1. Introduction

2. Preliminary Mathematical Constructs

2.2. The 4D Hyperchaotic System

A 3D hyperchaotic system was proposed for the first time by the authors of [33]. This system was later on improved by the authors of [34] who added a linear controller, resulting in the following 4D system of differential equations:

$$\left\{\begin{array}{c}\dot{u}=a(\nu -u)+e\nu w,\hfill \\ \dot{\nu }=cu+d\nu -uw+mp,\hfill \\ \dot{w}=-bw+u\nu ,\hfill \\ \dot{p}=-ku-k\nu ,\hfill \end{array}\right.$$

(2)

where $u,\nu ,w$ and p are the states of the system and $a,b,c,d,e,m,k$ are positive real parameters of the system and $c\in \mathbb{R}$. By letting $a=15,b=43,c=-1,d=16,e=5,m=5$ and varying k, such that $k\in [1.5,5.5]$, the authors of [34] show that the system in (2) possess hyperchaotic attractors and 2 positive Lyapunov exponents, providing the related plots that justify their claims (see Figure 1 for the Lyapunov exponents and Figure 2 for the bifurcation diagram, in [34]). In relation to image encryption, these properties make this system a good choice for utilization in PRNG generation and subsequent S-box construction. Furthermore, as a 4D system, the system in (2) has a large number of states and parameters, which allows it to provide an excellent expansion to the key space of the proposed image encryption algorithm.

Figure 4. Hyperchaotic attractors for the system (2) with $u_0=5.2,{\nu }_0=7.4,w_0=1.4,p_0=3.4$, shown for various 3D spaces.

Figure 4. Hyperchaotic attractors for the system (2) with $u_0=5.2,{\nu }_0=7.4,w_0=1.4,p_0=3.4$, shown for various 3D spaces.

3. Proposed Image Encryption Framework

3.1. The Encryption Process

The following sequence of steps outline the proposed 3-stage encryption process.

1.

A color image I of length M and width N is chosen and the pixel values of its RGB channels are arranged as a single bit-stream d, of length $L_d$ where:

$$L_d=M\times N\times 3\times 8.$$

(4)

2.

Encryption stage 1: The 7D hyperchaotic system encryption key and S-box.

(a)

The 7D hyperchaotic system is numerically solved for the S-box seeds, then the numerical solution is utilized in Algorithm 1, resulting in an S-box $S_{HC7D}$ (shown in Table 1).

(b)

A replacement process is applied to the image I, employing $S_{HC7D}$, and resulting in image $I_{11}$

$$I_{11}=S_{HC7D}\left(I\right).$$

(5)

(c)

The pixels of the encrypted image $I_{11}$ are transformed into a 1D bit-stream $d_{11}$.

(d)

The 7D hyperchaotic system is numerically solved for the key seeds. Algorithm 2 is utilized to generate an encryption key $k_{HC7D}$ from the obtained solution, such that the length of this key is equal to $L_d$.

(e)

The plaintext bit-stream $d_{11}$ is XORed with the encryption key $k_{HC7D}$

$$d_{12}=d_{11}\oplus k_{HC7D}.$$

(6)

(f)

The resulting bit-stream $d_{12}$ is transformed back into an image $I_{12}$.

3.

Encryption stage 2: The single neuron model encryption key and S-box.

(a)

The single neuron model is numerically solved for the S-box seeds, then the numerical solution is utilized in Algorithm 1, resulting in an S-box $S_{SNM}$ (shown in Table 2).

(b)

A replacement process is applied to the image $I_{12}$, employing $S_{SNM}$, and resulting in image $I_{21}$

$$I_{21}=S_{SNM}\left(I_{12}\right).$$

(7)

(c)

The pixels of the encrypted image $I_{21}$ are transformed into a 1D bit-stream $d_{21}$.

(d)

The single neuron model is numerically solved for the key seeds. Algorithm 2 is utilized to generate an encryption key $k_{SNM}$ from the obtained solution, such that the length of this key is equal to $L_d$.

(e)

The bit-stream of the encrypted image $d_{21}$ are XORed with the encryption key $k_{SNM}$

$$d_{22}=d_{21}\oplus k_{SNM}.$$

(8)

(f)

The resulting data bits $d_{22}$ are transformed back into an image $I_{22}$.

4.

Encryption stage 3: The 4D hyperchaotic system encryption key and S-box.

(a)

The 4D hyperchaotic system is numerically solved for the S-box seeds, then the numerical solution is utilized in Algorithm 1, resulting in an S-box $S_{HC4D}$ (shown in Table 3).

(b)

A replacement process is applied to the image $I_{22}$, employing $S_{HC4D}$, and resulting in image $I_{31}$

$$I_{31}=S_{HC4D}\left(I_{22}\right).$$

(9)

(c)

The pixels of the encrypted image $I_{31}$ are transformed into a 1D bit-stream $d_{31}$.

(d)

The 4D hyperchaotic system is numerically solved for the key seeds. Algorithm 2 is utilized to generate an encryption key $k_{HC4D}$ from the obtained solution, such that the length of this key is equal to $L_d$.

(e)

The bit-stream of the encrypted image $d_{31}$ are XORed with the encryption key $k_{HC4D}$

$$d_{32}=d_{31}\oplus k_{HC4D}.$$

(10)

(f)

The resulting data bits $d_{32}$ are transformed back into an image $I_{32}$. $I^{\prime }=I_{32}$ is the final output encrypted image.

A flow chart illustrative of the proposed 3-stage encryption process is provided in Figure 6.

Figure 6. Flow chart of the proposed 3-stage encryption process.

Figure 6. Flow chart of the proposed 3-stage encryption process.

3.2. The Decryption Process

The following sequence of steps outline the proposed 3-stage decryption process. They are in a reverse order to those applied in the encryption process.

• Beginning with the 3-stage encrypted image $I^{\prime }=I_{32}$ of length M and width N.
• Decryption stage 3: The 4D hyperchaotic system decryption key and inverse S-box.

  (a)

  The pixel values of the encrypted image $I_{32}$ are transformed into a bit-stream $d_{32}$.

  (b)

  The encrypted data bits $d_{32}$ are XORed with the decryption key $k_{HC4D}$

  $$d_{31}=d_{32}\oplus k_{HC4D}.$$

  (11)

  (c)

  The resulting encrypted data bits $d_{31}$ are converted into an image $I_{31}$.

  (d)

  A reverse replacement process is applied to the image $I_{31}$, employing $S_{HC4D}^{-1}$, and resulting in image $I_{22}$

  $$I_{22}=S_{HC4D}^{-1}\left(I_{31}\right).$$

  (12)

• Decryption stage 2: The single neuron model decryption key and inverse S-box.

  (a)

  The pixel values of the encrypted image $I_{22}$ are transformed into a bit-stream $d_{22}$.

  (b)

  The encrypted data bits $d_{22}$ are XORed with the decryption key $k_{SNM}$

  $$d_{21}=d_{22}\oplus k_{SNM}.$$

  (13)

  (c)

  The resulting encrypted data bits $d_{21}$ are converted into an image $I_{21}$.

  (d)

  A reverse replacement process is applied to the image $I_{21}$, employing $S_{SNM}^{-1}$, and resulting in image $I_{12}$

  $$I_{12}=S_{SNM}^{-1}\left(I_{21}\right).$$

  (14)

• Decryption stage 1: The 7D hyperchaotic system decryption key and inverse S-box.

  (a)

  The pixel values of the encrypted image $I_{12}$ are transformed into a bit-stream $d_{12}$.

  (b)

  The encrypted data bits $d_{12}$ are XORed with the decryption key $k_{HC7D}$

  $$d_{11}=d_{12}\oplus k_{HC7D}.$$

  (15)

  (c)

  The resulting encrypted data bits $d_{11}$ are converted into an image $I_{11}$.

  (d)

  A reverse replacement process is applied to the image $I_{11}$, employing $S_{HC7D}^{-1}$, and resulting in a plain image I

  $$I=S_{HC7D}^{-1}\left(I_{11}\right).$$

  (16)

A flow chart illustrative of the proposed 3-stage decryption process is provided in Figure 7.

Figure 7. Flow chart of the proposed 3-stage decryption process.

Figure 7. Flow chart of the proposed 3-stage decryption process.

3.3. Utilized Algorithms

Algorithm 1 describes the generation of an S-box given a pseudo-random bit-stream generated using one of the 3 PRNGs earlier proposed in Section 2. As discussed in SSection 2.4, towards treating the evaluation values of the S-box as part of the key space, Algorithm 1 attempts to achieve a certain set of provided values for S-box evaluations. The approach towards that is to generate an agreed upon number of S-boxes (which is a part of the key space as well), then each S-box is evaluated separately, and the chosen S-box is the one with performance evaluation values closer to the target values. Accordingly, it may come naturally to always provide optimal S-box evaluation values for every attempt of S-box generation. Nevertheless, fulfilling the need for complicating any cryptanalyses efforts, other, sub-optimal values may be provided resulting in the generation of different S-boxes. Such a decision would not affect the overall encryption process much as each generated S-box is only one component out of many stages. Therefore, performance evaluation of the encryption process is based on the integration of all the components in all 3 stages of the proposed encryption framework. Algorithm 2 elaborates the followed procedure in generating a PRNG bit-stream given the solution of a chaotic system.

Algorithm 1:Generate an S-box given a bit-stream $b_{PRNG}$, the number of S-box trials n, and target performance metric values $M=\{NL,SAC,BIC,LAP,DAP\}$, and a bit-stream $b_{PRNG}$ (an adaptation from that proposed in [14])

Algorithm 2:Generate a PRNG bit-stream given a chaotic system S of k dimensions, and the number of needed bits n (first proposed in [14])

4. Numerical Results and Performance Evaluation

4.2. Statistical Analyses

The mean squared error (MSE), peak signal-to-noise ratio (PSNR) and maximum absolute error (MAE) between input plain images and output encrypted images are reported in Table 5, Table 6 and Table 7, respectively. It is clear from each of the tables that the computed values for the proposed image encryption algorithm are comparable or superior to the state-of-the art, with high MSE and MAE values, and thus low PSNR values, indicating excellent scrambling and randomization performance of the proposed image encryption framework.

As for Shannon’s information entropy, Table 8 displays the computed values for various encrypted images, and compares those values with the state-of-the-art. Excellent entropy performance of $7.999$ is showcased for the proposed image encryption framework, being superior or comparable to the counterpart algorithms.

The pixel cross correlation coefficient $\rho$ is a measure of the linear dependency between neighboring pixel values in an image. It quantifies how well the encryption process has scrambled the pixels and eliminated any spatial relationships. For a well encrypted image, $\rho$ should have a value close to 0, indicating little or no correlation between adjacent pixel values. Table 9 provides the $\rho$ values for various images, each computed in 3 directions (horizontal, vertical and diagonal). While the plain images have $\rho$ values close to 1, their encrypted versions have $\rho$ values close to 0. A visual illustration of this metric is provided as a set of 2D plots in Figure 14 for the Lena image, as well as in Figure 15, Figure 16 and Figure 17, respectively, for each of its RGB channels. While sub-figures (c) and (f) of Figure 13 provide 3D plots of the same metric for the House image. It is clear that the plots of encrypted images exhibit a uniform distribution of values, unlike those of their plain versions. Table 10 and Table 11 display a comparison of $\rho$ with the state-of-the-art, utilizing the Lena image and each of its RGB channels, respectively. The computed $\rho$ values are shown to be near 0, as is the case in the state-of-the-art, in each of the tables.

Table 4. Performance evaluation metrics and their mathematical expressions.

Table 4. Performance evaluation metrics and their mathematical expressions.

Metric	Mathematical Expression
MSE	$$MSE=\frac{{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}{(I_{(i,j)}-I_{(i,j)}^{\prime })}^2}{M\times N},$$ (17) where I and $I^{\prime }$ are 2 images of dimensions $M\times N$.
PSNR	$$PSNR=10log\left(\frac{I_{max}^2}{MSE}\right),$$ (18) where $I_{max}=255$.
MAE	$$MAE=\frac{{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}|I_{(i,j)}-I_{(i,j)}^{\prime }|}{M\times N}.$$ (19)
Entropy	$$H\left(m\right)=\sum _{i=1}^{M}p\left(m_i\right){log}_2\frac{1}{p\left(m_i\right)},$$ (20) where $p\left(mi\right)$ is the probability of occurrence of symbol m while M is the total number of bits for each symbol.
DFT	$$F(k,l)=\sum _{i=0}^{N-1}\sum _{j=0}^{N-1}f(i,j)e^{-i2\pi (\frac{ki}{N}+\frac{li}{N})},$$ (21) $f(a,b)$ is the spatial domain representation of the image, where the exponential term is the basis function corresponding to each point $F(k,l)$ in the Fourier space.
CC	$$\rho (x,y)=\frac{cov(x,y)}{\sqrt{\sigma \left(x\right)}\sqrt{\sigma \left(y\right)}},$$ (22) $$\mathrm{where},cov(x,y)=\frac{1}{N}\sum _{i=1}^N(x_i-\mu \left(x\right))(y_i-\mu \left(y\right)),$$ (23) $$\sigma \left(x\right)=\frac{1}{N}\sum _{i=1}^N{(x_i-\mu \left(x\right))}^2,\mathrm{and}\mu \left(x\right)=\frac{1}{N}\sum _{i=1}^N\left(x_i\right).$$ (24)
NPCR	$$NPCR=\frac{{\sum }_{x=1}^M{\sum }_{y=1}^{N}D(x,y)}{M\times N}\times 100,$$ (25) $$\mathrm{where},D(x,y)=\left.\left\{\begin{array}{cc}0\hfill & I(x,y)=I^{\prime }(x,y)\hfill \\ 1\hfill & Otherwise\hfill \end{array}\right.\right|x\in [1,M]\&y\in [1,N].$$ (26)
UACI	$$UACI=\frac{1}{M\times N}\sum _{x=1}^M\sum _{y=1}^N\frac{|I(x,y)-I^{\prime }(x,y)|}{255}\times 100,$$ (27)

Another interesting manner of examining the pixel cross correlation among pixels would be to do so in the Fourier domain. By generating the discrete Fourier transform (DFT) of the plain and encrypted House image, as shown in sub-figures (b) and (e) of Figure 13 and visually examining them. The presence of a bright star-like shape at the center of the DFT plain image is characteristic of a normal plain image which has pixels depicting edges and corners. This is unlike the DFT encrypted image, which lacks any identifying visual characteristics.

Since randomness is an integral measure of any image encryption algorithm, an objective quantitative measure of it should be utilized. This is best carried out through a National Institute of Standards and Technology (NIST) SP 800 analysis [36]. In brief, a NIST analysis tests a bit-stream for a number of characteristics, including its randomness, as a well as quantifies its randomness and detects structural weaknesses. Such an analysis provides empirical evidence of the suitability of a PRNG’s use in cryptography applications. Table 12 presents the results of a NIST analysis carried out on a bit-stream depicting the data of an encrypted Girl image. All NIST tests are successfully passed, with scored $p-$values greater than $0.01$.

Table 5. MSE values for various images, for the proposed algorithm and the state-of-the-art.

Table 5. MSE values for various images, for the proposed algorithm and the state-of-the-art.

Image	Proposed	[7]	[16]	[37]	[17]	[6]	[21]	[14]
Lena	$8890.05$	$9112.1$	$8926.96$	$10869.73$	$4859.03$	$8888.88$	N/A	$8912.4$
Mandrill	$8345.25$	$8573.38$	$8290.84$	$10930.33$	$6399.05$	$8295.21$	N/A	$8320.41$
Peppers	$10074.0$	$10298.7$	$10045.1$	N/A	$7274.44$	$10092.3$	N/A	$10065.4$
House	$8361.44$	$8427.04$	$8351.64$	N/A	N/A	N/A	N/A	$8395.53$
House2	$9190.27$	$9374.65$	N/A	N/A	N/A	N/A	N/A	$9142.54$
Girl	$12152.8$	$12450.9$	N/A	N/A	N/A	N/A	N/A	$12104.2$
Sailboat	$10063.3$	N/A	N/A	N/A	N/A	N/A	N/A	$10071.9$
Tree	$9931.63$	N/A	N/A	N/A	N/A	N/A	N/A	$9873.24$
Average	$9626.09$	$9706.13$	$8903.64$	10900	$6177.51$	$9092.13$	N/A	$9610.65$

Table 6. PSNR values for various images, for the proposed algorithm and the state-of-the-art.

Table 6. PSNR values for various images, for the proposed algorithm and the state-of-the-art.

Image	Proposed	[7]	[16]	[37]	[17]	[6]	[21]	[14]
Lena	$8.64176$	$8.53462$	$8.6237$	$7.7677$	$11.3$	$8.64233$	$8.5674$	$8.63086$
Mandrill	$8.91641$	$8.79929$	$8.9448$	$7.7447$	$10.10$	$8.94253$	$10.0322$	$8.92936$
Peppers	$8.09877$	$8.00296$	$8.11128$	N/A	$9.55$	N/A	N/A	$8.10248$
House	$8.90799$	$8.87405$	$8.91309$	N/A	N/A	N/A	N/A	$8.89032$
House2	$8.49752$	$8.41125$	N/A	N/A	N/A	N/A	N/A	$8.52013$
Girl	$7.28403$	$7.17879$	N/A	N/A	N/A	N/A	N/A	$7.30144$
Sailboat	$8.10339$	N/A	N/A	N/A	N/A	N/A	N/A	$8.0997$
Tree	$8.1606$	N/A	N/A	N/A	N/A	N/A	N/A	$8.18621$
Average	$8.32631$	$8.30016$	$8.64822$	$7.7562$	$10.3167$	$8.79243$	$9.2998$	$8.33256$

Table 7. PSNR values for various images, for the proposed algorithm and the state-of-the-art.

Table 7. PSNR values for various images, for the proposed algorithm and the state-of-the-art.

Image	Proposed	[7]	[6]	[37]	[38]	[21]	[14]
Lena	$77.409$	$78.3564$	$77.3752$	87	$77.35$	$77.96$	$77.4877$
Peppers	$82.0156$	$82.3273$	$81.7740$	N/A	$74.71$	N/A	$81.9832$
Mandrill	$75.3335$	$81.913$	$75.1659$	92	$73.91$	$67.85$	$75.1632$
House	$75.3132$	N/A	N/A	N/A	N/A	N/A	$75.4983$
House 2	$78.5675$	N/A	N/A	N/A	N/A	N/A	$78.3327$
Girl	$90.1646$	N/A	N/A	N/A	N/A	N/A	$89.9807$
Sailboat	$82.0101$	N/A	N/A	N/A	N/A	N/A	$82.1003$
Tree	$81.4948$	N/A	N/A	N/A	N/A	N/A	$81.1623$
Average	$80.9993$	$80.8656$	$78.105$	$89.5$	$75.3233$	$72.905$	$80.2136$

Table 8. Entropy values for various images, for the proposed algorithm and the state-of-the-art.

Table 8. Entropy values for various images, for the proposed algorithm and the state-of-the-art.

Image	Proposed	[7]	[16]	[37]	[39]	[17]	[6]	[21]	[14]
Lena	$7.999$	$7.9856$	$7.999$	$7.999$	$7.997$	$7.996$	$7.997$	$7.9972$	$7.99887$
Mandrill	$7.999$	$7.9905$	$7.999$	$7.999$	$7.999$	N/A	$7.996$	$7.9969$	$7.99866$
Peppers	$7.999$	$7.9951$	$7.999$	$7.9991$	N/A	$7.997$	$7.9969$	N/A	$7.99834$
House	$7.999$	$7.9577$	$7.999$	N/A	N/A	N/A	N/A	N/A	$7.99729$
House2	$7.999$	$7.9847$	N/A	N/A	N/A	N/A	N/A	N/A	$7.99848$
Girl	$7.999$	$7.9789$	N/A	N/A	N/A	N/A	N/A	N/A	$7.99477$
Sailboat	$7.999$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	$7.99875$
Tree	$7.999$	N/A	N/A	N/A	N/A	N/A	N/A	N/A	$7.99713$
Average	$7.999$	$7.98208$	$7.999$	$7.999$	$7.99903$	$7.9965$	$7.99663$	$7.99705$	$7.99711$

Figure 13. House image alongside its Fourier transformation and 3D plot of its pixel cross-correlation matrix pre- and post-encryption.

Figure 13. House image alongside its Fourier transformation and 3D plot of its pixel cross-correlation matrix pre- and post-encryption.

Table 9. Correlation coefficient values for various plain and encrypted images.

Table 9. Correlation coefficient values for various plain and encrypted images.

Plain Image				Encrypted Image
Correlation Coefficient				Correlation Coefficient
Image	Horizontal	Diagonal	Vertical	Horizontal	Diagonal	Vertical
Lena	$0.938611$	$0.913175$	$0.96833$	$0.00180128$	$-0.000991502$	$-0.00018608$
Mandrill	$0.848778$	$0.750624$	$0.79088$	$0.00713777$	$-0.00152782$	$0.00491305$
Peppers	$0.959422$	$0.930426$	$0.966795$	$0.00301689$	$0.00419115$	$-0.00012237$
House	$0.978232$	$0.936044$	$0.952926$	$-0.00147997$	$-0.00286442$	$-0.0015624$
House2	$0.907075$	$0.850782$	$0.923091$	$0.000841531$	$-0.00214171$	$-0.00626431$
Girl	$0.974013$	$0.951471$	$0.965671$	$0.000841531$	$-0.00214171$	$-0.00626431$
Sailboat	$0.952381$	$0.0.919872$	$0.950138$	$0.00608092$	$0.00279574$	$0.00170383$
Tree	$0.968153$	$0.929967$	$0.94515$	$-0.00226616$	$0.00137505$	$0.00332063$

Table 10. Correlation coefficient values comparison with the state-of-the-art, for an encrypted Lena image.

Table 10. Correlation coefficient values comparison with the state-of-the-art, for an encrypted Lena image.

Algorithm	Horizontal	Diagonal	Vertical
Proposed	$0.00180128$	$-0.000991502$	$-0.000186079$
[6]	$0.002287$	$-0.00132$	$-0.00160$
[7]	$0.003265$	$-0.00413$	$0.002451$
[14]	$0.0064113$	$-0.0015143$	$0.000568333$
[19]	$0.00144$	$-0.00151$	$0.00795$
[21]	$-0.0061$	$-0.0018$	$0.0067$
[24]	$0.000199$	$0.003705$	$-0.000924$
[37]	$0.0054$	$0.0054$	$0.0016$

Table 11. Color channel separated correlation coefficient value comparison with the state-of-the-art, for the Lena image.

Table 11. Color channel separated correlation coefficient value comparison with the state-of-the-art, for the Lena image.

Channel	Direction	Plain image	Encrypted image	[25]	[26]	[27]	[6]
Red	Horizontal	$0.952474$	$0.00266725$	$0.001365$	$0.0021$	$0.9568$	$-0.00364$
	Diagonal	$0.928029$	$0.00564219$	$0.000232$	$-0.0026$	$0.0075$	$0.00016$
	Vertical	$0.975913$	$-0.0008351$	$0.004776$	$0.0018$	$-0.0376$	$0.000697$
Green	Horizontal	$0.935628$	$0.00307568$	$0.003294$	$-0.0006$	$0.0020$	$0.000118$
	Diagonal	$0.910534$	$-0.0020740$	$0.004807$	$0.0001$	$-0.0046$	$0.00177$
	Vertical	$0.966647$	$-0.0011823$	$-0.000579$	$0.0004$	$-0.0013$	$-0.0011$
Blue	Horizontal	$0.917439$	$-0.00046821$	$0.002060$	$-0.005$	$0.0071$	$-0.00164$
	Diagonal	$0.888482$	$-0.0025489$	$-0.004043$	$-0.0104$	$-0.0009$	$-0.00523$
	Vertical	$0.947961$	$0.0053944$	$0.000194$	$0.001$	$-0.0423$	$0.006041$

Figure 14. 2D plot of pixel cross-correlation matrices of the Lena image pre- and post-encryption.

Figure 14. 2D plot of pixel cross-correlation matrices of the Lena image pre- and post-encryption.

Figure 15. 2D plot of pixel cross-correlation matrices of the red channel of the Lena image pre- and post-encryption.

Figure 15. 2D plot of pixel cross-correlation matrices of the red channel of the Lena image pre- and post-encryption.

Figure 16. 2D plot of pixel cross-correlation matrices of the green channel of the Lena image pre- and post-encryption.

Figure 16. 2D plot of pixel cross-correlation matrices of the green channel of the Lena image pre- and post-encryption.

Figure 17. 2D plot of pixel cross-correlation matrices of the blue channel of the Lena image pre- and post-encryption.

Figure 17. 2D plot of pixel cross-correlation matrices of the blue channel of the Lena image pre- and post-encryption.

Table 12. NIST analysis on Girl encrypted image.

Table 12. NIST analysis on Girl encrypted image.

Test Name	Value	Remarks
Frequency	$0.425242$	Success
Block Frequency	$0.276783$	Success
Run	$0.714157$	Success
Longest run of ones	$0.932530$	Success
Rank	$0.138292$	Success
Spectral FFT	$0.846825$	Success
Non overlapping	$0.613084$	Success
Overlapping	$0.449023$	Success
Universal	$0.687511$	Success
Linear complexity	$0.241374$	Success
Serial	$0.850353$	Success
Approximate Entropy	$0.030357$	Success
Cumulative sum (forward)	$0.503876$	Success
Cumulative sum (reverse)	$0.783156$	Success

4.5. Histogram Dependency Tests

In this testing perspective, the histograms of the plain and encrypted images are statistically compared in order to show the lack of statistical dependence between the two. Table 17 shows the five utilized tests (Blomqvist $\beta$, Goodman-Kruskal $\gamma$, Kendall $\tau$, Spearman $\rho$, and Pearson r) and their mathematical expressions [45]. Table 18 demonstrates the results of applying these tests on a variety of images. As seen in the table, most of the resulting values are approaching 0, which shows the lack of any statistical dependency between every plain image and its encrypted version.

Table 17. Histogram dependency tests and their mathematical expressions.

Table 17. Histogram dependency tests and their mathematical expressions.

Metric	Mathematical Expression
Blomqvist	$$\beta =\{(X-\overline{x})(Y-\overline{y})>0\}-\{(X-\overline{x})(Y-\overline{y})<0\}.$$ (28)
Goodman-Kruskal	$$\gamma =\frac{n_c-n_d}{n_c+n_d}.$$ (29)
Kendall	$$\tau =\frac{n_c-n_d}{\frac{n(n-1)}{2}}.$$ (30)
Spearman	$$\rho =\frac{\sum (R_{ix}-{\overline{R}}_x)(R_{iy}-{\overline{R}}_y)}{\sqrt{\sum {(R_{ix}-{\overline{R}}_x)}^2\sum {(R_{iy}-{\overline{R}}_y)}^2}}.$$ (31)
Pearson	$$r=\frac{\sum (X_i-\overline{X})(Y_i-\overline{Y})}{\sqrt{\sum {(X_i-\overline{X})}^2\sum {(Y_i-\overline{Y})}^2}}.$$ (32)

Table 18. Tests of histogram dependency for various images.

Table 18. Tests of histogram dependency for various images.

Image	Color	$\mathit{\beta }$ (28)	$\mathit{\gamma }$ (29)	$\mathit{\tau }$ (30)	$\mathit{\rho }$ (31)	r (32)
Lena	Red	$0.00397832$	$0.0313687$	$0.0303408$	$0.0434341$	$0.0270663$
	Green	0	$0.0215872$	$0.014269$	$0.0166966$	$0.0193475$
	Blue	$-0.0793976$	$-0.0608106$	$-0.0572115$	$-0.0818634$	$-0.109206$
	Combined	$0.0277309$	$-0.0199151$	$-0.0198071$	$-0.0329345$	$-0.0468979$
Peppers	Red	$-0.0316218$	$-0.0270098$	$-0.0264574$	$-0.036404$	$-0.0223967$
	Green	$0.0355072$	$0.0185307$	$0.0183487$	$0.0317549$	$0.0375528$
	Blue	$-0.00396106$	$-0.0164391$	$-0.016167$	$-0.0263073$	$-0.0482875$
	Combined	$0.0948802$	$0.0436563$	$0.0433807$	$0.0645118$	$0.0666087$
Mandrill	Red	$-0.0198078$	$-0.0466956$	$-0.0462145$	$-0.0686459$	$-0.0672317$
	Green	$0.0676058$	$0.0704028$	$0.0691334$	$0.099575$	$0.108835$
	Blue	$-0.00394524$	$-0.027233$	$-0.0269739$	$-0.0414617$	$-0.032399$
	Combined	$0.0316238$	$-0.00105387$	$-0.00104776$	$-0.00335102$	$-0.00619468$
House	Red	$-0.078125$	$-0.0735036$	$-0.0713676$	$-0.10807$	$0.129594-0.124939$
	Green	$0.0278483$	$0.0505291$	$0.0499853$	$0.0757827$	$0.100048$
	Blue	$-0.0558677$	$-0.0186177$	$-0.0178087$	$-0.0270381$	$-0.0762734$
	Combined	$-0.03125$	$-0.0270421$	$-0.0268626$	$-0.0384522$	$-0.0603119$
House2	Red	$-0.122785$	$-0.11135$	$-0.109577$	$-0.159726$	$-0.187521$
	Green	$-0.0159414$	$-0.00680612$	$-0.00674228$	$-0.00405984$	$0.0230262$
	Blue	$0.0560937$	$0.0890006$	$0.0876046$	$0.131527$	$0.132005$
	Combined	$-0.109375$	$-0.0772136$	$-0.0768099$	$-0.110878$	$-0.0822007$
Girl	Red	$0.0830847$	$0.0581335$	$0.0489882$	$0.0663806$	$0.0108236$
	Green	$0.0336761$	$0.0192351$	$0.0159392$	$0.0191791$	$0.0103639$
	Blue	$-0.0232813$	$-0.0120988$	$-0.00979622$	$-0.0113034$	$-0.0309034$
	Combined	$0.15625$	$0.0462478$	$0.0441833$	$0.0606254$	$-0.00724911$
Sailboat	Red	$-0.0158755$	$0.00608236$	$0.00584084$	$0.00894156$	$0.015278$
	Green	$-0.0994093$	$-0.0609726$	$-0.060292$	$-0.0933644$	$-0.0969401$
	Blue	$-0.0357981$	$-0.0261754$	$-0.025861$	$-0.0370723$	$-0.104682$
	Combined	$0.031754$	$-0.0324742$	$-0.0322749$	$-0.0539398$	$-0.123518$
Tree	Red	$-0.0239129$	$-0.0587334$	$-0.0575335$	$-0.0886452$	$-0.116216$
	Green	$0.00401018$	$0.00607749$	$0.00597875$	$0.00638248$	$-0.0328477$
	Blue	$0.0236235$	$0.0389318$	$0.0374368$	$0.0573412$	$0.0216757$
	Combined	$0.015625$	$0.00158429$	$0.00157335$	$0.00789208$	$-0.0243032$

4.7. S-Box Performance Analysis

The proposed S-boxes are evaluated in this section independently from the proposed framework’s performance as a whole. This is because an S-box is a consistent element that is nearly always at the core of image encryption algorithms and is tasked with applying Shannon’s property of confusion. In order to evaluate the ability of an S-box to cause confusion, 5 metrics are commonly computed [46]. These are the non-linearity (NL), linear approximation probability (LAP), differential approximation probability (DAP), bit independence criterion (BIC) and strict avalanche criterion (SAC). They are calculated for the proposed S-boxes (shown in Table 1, Table 2 and Table 3) and compared to those reported in the state-of-the-art, as well as to the ideal values. While all 3 proposed S-boxes exhibit comparable performance to counterpart algorithms from the literature, it is clear that the S-boxes constructed from the hyperchaotic functions perform better than the S-box constructed from the SNM. This is an advantage of hyperchaotic functions [47]. As in [14], and as described earlier in SSection 3.3, upon designing and constructing each of the proposed S-boxes, the aim is not to reach the ideal values of the metrics, but rather to reach a set of target metrics that are in close proximity to the ideal ones. This was carried out so as to include this set of target values as part of the key space, expanding it by $5\times 3=15$ variables, and allowing it to reach a key space of $2^{2551}$, as explained earlier in SSection 4.4.

Table 21. Comparison of the numerical evaluation of the proposed S-boxes (displayed in Table 1, Table 2 and Table 3) among those provided in the literature.

Table 21. Comparison of the numerical evaluation of the proposed S-boxes (displayed in Table 1, Table 2 and Table 3) among those provided in the literature.

S-box	NL	SAC	BIC	LAP	DAP
Ideal values	112	$0.5$	112	$0.0625$	$0.015625$
Proposed, SNM (1)	106	$0.499268$	104	$0.09375$	$0.015625$
Proposed, HC 4D (2)	108	$0.500977$	108	$0.078125$	$0.015625$
Proposed, HC 7D (3)	108	$0.506592$	108	$0.078125$	$0.015625$
[14] MT	108	$0.503662$	92	$0.140625$	$0.015625$
[14] OpenSSL	108	$0.499023$	112	$0.0625$	$0.015625$
[14] Intel’s MKL	108	$0.499268$	104	$0.09375$	$0.015625$
[37]	111	$0.5036$	110	$0.0781$	$0.0234$
[48]	107	$0.497$	$103.5$	$0.1560$	$0.0390$
[49]	100	$0.5007$	$104.1$	$0.0390$	$0.1250$
[50] S4	112	$0.5$	112	$0.0625$	$0.0156$

5. Conclusions and Future Works

Abbreviations

References

1. Hosny, K.M.; Kamal, S.T.; Darwish, M.M. A color image encryption technique using block scrambling and chaos. Multimedia Tools and Applications 2022, pp. 1–21.
2. Alexan, W.; Elkandoz, M.; Mashaly, M.; Azab, E.; Aboshousha, A. Color Image Encryption Through Chaos and KAA Map. IEEE Access 2023. [CrossRef]
3. Alexan, W.; Mamdouh, E.; ElBeltagy, M.; Ashraf, A.; Moustafa, M.; Al-Qurashi, H. Social Engineering and Technical Security Fusion. 2022 International Telecommunications Conference (ITC-Egypt). IEEE, 2022, pp. 1–5.
4. Jasra, B.; Moon, A.H. Color image encryption and authentication using dynamic DNA encoding and hyper chaotic system. Expert Systems with Applications 2022, 206, 117861. [CrossRef]
5. Fang, J.; Jiang, M.; Yin, N.; Wei, D.; Zhang, Y. An image block encryption algorithm based on hyperchaotic system and DNA encoding. Multimedia Tools and Applications 2022, 81, 17245–17262. [CrossRef]
6. Alexan, W.; ElBeltagy, M.; Aboshousha, A. Rgb image encryption through cellular automata, s-box and the lorenz system. Symmetry 2022, 14, 443. [CrossRef]
7. Gabr, M.; Younis, H.; Ibrahim, M.; Alajmy, S.; Khalid, I.; Azab, E.; Elias, R.; Alexan, W. Application of DNA Coding, the Lorenz Differential Equations and a Variation of the Logistic Map in a Multi-Stage Cryptosystem. Symmetry 2022, 14, 2559. [CrossRef]
8. Gabr, M.; Alexan, W.; Moussa, K. Image Encryption Through CA, Chaos and Lucas Sequence Based S-box. 2022 Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA). IEEE, 2022, pp. 34–39.
9. Gabr, M.; Hussein, H.H.; Alexan, W. A Combination of Decimal-and Bit-Level Secure Multimedia Transmission. 2022 Workshop on Microwave Theory and Techniques in Wireless Communications (MTTW). IEEE, 2022, pp. 177–182.
10. Gabr, M.; Alexan, W.; Moussa, K.; Maged, B.; Mezar, A. Multi-Stage RGB Image Encryption. 2022 International Telecommunications Conference (ITC-Egypt). IEEE, 2022, pp. 1–6.
11. Shannon, C.E. Communication theory of secrecy systems. The Bell system technical journal 1949, 28, 656–715. [CrossRef]
12. Elqusy, A.S.; Essa, S.E.; El-Sayed, A. Key management techniques in wireless sensor networks. Commun Appl Electron (CAE) 2017, 7, 8–18.
13. Shariatzadeh, M.; Rostami, M.J.; Eftekhari, M. Proposing a novel Dynamic AES for image encryption using a chaotic map key management approach. Optik 2021, 246, 167779. [CrossRef]
14. Alexan, W.; Alexan, N.; Gabr, M. Multiple-Layer Image Encryption Utilizing Fractional-Order Chen Hyperchaotic Map and Cryptographically Secure PRNGs. Fractal and Fractional 2023, 7, 287. [CrossRef]
15. Gabr, M.; Younis, H.; Ibrahim, M.; Alajmy, S.; Alexan, W. Visual Data Enciphering via DNA Encoding, S-Box, and Tent Mapping. 2022 IEEE 5th International Conference on Image Processing Applications and Systems (IPAS). IEEE, 2022, pp. 1–6.
16. Alexan, W.; ElBeltagy, M.; Aboshousha, A. Image Encryption Through Lucas Sequence, S-Box and Chaos Theory. 2021 8th NAFOSTED Conference on Information and Computer Science (NICS). IEEE, 2021, pp. 77–83.
17. Younas, I.; Khan, M. A new efficient digital image encryption based on inverse left almost semi group and Lorenz chaotic system. Entropy 2018, 20, 913. [CrossRef]
18. Alexan, W.; ElBeltagy, M.; Aboshousha, A. Lightweight image encryption: Cellular automata and the lorenz system. 2021 International Conference on Microelectronics (ICM). IEEE, 2021, pp. 34–39.
19. Elkandoz, M.T.; Alexan, W. Image encryption based on a combination of multiple chaotic maps. Multimedia Tools and Applications 2022, pp. 1–22.
20. ElBeltagy, M.; Alexan, W.; Elkhamry, A.; Moustafa, M.; Hussein, H.H. Image Encryption Through Rössler System, PRNG S-Box and Recamán’s Sequence. 2022 IEEE 12th Annual Computing and Communication Workshop and Conference (CCWC). IEEE, 2022, pp. 0716–0722.
21. Iqbal, N.; Naqvi, R.A.; Atif, M.; Khan, M.A.; Hanif, M.; Abbas, S.; Hussain, D. On the Image Encryption Algorithm Based on the Chaotic System, DNA Encoding, and Castle. IEEE Access 2021, 9, 118253–118270. [CrossRef]
22. ur Rehman, A.; Liao, X.; Ashraf, R.; Ullah, S.; Wang, H. A color image encryption technique using exclusive-OR with DNA complementary rules based on chaos theory and SHA-2. Optik 2018, 159, 348–367. [CrossRef]
23. Yang, B.; Liao, X. A new color image encryption scheme based on logistic map over the finite field ZN. Multimedia Tools and Applications 2018, 77, 21803–21821. [CrossRef]
24. Wang, Y.; Wu, C.; Kang, S.; Wang, Q.; Mikulovich, V. Multi-channel chaotic encryption algorithm for color image based on DNA coding. Multimedia Tools and Applications 2020, pp. 1–26.
25. Zhang, Y.Q.; He, Y.; Li, P.; Wang, X.Y. A new color image encryption scheme based on 2DNLCML system and genetic operations. Optics and Lasers in Engineering 2020, 128, 106040. [CrossRef]
26. Jithin, K.; Sankar, S. Colour image encryption algorithm combining, Arnold map, DNA sequence operation, and a Mandelbrot set. Journal of Information Security and Applications 2020, 50, 102428. [CrossRef]
27. Rehman, A.U.; Firdous, A.; Iqbal, S.; Abbas, Z.; Shahid, M.M.A.; Wang, H.; Ullah, F. A Color Image Encryption Algorithm Based on One Time Key, Chaos Theory, and Concept of Rotor Machine. IEEE Access 2020, 8, 172275–172295. [CrossRef]
28. Li, B.; Liao, X.; Jiang, Y. A novel image encryption scheme based on logistic map and dynatomic modular curve. Multimedia Tools and Applications 2018, 77, 8911–8938. [CrossRef]
29. Hu, X.; Wei, L.; Chen, W.; Chen, Q.; Guo, Y. Color image encryption algorithm based on dynamic chaos and matrix convolution. IEEE Access 2020, 8, 12452–12466. [CrossRef]
30. Zhang, X.; Wang, L.; Wang, Y.; Niu, Y.; Li, Y. An image encryption algorithm based on hyperchaotic system and variable-step Josephus problem. International Journal of Optics 2020, 2020. [CrossRef]
31. Gong, L.; Qiu, K.; Deng, C.; Zhou, N. An image compression and encryption algorithm based on chaotic system and compressive sensing. Optics & Laser Technology 2019, 115, 257–267.
32. Li, C.; Chen, G. Coexisting chaotic attractors in a single neuron model with adapting feedback synapse. Chaos, Solitons & Fractals 2005, 23, 1599–1604.
33. Qi, G.; van Wyk, B.J.; van Wyk, M.A. A four-wing attractor and its analysis. Chaos, Solitons & Fractals 2009, 40, 2016–2030.
34. Cui, N.; Li, J. A new 4D hyperchaotic system and its control. AIMS Mathematics 2023, 8, 905–923. [CrossRef]
35. Yang, Q.; Zhu, D.; Yang, L. A new 7D hyperchaotic system with five positive Lyapunov exponents coined. International Journal of Bifurcation and Chaos 2018, 28, 1850057. [CrossRef]
36. Bassham, L.E.I.; Rukhin, A.L.; Soto, J.; Nechvatal, J.R.; Smid, M.E.; Barker, E.B.; Leigh, S.D.; Levenson, M.D.; Vangel, M.; Banks, D.L.; Heckert, A.; Dray, J.; Vo, S. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications. Revised 800-22, National Institute of Standards and Technology, 2010.
37. Khan, M.; Masood, F. A novel chaotic image encryption technique based on multiple discrete dynamical maps. Multimedia Tools and Applications 2019, 78, 26203–26222. [CrossRef]
38. Khan, M.; Shah, T. An efficient chaotic image encryption scheme. Neural Computing and Applications 2015, 26, 1137–1148. [CrossRef]
39. Liu, H.; Zhao, B.; Huang, L. Quantum image encryption scheme using Arnold transform and S-box scrambling. Entropy 2019, 21, 343. [CrossRef]
40. Wu, Y.; Noonan, J.; Agaian, S. NPCR and UACI randomness tests for image encryption. Cyber J. Multidiscip. J. Sci. Technol. J. Sel. Areas Telecommun. (JSAT) 2011, 1, 31–38.
41. Slimane, N.B.; Aouf, N.; Bouallegue, K.; Machhout, M. A novel chaotic image cryptosystem based on DNA sequence operations and single neuron model. Multimedia Tools and Applications 2018, 77, 30993–31019. [CrossRef]
42. Paul, L.; Gracias, C.; Desai, A.; Thanikaiselvan, V.; Suba Shanthini, S.; Rengarajan, A. A novel colour image encryption scheme using dynamic DNA coding, chaotic maps, and SHA-2. Multimedia Tools and Applications 2022, pp. 1–22.
43. Wu, X.; Kurths, J.; Kan, H. A robust and lossless DNA encryption scheme for color images. Multimedia Tools and Applications 2018, 77, 12349–12376. [CrossRef]
44. Alvarez, G.; Li, S. Some basic cryptographic requirements for chaos-based cryptosystems. International journal of bifurcation and chaos 2006, 16, 2129–2151. [CrossRef]
45. Asuero, A.G.; Sayago, A.; González, A. The correlation coefficient: An overview. Critical reviews in analytical chemistry 2006, 36, 41–59. [CrossRef]
46. Mahmood Malik, M.S.; Ali, M.A.; Khan, M.A.; Ehatisham-Ul-Haq, M.; Shah, S.N.M.; Rehman, M.; Ahmad, W. Generation of Highly Nonlinear and Dynamic AES Substitution-Boxes (S-Boxes) Using Chaos-Based Rotational Matrices. IEEE Access 2020, 8, 35682–35695. [CrossRef]
47. Hosny, K.M.; Kamal, S.T.; Darwish, M.M. Novel encryption for color images using fractional-order hyperchaotic system. Journal of Ambient Intelligence and Humanized Computing 2022, 13, 973–988. [CrossRef]
48. Zahid, A.H.; Arshad, M.J.; Ahmad, M. A novel construction of efficient substitution-boxes using cubic fractional transformation. Entropy 2019, 21, 245. [CrossRef]
49. Hayat, U.; Azam, N.A.; Asif, M. A method of generating 8× 8 substitution boxes based on elliptic curves. Wireless Personal Communications 2018, 101, 439–451. [CrossRef]
50. Siddiqui, N.; Yousaf, F.; Murtaza, F.; Ehatisham-ul Haq, M.; Ashraf, M.U.; Alghamdi, A.M.; Alfakeeh, A.S. A highly nonlinear substitution-box (S-box) design using action of modular group on a projective line over a finite field. PLOS one 2020, 15, e0241890. [CrossRef]