Memristive chaotic system-based hybrid image encryption application with AES and RSA algorithms

In image encryption applications, chaos theory has frequently been utilized as a means of enhancing the security of encrypted images, primarily due to the inherent properties of chaotic systems, such as their unpredictability and sensitivity to initial conditions. In this study, memristor-based chaotic systems are integrated with various chaotic systems to further augment the security of encrypted images. The study explores different chaotic systems described in the literature, which are then combined with different encryption methods, including AES and RSA. The use of memristor-based chaotic systems provides an additional layer of security by exploiting the non-linear characteristics of memristor to generate complex and unpredictable patterns, which can further enhance the encryption process.

Many chaotic systems have been applied to image encryption algorithms in the literature. While chaotic sequences obtained from these systems exhibit random-like behavior, they are sensitively dependent on the initial conditions of the system. These chaotic generators can exist in both discrete and time-dependent systems. Some generators and system dynamics used in image encryption are given below. All chaotic systems are designed on the MATLAB platform and produce chaotic sequences for image encryption. In this study, the logistic map, Lorenz chaotic system, and memristor-based chaotic system are used to obtain chaotic sequences for image encryption.

2.2.1. Logistic map

The logistic map is given as follows in equation (1) [35]:

$$\begin{eqnarray}&&f\left(x\right)=Rx(1-x)\end{eqnarray} \tag{ 1 }$$

where R is limited to the range (0.4]. The solution of function with the iteration method is as follows in equation (2):

$$\begin{eqnarray}&&{x}_{n+1}=R{x}_{n}(1-{x}_{n})\end{eqnarray} \tag{ 2 }$$

It is used to generate a chaotic signal with an initial value of ${x}_{0}.$ Parameters are ${x}_{0}=0.32\,{\rm{and}}\,R=3.99.$

2.2.2. Lorenz chaotic system

The chaotic system which is coined by Lorenz in the literature consists of three nonlinear, first-order differential equations (equation (3)),

$$\begin{eqnarray}\begin{array}{rcl}\dot{x} & = & {\rm{\sigma }}(y-x)\\ \dot{y} & = & ax-y-xz\\ \dot{z} & = & xy-bz\end{array}\end{eqnarray} \tag{ 3 }$$

Where x, y and z are state variables, σ, a, and b are scalar parameters. For the solution of this system, Lorenz drew the attractors of the x, y, and z variables (X, Y, and Z) in three-dimensional space by computer when the time changes by Δt ≈ dt. Lorenz performed an analysis as in equation (4), assuming $dt=0.02,\,\sigma =5,\,a=15,\,b=1.$

$$\begin{eqnarray}\begin{array}{rcl}{X}_{i+1} & = & \left({X}_{i}+\left({\rm{S}}\left({Y}_{i}-{X}_{i}\right){\rm{DT}}\right)\right)\\ {Y}_{i+1} & = & \left({Y}_{i}+\left(\left({X}_{i}\left({\rm{P}}-{Z}_{i}\right)-{Y}_{i}\right){\rm{DT}}\right)\right)\\ {Z}_{i+1} & = & \left({Z}_{i}+\left(\left({X}_{i}{Y}_{i}-{\rm{B}}{Z}_{i}\right){\rm{DT}}\right)\right)\end{array}\end{eqnarray} \tag{ 4 }$$

The used parameters and initial conditions are as follows ; $S=10,\,P=28,$ $B=2.667,\,DT=0.01,$ ${x}_{0}=0.6,\,{x}_{1}=0.6,\,{x}_{2}=3.09.$ In this study, image encryption is also used by obtaining chaotic sequences from the third state variable of the Lorenz chaotic.

2.2.3. Memristor based chaotic system

In recent years, the rapid development of dynamic analysis, image encryption, and circuit applications has been facilitated by the versatile and effective properties of memristor-based chaotic systems. These systems have found wide usage in these areas [29, 38, 39]. In this part, image encryption procedures are conducted using the flux-controlled memristor-based 5D hyperchaotic system, which is derived from Wang's 5D hyperchaotic system as described in the literature. The corresponding equations are provided in equation (5) [40].

$$\begin{eqnarray}\begin{array}{rcl}\dot{x} & = & {\rm{a}}\left(y-x\right)+4yz-kxW(u)\\ \dot{y} & = & -x+16y-xz+w\\ \dot{z} & = & -bz+xy-xu-yw\\ \dot{w} & = & -10y+0.15xz-gz-gzu\\ \dot{u} & = & -x\end{array}\end{eqnarray} \tag{ 5 }$$

where $W\left(u\right)=m+3n{u}^{2}$ and k, m, n, g are parameters are; $a=14;b=78;c=30;m=0.1;n\,=0.01;k=0.02;g=0.3$ and initial conditions are; [x y z w u] = [4 1.2 0.5–3.6 6]. In this study, image encryption is also used by obtaining chaotic sequences from the third state variable of the memristor-based hyperchaotic system.

2.3.1. AES

2.3.2. RSA

In the literature, several security tests have been presented to measure the reliability of image encryption systems and whether attacks in different ways weaken the system. Some of tests are described in the following sections.

2.4.1. Correlation analysis

There is a relationship between the pixels of any original image. The correlation of two adjacent pixels in an encrypted image should be minimal to deal with statistical attacks. The correlation coefficient x, y of each even pixel in the encrypted image is calculated by equations (6)–(8) given below [43]:

$$\begin{eqnarray}&&{r}_{xy}=\displaystyle \frac{cov(x,y)}{\sqrt{D(x)}\sqrt{D(x)}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&cov\left(x,y\right)=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left({x}_{i}-E(x)\right)\left({y}_{i}-E(y)\right)\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&E(x)=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left({x}_{i}\right),\,D=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-E(x)\right)}^{2}\end{eqnarray} \tag{ 8 }$$

Here x and y are the values of two adjacent pixels in the image on the gray legend chart. N represents the number of pixels selected from the image. If the correlation coefficient calculated for any pixel group selected from the encrypted image is close to or below 0, it indicates that the correlation between the pixels is very low. This makes the encryption system resistant to statistical attacks.

2.4.2. Histogram analysis

2.4.3. Differential attack

The differential attack is essentially a selected plaintext attack. In this attack, the attacker continuously modifies the plaintext images and analyzes the corresponding encrypted images in an attempt to retrieve the secret key. It is desirable for a minimal change in the plaintext image to result in a completely different encrypted image to assess the resistance against differential attacks.

In the literature, NPCR (number of pixels change rate) [46] and UACI (unified average changing intensity) [47] are used to measure this difference, in other words, to analyze differential attack. How to calculate the NPCR and UACI values is expressed with the following formula in equations (9)–(11) with 1C and 2C being the encrypted state of the plain image and the encrypted state of the plain image with a very slight change, and the width of the image is W and the height is H:

$$\begin{eqnarray}&&NPCR=\displaystyle \frac{{\sum }_{i,j}D(i,j)}{WxH}x100\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&UACI=\displaystyle \frac{1}{WxH}\left[\displaystyle {\sum }_{i,j}\displaystyle \frac{{C}_{1}\left(i,j\right)-{C}_{2}(i,j)}{255}\right]x100 \% \end{eqnarray} \tag{ 10 }$$

$D\left(i,j\right)$ is calculated as follows:

$$\begin{eqnarray}&&D\left(i,j\right)=\left\{\begin{array}{c}1,\,{C}_{1}\left(i,j\right)\ne {C}_{2}\left(i,j\right)\,\\ 0,\,{C}_{1}\left(i,j\right)={C}_{2}\left(i,j\right)\end{array}\right.\end{eqnarray} \tag{ 11 }$$

2.4.4. Information entropy (IE)

Information entropy, also known as Shannon entropy, is a measure of the uncertainty or randomness in a set of data [48, 49]. In the context of digital images, it can be used to quantify the degree of randomness in the distribution of pixels. . A higher entropy value suggests a more random distribution of pixel values, while a lower entropy value signifies a more uniform distribution. The equation for calculating information entropy, denoted as IE, is expressed in equation 13:

$$\begin{eqnarray}&&IE=\displaystyle \sum _{i=0}^{{2}^{n}-1}p(i){\mathrm{log}}_{2}\displaystyle \frac{1}{p(i)}\end{eqnarray} \tag{ 12 }$$

where IE is the entropy, $p(i)$ is the probability of a pixel having a particular value (ranging from 0 to 255 for 8-bit images), and $lo{g}_{2}$ is the base-2 logarithm. In practice, information entropy is often used as a measure of the quality of encryption in image processing. A well-encrypted image should have a high entropy value, indicating a significant degree of randomness in the pixel distribution.

• 1.  
  Choose a chaotic system and set the initial conditions. This could be a map, such as the logistic map or a continuous-time system, such as the Lorenz system or memristive system. Set the initial conditions for the system, such as the initial state vector or the initial value of the system variable.
• 2.  
  Iterate the chaotic system for a large number of time steps to generate a long sequence of real-valued numbers. This sequence should have the desired statistical properties of a chaotic system, such as sensitivity to initial conditions and a broad range of values.
• 3.  
  Choose a threshold value to use for binarization. This value will determine the boundary between the two binary values, such as 0 and 1.
• 4.  
  Apply the threshold value to the real-valued sequence to create a binary sequence. Each element of the binary sequence will be 0 if the corresponding element of the real-valued sequence is less than the threshold value, and 1 if it is greater than or equal to the threshold value.
• 5.  
  Optionally, post-process the binary sequence to remove any patterns or biases that may be present. This could involve techniques such as filtering or shuffling the sequence.

Statistical tests are employed to confirm the randomness and safety of generated random number sequences. In general, the randomness of the number sequences is accepted when they pass all statistical tests. NIST 800–22 and FIPS 140–2 are the most commonly employed randomization tests in the literature [50]. In this work, the NIST 800–22 test is utilized to assess the unpredictability of the chaotic sequences produced by the proposed encryption scheme. Table 1 provides a comparison of the chaotic sequences' test results. NIST test results for three different datasets are given in table 2. Symbols S and F in the table indicate successful and unsuccessful test results, respectively. The symbol P is the calculated probability value of the test. Examining the table, the logistic and memristive systems passed seven tests while the Lorenz system only passed six. For the NIST 800–22 test, it is recommended to execute randomization testing using at least 1 million bits of data to ensure accurate and dependable results. The NIST 800–22 exam is comprised of fifteen distinct subtests, and a p-value is calculated for each subtest. Each p-value must be greater than 0.01 in order for the tests to be judged successful. Comparing the p-values obtained from the proposed chaotic systems to the threshold value of 0.01 demonstrates the randomness and safety of the created chaotic sequences.

A 128-bit array is created using the logistic map, Lorenz, and the memristive system. This array is used for key expansion and encryption. The image is encrypted in 128-bit chunks because a 128-bit AES is used, and normally ten rounds of AES are reduced to one round for faster encryption. Encryption and decryption block diagrams with the AES algorithm are given in figures 2 and 3.

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In this part, encryption is applied to the images shown in figure 4, namely 1.jpg and 2.jpg, using the logistic map, Lorenz and memristive hyperchaotic systems in combination with the Advanced Encryption Standard (AES) algorithm.

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Figure 5 displays the encrypted images using the AES algorithm with different chaotic systems. A histogram, which represents the distribution of colour values in a numerical image, is used to obtain brightness or tone information about the image. This graph provides insights into the image characteristics. Figures 6–11 depict the logistic maps obtained by blending the 256 × 256 1.jpg and 2.jpg images, which are created using the Lorenz and memristive chaotic systems, through bit-based and pixel-based approaches. These figures display the original, encrypted, and decrypted versions of the 1.jpg and 2.jpg images.

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As can be seen from the histogram graphs, although there is no change in the histogram information of the plain image in pixel-based mixing, the histogram graph of the encrypted image obtained by bit-based mixing is close to linear character and regularity. Mixing only the pixels in the confusion phase does not remove some of the statistical properties of the original image. In bit-based hashing, since the pixel values change, secure encryption can be achieved with fewer cycles during the confusion and diffusion stage. It is understood that the image encryption method in this study with bit-based hashing is more effective in the diffusion stage. In order to completely remove the statistical properties of the plain image, encryption systems apply the sequential diffusion on the image pixels. With a general expression, it is understood from the graphics that the methods compared are hiding the histogram information of the original image.

The figures (figures 12–17) depict correlation scatter plots for two different images, 1.jpg and 2.jpg, which were encrypted and decrypted using three different chaotic systems: the logistic map, the Lorenz system, and the memristive system. The purpose of these plots is to visually demonstrate the degree of correlation between adjacent pixels in the original and encrypted images. Upon examining the correlation scatter plots for the original photos, as mentioned in the section, it becomes apparent that neighboring pixels in the photographs exhibit a very strong correlation with each other. This can be observed by analyzing the plots. This strong correlation is likely due to the presence of uniform colour or texture regions in the photographs, causing nearby pixels to have similar values.

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However, when analyzing the correlation scatter plots for the encrypted images, it is evident that the correlation between any two adjacent pixels is extremely low, almost non-existent. This is because the encryption process utilizing the chaotic systems introduced significant randomness and unpredictability to the pixel values, effectively disrupting any patterns or correlations that are present in the original image. Finally, the section also includes visual examples of the original, encrypted, and decrypted versions of the images, which are shown in the plots as vertical, horizontal, and diagonal images, respectively. This helps to give a sense of how the encryption process affects the appearance of the original image, and how the decrypted image compares to the original.

In the encryption phase of the RSA algorithm, two prime numbers, seventeen and nineteen, are selected as the public key. These numbers are used to generate a private key, which is kept secret by the sender. Procedure of encryption with RSA algorithms is given in table 3. The value of 'e', which is used to encrypt the message, is chosen as five in this study. After generating a 256 × 256 chaotic array using the selected chaotic system, the XORing process is applied to the image. This process involved performing an XOR operation between each pixel value of the image and the corresponding pixel value of the chaotic array. Encryption block diagram with RSA algorithm is shown in figure 18.

Table 3. Procedure of encryption with RSA algorithms.

Encryption
Step 1: Calculate the value of n by choosing the prime numbers p and q
Step 2: For RSA, the public key (e, n) and private key (d, n) are generated
Step 3: In this section, 6 positive integers are randomly selected (a1, a2, a3, a4, a5, a6) and a new value is generated for each (b1, b2, b3, b4, b5, b6) using the public key (e, n).
Step 4: Parameters for chaotic maps are calculated using equation (1).
$\left\{\begin{array}{c}{x}_{1}=\sqrt{\mathrm{log}({b}_{1}+{a}_{1})}\\ {x}_{2}=\,\sqrt{\mathrm{log}({b}_{2}+{a}_{2})}\\ {x}_{3}=\,\sqrt{\mathrm{log}({b}_{3}+{a}_{3})}\\ {x}_{4}=\sqrt{\mathrm{log}({b}_{4}+{a}_{4})}\\ {x}_{5}=\sqrt{\mathrm{log}({b}_{5}+{a}_{5})}\\ {x}_{6}=\,\sqrt{\mathrm{log}({b}_{6}+{a}_{6})}\end{array}\right.$
Step 5: The generated values (x1, x2, x3, x4, x5, x6) are given to chaotic maps and chaotic sequences are created.
Step 6: Arrays (k1, k2, k3) produced from the chaotic map are ordered from smallest to largest and their indexes are obtained. The image is mixed (confusion) using indices.
Step 7: Then the obtained values are converted to integers between 0–255 using equation given below.
$\left\{\begin{array}{c}nk1=mod(floor\left(k1\times {10}^{6}\right),\,256)\,\\ nk2=\,mod(floor\left(k2\times {10}^{6}\right),\,256)\\ nk3=mod(floor\left(k3\times {10}^{6}\right),\,256)\end{array}\right.$
Step 8: The resulting new sequences (nk1, nk2, nk3) are XORed with the scrambled image (confusion) for the diffusion step. Finally, the encrypted image is obtained.

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In figures 19(a) and (e) are the original grayscale images of 1.jpg and 2.jpg, respectively; (b) and (f) are images encrypted using the logistic map and the RSA algorithm; (c) and (g) are images encrypted using the Lorenz system and the RSA algorithm, and (d) and (h) are images encrypted using the RSA algorithm and the memristive chaotic system.

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Figures 20–25 logistic map is obtained by bit-based and pixel-based mixing of 256 × 256 size 1.jpg and 2.jpg images created using Lorenz and memristive chaotic systems. Histogram graphs of encrypted images are given. The original, encrypted and decrypted images of the used 1.jpg and 2.jpg are shown in these figures.

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Figures 26–31 display correlation scatter plots for 1.jpg and 2.jpg using the logistic map, Lorenz system, and memristive system. These figures showcase the original, encrypted, and decrypted versions of both 1.jpg and 2.jpg images, represented vertically, horizontally, and diagonally.Upon analyzing the graphics, it becomes evident that adjacent pixels in the original image exhibit a significantly high correlation with each other. In contrast, the correlation between neighboring pixels in the encrypted image is notably small and almost negligible. Furthermore, table 4 presents a comparison of the correlation coefficients calculated based on the AES encryption standard for the horizontal, vertical, and diagonal neighborhoods of the encrypted images. According to the correlation results you provided, it appears that for the original image, all three systems yield similar correlation values in the horizontal, vertical, and diagonal directions. However, for the encrypted image, the correlation values are considerably lower and closer to zero across all three systems and directions. This observation suggests that the encryption systems have effectively reduced the correlation between adjacent pixels in the encrypted image.

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When examining the NPCR (Number of Pixel Change Rate) and UACI (Unified Average Change Intensity) values presented in table 5, it can be observed that the memristive system performs better for 1.jpg, while the logistic map demonstrates better results for 2.jpg. Information entropy serves as a valuable metric for assessing the effectiveness of encryption techniques in digital image processing, ensuring the security and privacy of sensitive image data. Table 6 provides the Information Entropy (IE) values for the original images, while table 7 presents the IE values for the encrypted images obtained using chaotic systems with the AES algorithm. According to current results, the IE values of the encrypted images are nearly equal to eight, indicating a high level of security. Among the chaotic systems, the memristive chaotic system exhibits the highest IE value, implying a higher degree of information randomness and encryption effectiveness.

At the same time, the comparison of the correlation coefficients of the RSA encryption standard calculated according to the horizontal, vertical and diagonal neighborhoods of the encrypted image are shown in table 8. In the table 9, the NPCR and UACI values by using chaotic systems for RSA are given. In this part, while the NPCR values of the Lorenz system are better for the 1.jpg image, the memristive system for the 2.jpg outperforms the other chaotic systems.

The information entropy of encrypted images of chaotic systems by using RSA algorithm are given in table 10. The proposed system's information entropy value of nearly eight suggests that it has a high degree of randomness and unpredictability. This is an essential characteristic of an effective encryption system as it ensures that the encrypted data remains confidential and secure from potential attackers.