Synchronization of Chaotic Extremum-Coded Random Number Generators and Its Application to Segmented Image Encryption

Abstract

This paper proposes a highly secure image encryption technique based on chaotic synchronization. Firstly, through the design of a synchronization controller, we ensure that the master–slave chaotic extremum-coded random number generators (ECRNGs) embedded in separated transmitters and receivers are fully synchronized to provide synchronized dynamic random sequences for image encryption applications. Next, combining these synchronized chaotic sequences with the AES encryption algorithm, we propose an image segmentation and multi-encryption method to enhance the security of encrypted images and realize a secure image transmission system. Notably, in the design of the synchronization controller, the transient time before complete synchronization between the master and slave ECRNGs is effectively controlled by specifying the eigenvalues of the matrix in the synchronization error dynamics. Research results in this paper also show that complete synchronization of ECRNGs can be achieved within a single sampling time, which significantly contributes to the time efficiency of the image transmission system. As for the image encryption technique, we propose the method of image segmentation and use the synchronized dynamic random sequences generated by the ECRNGs to produce the keys and initialization vectors (IVs) required for AES-CBC image encryption, greatly enhancing the security of the encrypted images. To highlight the contribution of the proposed segmented image encryption, statistical analyses are conducted on the encrypted images, including histogram analysis (HA), information entropy (IE), correlation coefficient analysis (CCA), number of pixels change rate (NPCR), and unified average changing intensity (UACI), and compared with existing literature. The comparative results fully demonstrate that the proposed encryption method significantly enhances image encryption performance. Finally, under the network transmission control protocol (TCP), the synchronization of ECRNGs, dynamic keys, and IVs is implemented as well as segmented image encryption and transmission, and a highly secure image transmission system is realized to validate the practicality and feasibility of our design.

1. Introduction

With the advancement of wireless network communication, mobile technologies, and the Internet of Things (IoT), data transmission is experiencing exponential growth. This data includes various forms of information, comprising sensitive personal information such as medical images and processing parameters of smart factories [1]. Due to the prevalence of hacker attacks, there is an urgent need to enhance the security of data transmission to safeguard against these threats. As a result, substantial research is now focused on creating efficient and secure techniques for data and image encryption. These approaches include symmetric, asymmetric, and hybrid encryption methods, alongside deep learning-based encryption for enhanced data protection. By utilizing complex encryption algorithms and static keys, these techniques transform data and images into noise-like, unintelligible forms, ensuring their security during transmission and storage [2,3,4]. However, with the increasing speed of computer processing and advancements in quantum computing technology, quantum computers have become capable of solving mathematical problems that were traditionally encrypted using algorithms like Shor’s or Grover’s quantum computing algorithms [5,6]. If future quantum computers possess a sufficient number of qubits, their computational speed will significantly increase and pose a risk to current cryptographic systems [7,8].

Chaotic systems are highly complex nonlinear systems that exhibit dynamics resembling quasi-random and aperiodic behavior. Moreover, the state responses of chaotic systems differ from those of typical linear or nonlinear systems; they feature specific strange attractors within which states dynamically and randomly evolve without convergence or divergence. Additionally, chaotic systems demonstrate extreme sensitivity to initial conditions, a phenomenon known as the butterfly effect. Due to these characteristics, chaotic systems have been widely applied in applications such as random number generation (RNG) and image encryption [9,10,11,12,13]. In related works [14,15], researchers leveraged the butterfly effect and unpredictability of chaotic systems to design dynamic encryption keys. Nevertheless, the design of chaos synchronization control was not considered. Consequently, it is imperative to synchronize both master and slave chaotic systems with identical initial conditions to maintain cryptographic system integrity. However, practical communication may introduce disturbances resulting in minor state differences. Due to the butterfly effect, this leads to disparate chaos-based dynamic keys at the transmitter and receiver, degenerating system effectiveness. To solve this problem, researchers in chaos cryptography have devised synchronization controllers to suppress the butterfly effect between master and slave chaotic systems for ensuring synchronization and generating uniform dynamic keys [16,17,18,19]. Yet, these efforts [16,17,18,19] have overlooked pre-synchronization transient periods, which can significantly impact the efficacy of communication security systems. Encryption/decryption processes can only commence once dynamic keys achieve complete synchronization. Hence, this study focuses on real-time synchronization by specifying the eigenvalues of a matrix in the synchronization error dynamics to control synchronization transient times. This approach facilitates the design of real-time synchronized dynamic keys to ensure efficiency. On the other hand, AES is widely recognized for its strong mathematical security, but in actual hardware implementation, it is vulnerable to side-channel attacks (SCA), such as power consumption analysis attacks and electromagnetic radiation attacks. In particular, differential power analysis, by running AES multiple times and monitoring its physical characteristics (such as power consumption), attackers might successfully speculate on encryption keys [20]. In order to improve the resistance of AES to SCA, recent research shows that combining the key generated by the chaotic system with AES is an effective solution [21]. The keys generated by chaotic systems are highly unpredictable and can change dynamically in each encryption operation such that it is difficult for attackers to extract dynamic keys through multiple measurements [22]. In addition, the keys generated by chaotic systems have high entropy, which reduces the possibility of attackers inferring key patterns through statistical analysis, and the randomization of chaotic keys makes the encryption process more unpredictable, effectively reducing the chance of successful side-channel attacks.

Motivated by the aforementioned works, this paper aims to realize a highly secure chaos-based image transmission system. Firstly, by integrating high-quality random numbers generated by ECRNGs [10] with the AES encryption algorithm, we propose a novel approach for encrypting segmented images using multiple keys and initialization vectors (IVs). To evaluate the performance of the proposed method, statistical analyses including HA, CCA, IE, NPCR, and UACI are conducted. The results are then compared with previous studies on image encryption that employed various techniques, such as chaos encryption, DNA coding, cross-plane joint scrambling-diffusion, and bit-level permutation [23,24,25,26,27,28,29,30,31,32,33], highlighting the advantages of the encryption method presented in this paper. Secondly, to relax the constraint of requiring identical initial states for master–slave chaotic systems in practical applications, this paper designs synchronization controls. These controls ensure that master–slave chaotic systems embedded in the transmitter and receiver can synchronize and generate identical dynamic keys for completing image encryption and decryption. Additionally, to reduce transient synchronization times, real-time synchronization is also discussed to effectively enhance system performance. Finally, to validate all inference results, a highly secure image capture and transmission system is implemented under the TCP (transmission control protocol) network protocol. In this system, the synchronization of chaotic systems and dynamic keys between the transmitter and receiver, as well as the implementation of image segmentation encryption methods, are thoroughly discussed to confirm the feasibility of this design. The subsequent sections are structured as follows: Section 2 describes the main structure and core technologies of segmented image encryption and secure transmission systems. Section 3 integrates synchronization controller design, extreme value coding, and SHA-256 to achieve synchronized dynamic key generation, applied to image encryption applications. Statistical analysis and comparisons with literature results demonstrate the significant enhancement of image encryption effectiveness with this method. Section 4 proposes the implementation of an image information encryption transmission system, including synchronization of dynamic keys and IVs within the TCP framework and system realization. Section 5 describes future research work to address the security and robust synchronization in network environments where noise interference is present. Section 6 summarizes the research findings and further works.

2. Design of Image Secure Transmission System and Problem Formulation

As mentioned, the goal of this paper is to propose an image secure transmission system, and the architecture of the system is illustrated in Figure 1 below. In Figure 1, we employ the TCP communication protocol for data transmission between the receiver and the transmitter. Encryption of segmented plain images is achieved by integration AES algorithm with the synchronized chaotic dynamic keys and IVs proposed in this study. To reduce the transmission time, we combine all encrypted sub-graphics with partial information of synchronization controller into a packet for transmission at once. Upon receiving all encrypted sub-graphics, the receiver decrypts the cipher sub-images using corresponding synchronized dynamic keys and IVs corresponding to each segmented subgraph, thereby obtaining the decrypted graphics. To implement this system, core technologies include the synchronization of the chaotic extremum-coded RNGs and the design of AES encryption algorithm with chaotic dynamic keys and IVs. The detailed descriptions of the main technical contents are elaborated below.

Synchronization of Extremum-Coded Random Number Generators

In AES encryption algorithm, the key is used for both encryption and decryption, while the initialization vector (IV) is a fixed-length random number used to enhance the security of the encryption process. The combination of IV and the key ensures that each encryption/decryption operation is unique. In this study, random dynamic keys and IVs are employed and randomly and dynamically updated for each encryption/decryption operation to effectively enhance the security of AES encryption. The extremum-coded RNG [10] is used in this study to generate the keys and IVs for AES encryption/decryption. In the extremum-coded RNG, chaotic states combined with extremum encoding dynamically generate high-quality random number sequences. Therefore, to apply it in the image secure transmission system, extremum-coded RNGs need to be deployed at both the transmitter and receiver and ensure that they can synchronously generate the same random number sequences dynamically and synchronously to perform encryption and decryption for AES algorithm. Surveying the mechanism of extremum-coded RNG demonstrates that if the chaotic systems are synchronized, they can generate the same random number sequences synchronously. Therefore, to address this issue, the most important task is to achieve synchronization control of the chaotic systems in the master–slave extremum-coded RNGs. The following discusses the design of synchronization controller primarily located on the slave end (receiver). The embedded master–slave chaotic systems in the transmitter and receiver are generally described in (1) and (2), respectively:

$$x_m\left(\left(k+1\right)T\right)=A{x}_m\left(kT\right)+Hg\left(x_m\left(kT\right)\right)$$

(1)

$$x_s\left(\left(k+1\right)T\right)=A{x}_s\left(kT\right)+Hg\left(x_s\left(kT\right)\right)+u\left(kT\right),$$

(2)

where $A\in R^{n\times n},H\in R^{n\times r}$ are system matrices, $g\left(.\right)\in R^{r\times 1}$ is the nonlinear function vector of system, $x_m, x_s\in R^n$ are the states of the master and slave chaotic systems, respectively, for the random generation of extreme value encoding. $u\left(kT\right)\in R^{l\times 1}$ is the synchronization controller to ensure the synchronization between the master and slave systems.

The following describes the design of the synchronization controller $u\left(kT\right)$. First, define the state error $e\left(kT\right)$ as follows:

$${e\left(kT\right)=x}_s\left(kT\right)-x_m\left(kT\right)$$

(3)

From Equations (1)–(3), we can derive the dynamic equation for the state error as follows:

$$e\left(\left(k+1\right)T\right)=Ae\left(kT\right)+H\left[g\left(x_s\left(kT\right)\right)-g\left(x_m\left(kT\right)\right)\right]+u\left(kT\right)$$

(4)

The controller input is designed as shown in Equation (5).

$$u\left(kT\right)={Hu}_w\left(kT\right)+{Bu}_f\left(kT\right)$$

(5)

In (5), $B\in R^{n\times l}$ is the coefficient matrix to be designed, which can be freely chosen, but it must satisfy that $\left(A,B\right)$ is a controllable matrix pair and

$$u_w\left(kT\right)=g\left(x_m\left(kT\right)\right)-g\left(x_s\left(kT\right)\right)$$

(6)

$$u_f\left(kT\right)=-Ke\left(kT\right)$$

(7)

The matrix K is the designed feedback matrix satisfying $\left|{\lambda }_i\right|<1,i=\mathrm{1,2},..,n$ for all eigenvalues of matrix $\left(A-BK\right)$. Substituting Equations (6) and (7) into (4), we can obtain:

$$e\left(\left(k+1\right)T\right)=\left(A-BK\right)e\left(kT\right)$$

(8)

Since $\left(A,B\right)$ is controllable, the pole placement method can be utilized to find the matrix $K$ to arbitrarily specify the eigenvalues of $\left(A-BK\right)$ such that $\left|{\lambda }_i\right|<1$ and $e\left(kT\right)$ can converge to zero. Moreover, when the rank of B is $v$, then at most $v$ eigenvalues of $\left(A-BK\right)$ can be specified to be the same. Since the convergence speed of the error state $e\left(kT\right)$ depends on the eigenvalues of $\left(A-BK\right)$. When the eigenvalues $\left|{\lambda }_i\right|$ are closer to 0, it indicates a faster synchronization speed for $e\left(kT\right)$. When the rank of B is $v=n$ and all the eigenvalues of $\left(A-BK\right)$ are assigned to be at 0, the master–slave chaotic system can achieve complete synchronization with just one sampling time $T$, termed as real time synchronization. Next, we take a 4th-order discrete chaotic system as an example. Of course, this method is applicable to other chaotic systems with appropriate modifications. The chaotic system to be used is as follows [10]

$$\begin{gathered} x_1\left(\left(k+1\right)T\right)=0.9699x_1\left(kT\right)+0.0315x_2\left(kT\right)+0.0014x_4\left(kT\right) \\ {x}_2\left(\left(k+1\right)T\right)=0.0845x_1\left(kT\right)+0.9989x_2\left(kT\right)+0.0845x_4\left(kT\right) \\ -0.002x_1\left(kT\right)x_3\left(kT\right)-0.002x_3\left(kT\right)x_4\left(kT\right) \\ {x}_3\left(\left(k+1\right)T\right)=0.992x_3\left(kT\right)+0.002x_1\left(kT\right)x_2\left(kT\right)+0.002x_2\left(kT\right)x_4\left(kT\right) \\ {x}_4\left(\left(k+1\right)T\right)=0.0014x_1\left(kT\right)+0.0315x_2\left(kT\right)+0.9699x_4\left(kT\right) \\ -0.0001x_1\left(kT\right)x_3\left(kT\right) \end{gathered}$$

(9)

Obviously, (9) can be expressed in the form of Equation (1).

$$x\left(\left(k+1\right)T\right)=Ax\left(kT\right)+Hg\left(x\left(kT\right)\right),$$

(10)

where $x\left(kT\right)=[x_1\left(kT\right)x_2\left(kT\right)x_3\left(kT\right)x_4\left(kT\right){]}^T$ and

$$g\left(x\left(kT\right)\right)=[x_1\left(kT\right)x_3\left(kT\right) x_3\left(kT\right)x_4\left(kT\right) x_1\left(kT\right)x_2\left(kT\right) x_2\left(kT\right)x_4\left(kT\right){]}^T$$

$$A=\left[\begin{array}{cccc}0.9699& 0.0315& 0& 0.0014\\ 0.0885& 0.9989& 0& 0.0885\\ 0& 0& 0.992& 0\\ 0.0014& 0.0315& 0& 0.9699\end{array}\right]; H=\left[\begin{array}{cccc}0& 0& 0& 0\\ -0.002& -0.002& 0& 0\\ 0& 0& 0.002& 0.002\\ -0.0001& 0& 0& 0\end{array}\right]$$

According to the system described in (9), we design the master–slave chaotic systems as shown in Equations (1) and (2). Following the earlier discussion, we design the control input as $u\left(kT\right)={Hu}_w\left(kT\right)+{Bu}_f\left(kT\right)$, $u_f\left(kT\right)=-Ke\left(kT\right)$, where $u_w\left(kT\right)=g\left(x_m\left(kT\right)\right)-g\left(x_s\left(kT\right)\right)$. In the following, we focus on the design of matrices $B \mathrm{a}\mathrm{n}\mathrm{d} K$ mentioned earlier, aiming to ensure all eigenvalues of $\left(A-BK\right)$ satisfying $\left|{\lambda }_i\right|<1$. Additionally, we analyze the effects of the selection of matrix $K$ and variations in the eigenvalues $\left|{\lambda }_i\right|$, as well as the rank $v$ of matrix $B$, on the convergence speed of synchronization. We conduct analysis for two cases as (1)$B=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right], v=3$ and $\lambda =\left[\begin{array}{ccc}0.5& 0.5& \begin{array}{cc}0.5& -0.4\end{array}\end{array}\right]$; (2)$B=\left[\begin{array}{cccc}1& -1& 0& 0\\ 1& 1& 0& 0\\ 0& 2& 1& 1\\ 0& 0& 0& 1\end{array}\right], v=4$ and $\lambda =\left[\begin{array}{ccc}0& 0& \begin{array}{cc}0& 0\end{array}\end{array}\right]$.

By using the pole assignment approach, we easily obtain K corresponding to Cases (1) and (2), respectively, as $\left[\begin{array}{cc}\begin{array}{cc}20.5096& 1.8660\\ 0& 0\\ 0.9156& 0.0927\end{array}& \begin{array}{cc}0& 0.1497\\ 0.4920& 0\\ 0& 0.4727\end{array}\end{array}\right]$; $\left[\begin{array}{cccc}0.5292& 0.5152& 0& 0.0449\\ -0.4407& 0.4837& 0& 0.0435\\ 0.8800& -0.9989& 0.9920& -1.0570\\ 0.0014& 0.0315& 0& 0.9699\end{array}\right]$.

Then simulation results are given in Figure 2. From Figure 2, we can verify that when the eigenvalues specified by $\left(A-BK\right)$ are closer to 0, the synchronization speed is faster. Additionally, when all eigenvalues are specified as 0, synchronization can be achieved just within one sampling time. Such design is beneficial for practical applications as it reduces transient synchronization time, thus significantly improving the system’s performance.

3. Image Encryption Performance Analysis

In our previously published research paper [10], the security of the ECRNGs used in this paper has been verified. Therefore, we adopt the dynamically generated binary random numbers from ECRNGs as the keys and initialization vectors (IVs) for AES-CBC image encryption. For image encryption analysis, this paper proposed a new approach of image segmentation encryption. The image is segmented into $2^N$ sub-images, and each segmented image is independently encrypted using different key and IV to enhance encryption quality. After encryption, all images are merged into an encrypted image. Figure 3 illustrates the image segmentation encryption for $2^2$ segments.

In recent years, research papers related to image encryption have often utilized 256 × 256 and 512 × 512 color Lena images for encryption and analysis. To compare the performance of image segmentation encryption in this paper, we also use Lena images of the same resolutions. Both 256 × 256 and 512 × 512 color Lena images are encrypted using 1, 4, and 16 pairs of keys and IVs, respectively. The encrypted images are then analyzed using techniques such as HA, CCA, IE, NPCR, and UACI. The obtained results are compared with those reported in existing papers.

Histogram analysis (HA): HA is mainly conducted by statistically analyzing the values of each pixel to obtain the histogram of the image. Generally, in unencrypted images, distinct peaks and valleys are observed in the histogram. However, in encrypted images, these peaks and valleys may be blurred or concealed due to the encryption process which resulting in a more uniform and difficult-to-interpret distribution in the histogram. The histogram analysis results of encrypted images under different keys and IVs using the proposed approach are shown in Figure 4 and Figure 5. From Figure 4 and Figure 5, it can be observed that in unencrypted images, the distribution tends to be concentrated, highlighting specific texture features of the image. In contrast, in encrypted images, the distribution appears to be randomly dispersed and evenly distributed. Therefore, the encrypted images exhibit good encryption effectiveness and information protection capabilities.

Canonical correlation analysis (CCA): CCA is primarily used to analyze the relationships between various pixels in an image. It calculates the correlation coefficients for horizontal, vertical, and diagonal directions, with the results typically ranging from 0 to 1. In unencrypted images, adjacent pixels exhibit extremely high correlation, so the values from CCA tend to approach 1. Higher correlation indicates weaker encryption effectiveness, while lower correlation indicates better encryption effectiveness. The calculation formula for CCA is as follows [23]:

$$CCA={\displaystyle \frac{\frac{1}{N}\sum _{i=1}^N\left(x_i-\frac{1}{N}\sum _{i=1}^N{x}_i\right)-\left(y_i-\frac{1}{N}\sum _{i=1}^N{y}_i\right)}{\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(x_i-\frac{1}{N}\sum _{i=1}^N{x}_i\right)}^2}\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(y_i-\frac{1}{N}\sum _{i=1}^N{y}_i\right)}^2}}}$$

(11)

where $x_{i }$ and $y_i$ denote the pixel values of the two adjacent pixels and N is the number of pairs $(x_i,y_i)$. Continuing with N set to 10,000, the unencrypted Lena image and the encrypted Lena image are subjected to CCA computation using Formula (11) above. Finally, the results are presented numerically and in scatter plots, and the CCA values obtained from the encrypted Lena images are compared with those reported in in the previous reports [23,24,25,26,27,28,29,30,31,32,33]. In the analysis results shown in Figure 6 and Figure 7, it can be observed that the points in the encrypted CCA are uniformly dispersed compared to the original image. Therefore, the encrypted image exhibits good encryption effectiveness and information protection capabilities.

Continuing, the CCA values calculated for different numbers of keys and IVs are presented in

Table 1. CCA analysis (a) Lena color (256 × 256). (b) Lena color (512 × 512).

Image: Lena Color (256 × 256), N = 10,000
Channels	Direction	Plain Image	1 Pair	4 Pairs	16 Pairs
R	Horizontal	0.9156	0.0056	−0.0023	0.0008
	Vertical	0.9470	0.0125	0.0120	−0.0007
	Diagonal	0.8979	−0.0037	−0.0019	−0.0012
	Total	2.7605	0.0218	0.0162	−0.0027
G	Horizontal	0.9413	−0.0104	0.0030	−0.0015
	Vertical	0.9686	0.0014	−0.0018	−0.0006
	Diagonal	0.9101	−0.002	0.0061	0.0005
	Total	2.8200	0.0138	0.0109	0.0027
B	Horizontal	0.9522	−0.0115	0.0052	0.0001
	Vertical	0.9780	−0.0033	−0.0052	−0.0025
	Diagonal	0.9337	−0.0043	0.0059	−0.0009
	Total	2.8639	0.0191	0.0163	0.0035
(a)
Image: Lena color (512 × 512), N = 10,000
Channels	Direction	Plain image	1 pair	4 pairs	16 pairs
R	Horizontal	0.9156	0.0012	0.0009	0.0003
	Vertical	0.9470	−0.0059	−0.0035	0.0003
	Diagonal	0.8979	0.0035	0.0005	−0.0008
	Total	2.7605	0.0106	0.0049	0.0014
G	Horizontal	0.9413	0.0014	−0.0042	0.0005
	Vertical	0.9686	−0.0077	0.0007	−0.0002
	Diagonal	0.9101	−0.0041	0.0007	0.0021
	Total	2.82	0.0132	0.0056	0.0028
B	Horizontal	0.9522	−0.0002	0.0021	−0.0018
	Vertical	0.9780	−0.00001	0.0047	−0.0004
	Diagonal	0.9337	−0.0096	0.0007	0.0014
	Total	2.8639	0.0098	0.0075	0.0036
(b)

and

Table 2. Encryption and decryption time.

Number of Key and IV Pairs	Lena Color (256 × 256)			Lena Color (512 × 512)
	Key and IV Generation Time (s)	Encryption Time (s)	Decryption Time (s)	Key and IV Generation Time (s)	Encryption Time (s)	Decryption Time (s)
1 pair	0.00199	0.00205	0.00601	0.00101	0.00501	0.00699
4 pairs	0.06172	0.03002	0.01709	0.06042	0.04649	0.01821
16 pairs	0.08012	0.03497	0.05683	0.10475	0.07515	0.05760

CPU: 12th Gen Intel(R) Core (TM) i7-12700H 2.70 GHz RAM: 32GB.

. When the CCA value is closer to 0, it indicates better encryption effectiveness. Therefore, from Table 1 and Table 2, it can be observed that as the number of dynamic keys and IVs increases, the CCA of the encrypted images approaches 0. This implies that increasing the number of keys and IVs leads to better encryption effectiveness.

Encryption and decryption time: To demonstrate the efficiency of the image encryption process, we measured the time taken for key generation, encryption, and decryption using different numbers of key and IV pairs. The results are presented in Table 2 below. As shown, the time required for key generation, encryption, and decryption increases as the number of key and IV pairs grows.

Continuing, the Lena image encrypted using 16 pairs of dynamic keys and IVs is compared to papers using different encryption algorithms [24,25,26,27,28,29,30,31,32,33]. The results shown in

Table 3. Comparison results of CCA analysis (a) Lena color (256 × 256). (b) Lena color (512 × 512).

Image: Lena Color (256 × 256), N = 10,000
Channels	Direction	[24]	[25]	[26]	[27]	[28]	Our 16 Pairs
R	Horizontal	0.0071	−0.0024	0.0002	0.0001	−0.0025	0.0008
	Vertical	0.0089	−0.0017	0.0018	−0.0091	−0.0020	−0.0007
	Diagonal	−0.0006	0.0024	−0.0015	−0.0023	0.0027	−0.0012
	Total	0.0016	0.0065	0.0035	0.0015	0.0072	0.0027
G	Horizontal	−0.0012	−0.0015	−0.0010	−0.0025	−0.0029	−0.0015
	Vertical	−0.0018	−0.0007	0.0017	−0.0061	−0.0036	−0.0006
	Diagonal	−0.0043	0.0015	0.0013	0.0058	−0.0015	0.0005
	Total	0.0073	0.0037	0.0040	0.0144	0.0080	0.0027
B	Horizontal	−0.0015	−0.0016	−0.0018	−0.0074	0.0027	0.0001
	Vertical	0.0041	0.0024	0.0001	−0.0059	−0.0018	−0.0025
	Diagonal	−0.0041	−0.0005	−0.0017	0.0015	−0.0024	−0.0009
	Total	0.0097	0.0045	0.0036	0.0148	0.0069	0.0035
(a)
Image: Lena color (512 × 512), N = 10,000
Channels	Direction	[29]	[30]	[31]	[32]	[33]	Our 16 pairs
R	Horizontal	0.0035	−0.00028054	−0.0022	−0.00215	−0.0040	0.0003
	Vertical	−0.0014	−0.00236150	0.0009	0.00276	0.0015	0.0003
	Diagonal	−0.0092	−0.00215030	0.0013	−0.00032	0.0025	−0.0008
	Total	0.0141	0.0048	0.0044	0.0052	0.0080	0.0014
G	Horizontal	−0.0052	−0.00029569	0.0057	0.00179	0.0074	0.0005
	Vertical	0.0006	−0.00432950	−0.0041	0.00232	−0.0016	−0.0002
	Diagonal	0.0002	0.000777590	0.0017	0.00102	−0.0024	0.0021
	Total	0.0060	0.0054	0.0115	0.0051	0.0014	0.0028
B	Horizontal	−0.0038	−0.00743010	0.00007	0.00121	−0.0002	−0.0018
	Vertical	0.0105	−0.00106270	0.00004	−0.00113	−0.0041	−0.0004
	Diagonal	0.0061	−0.00070684	0.0104	0.00069	0.0011	0.0014
	Total	0.0204	0.0092	0.0105	0.0030	0.0054	0.0036
(b)

indicate that the sum of CCA values for the three channels obtained using our method is closest to 0, which demonstrates that our algorithm outperforms the compared papers in terms of encryption effectiveness.

Information entropy analysis (IE): The IE analysis is used to examine the uncertainty within encrypted images. When uncertainty is higher, the calculated entropy value is larger. A larger entropy value indicates a better encryption effectiveness. The calculation method is given as follows:

$$H=-{\displaystyle \sum _{i=1}^{255}{p}_i }{\log }_2 p_i,$$

(12)

where $p_i$ is the frequency of each greyscale. For a grayscale image, the pixel has a data field of [0, 255], and the maximum value of IE will be 8. When the calculated results approach the ideal value, it indicates a better encryption effect.

Table 4. Information entropy (a) Lena color (256 × 256). (b) Lena color (512 × 512).

Image: Lena Color (256 × 256)
		R	G	B	Average
Plain image		6.3712	6.6362	6.9176	6.6417
1 pair		7.9972	7.9971	7.9968	7.9970
4 pairs		7.9973	7.9967	7.9974	7.9971
16 pairs		7.9972	7.9976	7.9977	7.9975
(a)
Image: Lena color (512 × 512)
		R	G	B	Average
Plain image		6.87920	6.92623	6.96843	6.924619
1 pair		7.99920	7.99926	7.99925	7.999237
4 pairs		7.99933	7.99931	7.99934	7.99324
16 pairs		7.99932	7.99933	7.99936	7.999334
(b)

a,b show the calculated results of information entropy after encryption with different pairs of keys and IVs in each channel.

From Table 4, it can be observed that as the number of keys increases, the entropy value gets closer to the ideal value 8. Therefore, when the number of pairs increases, it indicates a better encryption effect. Following that, we will encrypt using 16 pairs of keys and IVs and compare the results with the previous literature. The results are shown in

Table 5. Comparison results of information entropy (a) Lena color (256 × 256). (b) Lena color (512 × 512).

Image: Lena Color (256 × 256)
Ref.	R	G	B	Average
[24]	7.9970	7.9972	7.9967	7.9970
[25]	7.99739	7.99738	7.99736	7.9973
[28]	7.9974	7.9971	7.9975	7.9973
[34]	7.9972	7.9972	7.9975	7.9973
[35]	7.9974	7.9977	7.9970	7.9973
Our	7.9972	7.9976	7.9977	7.9975
(a)
Image: Lena color (512 × 512)
Ref.	R	G	B	Average
[31]	7.999279	7.999264	7.999353	7.999299
[32]	7.99942	7.99929	7.99929	7.999233
[33]	7.9991	7.9993	7.9993	7.999233
[36]	7.9992	7.9993	7.9993	7.999267
[37]	7.9980	7.9979	7.9978	7.997900
Our	7.99932	7.99933	7.99936	7.999334
(b)

. From the comparison results in Table 5, it can be observed that the information entropy analysis in this study is closest to the ideal value of 8. Therefore, this indicates that the method proposed in this paper is superior to the previous literature [24,25,28,31,32,33,34,35,36,37].

Sensitivity analysis: In encrypted image analysis, sensitivity analysis aims to detect whether encrypted images can resist differential attacks. To analyze whether encrypted images can withstand such attacks, two methods are commonly referenced: NPCR (normalized pixel change rate) and UACI (unified average change intensity). The formulas for these methods are defined as follows [24].

$$\mathrm{N}\mathrm{P}\mathrm{C}\mathrm{R}={\displaystyle \frac{{\sum }_{i,j}D(i,j)}{M\times N}}\times 100\%$$

(13)

$$D\left(i,j\right)=f\left(x\right)=\left\{\begin{array}{c}1,C_1(i,j)\ne C_2(i,j)\\ 0,C_1(i,j)=C_2(i,j)\end{array}\right.$$

(14)

$$\mathrm{U}\mathrm{A}\mathrm{C}\mathrm{I}={\displaystyle \frac{1}{M\times N}}\times {\displaystyle \frac{\sum \left(C_1\left(i,j\right)-C_2(i,j)\right)}{255}}\times 100\%$$

(15)

In the formulas above, M and N, respectively, represent the number of rows and columns in the image, while $C_1\left(i,j\right)$ and $C_2\left(i,j\right)$ represent the pixel values at corresponding positions in two images. In NPCR and UACI, their ideal values are 99.6094% and 33.4635%, respectively. Therefore, when the calculated values approach these ideal values, it indicates a better resistance against differential attacks. In this paper, encrypted images are randomly chosen to have one pixel value incremented or decremented by 1, and sensitivity analysis is performed by comparing these images with the original ones. The results are presented in

Table 6. NPCR and UACI (a) Lena color (256 × 256). (b) Lena color (512 × 512).

Image: Lena Color (256 × 256)
		NPCR (%)				UACI (%)
		R	G	B	Average	R	G	B	Average
1 pair		99.6384	99.5392	99.5911	99.5896	33.3101	33.3919	33.0359	33.3360
4 pairs		99.6170	99.6155	99.6292	99.6206	33.5197	33.5550	33.4046	33.4931
16 pairs		99.6490	99.5514	99.6246	99.6083	33.4653	33.3758	33.5449	33.4620
(a)
Image: Lena color (512 × 512)
		NPCR (%)				UACI (%)
		R	G	B	Average	R	G	B	Average
1 pair		99.5941	99.5964	99.6037	99.5981	33.4401	33.4877	33.4118	33.4465
4 pairs		99.5834	99.6197	99.6044	99.6025	33.4103	33.4490	33.5045	33.4546
16 pairs		99.5846	99.5941	99.6391	99.6059	33.5613	33.3647	33.4625	33.4628
(b)

below.

From Table 6, it can be observed that as the pair number of keys and IVs increases, the analysis results approach the ideal values more closely. Next, the results using 16 keys are compared with different encryption algorithms in the literature and the comparison results are shown in

Table 7. Comparison results of NPCR and UACI (a) Lena color (256 × 256). (b) Lena color (512 × 512).

Image: Lena Color (256 × 256)
		NPCR (%)				UACI (%)
		R	G	B	Average	R	G	B	Average
[24]		99.6033	99.6292	99.5850	99.6058	33.4740	33.4457	33.5339	33.4845
[26]		99.6066	99.6040	99.6046	99.6051	33.4887	33.4987	33.4907	33.4927
[38]		99.6216	99.6277	99.6189	99.6277	33.4032	33.5397	33.4912	33.4780
[35]		99.6048	99.6323	99.6063	99.6145	33.4115	33.4454	33.4640	33.4403
[39]		99.5925	99.5921	99.5917	99.5921	33.0371	33.3102	33.0319	33.1264
Our		99.6490	99.5514	99.6246	99.6083	33.4653	33.3758	33.5449	33.4620
(a)
Image: Lena color (512 × 512)
		NPCR (%)				UACI (%)
		R	G	B	Average	R	G	B	Average
[29]		99.6101	99.6288	99.6147	99.6178	33.4632	33.4756	33.4623	33.4670
[30]		99.6052	99.6120	99.6303	99.6158	33.4025	33.4428	33.5029	33.4494
[31]		99.6351	99.6522	99.6518	99.6523	33.4572	33.4715	33.4384	33.4557
[40]		99.6404	99.6334	99.6470	99.6403	33.4885	33.4930	33.5089	33.4968
[41]		99.6969	99.6278	99.6268	99.6505	33.4299	33.4982	33.4685	33.4863
Our		99.5846	99.5941	99.6391	99.6059	33.5613	33.3647	33.4625	33.4628
(b)

. From the sensitivity analysis results in Table 7, it can be observed that in NPCR and UACI analysis, the method proposed in this paper is closest to the ideal values—NPCR: 99.6094% and UACI: 33.4635%—compared to the previous literature [24,26,29,30,31,35,38,39,40,41]. Therefore, the encryption method proposed in this paper exhibits high security and can effectively resist differential attacks.

4. Implementation of the High-Security Image Transmission System

Based on the discussions and comparisons above, it can be confirmed that the encryption method proposed in this paper exhibits excellent security. The RNGs embedded in the transmitter can achieve real-time synchronization and both RNGs at the transmitter and receiver, respectively, dynamically generate identical dynamic keys and IVs for AES encryption/decryption. The following will integrate the aforementioned RNGs, synchronization controller, and AES encryption algorithm to realize a high-security image transmission system with the TCP communication protocol. The flowchart in Figure 8 depicts the proposed algorithm and outlines the steps involved in implementing the high-security image transmission system.

In Figure 8, master–slave chaotic systems are embedded at the transmitter and receiver respectively, and according to the previously designed synchronization controller $u\left(kT\right)={Hu}_w\left(kT\right)+{Bu}_f\left(kT\right)$, the state synchronization of the master–slave chaotic systems can be ensured. In the implementation of the system, since the master–slave chaotic systems are embedded in the transmitter and receiver respectively, the controller $u\left(kT\right)$ needs to be decomposed into two parts as $u\left(kT\right)=u_m\left(kT\right)-u_s\left(kT\right)$. Each part is separately calculated at the transmitter and receiver. Then, $u_m\left(kT\right)$ is transmitted from the transmitter to the receiver. When receiving $u_m\left(kT\right)$ at the receiver, $u_s\left(kT\right)$ and $u_m\left(kT\right)$ are combined to form the complete control input signal using $u\left(kT\right)=u_m\left(kT\right)-u_s\left(kT\right)$ to enable the system to achieve synchronization. The calculation of $u_s\left(kT\right)$ and $u_m\left(kT\right)$ is explained as follows. Let

$${Hu}_w\left(kT\right)=\stackrel{u_{wm}\left(kT\right)}{\overbrace{Hg\left(x_m\left(kT\right)\right)}}-\stackrel{u_{ws}\left(kT\right)}{\overbrace{Hg\left(x_s\left(kT\right)\right)}}$$

(16)

$${Bu}_f\left(kT\right)=-BKe\left(kT\right)=\stackrel{u_{fm}\left(kT\right)}{\overbrace{BK{x}_m\left(kT\right)}}-\stackrel{u_{fs}\left(kT\right)}{\overbrace{BK{x}_s\left(kT\right)}}$$

(17)

Whereby, the transmitter calculates $u_m\left(kT\right)=u_{wm}\left(kT\right)+u_{fm}\left(kT\right)$ and the receiver calculates $u_s\left(kT\right)=u_{ws}\left(kT\right)+u_{fs}\left(kT\right)$. Through the TCP protocol, $u_m\left(kT\right)$ can be smoothly transmitted to the receiver. Then, it is subtracted from the calculated control component $u_s\left(kT\right)$ at the receiver. This process successfully implements the synchronization controller ${u\left(kT\right)=Hu}_w\left(kT\right)+{Bu}_f\left(kT\right)=u_m\left(kT\right)-u_s\left(kT\right)$ at the receiver end, thus achieving synchronization. Upon achieving the synchronization, the extremum-coded RNGs at the transmitter and receiver can generate synchronized dynamic keys and IVs. Combined with AES-CBC, image encryption and decryption are successfully accomplished. In the system implementation, we set N = 2, dividing each original image into four sub-images. Through the synchronized RNGs, four pairs of dynamic keys and IVs are generated, thus completing this high-security image transmission system. In the realization, the same matrices B and K for Figure 2b are used to ensure real time synchronization. The completed system and execution interface are shown in Figure 9. On the transmitter side, to facilitate the observation of changes and synchronization of dynamic keys and IVs, we designed three buttons with different speeds. In Figure 9, the dynamic keys, IVs, and encryption/decryption images can be observed to confirm that the system can transmit, encrypt, and decrypt normally. For detailed system operation, please visit the provided website link (https://youtu.be/bU7RdOmvrmY, accessed on 22 September 2024). From Figure 9 and the video, it illustrates that each image is divided into 4 sub-images, and utilizing synchronized dynamic keys and IVs, image encryption and decryption are successfully performed at both the transmitter and receiver, confirming the feasibility of this research.

5. Future Research Work

This article presents an encryption method utilizing multiple keys and initialization vectors (IVs) for image encryption and secure communication systems. Future research should prioritize ensuring robust synchronization in network environments where noise interference is present, as this poses a significant challenge worthy of further exploration. Traditional encryption methods based on mathematical algorithms often require extensive mathematical proofs to handle noise or interference in communication channels, making system implementation more complex and difficult. In recent years, numerous studies have highlighted the significant contributions of deep learning (DL) and neural networks, particularly brain-like learning, to addressing noise issues in communication channels and enhancing security, especially in the context of image encryption and transmission [4,42,43,44]. These technologies offer efficient noise-reduction capabilities and adaptability. By training neural networks, systems can automatically detect and correct errors caused by noise and adjust parameters dynamically according to varying channel conditions to achieve optimal noise resistance. Moreover, deep learning models can generate nonlinear encryption structures through multi-layer neural networks, thereby strengthening the security of image encryption and effectively defending against threats such as chosen plaintext attacks. Future research directions will likely involve combining the sensitivity to initial conditions and the inherent randomness of chaotic systems with the learning capabilities of deep learning to generate highly random encryption keys, establish multiple encryption paths, and enhance resilience against attacks and disturbances. This approach might ultimately lead to more efficient and secure image encryption solutions.

6. Conclusions

This article introduces an encryption method using multiple keys and IVs for image encryption and secure communication systems. By comparing quantitative performance data including HA, IE, CCA, NPCR, and UACI, with previous studies, it is evident that the proposed method significantly improves image encryption effectiveness. Additionally, a synchronization controller for the image transmission system was designed, allowing for control of the master–slave RNG’s transient synchronization time by setting the eigenvalues of a specified matrix, thereby achieving real-time synchronization and enhancing performance. The TCP network protocol was also employed to synchronize dynamic keys and IVs, as well as to implement segmented image encryption, resulting in a highly secure image capture and transmission system. This confirms the practicality and feasibility of the proposed design. Future research should focus on ensuring key robust synchronization in network environments where noise interference occurs, a challenge that merits further investigation.