Color Image Encryption by Using Modified E0 Keystream Generator with Peter De Jong Chaotic Map

The studies were chosen because they share a lot of similarities with the subject matter of this study. Although some studies rely on old references, they remain of scientific value due to the lack of use of these methods in modern research (Table 1). It is still an important point for developing new solutions, which requires the use of these references.

El-Fishawy et al. [10] modified a version of the Bluetooth standard encryption technique, known as E0 stream encryption, was employed. The upgraded model is controlled by incorporating a 5th Linear Feedback Shift Register (LFSR) into the E0 encoder's master key flow generator. By encrypting various images, it has also been shown that the application of certain measuring criteria, such as the highest deviation, non-uniform deviation, and correlation coefficient, improves the coding quality. The higher level of encryption has been proved by the parameter measured on the image encryption attained through increased encryption. This method was limited to grayscale images and did not extend the solution to color images or enhance the complexity of key generation.

Saikia et al. [11] presented a partial bit-plane encryption method that breaks down an image into its component bit-planes. The purpose of the chaotic map Peter De Jong is to encrypt these bit-planes. Bit-plane component coordinates serve as the chaotic map's initial input. Coordinates are changed after a number of iterations, but the bits in the bit-plane are not changed immediately. By permuting the bit-planes, bits can be modified, changing the pixel values in an image. When encryption is done over partial bit-planes, correlation analysis not only decreases computing time but also demonstrates that no information is wasted. This method it focused only on partial bit encryption, which could compromise overall security. The method was limited to grayscale images. Hanchinamani and Kulkarni [12] presented an effective image encryption technique according to a chaotic map of Peter De Jong, RC4 is the proposed stream cipher in this work. Using a Peter De Jong map, the RC4 stream generator's initial keys are produced, it also employed throughout stage of permutation. The values that are pseudo-random for the diffusion and rotation of pixel values are produced using RC4 stream generator. Three steps make up each encryption round diffusion, pixel value rotation, and permutation. In addition to rotating the rows and columns in different directions, confusion between the rows and columns is the basis for the permutation. This method focused on grayscale images. Giesl et al. [13] provided an image encryption method depending on Peter-de Jong attractor map of chaos. Upon moving image into wavelet domain, the wavelet coefficients are modified as necessary. The wavelet transformation is primarily used to decrease the amount of calculation time required for the method of encryption and to achieve a greater or comparable level of the encrypted image's security. This method limited scope as it exclusively targeted grayscale images and required wavelet transformation. Altaay et al. [14] used a lightweight encryption technique in conjunction with the chaotic Peter De Jong map, they suggest an images encryption method with exceptional security. they used Peter De Jong map for the Lilliput algorithm, a lightweight encryption technique, is used to create keys. The proposed method was able to match the historical requirements for transmission images in terms of complexity. The lightweight approach may limit robustness in highly sensitive applications compared to more complex encryption methods and it limited on grayscale image.

Table 1. Comparison between the proposed algorithm and related works

The difference equations that Peter-De-Jong proposed and named for him are known as the Peter-De-Jong map. Despite its seemingly straightforward appearance, this chaotic system exhibits a number of intricate attractors that correlate to various parameter choices. The map is described as [19]:

$X_{n+1}=\sin \left(a_{y n}\right)-\cos \left(b_{x n}\right)$                 (5)

$Y_{n+1}=\sin \left(a_{y n}\right)-\cos \left(d_{y n}\right)$                  (6)

Using the parameters control parameters for a system that are chaotic are a, b, c, and d the values of xn and yn give the following two values: xn+1 and yn+1. The first step is to identify a rectangular portion of the plane that spans the image. It is necessary to choose a collection of distinct pixels. After that, calculate a number of map iterations and determine how long it takes to reach each pixel. Next, the mean and self-correlations of the chaotic output sequence should be computed. A selection of shapes from the map by Giesl et al. [13]. In Figure 2, Peter De Jong found this attractor, and the dynamic system where 1.4 for a, -2.3 for b, 2.4 for c, and -2.1 ford has been used for encryption.

2.png

Figure 2. Peter De Jong discovered this attractor, and the dynamical system where a=1.4, b=-2.3, c=2.4, and d=-2.1 is used for encryption

The proposed method is calculated and compared with the results of previous research, despite the previous research that used Peter De Jong in the encryption process for grayscale images with map encryption research using Peter De Jong due to the complexity of the color image encryption process in that it contains 3 color channels, and will compare the efficiency of the method. Where used Python 3.9 for Dell laptop Intel(R) Core (TM) 64-bit Windows 11 Pro i7-9850H 16.0 GB of RAM with a 2.60GHz CPU.

6.1 Data set

From images in the database of the USC-SIPI (sipi.usc.edu/db/), the test images were selected. from the Miscellaneous volume. chosen color images that has 24 bits per pixel in different sizes (256*256, 512*512). similar to the one used in previous study [12]. This dataset's images are available in ".tiff" format. The dataset's samples are displayed in Table 2.

Table 2. Data set samples

Image	Dimension	Title
2-1.png	256*256	Female
2-2.png	256*256	House
2-3.png	512*512	Pepper

6.2 NIST standard test

The results of the NIST randomness test all showed positive results, which indicates the randomness of the key utilized throughout the encryption procedure, which indicates ability to use the key in different applications. It appears random and unpredictable, as 0.01 is the minimum randomness in the test. Table 3 shows the results of the randomness that was tested on the key.

Table 3. NIST results

6.3 Key space analysis

One important metric is the key space for assessing the extent to which the method recommended endures brute force attacks [21]. The key generated by the E0 algorithm is 128 bits. After adding the new registers in the proposed method, the key size is 2²¹², which is a large size that is resistant to brute-force attacks.

6.4 Information entropy analysis

The most significant indicator of a cryptosystem's strength is its information entropy. Eq. (13) defines a plain image's information entropy. Information entropy should be 8 for an 8-bit truly random image, which would prevent attackers from learning anything of value from the image. According to Eq. (13), the technique achieves better entropy through uniform distribution [22]. The proposed method showed good results close to 8 as shown in Table 4, where it was compared with the results of the study [12]. The proposed method consistently shows slightly higher entropy values, indicating better uniformity and stronger encryption.

Entropy $E(X)=\sum_{i=1}^x p\left(X_i\right) \log \frac{1}{p\left(X_i\right)}$                         (13)

Table 4. Information entropy analysis

6.5 Differential attack analysis

An effective encryption technique should prevent differential attack by making sure that even little changes to the ciphered image differ noticeably from source image. evaluating and analyzing differential attacks with UACI and NPCR. The NPCR measurement establishes the impact of a single pixel alteration on the entire image [23]. The NPCR and UACI value calculation formulas are provided in (14) and (15). The source image (plain image) and the ciphered image of size m×n is compared to calculate the results.

$\mathrm{NPCR}=\frac{\sum_{\mathrm{i}, \mathrm{j}} \mathrm{D}(\mathrm{i}, \mathrm{j})}{\mathrm{W} \times \mathrm{H}} \times 100 \%$                         (14)

$\mathrm{UACI}=\frac{1}{W X H}\left[\sum_{\mathrm{i}, \mathrm{j}} \frac{\left|\mathrm{C}_1(\mathrm{i}, \mathrm{j})-\mathrm{C}_2(\mathrm{i}, \mathrm{j})\right|}{255}\right] \times 100 \%$                         (15)

where, $\mathrm{f}=\left\{\begin{array}{l}0, \mathrm{ifC}_1(\mathrm{i}, \mathrm{j})=\mathrm{C}_2 \\ 1, \mathrm{ifC}_1(\mathrm{i}, \mathrm{j}) \neq \mathrm{C}_2\end{array}\right.$

Table 5. Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI)

The results are compared with the study [12], where in the study [12] the average values to NPCR and UACI 99.616815% and 33.465988% respectively, where our proposed method the UACI is 33.50% and the NPCR is 99.60%. These results indicate that the proposed method performs similarly to the research [12], with a slight improvement in NPCR and a very close UACI value (Table 5). This means that it is resistant to attacks.

6.6 Histogram analysis

Analyzing the relative occurrence frequency of various pixel values is done via histogram analysis. Table 6 displays the histograms for the encrypted and plain images. A plain image has a particular histogram pattern. The encrypted image's histogram indicates that the pixel values are distributed as uniformly as possible. This suggests that the proposed method is difficult to attack statistical attack. The results are compared to the study [12], where they used grayscale images with the color images that encrypted using the proposed method. In both methods, the histograms of the encrypted images show a similar distribution pattern, indicating that the proposed method performs well to that in the research [12] for color images.

Table 6. Histogram analysis

Image	Original Image Histogram	Proposed Method	Ref. [12]
Female	61.png	64.png	67.png
House	62.png	65.png	biao_6_zui_you_lie_2.png
Pepper	63.png	66.png	69.png

6.7 Time analysis

Time in seconds is used to assess the speed execution for the proposed method that required for the encryption, decryption and key generation procedures, and the proposal shows different performance levels over time, as shown in Table 7.

Table 7. Time analysis

According to previous study [12], the time required to encrypt the image is 0.000006 While encrypting color images takes longer than grayscale images, as noted in previous study [12], the execution times of encryption and decryption remain efficient, demonstrating the feasibility of the proposed method for practical applications despite the increasing complexity of dealing with color images. Future work could focus on improving the key generation and encryption processes to further reduce the computation time, perhaps through parallel processing techniques or lightweight encryption algorithms, while maintaining the high levels of security achieved.

6.8 MSE and PSNR

This study uses the MSE and PSNR, which are calculated using Eqs. (16) and (17) to determine the difference between the plain and encrypted images.

$M S E=\frac{1}{\mathrm{MN}} \sum_{\mathrm{i}=1}^{\mathrm{M}} \sum_{\mathrm{j}=1}^{\mathrm{N}}(|I(i, j)-K(i, j)|)^2$                         (16)

$P S N R=20 X \log _{10}\left(\frac{255}{\sqrt{M S E}}\right) d b$                         (17)

where, the pixel value of the source (original image) is I (i, j), and the pixel value of the cipher image (encrypted image) at position (i, j) is K (i, j) [24].

MSE values of the encrypted images indicate a significant difference between the encrypted and original images. The proposed method achieves high MSE values, which is expected in a strong encryption technique, as higher MSE values indicate that the encrypted image differs significantly from the original image, making it resistant to cryptographic attacks (Table 8).

Table 8. MSE result for proposed method between encrypted image and original

Table 9. PSNR result for proposed method between encrypted image and original image compare with Ref. [12]

Table 9 shows that the proposed method achieves lower PSNR values compared to the study [12], indicating stronger encryption. Future work could explore methods to optimize PSNR while preserving security.

6.9 Correlation coefficient analysis

An original image's high the correlation relationship between neighboring pixels in the diagonal, vertical, and horizontal directions is one of most crucial features. Two pixels' degree of resemblance is assessed by the correlation test [25]. Estimating the similarities between the encrypted image and the original is made easier with using the correlation coefficient analysis. When both the original and encrypted images match exactly, it should be 1 for the correlation coefficient. The color bytes' correlation coefficients are computed using Eq. (18). The correlation coefficient values in Tables 10 and 11 are becoming closer to "0" and negative, indicating that the plain and cipher images are unrelated [26].

$Y_{\mathrm{x}, \mathrm{y}}=\frac{\sum\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)\left(\mathrm{y}_{\mathrm{i}}-\overline{\mathrm{y}}\right)}{\sqrt{\sum\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^2} \sqrt{\sum\left(\mathrm{y}_{\mathrm{i}}-\overline{\mathrm{y}}\right)^2}}$                          (18)

Table 10. Correlation coefficient analysis for encrypted image and original image compare with Ref. [12]

Table 11. Correlation coefficient analysis for decrypted image and original image compared with Ref. [12]

The results in Table 10 show that the correlation values of the encrypted images in the proposed method are slightly higher than those in Reference [12] in some directions. While this indicates a slightly lower spread compared to Reference [12], the values remain close to zero, ensuring sufficient encryption security. On the other hand, Table 11 highlights that the proposed method achieves higher correlation values for the encrypted images, confirming better accuracy in restoring the original image.