Mixed Multi-Chaos Quantum Image Encryption Scheme Based on Quantum Cellular Automata (QCA)

Abstract

The advent of quantum computers could enable the resolution of complex computational problems that conventional cryptographic protocols find challenging. As a result, the formidable computing capabilities of quantum computers may render all present-day cryptographic schemes that rely on computational complexity ineffectual. Inspired by these possibilities, the primary purpose of this paper is to suggest a quantum image encryption scheme based on quantum cellular automata with mixed multi-chaos hybrid maps and a hyperchaotic system with quantum operations. To achieve desirable encryption outcomes, we designed an encryption scheme involving two main operations: (1) pixel-level diffusion and (2) pixel-level permutation. Initially, the secret keys generated using the hyperchaotic system were closely tied to the original image. During the first phase, the establishment of correlations among the image pixels, in addition to the three chaotic sequences obtained from the hyperchaotic system, was achieved with the application of a quantum-state superposition and measurement principle, wherein the color information of a pixel is described using a single qubit. Therefore, the three channels of the plain image were subjected to quantum operations, which involve Hadamard transformation and the quantum-controlled NOT gate, before the diffusion of each color channel with the hyperchaotic system. Subsequently, a quantum ternary Toffoli gate was used to perform the diffusion operation. Next, the appropriate measurement was performed on the three diffused channels. To attain the confusion phase, a blend of mixed multi-chaos hybrid maps and a two-dimensional quantum cellular automaton was used to produce random and chaotic sequence keys. Subsequently, the circular shift was utilized to additionally shuffle the rows and columns of the three diffused components, in order to alter the positions of their pixel values, which significantly contributes to the permutation process. Lastly, the three encoding channels, R, G, and B, were merged to acquire the encrypted image. The experimental findings and security analyses established that the designed quantum image encryption scheme possesses excellent encryption efficiency, a high degree of security, and the ability to effectively withstand a diverse variety of statistical attacks.

1. Introduction

With the acceleration of cybercrimes due to the rapid growth of the Internet and recent technologies [1], multimedia information, such as digital images, audio data, and video, is increasingly transmitted online, where people can obtain various types of data at any time and in any location, as well as sharing resources and exchanging information [2,3].

In addition, with the birth and development of fifth-generation mobile communication technology and with recent advancements in social media platforms [3,4,5,6,7], the transmission of digital images has become normalized since digital images are highly visual and informative. In addition, various digital images are transmitted over the web daily, which is considered an insecure line of communication. Since some of these transmitted digital images may contain individual secret information, military secrets, and even national strategies, the illegal sharing of these digital images through any digital medium causes serious damage to any organization. Hence, transmitted digital images pose high levels of risk once they are lost and intercepted (i.e., copied and distributed illegally), and they can be altered maliciously.

Securing such digital images against unwanted disclosure and other malicious behavior is considerably more critical for users in various fields that involve unique security issues [5]. Therefore, information security has consistently commanded extensive attention from academia and industry, especially image information security, which has been a major focus of research [8]. To meet the above needs in practice, with regard to guaranteeing the security and protection of digital images, researchers have developed many techniques, including image encryption, watermarking, and steganography. Of these techniques [9], image encryption is considered the most direct and effective method since, in this method, information is encoded in such a way that the intended receiver is the only user who can retrieve its contents.

Digital images [10,11] are different from textual information due to their vivid and intuitive characteristics, such as bulk data capacity, high redundancy, and strong correlations between adjacent pixels. All these defects make conventional encryption algorithms [12,13], such as DES, AES, RSA, ECC, and IDEA, appear to be inappropriate for digital-image encryption due to their inefficiency and computational expenses. Therefore, image-information encryption [14] has become a major focus of research in the field of information security. To satisfy the emerging demand, a significant number of useful image-encryption algorithms based on optical transformation, DNA-sequence operations, wave motion, Brownian motion, cellular automata, compressive sensing, and chaotic theory were developed in the literature to secure these digital images [15]. One very promising direction is the application of chaotic maps and systems to image encryption. This is because cryptography places a particular emphasis on the introduction of nonlinear transformations, which is a distinctive feature of chaotic systems [16].

Chaos is a complex physical phenomenon, which can be represented in many dynamical systems [17,18]. It therefore penetrates various scientific fields, such as physics, biology, economics, sociology, stochastic computing, cryptography, etc. Chaotic systems have many inherent characteristics, such as ergodicity, extreme sensitivity to initial conditions and control parameters, unpredictability, good pseudorandom behavior, non-periodicity, and highly complex nonlinearity, which are very similar to the properties required by cryptography [19]. Thus, the research direction of cryptography based on chaos was born. Accordingly, image encryption based on chaos has become one of the most important image protection methods [12,20]. Some examples of typical image encryption algorithms are the following. In [13], a fast image encryption algorithm based on logistics–sine–cosine mapping was proposed, in which the algorithm first generates five sets of encrypted sequences from the logistics–sine–cosine mapping, and then uses the order of the encryption sequence to scramble the image pixels and designs a new pixel-diffusion network to further improve the key sensitivity and plain-image sensitivity of the encryption algorithm. Nehal Abd El-Salam Mohamed et al. [15] proposed an image encryption algorithm based on combining a hyperchaotic system and quantum 3D logistic map, which have the advantage of excellent random sequence to expand the key space, stronger randomness, and enhance the performance of resisting common attacks. The authors [21] presented a meaningful data encryption scheme by combining a P-tensor product compressive sensing model, a newly designed chaotic map, and data embedding technology, which ensures sufficient security and improves the proposed encryption scheme in terms of visual security, compressibility, and encryption efficiency greatly. On the other hand, a 1D sinusoidal polynomial composite chaotic system (SPCCS) combining the sine function with some special polynomial functions was proposed and proved that SPCCS satisfies Devaney’s definition of chaos in Ref. [22], wherein its detailed simulation results verify superiority and effectiveness of the proposed image encryption algorithm.

In the literature, some authors used cellular automata in collaboration with different chaotic maps to enhance the security of image encryption schemes. Mondal et al. [23] present a highly secure image encryption scheme for secure image communication and storage. The scheme is based on a chaotic skew tent map and cellular automata (CA), where the combination of both a chaotic map and CA gives a system with higher key space and a faster PRNS generator and is capable of resisting any kind of known attacks. In 2022, the authors [24] used the Lorenz system as the first encryption stage in a three-stage encryption process. The other two stages involve an S-box and Rule 30 cellular automaton. The authors apply different scan patterns to each of the RGB color channels of the image to be encrypted. Next, they separate the Lorenz equations and apply the use of each equation on a different color channel.

Meanwhile, despite the significant role of chaotic maps in designing modern cryptographic mechanisms, and as the traditional computer approaches its physical limits [25,26,27,28,29], researchers are beginning to study a new computing model, and quantum computers have produced a series of satisfactory encryption results. Given the quantum outstanding advantages of entanglement, superposition, and parallelism, quantum computing is widely used in all aspects of information science. With quantum computation’s rapid development, studies on image processing have been developed from classical image processing to quantum image processing (QIP), which has triggered researchers’ interest and is considered a new interdisciplinary subject that integrates quantum computing and digital image processing. Since one of the merits of quantum computers and quantum algorithms is that qubits obey the no-cloning theorem [30,31], which means no qubit can be cloned, quantum mapping theory has been introduced in image encryption schemes, aiming to improve the security level and practical value of image cryptography in the quantum era. Furthermore [32], quantum encryption protocols also include scrambling the pixel positions and/or changing the pixel value using chaos theory or any other quantum transformations.

For example, the authors [33] presented an encryption scheme based on discrete cosine transform (DCT) and alternate quantum walks (AQW) which explores the integration of quantum walks into an optical image encryption framework and uses different encryption matrices to encrypt images in the spatial domain and DCT domain. Ref. [34] introduced a self-adaptive encryption scheme to protect quantum images efficiently with minimal storage requirements which uses two rounds of encryption with two different pseudorandom number sequences which are obtained from a newly designed pseudorandom number generator (PRNG) and achieves less time complexity than many recently published quantum image encryption schemes. Zhang and Wang [35] proposed an image encryption scheme based on controlled zigzag transform and bit-level encryption under the quantum walk environment, which has good key security to resist attacks effectively such as statistical analysis, brute force search, and noise influence, as well as having better key sensitivity and dislocation effect.

Motivated by the better dynamics of hyperchaotic systems, the benefits of using many chaotic systems, and based on the physical properties of quantum computing, this study investigates a simple but further secure and efficient quantum image encryption algorithm. The principal contributions of this presented research work can be recapped as given below:

(1)

From the perspectives of cryptography and information theory, quantum transformation theory is applied as a new tool to quantum image encryption technology. Hence, based on Hadamard transformation theory and CNOT gate (entangle qubits procedure), we design a safe and efficient quantum image encryption with a new algorithm.

(2)

The proposed quantum image encryption scheme uses both confusion and diffusion to achieve a higher level of security, which is developed in accordance with Shannon’s theories.

(3)

Two different types of chaos, namely, a Lü system as a hyperchaotic system or a quantum logistic map as a multi-dimensional chaotic map (pixel-level diffusion), are utilized in designing a new visually meaningful quantum image cryptosystem.

(4)

The proposed scheme uses a secret key generated dependent on the original image to determine the number of cycles for both chaotic systems. Hence, it verifies that the proposed scheme has a strong resistance to chosen-plaintext attacks.

(5)

Concretely, this study investigates the integration of quantum cellular automata (QCA) as a quantum-inspired model and mixed multi-chaos maps as hybrid chaotic maps (pixel-level permutation).

(6)

To accomplish the permutation process, a circular shift is utilized to additionally shuffle the rows and columns of the diffused R, G, and B channels.

(7)

The key size of the proposed algorithm is sufficiently large to withstand brute-force attacks.

(8)

Statistical analysis, differential analysis, key analysis, and robustness analysis confirm that the designed scheme achieves a high level of security against most various attacks.

(9)

The proposed quantum scheme attains a reduced time complexity, leading to faster encryption and decryption processes. Additionally, a lower time complexity can make the proposed scheme more resistant to attacks that rely on brute-force methods as there are fewer possible combinations to test within a given amount of time.

(10)

The efficacy of the proposed method in safeguarding quantum image information is validated using numerical simulation and performance comparison.

The remainder of this article is structured as follows. Section 2 outlines the fundamental concepts utilized in the proposed scheme and analyses their dynamic characteristics. Section 3 provides a detailed explanation of the presented quantum image encryption and decryption scheme. Section 4 showcases the experimental and numerical outcomes of the proposed encryption scheme concerning diverse visual, statistical, differential, and brute-force analyses. In conclusion, Section 5 summarizes the main findings of this paper.

2. Preliminary Knowledge

2.1. Hyperchaotic Systems

Hyperchaotic systems are a valuable tool for exploring the behavior of nonlinear systems and developing new techniques for their analysis and control [36,37,38,39,40,41]. These systems are characterized by having more than one chaotic attractor, which means that they exhibit more complex and unpredictable behavior than chaotic systems. In addition, hyperchaotic systems have several advantages over lower-dimensional chaotic systems that make them useful for a wide range of applications in science, technology, and engineering, which can be summarized as follows: (1) increased complexity; (2) more sensitivity to small changes in initial conditions; (3) richer mathematical properties, where these systems also exhibit a rich variety of bifurcations and strange attractors; and (4) better performance in certain applications where hyperchaotic systems can be used to generate multiple secure encryption keys.

Hence, researchers continue to study hyperchaotic systems because of their intrinsic properties and their potential for applications in a wide range of fields, including secure communication, random number generation, cryptography, and control theory.

In addition, most existing encryption algorithms are still using a single chaotic system for encryption operation. These schemes cannot resist brute-force attacks efficiently as their key space is small. Moreover, a single chaotic system only has a few chaotic sequences that can be applied. The use of multi-chaotic systems has many benefits, which can be listed as follows: (1) multi-chaotic sequences; (2) expanded key space; (3) increased level of security; and (4) resisting brute-force attack.

In this section, we briefly present the two hyperchaotic systems considered in this work. One example of a hyperchaotic system is the Lü system, which has three nonlinear differential equations that describe the evolution of its state variables. Another example is the quantum logistic map, which has three nonlinear differential equations that describe the evolution of its state variables. Both systems have been extensively studied, and their properties are well-understood.

2.1.1. Hyperchaotic Lü System

In 2002 [42,43], Lü et al. found a new three-dimensional autonomous chaotic system, namely–Lü system which is given by Lü and Chen in the chaotic anti-control method. This system is considered an extension of the well-known Lü system.

The hyperchaotic Lü system has been studied extensively in recent years and has been found to have important applications in cryptography, chaos-based communications, and other fields. For example, the system can be used to generate random numbers, which are essential for many cryptographic applications.

It governs the evolution of its state variables x, y, and z. These equations contain various nonlinear terms, such as cubic and quadratic terms, that give rise to the system’s hyperchaotic behavior. This system is described by three-dimensional nonlinear differential equations that exhibit hyperchaotic behavior as follows [44]:

$$\begin{gathered} \dot{x}=a\left(y-x\right) , \\ \dot{y}=cy-xz , \\ \dot{z}=xy-bz \end{gathered}$$

(1)

where $a, b, \mathrm{and} c$ are system control parameters. The Lü system is in a chaotic state [42] when the parameter set $\left\{a, b\right\}=\left\{36, 3\right\}$ is fixed while parameter $c$ varies and, likewise, it may not be chaotic for any other parameter. This system is hyperchaotic with four positive Lyapunov exponents ${\lambda }_1=1.5046$, ${\lambda }_2=0, {\lambda }_3=-22.5044,$ and Lyapunov dimension $d_L=2.0669,$ which indicates that it has multiple unstable directions in phase space [45], as displayed in Figure 1. The hyper-chaos attractors and the bifurcation diagram of this hyperchaotic system are illustrated in Figure 2a,b, respectively.

2.1.2. Hyperchaotic Quantum Logistic Map

The quantum 3D logistic map [46,47] was first proposed by Muthuswamy and Lakshmanan in 2005 to explore the behavior of quantum systems that exhibit classical chaotic behavior. It exhibits complex behavior that is similar to that of the classical 3D logistic map, including bifurcations and chaos. However, the quantum version of the system also exhibits unique quantum properties, such as quantum interference and entanglement, which arise from the system’s quantum nature. Researchers continue to explore its properties and applications to develop new insights and techniques. One promising direction of research is the use of the system for quantum chaos-based cryptography, which relies on the chaotic behavior of the system to generate secure cryptographic keys.

The quantum 3D logistic map [15] is described as a three-dimensional quantum map that is derived from the classical 3D logistic map. The map can be expressed as follows:

$$\begin{array}{l}{x}_{n+1}=r\left(x_n-{\left|x_n\right|}^2\right)-r{y}_n,\\ y_{n+1}=-y_n{e}^{-2\beta }+e^{-\beta }r\left[\left(2-x_n-x_n^{*}\right)y_n-x_n{z}_n^{*}-x_n^{*}{z}_n\right],\\ z_{n+1}=-z_n{e}^{-2\beta }+e^{-\beta }r\left[2\left(1-x_n^{*}\right)z_n-2x_n{y}_n-x_n\right]\end{array}$$

(2)

where β is the dissipation parameter, $r$ represents the control parameter, and $x_n^{*} \mathrm{and} z_n^{*}$ are the complex conjugates of $x_n \mathrm{and} z_n$, respectively. However, the initial conditions $\left(x_n, y_n, z_n\right)$ are set as real numbers to meet the requirement of communication.

The quantum 3D logistic map is analyzed in terms of hyper-chaos attractors of the quantum logistic map and its bifurcation diagram, respectively, as shown in Figure 2c,d. Whereas Figure 3 depicts the Lyapunov exponents spectrum.

2.2. Hadamard Gate

In quantum mechanics [48], a particle can exist in a superposition state, which means that it is in multiple states simultaneously. For example, an electron can exist in a superposition of two energy levels, or a photon can be in a superposition of two polarization states. In contrast, bringing a qubit into a superposition state is a relatively simple task for quantum computing developers. The process simply requires applying a specific gate to the qubit. The Hadamard gate, for example, is a fundamental single-qubit gate in quantum computing that plays an important role in creating superposition states and other quantum algorithms and protocols. The quantum circuit of the Hadamard gate is shown in Figure 4, and its matrix representation is given by [49]:

$$H=\frac{1}{\sqrt{2}} \left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$$

(3)

The Hadamard gate has the following action on both the computational basis states |0〉 and |1〉:

$$\begin{gathered} H\left|0\rangle =\right.\frac{\left|0\rangle +\left|1\rangle \right.\right.}{\sqrt{2}} \\ H\left|1\rangle =\right.\frac{\left|0\rangle -\left|1\rangle \right.\right.}{\sqrt{2}} \end{gathered}$$

Finally, as depicted in Figure 5 [50], when a qubit initially in the state |0⟩ undergoes the Hadamard gate operation, it enters a superposition state, where the chances of measuring 0 and 1 are equal amplitudes. The Hadamard gate is also important because it has its own inverse. That is, applying the Hadamard gate twice returns the qubit to its original state. This property is useful in quantum algorithms that require the ability to create and undo superposition states.

2.3. Quantum Cellular Automata

Quantum cellular automata (QCA) is the quantum version of cellular [51,52,53]. It is a theoretical model of computation that combines principles of quantum mechanics and cellular automata. In QCA, the state of each cell is a quantum superposition of two or more states, and the evolution of the system is governed by quantum mechanical laws. The evolution of the system is typically described using a unitary operator, which represents a quantum gate that acts on the state of each cell, and the evolutionary pattern of quantum cellular automata is displayed in Figure 6.

The basic idea behind QCA is to use the principles of quantum mechanics to perform certain computations more efficiently than classical methods. Quantum cellular automata (QCA) has several potential advantages over classical cellular automata (CA), including (1) increased speed, where QCA has the potential to provide a significant speedup over classical CA for certain computations that are inherently quantum in nature; (2) increased capacity, where QCA has the potential to store and process more information than classical CA; (3) increased security, where QCA has the potential to provide enhanced security for communication and data storage; and (4) novel applications, where QCA has the potential to provide new insights into physical and biological systems that are difficult to study using classical methods. However, the design and implementation of QCA circuits can be complex and require specialized knowledge of quantum mechanics and circuit design.

2.4. Hybrid Chaotic Maps

Hybrid chaotic maps [54,55] are mathematical models that combine multiple chaotic maps or systems to create intricate and versatile dynamical behaviors. By interconnecting individual chaotic maps with techniques like sequential coupling, parallel coupling, or feedback coupling, hybrid chaotic maps exhibit enhanced complexity and unpredictability compared with their constituent maps. In essence, the purpose of combining these maps is to generate more intricate dynamics, increase randomness, or improve certain characteristics of chaotic behavior.

Three hybrid chaotic systems are utilized in this work. Each chaotic system comprises a nonlinear mixture of two different one-dimensional chaotic maps, i.e., the logistic map, tent map, and sine map, which are supposed to be seed maps. These hybrid chaotic maps are the logistic–tent system, logistic–sine system, and tent–sine system, which are defined, respectively, as:

$$x_{n+1}=mod\left(\left(\left(r\ast x_n\right)\ast \left(1-x_n\right)+\left(\left(4\ast r\ast x_n\right)/2\right)\right),1\right)$$

(4)

$$x_{n+1}=mod\left(\left(\left(r\ast x_n\right)\ast \left(1-x_n\right)+\left(4\ast r\ast sin\left(\pi \ast x_n\right)/4\right)\right),1\right)$$

(5)

$$x_{n+1}=mod\left(\left(\left(\mu \ast x_n\right)+\left(4\ast \mu \ast sin\left(\pi \ast x_n\right)/2\right)\right),1\right)$$

(6)

These maps find applications in various fields such as cryptography, secure communications, random number generation, image encryption, and chaotic control systems. With their diverse and dynamic behaviors, hybrid chaotic maps provide a powerful tool for generating secure encryption algorithms, optimizing complex functions, and exploring the fascinating realm of chaotic dynamics.

3. Proposed Quantum Image Encryption and Decryption Scheme

This section provides a detailed presentation of the proposed quantum image encryption scheme, which utilizes quantum cellular automata (QCA) and mixed multi-chaos hybrid systems to safeguard both grayscale and color images.

3.1. Encryption Procedure

The fundamental C. E. Shannon’s permutation–diffusion model [56] serves as the basis for the entire scheme, which comprises two primary operations, namely, pixel-level diffusion and pixel-level permutation. The detailed steps of these operations can be listed as follows, and the framework of the designed encryption scheme is shown in Figure 7.

3.1.1. Pixel Plane Diffusion

Step 1. Import the color plain image $P$, then acquire the dimensions $M$ and $N$ of the input color plain image $P$.

Step 2. From the input plain image $P$ of size $M\times N$, crop the first row and first column from each component.

Step 3. Meanwhile, while carrying out Step 2, extract the R, G, and B components of the color image, which has dimensions $\left(M-1\right)\times \left(N-1\right)$.

Step 4. Preceding the process of image encryption, a parameter $M$ is generated based on a plain image by determining the mean pixel values of any cropped row or column in any of the three components of the plain image.

Step 5. The user can choose to apply one of the hyperchaotic systems, either a quantum logistic map or Lü system at the level of pixel diffusion. Afterward, the calculated mean value is taken to pre-iterate the chosen hyperchaotic system $M$ times, with the aim of eliminating any transient effects and ensuring that the system enters a fully chaotic state.

Step 6. Set the initial values and control parameters of the chosen hyperchaotic system by applying calculation Formula (1) for the Lü system or calculation Formula (2) for the quantum logistic map as outlined in Section 2.1.1 and Section 2.1.2, respectively.

Step 7. If the selected hyperchaotic system is the quantum logistic map, it will undergo $\left(M-1\right)\times \left(N-1\right)$ iterations to produce three chaotic matrices $\left(Q_i^R, Q_i^G, Q_i^B\right)$ of length $\left(M-1\right)\times \left(N-1\right)$. These matrices act as the diffusion keys and can be determined using the provided formula.

$$\begin{gathered} Q_i^R=mod\left(floor\left({\epsilon }_1\ast x_{i+1}+{\epsilon }_2\right),256\right) i=1, 2, \dots ,M-1\times N-1 \\ {Q}_i^G=mod\left(floor\left({\epsilon }_1\ast y_{i+1}+{\epsilon }_2\right),256\right) i=1, 2, \dots , M-1\times N-1 \\ {Q}_i^B=mod\left(floor\left({\epsilon }_1\ast z_{i+1}+{\epsilon }_2\right),256\right) i=1, 2, \dots ,M-1\times N-1 \end{gathered}$$

(7)

where $({\epsilon }_1, {\epsilon }_2$) are two large prime numbers and $\left(x_{i+1}, y_{i+1}, \mathrm{and} z_{i+1}\right)$ are random sequences, which are generated using 3D quantum logistic map Equation (2).

Step 7.1. Whereas, if the chosen hyperchaotic system is the Lü system, $\left(M-1\right)\times \left(N-1\right)$ iterations will be performed to generate three chaotic matrices $\left(L_i^R, L_i^G, L_i^B\right)$ of length $\left(M-1\right)\times \left(N-1\right)$. These matrices serve as the diffusion keys and can be calculated using the given formula.

$$\begin{gathered} L_i^R\left(i, j\right)=round\left(mod\left(\left(abs\left(x\right)-floor\left(abs\left(x\right)\right)\right)\times {10}^{14}, 256\right)\right) \\ {L}_i^G\left(i, j\right)=round\left(mod\left(\left(abs\left(y\right)-floor\left(abs\left(y\right)\right)\right)\times {10}^{14}, 256\right)\right) \\ {L}_i^B\left(i, j\right)=round\left(mod\left(\left(abs\left(z\right)-floor\left(abs\left(z\right)\right)\right)\times {10}^{14}, 256\right)\right) \end{gathered}$$

(8)

where $i=1, 2, \dots ., M-1\times N-1$ and $\left(x, \mathrm{y}, \mathrm{and} \mathrm{z}\right)$ are random sequences that are generated using Lü system Equation (1).

Step 8. The generated three different diffusion keys either with a quantum logistic map or Lü system as well as the original separated R, G, and B components of the color image are subjected to Hadamard transformation and the CNOT gate together in one quantum circuit to produce entangled qubits. Using a Hadamard gate followed by a CNOT gate will bring all values in all the resulted matrices into the zone of the superposition state and create entanglement. Hadamard gate H can be defined using Equation (3) in Section 2.2 and the action of the CNOT gate can be represented by its truth table alongside its quantum circuit, as shown in Figure 8 [57]:

Step 9. Once all values of the three channels of the plain image and the three chaotic matrices obtained from either mathematical Formula (4) or mathematical Formula (5) are brought into a superposition state zone, a ternary Toffoli quantum gate is executed to modify the value of each pixel qubit. To effectively accomplish pixel-level diffusion, an XOR operation with the corresponding qubits of the three chaotic matrices obtained from either mathematical Formula (4), or mathematical Formula (5) must be performed. The quantum gate form of the Toffoli gate is often called CCNOT. Its quantum circuit diagram and its matrix can be illustrated as follows in Figure 9 [57].

Step 10. Perform the quantum measurement process, which collapses the quantum state of the diffused red, green, and blue components of the image into a classical state their classical versions $\left(C_i^R, C_i^G, C_i^B\right)$.

3.1.2. Pixel Plane Confusion

The objective of pixel plane confusion is to disrupt the positions of image pixels. To achieve this, a combination of quantum cellular automata and mixed multi-chaos hybrid maps is utilized to generate chaotic sequences. The detail of each step of the pixel plane confusion is given below.

Step 1. Multiple hybrid chaotic maps are generated, namely, the Logistic–Tent chaotic map, the Logistic–Sine chaotic map, and the Tent–Sine chaotic map. Each uses a mix of one-dimensional chaotic maps, as outlined in Section 2.4.

Step 2. Initialize the control parameter $r$ and the original variable $x_0$ for both Logistic–Tent Equation (4) and Logistic–Sine Equation (5) hybrid chaotic maps, whereas set parameter coefficient µ and the original variable $x_0$ for Tent–Sine Equation (6) hybrid chaotic map, to acquire two chaotic sequences. The calculation procedure for obtaining these sequences is as follows.

$$\begin{gathered} S_1=round \left(mod\left(\left(abs\left(x\right)-floor\left(abs\left(x\right)\times {10}^{14}\right), 256\right)\right)\right) \\ {S}_2=round \left(mod\left(\left(abs\left(x\right)-floor\left(abs\left(x\right)\times {10}^{16}\right), 256\right)\right)\right) \end{gathered}$$

(9)

where $x$ is the chaotic sequence generated from each of the hybrid chaotic maps.

Step 3. By selecting two resulting chaotic sequences of one of the hybrid chaotic maps to use as the parameters for the two-dimensional cellular automata (2D-CA), one chaotic sequence $CA$ is generated using the 2D-CA. This can be expressed as follows.

$$\begin{gathered} C_1=mod\left(S_1, \mathrm{n}\_\mathrm{states}\right) \\ {C}_2=mod\left(S_2, \mathrm{n}\_\mathrm{states}\right) \end{gathered}$$

(10)

where $S_1 \mathrm{and} S_2$ are the chosen chaotic sequences generated from one of the hybrid chaotic maps and n_states refer to the number of possible values that each pixel in the image can take on.

Step 4. Reshape the generated chaotic sequence $CA$ into a matrix with size $3\times N$.

Where $N$ represents the column numbers in the image.

Step 5. For the diffused R, G, and B components, select one of the hybrid chaotic maps to be applied on each one of them in the permutation process.

Step 6. The chaotic sequence $CA$ produced by the cellular automaton has a size of $3\times N$, indicating that it comprises three chaotic sequences, each with a length of $N$. This also allows the user to choose which sequence from the $CA$ to utilize in the permutation process.

Step 7. Using the selected sequence from the chaotic sequence $CA$ generated by the cellular automaton in the former step, the pixels positions in each row of the diffused channels are shuffled. The confusion operation is achieved by using the circular shift operation in the subsequent manner.

$$\begin{gathered} F{C}_i^R\left(k, :\right)=circshift\left(C_i^R\left(k, :\right), CA\left(key2\left(1\right), k\right)\right) k=1, 2, \dots , N-1 \\ F{C}_i^G\left(k, :\right)=circshift\left(C_i^G\left(k, :\right), CA\left(key2\left(1\right), k\right)\right) k=1, 2, \dots , N-1 \\ F{C}_i^B\left(k, :\right)=circshift\left(C_i^B\left(k, :\right), CA\left(key2\left(1\right), k\right)\right) k=1, 2, \dots , N-1 \end{gathered}$$

(11)

where $key2$ refers to the selected vector from the chaotic sequence generated by the $CA$.

Step 8. Transpose the red, green, and blue components of the scrambled image to randomly permute their rows and columns as represented as follows.

$$\begin{gathered} F{C}^R={FC}^{\prime R} \\ F{C}^G={FC}^{\prime G} \\ F{C}^B={FC}^{\prime B} \end{gathered}$$

Step 9. To obtain the desired final versions of the three confused channels $\left(F{C}^R, F{C}^G, F{C}^B\right)$, execute steps 7 and 8 one more time.

Step 10. The first cropped original row and column are added to each of the three encoding components. Afterward, at the last, recompose the three cipher layers into one RGB image.

3.2. Decryption Procedure

Due to the quantum operations being invertible, the decryption process involves executing the steps of the encryption process in reverse order and utilizing the identical set of keys generated earlier. This means that the encryption and decryption are reciprocal processes in the presented scheme.

The process of decryption comprises two stages: (1) reversing the pixel plane confusion and (2) reversing the pixel plane diffusion. Figure 10 depicts a diagram of the designed decryption process, which demonstrates that the proposed scheme enables the retrieval of the original image precisely from the encoded cipher image.

4. Experimental Results and Numerical Analysis

4.1. Experiment Platform

The efficiency of the proposed scheme is assessed by conducting simulations and comparing the results against security parameters. Quantum computing is a relatively new field, and while there have been significant advances in recent years, a practical and fully functional quantum computer is not yet available. As a result, all the experimental results of the proposed quantum image encryption scheme are tested using the MATLAB R2017b platform on a classical computer to authenticate the security and efficacy of the quantum image encryption algorithm that is presented.

The numerical simulation of the proposed encryption scheme is run in a Windows 11 Home Single-Language environment. The specification of the utilized computer system includes 11th Gen Intel(R) Core (TM) i7-11800H applied as the central processing unit (CPU) @ 2.30 GHz and the random-access memory (RAM) is 16 GB.

4.2. Experiment Results

Numerous experiments were conducted to evaluate the efficiency of the proposed image encryption algorithm and demonstrate its capability and effectiveness in the encryption and decryption process. In these experiments, six diverse images (including grayscale and colored images) are selected from the CVG-UGR-Image Database [58] and from the USC-SIPI (University of Southern California Signal and Image Processing Institute) Image Database [59] as standard test images for this method.

Table 1. Selected test images.

Image	Male	Baboon	Airport	Lena	Splash	Airplane
Size	256 × 256	512 × 512	1024 × 1024	256 × 256	512 × 512	1024 × 1024
Type	Grayscale	Grayscale	Grayscale	Color	Color	Color

contains a comprehensive list of their detailed information.

Figure 11 illustrates the simulation results of the proposed image encryption and decryption approach. It showcases the plain images (PI), namely, “Male”, “Baboon”, “Airport”, “Lena”, “Splash”, and “Airplane” in Figure 11a–f. Their corresponding cipher images (CIs) are presented in Figure 11g–l, whereas the decryption images (DIs) are shown in Figure 11m–r, which are consistent with the original images before encryption.

As illustrated in Figure 11, the proposed encryption scheme is not limited by the number and size of the images, it is implemented on multiple images with different sizes and different numbers of images. In addition, the proposed scheme obfuscates the clear patterns in the plain image and generates a ciphered image that appears to have a random noise-like pattern, thereby making it difficult for an intruder to discern any meaningful information. There is no correlation between the encrypted and decrypted images, and the decryption algorithm effectively produces a visually indistinguishable decrypted image from the original image. As a result, the proposed encryption algorithm achieves a high level of encryption and decryption effectiveness, ensuring both the security and visual integrity of the image data.

4.3. Security Analysis

Cryptography is vulnerable to various types of attacks, such as entropy attacks, chosen-plaintext/ciphertext attacks, and brute-force attacks. Therefore, an image encryption method must possess sufficient strength to withstand such attacks. It is important to note that no encryption algorithm can provide perfect security, but a robust algorithm should provide a high level of security against a wide range of attacks.

Several well-known encryption evaluation metrics are commonly used to assess the security and the quantitative performance of the proposed image encryption algorithm. Some of these metrics are (1) key security analysis; (2) statistical attack analysis; and (3) differential attack analysis. In addition to the measures mentioned, there are other techniques used to gauge the algorithm’s ability to determine its effectiveness in protecting image data including robustness to noise and robustness to attacks. A detailed discussion of each of these measures is provided in the accompanying subsections. The last subsection also includes a discussion on the speed of the proposed algorithm.

4.3.1. Key Security Analysis

Key Space Analysis

In modern cryptography [60,61], the size of the key space is a crucial factor in determining the cryptosystem’s effectiveness against brute-force attacks. To protect against brute-force attacks, it is important to use strong passwords or passphrases and to use encryption algorithms with sufficiently large key spaces. While there is no universal standard for an ideal key space, it is generally recommended that the key space should be at least $2^{100}$ to withstand a brute-force attack, which is approximately ${10}^{30}$. Generally, the larger the key space, the more difficult it is to guess the correct key using brute force, and the more secure the scheme is. The key space is computed based on the number of distinct keys used in both the diffusion and confusion stages [62]. The private key for the proposed quantum image encryption scheme consists of a number of initial conditions and control parameters in total, listed as follows: (1) The parameters of Lü’s hyperchaotic system are a, b, and c, and each of the original variables $\left(x_1, x_2, x_3\right)$ has two decimal places; there exist ${10}^2$ possible values for each value. This applies to $\left(a, b, c, x_0, y_0, \mathrm{and} z_0\right)$ as well. Thus, there are ${10}^{12}$ possible values for each value. (2) The Lü system has the capacity to produce numerous distinct values, estimated at ${10}^{42}$, for all of its generated keys $\left(L_i^R, L_i^G, L_i^B\right)$ as they can be represented up to 14 decimal places. Overall, the Lü’s hyperchaotic system has a key space of ${10}^{54}\approx 2^{179}$. (3) Parameters β and r are used in the quantum logistic chaotic map, where β consists of four decimal places; there exist ${10}^4$ possible values for β and r consists of two decimal places, and there exist ${10}^2$ possible guesses of its value. Thus, there are ${10}^6$ possible values of β and r. (4) Each initial value of quantum logistic map $\left(x_0, y_0, z_0\right)$ consists of 12 decimal places with a range between 0 and 1; there exist ${10}^{12}$ possible values for each value. Thus, there are ${10}^{36}$ possible values of $x_0, y_0, \mathrm{and} z_0$. (5) Two large prime numbers $\left({\epsilon }_1, {\epsilon }_2\right)$ consist of eight decimal places; there exist ${10}^8$ possible values for each value. Thus, there are ${10}^{16}$ possible values of $\left({\epsilon }_1, {\epsilon }_2\right)$. Overall, the quantum logistic chaotic map possesses a key space that is equivalent to ${10}^{34}\approx 2^{113}$. (6) The cellular automata Logistic–Sine hybrid map has one control parameter r, and an original variable $x_0$, wherein r consists of four decimal places; there exist ${10}^4$ possible values for r, while $x_0$ has two decimal places; there exist ${10}^2$ possible values for its value. Thus, there are ${10}^6$ possible values of r and $x_0$. (7) Likewise, the cellular automata Logistic–Tent hybrid map possesses a total of ${10}^6$ potential values for its parameter and initial value r and $x_0$, respectively. (8) The initial condition and control parameter of the cellular automata Logistic–Tent hybrid map are $x \mathrm{and} \mu$, wherein µ consists of three decimal places; there exist ${10}^3$ possible values for µ, while $x_0$ has two decimal places; there exist ${10}^2$ possible values for its value. Thus, there are ${10}^5$ possible values of $x \mathrm{and} \mu$. (9) Each hybrid chaotic map produces two chaotic sequences, $S_1 \mathrm{and} S_2$, which can be represented with up to 14 and 16 decimal places, respectively. As a result, this provides for a total of ${10}^{30}$ feasible value guesses for each map. (10) The key value utilized for circular shift is derived from any of the former-mentioned cellular automata hybrid maps, which has a size of $1024\times 3$ or approximately ${10}^{3.4878}$.

As a result, the estimated overall key space for the suggested quantum image encryption scheme comes out as follows:

$\langle {10}^{54}|{10}^{34}\rangle \times \langle {10}^6|{10}^6|{10}^5\rangle \times \langle {10}^{30}|{10}^{30}|{10}^{30}\rangle \times {10}^{3.4878}={10}^{93.4878}\cong 2^{310.2344}$, which is greater than $2^{100}.$ Consequently, the proposed cryptosystem has a key space that is adequately large to withstand all possible types of brute-force attacks and even potential attacks from quantum computers, and hence, it is able to ensure a high degree of security.

Table 2. Security key space comparison of proposed and recent encryption schemes.

Encryption Scheme	Ref. [2]	Ref. [10]	Ref. [20]	Ref. [60]	Ref. [63]	Proposed
Key Space Size	$2^{128}$	$2^{186}$	$2^{249}$	$2^{256}$	$2^{256}$	$2^{310.2344}$

highlights the key space comparison of the proposed quantum image encryption scheme with other recently presented schemes [2,10,20,60,63]. Based on the comparison results, it can be concluded that the proposed scheme provides better security than its competitors when transmitting images through unsecured communication channels.

Key Sensitivity Analysis

In an ideal image encryption scheme [27,64,65,66], the encryption and decryption processes are dependent on a secret key. Each bit of the key plays a crucial role in determining the output of the encryption and decryption process. The sensitivity of the key refers to the degree of protection required for the key to prevent unauthorized access to the encrypted data. If any bit of the key is changed, even slightly, the result for both the encryption and decryption output would be drastically different, making it impossible to access the original data without the correct key. The more sensitive the encryption algorithm is to the key, the more robust and secure it is. To test the sensitivity of the proposed quantum image encryption scheme, the color image “Lena” with dimensions of $512\times 512$ is used as an example, as shown in Figure 12a. A minor modification was made to one of the original secret keys, which is used to encrypt the original image using the same encryption algorithm during the testing phase, while the rest of the keys remained unchanged. In the proposed cryptosystem, using the initial set of the control parameters and original variables of Lü’s system $\left\{a, b, c, x_0, y_0,z_0\right\},$ which are denoted as $K_1$, produces the encrypted image $E_1$, as shown in Figure 12b. Here, the values of $\left(c,x_0,y_0,z_0\right)$ serve as the private key, and $\left(a, b\right)$ are the public key. Assuming a minor modification is made only to the value of $x_0$ in $K_1,$ $K_2$ can be represented as $\left\{a-1, b, c, x_0, y_0,z_0\right\}.$ Utilizing the same encryption algorithm, the plain image is then encrypted with the $K_2$ algorithm to produce another encrypted image $E_2$, as depicted in Figure 12c. From the results, there is a considerable difference between the image encrypted with $K_1$ and the image encrypted with $K_2$, as illustrated in Figure 12d. In addition, Figure 12e displays the decryption outcome when the incorrect decryption key, $K_2$, is used to decrypt the image in Figure 12b. The correct decrypted image is presented in Figure 12f. It is apparent that the process of decrypting the encrypted image is feasible only when the same key is used. In conclusion, the proposed encryption and decryption scheme is extremely sensitive to the secret key. This contributes to achieving superior levels of security and performance. Moreover, it ensures that the cryptosystem can withstand attacks and maintain its robustness against various threats.

4.3.2. Statistical Attack Analysis

The assessment of an image encryption scheme’s security using statistical attack analysis involves analyzing its susceptibility to statistical attacks by focusing on analyzing the statistical characteristics of the encrypted image to gather insights about the encryption key or the original image. Various metrics are available to assess an image encryption scheme’s resistance to statistical attacks. These metrics comprise the correlation coefficient, pixel distribution, and entropy of the encrypted image, which aid in measuring the level of randomness and complexity in the encrypted image. The succeeding subsections provide a detailed description of these metrics.

Histogram Analysis

Histogram analysis is a statistical metric in image processing and analysis, which is also widely used in image encryption. In image encryption [67], histogram analysis is one of the fundamental indicators for evaluating the security of an encryption scheme against statistical attacks. A histogram [68,69,70] is a visual depiction that illustrates the frequency distribution of the pixel values in an image, and it displays the number of pixels that have a particular intensity level. The horizontal axis of the histogram represents the intensity levels of the pixels, while the vertical axis represents their frequency or count. A histogram can be created for each color channel of an image, such as red, green, and blue. Histogram analysis uncovers that the original images have non-uniform histograms, making them more susceptible to attacks. In the case of cipher images, the histogram should be as consistently uniform and flat as possible to achieve maximum security.

Figure 13a,c illustrates the histogram analysis of the standard test grayscale image “Baboon” and color image “Splash” for red (R), green (G), and blue (B) channels before the encryption process, both of dimensions 512 × 512, and Figure 13b,d presents their corresponding encrypted versions, respectively. As shown in these figures, the histogram distribution is more uniform in the cipher image, which means a stronger resistance against various statistical analysis attacks, such as known-plaintext attacks and chosen-plaintext attacks.

In quantitative analysis of the image histogram [71,72], another important metric called the histogram variance (var) is used to demonstrate the degree of variation or diversity in the pixel intensity distribution of the encrypted image’s histogram. A higher histogram variance indicates a greater diversity of pixel values in the image, while a lower variance indicates a more uniform pixel distribution. The variance in the histogram can be computed as follows [15]:

$$var\left(H\right)=\frac{1}{n^2}{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^n\frac{1}{2}{\left(h_i-h_j\right)}^2$$

(12)

where $H=\left\{h_1,h_2,\dots .,h_{256}\right\}$ is a one-dimensional array of the histogram values, with pixel values $i \mathrm{and} j$ recorded as $h_i \mathrm{and} h_j$, respectively.

Table 3. Histogram variance analysis for greyscale images.

Image Name	Variance Value
	Plain Image	Cipher Image
Male	36,436.4941	251.1294
Baboon	1,185,695.4275	1332.502
Airport	31,720,325.7098	4023.2078

and

Table 4. Histogram variance analysis for color images.

Image Name	Variance Value
	Plain Image			Cipher Image
	R	G	B	R	G	B
Lena	62,616.6039	29,467.3176	88,387.1922	240.5804	298.9725	248.9176
Splash	2,432,151.2863	3,095,990.0627	5,940,168.3608	1124.4157	2305.9843	1175.0431
Airplane	43,315,434.4549	43,368,407.9686	71,618,941.898	3903.5608	8508.5098	4671.0588

display the intensity channels of the experimented grayscale and color plaint and cipher images, respectively, and illustrate that the variance in the images after encryption is greatly reduced when compared with the variance in those images before encryption.

By observing the former figures and tables of the current test, it is apparent that the distribution of grayscale and RGB components in plain images contains numerous peaks and valleys, whereas the distribution of grayscale and RGB components in cipher images is highly uniform, and their histogram is entirely dissimilar to that of the plain images. This establishes the arduousness of extracting meaningful information from the encrypted image; thereby, the proposed quantum image encryption scheme is secure enough to overcome statistical attacks.

Information Entropy Analysis

In image encryption, entropy analysis is used to evaluate the effectiveness of an encryption algorithm in producing encrypted images that are resistant to attacks. Information entropy analysis [73,74] is a fundamental metric in image encryption that assesses the degree of randomness and unpredictability in the distribution of pixel values within an image. It is based on the concept of entropy, which is a measure of the amount of uncertainty or randomness in a system.

The entropy of an image is computed before and after the encryption process. As the level of disorder in a cipher image increases, so does its information entropy, while the amount of information conveyed decreases. This correlation leads to an improved encryption outcome making it more difficult for unauthorized parties to access the information. The entropy (E) of an image can be calculated using the mathematical formula given below [75]:

$$E=-{\displaystyle \sum }_{i=0 }^{255}P\left(i\right){\log }_{2}P\left(i\right)$$

(13)

where $P\left(i\right)$ is the probability of having the ith gray level in the image. Due to the uniform distribution of values in grayscale images, which have $2^8$ identical values among the 256 possible values, an ideal entropy value suitable for grayscale images is equal to 8 bits. To verify the effectiveness of the presented image encryption scheme, the entropy values for each component in the cipher image ought to be very close to 8.

Nevertheless [76], the overall entropy may not accurately capture the distribution of entropy within encrypted images. Therefore, local Shannon entropy is used to enhance the security and randomness of the encrypted image. It is derived by computing the global entropy for K non-overlapping blocks, each comprising a fixed number of pixels.

Table 5. Information entropy of both plain and cipher images.

Image Name	Dimension	Information Entropy		Local Shannon Entropy
		Plain Image	Cipher Image	Plain Image	Cipher Image
Male	256 × 256	7.5341	7.9972	6.9856	7.9542
Baboon	512 × 512	7.0800	7.9991	6.8492	7.9888
Airport	1024 × 1024	6.8303	7.9998	6.5529	7.9972
Lena	256 × 256 × 3	7.7508	7.9990	7.3532	7.9846
Splash	512 × 512 × 3	7.2428	7.9996	6.5284	7.9958
Airplane	1024 × 1024 × 3	6.6654	7.9999	6.1091	7.9989

displays the results for both global entropy and local Shannon entropy of several standard test images alongside their encrypted versions. From the stated data in Table 5, the information entropy values of the cipher images generated with the proposed approach are actually very near to the ideal value of 8. This proves that the proposed approach provides a high level of security against various types of attacks, including statistical analysis and brute-force attacks.

In addition,

Table 6. Information entropy comparative analysis.

Image Name	Dimension	Information Entropy
		Proposed Scheme	Ref. [15]	Ref. [67]	Ref. [77]	Ref. [78]	Ref. [79]
Airplane	256 × 256	7.9977	-	-	-	7.9236	-
Baboon	512 × 512	7.9991	-	7.9985	-	-	-
Male	1024 × 1024	7.9998	7.9998	7.9996	-	-	-
Lena	256 × 256 × 3	7.9990	-	-	7.9971	-	-
Splash	512 × 512 × 3	7.9996	-	-	-	-	7.9992
Mandrill	1024 × 1024 × 3	7.9999	7.9999	-	-	-	-

exposes a comparison of the information entropy values resulting from the proposed scheme with those obtained from other recent schemes. It can be seen that its overall performance is thus the same as that of the approach proposed in [15], and better than that of the other tested methods, which indicates the superiority of the proposed quantum image encryption scheme over existing schemes implies that it provides better protection against statistical attacks of any kind.

Correlation Distribution Analysis

Due to the nature of digital images [80,81], the pixels within the image are not independent entities, resulting in a significant degree of correlation between adjacent pixels. In image encryption, a high correlation between adjacent pixels can make the encrypted image susceptible to various attacks, such as differential attacks and linear attacks. Whereas a low correlation coefficient indicates that the encryption algorithm has successfully randomized the pixel values in the image and reduced the correlation between adjacent pixels, making it more challenging for an attacker to extract information from the encrypted image. Therefore, correlation coefficient analysis is a vital metric for evaluating the effectiveness of image encryption algorithms in reducing the correlation between adjacent pixels and improving the security of the encrypted image.

Correlation coefficient analysis provides a quantitative measure of the degree of correlation between adjacent pixels in an encrypted image. It ranges from −1 to 1, where −1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation [82]. Thus, a low correlation coefficient is desirable in image encryption to enhance the security and resistance of the encryption scheme against various attacks.

The correlation coefficient among adjacent pixels can be represented mathematically as [12]:

$$\left\{\begin{array}{c}{r}_{xy}=\frac{cov\left(x,y\right)}{\sqrt{D\left(x\right)}\times \sqrt{D\left(y\right)}}\\ cov\left(x,y\right)=\frac{1}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}\left(x_i-E\left(x\right)\right)\left(y_i-E\left(y\right)\right)\\ D\left(x\right)=\frac{1}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}{\left(x_i-E\left(x\right)\right)}^2\\ E\left(x\right)=\frac{1}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}{x}_i\end{array} \right.$$

(14)

where $x_i$ and $y_i$ are randomly chosen pixels values of two adjacent pixels, N represents the total number of selected samples in each direction,$cov\left(x,y\right)$ is the covariance of x and y, and $E\left(x\right)$ and $D\left(x\right)$ denote the mean value and the variance of pixel value x, respectively.

Moreover, correlation coefficient analysis can be performed in different directions, such as horizontal, vertical, and diagonal, to evaluate the degree of correlation between adjacent pixels in different directions. Thus, 2500 pairs of two adjacent pixels in both the plain and cipher images in each direction were randomly selected to evaluate the effectiveness of the proposed quantum image encryption scheme in reducing the correlation between adjacent pixels. Compared with the histogram and data entropy, the correlation coefficient provides a more discernible visual representation of the dissimilarity between the plain and cipher images. To figure out this differentiation, the grayscale “Baboon” image and color “Splash” image, with dimensions of $512\times 512$, were used before and after applying the image encryption technique, as shown in Figure 14 and Figure 15, respectively. From these figures and

Table 7. Correlation coefficients results.

Image Name	Plain Image			Cipher Image
	Vertical	Horizontal	Diagonal	Vertical	Horizontal	Diagonal
Male	0.9630	0.9517	0.9240	−0.0039	0.0042	−0.0055
Baboon	0.8910	0.9007	0.8240	0.0009	0.0036	−0.0007
Airport	0.9034	0.9099	0.8590	0.0034	0.0019	−0.0016
Lena	0.9725	0.9479	0.9231	0.0045	−0.0019	0.0037
Splash	0.9951	0.9936	0.9894	0.0040	−0.0017	−0.0031
Airplane	0.9904	0.9937	0.9845	0.0018	0.0001	0.0003

, you can clearly notice that the correlation among adjacent pixel pairs in the plain image is highly concentrated, whereas in the cipher image, it appears to be randomly dispersed, resulting in a more uniform distribution and a significant reduction in the correlation. In addition, the correlation coefficient between adjacent pixels is close to 0.

4.3.3. Differential Attack Analysis

Differential attack analysis [83,84,85] is considered another type of widely used cryptanalysis technique for evaluating the security of image encryption schemes. It is based on the idea that small changes in the plaintext should result in large changes in the ciphertext. In this analysis, an attacker studies the differences between two images that are similar, such as an original image and a slightly modified version of that image. The attacker then compares the two images pixel-by-pixel to determine the differences in pixel values between the two images. By examining these differences, the attacker can attempt to identify patterns or relationships that could be exploited to break the encryption and hence, successfully recover the original image or the encryption key.

Assuming an original image $I$ with dimension $M\times N$ is available, along with a set of encryption keys, the encrypted images $C$ and $C^{\prime }$ can be generated either with a slight modification to one of the encryption keys in the set or a single pixel in one of the intensity channels in the same original image $I$. A bipolar array $D\left(i, j\right)$, with sizes similar to those of $C_1$ and $C_2$ resulting from this modification can be defined as follows:

$$D\left(i,j\right)=\left\{\begin{array}{c}0, C\left(i,j\right)=C^{\prime }\left(i,j\right)\\ 1, C\left(i,j\right)\ne C^{\prime }\left(i,j\right)\end{array}\right.$$

(15)

To protect against differential attacks, encryption schemes must be designed so that small changes in the plaintext result in significant changes in the ciphertext, and this can be achieved using techniques such as confusion and diffusion. The plain image sensitivity of the proposed quantum image encryption scheme can be evaluated by utilizing differential attack test metrics, such as the number of pixels changing rate (NPCR) and unified average changing intensity (UACI). The proposal of these metrics aims to improve the evaluation of the diffusion and confusion properties of the encryption scheme by providing more accurate measurements for each property.

NPCR measures the diffusivity of the encryption scheme. It calculates the percentage of pixels that change between two encrypted images generated from two plain images that differ by only one pixel. Meanwhile, UACI measures the confusion property of the encryption scheme. It calculates the average difference between the pixel values of the two encrypted images generated from two plain images that differ by only one pixel. The NPCR and UACI are defined as follows:

$$\mathrm{NPCR}=\frac{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^{N}D\left(i,j\right)}{M\times N}\times 100\%$$

(16)

$$UACI=\frac{1}{M\times N}\left({\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^N\frac{\left|C_1\left(i,j\right)-C_2\left(i,j\right)\right|}{255}\right)\times 100\%$$

(17)

A secure encryption scheme should produce a high NPCR and a low UACI, indicating that even small changes in the plaintext image result in significant changes in the encrypted image, and the pixel values in the encrypted image are more random and less correlated with the pixel values in the original image. For a grayscale image, the NPCR and UACI have theoretical ideal values of 99.6094% and 33.4635%, respectively.

Table 8. NPCR and UACI results of cipher grayscale images.

Image Name	Image Size	NPCR (%)	UACI (%)
Male	256 × 256	99.6586	33.6368
Baboon	512 × 512	99.6243	33.4917
Airport	1024 × 1024	99.6149	33.4560

and

Table 9. NPCR and UACI of cipher color images.

Image Name	Dimension	NPCR (%)			UACI (%)
		Red	Green	Blue	Red	Green	Blue
Lena	256 × 256 × 3	99.6078	99.6432	99.6386	33.5419	33.4985	33.5333
Splash	512 × 512 × 3	99.6147	99.6446	99.6082	33.4492	33.5153	33.4453
Airplane	1024 × 1024 × 3	99.6128	99.6099	99.6131	33.4585	33.4692	33.4790

present the results of differential attack analysis on grayscale and color images of various sizes, respectively.

Table 8 and Table 9 demonstrate that the NPCR and UACI simulation outcomes for all components on the testing images are near their ideal values. These results indicate that the proposed quantum image encryption scheme is highly immune to differential attacks and provides a high level of security, as it exhibits a high degree of sensitivity even to slight pixel variations in the intensity channels of the plaintext images.

4.3.4. Robustness to Attacks Analysis

Chosen/Known-Plaintext Attacks Analysis

The chosen-plaintext attack and the known-plaintext attack [86,87] are two of the most common types of attacks used in cryptanalysis to break encryption algorithms, including those used for image encryption. In the context of image encryption, these attacks can be particularly challenging, as images often contain large amounts of data that can be difficult to protect against attack. By utilizing these attacks, assailants attempt to uncover the connection between plain and cipher images, with the intention of extracting recurring patterns that aid in deducing the private encryption key or recovering the original image.

Since the attacker has more control over the data being encrypted, the chosen-plaintext attack is considered more potent than the known-plaintext attack. Consequently, an image encryption algorithm that is capable successfully of withstanding a chosen-plaintext attack is also able to resist a known-plaintext attack [60]. To defend against chosen/known-plaintext attacks in image encryption, multiple plain images including “Black”, “White”, and “Gray Bars” are chosen as test images to undergo encryption using the proposed quantum image encryption scheme, to break the scheme, as shown in Figure 16.

From Figure 16, it is evident that the cipher images generated using the proposed quantum image encryption scheme exhibit a noise-like pattern and possess uniformly distributed pixel values, thus providing the proposed scheme with the ability to effectively resist both chosen-plaintext and known-plaintext attacks.

Noise Attack Analysis

The conveying of cipher images via physical communication channels is prone to noise or interference, making it necessary for the image decryption scheme to possess the strength to decrypt the cipher images even if noise accumulates during transmission [88]. A noise attack is a type of attack in which an attacker introduces random noise into the encrypted image, with the hope of discovering information about the original image. To evaluate the security of an image encryption scheme against noise attacks, an attacker may use various types of noise attacks to try to retrieve the original image.

For the current test, salt-and-pepper noise is utilized to assess the effectiveness of the image’s anti-noise capabilities. The noise gets its name from the fact that it randomly changes some pixels in the original image to either black or white, simulating the effect of random errors in the transmission. The mean square error (MSE) and the peak signal-to-noise ratio (PSNR) are widely used metrics for evaluating the robustness of an image encryption scheme against salt-and-pepper noise. The MSE measures the average squared difference between the original image and the decrypted image, while the PSNR measures the ratio of the maximum possible power of a signal to the power of the noise that corrupts the signal.

A lower MSE value indicates that the decrypted image is closer to the original image, whereas a higher PSNR value indicates that the decrypted image has less noise and is closer in quality to the original image. Therefore, these metrics are important indicators for measuring the ability of an encryption system to resist noise and maintain the quality of the image. These metrics are defined below [64]:

$$MSE=\frac{1}{M\times N} {\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=1}^N{\left(O\left(i, j\right)-D\left(i, j\right)\right)}^2$$

(18)

$$PSNR=10{\log }_{10}\left(\frac{{255}^2}{MSE}\right)$$

(19)

In this experiment, to evaluate the resilience of this algorithm over its ability to withstand noise, the “Lena” color image of dimension $512\times 512$ is subjected to different levels of added noise, specifically 0.001, 0.005, 0.01, and 0.05, as shown in Figure 17. Figure 17 illustrates that the proposed quantum image encryption scheme maintains the decrypted image’s clarity and recognizability, even when the image is affected by noise data. This finding indicates that the encryption scheme provides a high level of security and is capable of effectively resisting noise attacks.

Moreover,

Table 10. Noise attack analysis.

Attack Intensity	PSNR (db)
	R	G	B
0.001	35.7982	35.6893	33.6509
0.005	33.0676	33.5090	32.4071
0.01	31.2846	31.8355	31.2754
0.05	25.3220	25.8874	26.5529

presents the experimental findings, which demonstrate that despite a decrease in the PSNR of the encrypted image following decryption in the presence of an attack, the decrypted image remains visually distinguishable. These results confirm the efficacy of the proposed scheme in effectively combating salt-and-pepper noise.

Occlusion Attack Analysis

In image encryption, occlusion attacks are used to compromise the integrity of an encrypted image, with the intention of impeding the decryption process or revealing its content [89]. The idea behind occlusion attacks is that by obscuring parts of the encrypted image, an adversary tries to decrypt an encrypted image by partially or completely obscuring some parts of the encrypted image. The adversary may perform this by covering parts of the image with opaque objects, adding random pixels or patterns, or using other techniques to mask parts of the image.

To counteract occlusion attacks, the proposed approach encrypts and decrypts the image of “Lena” subsequent to the loss of 1/16, 1/8, 1/4, and 1/2 of its information, respectively, as depicted in Figure 18.

The findings presented in Figure 15 reveal that, despite considerable data loss from the encrypted image due to occlusion attacks, the decryption process can recover a substantial amount of the original information. These outcomes suggest that the designed quantum image encryption scheme is highly resistant to occlusion attacks.

4.3.5. Time Complexity Analysis

Time complexity plays a crucial role in determining the efficiency and performance of image encryption schemes. In image encryption schemes, time complexity refers to the measure of the computational time required to encrypt or decrypt an image and is typically expressed as a function of the size of the input image, which is represented by the number of pixels in the image [90,91].

The time consumption of the proposed quantum image encryption scheme primarily can be divided into two components: the time required for pixel-level substitution operations and the time needed for pixel-level permutation operations.

In the pixel-level substitution stage, the mean operation on any cropped row or column in the R, G, and B channels of the original image has a complexity of O(1); the complexity of the pre-iterate process, which is performed M times for the chosen hyperchaotic system, is O(n); the application of either the quantum logistic map or Lü system hyperchaotic systems requires a complexity of O(n²); the computational complexity associated with utilizing a Hadamard gate, followed by a CNOT gate, and then applying a ternary Toffoli quantum gate is O(n²); executing the quantum measurement operation incurs a time cost of O(n²). In the pixel-level permutation stage, using a hybrid of mixed multi-chaos hybrid maps and a two-dimensional quantum cellular automaton has a computational complexity of O(n²); a circular shift operation has a computational complexity of O(n). Based on this analysis, the computational complexity of the proposed quantum scheme is O(n²).

In addition,

Table 11. Execution time analysis.

Image Name	Image Size	Encryption/Decryption Time (s)
Grayscale Male	256 × 256	1.7813
Color Lena	256 × 256	3.1094
Grayscale Baboon	512 × 512	4.1094
Color Splash	512 × 512	9.5469
Grayscale Airport	1024 × 1024	13.7188
Color Airplane	1024 × 1024	35.7031

showcases the performance of the proposed quantum encryption scheme in terms of execution time on a variety of standard images with different sizes.

Table 12. Encryption speed comparison of proposed and recent encryption.

Encryption Scheme	Ref. [92]	Ref. [93]	Ref. [94]	Proposed
Encryption Time (s)	3.321	5.932	22.223	3.1094

compares the execution time of the proposed quantum encryption scheme with other relevant schemes considering the “Lena” image with dimensions of $256\times 256$ as an instance. According to the results listed in Table 12, the proposed scheme takes less time than the others, which means that the proposed scheme exhibits the highest encryption speed, which makes it a suitable candidate for deployment in encryption systems that require both fast communication and high security.

5. Conclusions

The efficiency and security of the quantum image encryption algorithm are enhanced by leveraging the properties of dynamics of hyperchaotic systems and quantum computation. This study introduces an efficacious and safe quantum image encryption scheme that is grounded on the utilization of quantum operations with hyper-chaos systems and a fusion of quantum cellular automata and mixed multi-chaos maps.

A stronger correlation between the plain image and the secret key can be achieved by computing the average of the first row/column of the original image. The presented scheme is essentially made up of two phases, which are pixel-level diffusion and pixel-level permutation. In this work, the emphasis of the diffusion process is on generating and encrypting superposition states using hyper-chaos systems and quantum operations. On the other hand, the confusion phase is conducted following the application of suitable measurements on the three diffused channels. Subsequently, a chaos row/column cyclic shift is applied to further modify the positions of the pixel values in the three diffused components of the image. These phases are explored in detail throughout this paper.

The scheme was implemented and analyzed with various parameters, such as key sensitivity and key space analyses. In addition, various statistical and differential analyses covering histograms, histogram variance, global entropy, Shannon entropy, correlation coefficients, NPCR and UACI, chosen/known-plaintext attacks, MSE, and PSNR are performed in this work.

Using these several tests and experiments, the security and robustness evaluation of the designed quantum image encryption scheme demonstrated its high sensitivity to even minor modifications in either the plain image or the secret key, and its resilience against various common attacks. It is characterized by computationally low cost. Moreover, the results of such analysis are compared with those of the existing scheme, which demonstrates that the proposed scheme works more effectively than the existing schemes.

Although the proposed scheme offers several advantages, there is still room for potential improvements in the future. It may be beneficial to provide a further combination of a novel enhanced quantum representation (NEQR) model for digital images with hyperchaotic systems. Another limitation of this study is that it may not be suitable for real-time applications due to it consuming long encryption and decryption time for the digital images with dimensions $1024\times 1024$. In upcoming research, we aim to address these limitations and further improve the encryption scheme.