹²⁸), making it resistant to brute-force attacks.

5.8 Comparison results with other encryption methods

In Table 10, the work is compared with previously published works [2, 18], where better results were obtained, especially in entropy, as well as differences in the values reported in the previous studies [2, 18]. The Sailboat image was initially used at a small size, but in this work, it was used at 512 × 512. This resulted in better performance in terms of Entropy, NPCR, and UACI, with close correlations. As for comparing the Pepper image with the study by Shakir et al. [18], it is the same size, but in this work, it is colored, which means more complexity, and the results also show our work, especially in entropy.

Table 9. PSNR and MSE for plain and decrypted images

Table 10. Proposed algorithm comparison results with other encryption methods for the Sailboat and Pepper images