Chaotic Image Encryption: State-of-the-Art, Ecosystem, and Future Roadmap

Abstract

Recently, many researchers have been interested in the application of chaos in cryptography. Specifically, numerous research works have been focusing on chaotic image encryption. A comprehensive survey can highlight existing trends and shed light on less-studied topics in the area of chaotic image encryption. In addition to such a survey, this paper studies the main challenges in this field, establishes an ecosystem for chaotic image encryption, and develops a future roadmap for further research in this area.

1. Introduction

Image processing is used in various computing environments [1,2]. Image processing techniques take advantage of different security mechanisms. Among these mechanisms, in this paper, we focus on encryption, which has been of critical importance in image processing [3], as well as many other areas [4,5,6].

In recent years, the cryptography research community has taken advantage of the advancements in different technologies and theories including information theory [7], quantum computing [8], neural computing [9], Very Large Scale Integration (VLSI) technology [10], and especially, chaos theory [11].

All the above-mentioned theories have especially affected image encryption. However, in this paper, we are specifically interested in the applications of chaos theory in image encryption. Chaos is the characteristic of a system whose current state is guaranteed to be highly sensitive to the previous state (spatial chaos), the initial conditions (temporal chaos), or both (spatio-temporal chaos). Such a sensitivity makes the output or the behavior of a chaotic system difficult to predict. Chaos theory justifies and formulates the apparent disorder of chaotic systems on the basis of orderly patterns, structured feedback loops, iterative repetitions, self-organization, self-similarity, fractals, etc. Chaotic maps, attractors, and sequences all refer to the mathematical structures used for this formulation. Chaotic systems, maps, attractors, and sequences have been of great interest to the research community in recent years [12,13]. They have been used for security purposes in a broad variety of applications ranging from smart grids [14] to communication systems [15]. Especially, chaotic encryption has been used for encrypting a variety of content types in addition to images [1,2].

Figure 1 illustrates how image encryption converges with chaos theory at chaotic image encryption.

Figure 1 first of all introduces the icons we will use in the rest of this paper to represent image processing, encryption, image encryption, chaos, and chaotic image encryption. Furthermore, this figure shows how image processing joins encryption and then chaos theory to build chaotic image encryption as a branch of science and a field of research.

A comprehensive survey on research works focusing on chaotic image encryption can pave the way for further research in this area via highlighting current trends, shedding light on less-studied related topics, and developing directions for future research in this field. In addition to presenting such a survey, this paper establishes an ecosystem for chaotic image encryption. The ecosystem contains the following items.

• Challenges: The problems that can potentially make chaotic image encryption difficult, costly, or challenging.
• Applications: Environments or areas wherein chaotic image encryption has been demonstrated to be of assistance or efficiency.
• Enablers: The technologies or branches of science that can support chaotic image encryption via improving its feasibility, security, performance, or cost efficiency.

The literature comes with several surveys somehow relevant to the work of this paper. However, some of them are too outdated for such a fast-moving research area. Some existing surveys do not focus on chaotic image encryption, and others fail to develop an ecosystem for chaotic image encryption or a future roadmap for further research in this area. These shortcomings motivate our work in this paper.

The rest of this paper is organized as follows. Section 2 studies existing surveys and their shortcomings to highlight our motivations for the work of this paper. Section 3 discusses the state-of-the-art of chaotic image encryption in chaos, image, and encryption aspects. Section 4 studies the ecosystem of chaotic image encryption. Section 5 develops the future roadmap, and lastly, Section 6 concludes the paper.

2. Existing Surveys

As suggested by the topic of this survey, as well as the future predicted in Section 5, relevant surveys focusing on the following topics are deemed relevant:

• Surveys on Image encryption;
• Surveys on chaotic image encryption;
• Surveys on AI-assisted image processing;
• Surveys on AI-assisted image encryption.

In the following, each of the above categories are briefly reviewed. In each category, surveys are studied ordered by their publication year.

2.1. Surveys on Image Encryption

Image files are increasingly distributed across the Internet. This distribution requires security techniques that are different from traditional practices to manage confidentiality. The reason is that images can be vulnerable to several attacks, particularly if these files are sent through insecure channels. Medical images, for example, contain highly sensitive data, and thus, sending these images over the network requires a strong encryption algorithm that protects against these attacks [16].

In recent years, reviewing the literature of image encryption has been of interest to researchers [17,18]. Moreover, different related topics have been reviewed. For example, some researchers have conducted surveys on the techniques for encrypting plaintext into images through an algorithm that calculates the RGB value [19]. Furthermore, some related techniques such as image steganography have been studied along with image encryption [16]. As another topic of interest, some surveys have focused on the applications of image encryption in specific areas [20].

2.2. Surveys on Chaotic Image Encryption

There are some reviews directly focusing on the applications of chaos theory in image encryption. However, some of the surveys studied above (including the one reported in [21]) are too outdated. Moreover, although a few of them develop a future roadmap, all of them fail to establish an ecosystem for chaotic image encryption.

Deepa and Sivamangai [22] argued that a maliciously modified medical image makes it more difficult to diagnose an actual disease. This raises a critical need for the confidentiality of clinical images. On the other hand, the encryption time can pose a heavy overhead on the medical communication and processing systems. They claimed that this tradeoff is best resolved by DNA cryptography and chaotic cryptography. In their review, they reported some qualitative and quantitative measurements extracted from existing relevant research works to show how the tradeoff is resolved by the mentioned technologies. Moreover, they established some guidelines for further research in this area.

Yadav and Chaware [23] believe that despite existing encryption and information hiding techniques, information can be stolen and copyrights can be infringed because of vulnerabilities in available methods. They first presented a review of state-of-the-art image encryption methods. They especially focused on joint encoding (error correction) encryption methods. Then, they proposed a novel method based on Low-Density Parity-Check (LDPC) code and chaotic maps with the support of the Advanced Encryption Standard (AES) and Substitution boxes (S-boxes).

Some existing reviews take a comparative approach. For example, the advantages and disadvantages of existing chaotic image encryption methods were compared in [24]. Another relevant survey was reported in [25], where the authors reviewed and compared some one-dimensional chaotic maps with some hyper-dimensional ones with respect to their applications in image encryption. As another example, the authors of [26] highlighted chaotic encryption as a promising solution for encrypting images and videos, wherein neighboring pixels are highly correlated. They presented a review on existing chaotic methods for image encryption with the goal of identifying the most proper chaotic map. They studied tent map, logistic map, sine map, etc., and suggested Arnold’s cat map as the most promising chaotic map for this purpose. Moreover, in [27], the authors reviewed and compared image encryption methods based on five traditional algorithms, namely Blowfish, RSA, El-Gamal, AES, and DES with some chaos-based methods in terms of performance.

2.3. Surveys on AI-Assisted Image Processing

Reviewing existing AI-assisted image processing methods has been of interest to many researchers. For example, a survey reported in [28] focused on the interactions between machine learning and binocular stereo for depth estimation from images. Depth estimation has many practical purposes in fields such as 3D image reconstruction and autonomous driving. Included in the many techniques for estimating depth, stereo matching compares two images for pixel disparity and utilizes triangulation to determine the depth of the pixel. Data-driven and learning-based techniques have been applied to stereo matchingwith outstanding success, but the reverse has also yielded promising advances in using stereo matchingto develop new methodologies based on deep networks.

Another relevant review studied deep learning-based Multi-Focus Image Fusion (MFIF) methods [29]. MFIF is an image processing technique for fusing multiple images with differing depths of fields to create a single in-focus image. Propositions for solving the MFIF problem using deep learning techniques have been growing at a rapid rate since 2017, although none yet have shown any advantages or performance improvements over traditional methods. The applications of deep learning in image segmentation were studied in another survey [30]. Image segmentation, the process of partitioning an image into two or more segments, has a wide range of use cases in fields such as video surveillance, image compression, augmented reality, and scene interpretation. Algorithms based on deep learning models have demonstrated very impressive results, often outperforming traditional segmentation algorithms on many popular benchmarks.

2.4. Surveys on AI-Assisted Image Encryption

As a branch of AI-assisted image processing, AI-assisted image encryption has received a research focus in recent years. A few researchers have conducted surveys on existing research works in this area. As an example, one may refer to [31], wherein the applications of neural networks in image encryption for optical security in the healthcare sector were studied. Image encryption is an important component in the healthcare sector for improving the security of patient images gathered from sources such as ultrasounds, MRI scans, and X-rays. Neural networks are heavily used to provide security and privacy through encryption, although the algorithms are currently limited by their complexity and speed, and therefore, much research in the field is focused on optimization.

Table 1. Summary of Existing Surveys.

Survey	Year	Chaotic	Ecosystem	Roadmap
[17]	2021	No	No	No
[19]	2020	No	No	No
[32]	2020	No	No	No
[18]	2020	No	No	No
[16]	2018	No	No	No
[20]	2018	No	No	No
[33]	2015	No	No	No
[21]	2014	Yes	No	No
[22]	2022	Yes	No	Yes
[23]	2021	Yes	No	No
[24]	2019	Yes	No	No
[25]	2021	Yes	No	Yes
[26]	2021	Yes	No	No
[27]	2017	Yes	No	No
[28]	2021	No	No	No
[29]	2021	No	No	No
[30]	2021	No	No	No
[34]	2020	No	No	No
[31]	2020	No	No	Yes

summarizes the surveys discussed above in order to highlight their shortcomings, which motivated the work of this paper.

In Table 1, each entry in the first column contains a survey. The second column states whether or not the survey is focused on chaotic image encryption. The third column demonstrates whether or not the survey establishes an ecosystem for chaotic image encryption. Finally, the fourth column contains “Yes” if the survey develops a future roadmap for the field. It contains “No” otherwise.

3. State-of-the-Art

Research on chaotic image encryption is going on in three aspects; chaos, image, and encryption. These aspects are shown in Figure 2. The state-of-the-art in each of the mentioned aspects is reviewed below.

As seen in Figure 2, the chaos aspect is about chaos domains, sources, and dimensions. The image aspect studies 2D versus 3D images, gray-scale versus color images, and image transforms. Moreover, the encryption aspect is related to symmetric versus asymmetric encryption, block ciphers versus stream ciphers, and cryptographic primitives. The research works reviewed in this section are categorized according to these aspects.

3.1. State-of-the-Art in Chaos Aspect

Researchers focusing on the chaos aspect have tried different chaos domains, sources, and dimensions. These concepts are studied below along with related research works.

3.1.1. Chaos Domains

Chaos is studied in three domains; space domain (spatial chaos), time domain (temporal chaos), and space–time domain (spatio-temporal chaos). All of these domains play roles in chaotic image encryption. These roles are reviewed in the following.

Spatial Chaos

Spatial chaotic systems and maps are functions that only depend on an input value to determine the state. They have many uses in image encryption; for example, ShuTang et al. [35] utilized a 2D spatial map in a novel image encryption algorithm that exhibits strong security after applying key sensitivity tests, adjacent pixel correlation analysis, keyspace analysis, and testing against various attacks. Other works in the spatial domain include that performed by Faragallah et al. [36], where they compiled a report investigating the effectiveness of several chaotic maps in the spatial domain, those being the Arnold cat map, baker map, and logistic map. The report describes the analysis of the maps’ effectiveness in a novel encryption scheme using visual, entropy, histogram, encryption quality, differential, Known Plain Text (KPA), and Chosen Plain Text (CPT) analysis.

Temporal Chaos

A temporal system only depends on a time index and the state of the system at the previous index to determine the current state. Once such pure temporal chaos system, referred to as a “super-chaotic” map, was utilized by Wang et al. [37] in a proposed image encryption algorithm that exhibited strong security properties such as a large keyspace, high key sensitivity, and statistical analysis resistance.

Spatiotemporal Chaos

A spatiotemporal chaotic system depends on both the spatial domain (input) and the time index. Encryption schemes using spatiotemporal chaos have been proposed by Xin et al. [38] and Luo et al. [39] where the chaos systems were paired with the Discrete Cosine Transform (DCT). The former additionally employs the Propagating Cipher-Block Chaining (PCBC) mode to achieve the image encryption, which contrasts with the work performed by He et al. [40], where the basic Cipher Block Chaining (CBC) mode was opted for instead. All three algorithms exhibit strong security when analyzed using encryption analysis methods.

Another work in the spatiotemporal domain was performed by Xingyuan et al. [41]. In their paper, they proposed a novel spatiotemporal chaos model called the Logistic-Dynamic Coupled Logistic Map Lattice (LDCML). Analysis of the proposed map demonstrated strong chaotic properties, and when applied to image encryption, the further experimental analysis showed high levels of effectiveness.

3.1.2. Chaos Sources

Chaos can be created using mathematical or physical sources. In the following, we discuss the role of both types in the state-of-the-art of chaotic image encryption.

Mathematical Sources

Well-known mathematical chaos sources commonly used for image encryption purposes are studied below:

• Chaotic systems and maps
  These are functions originally designed for creating chaos. Chaotic systems and maps play a critical role in chaotic image encryption. To mention a few, one may refer to the following:

  –

  Fractional-order chaotic system

  Fractional calculus goes back more than 300 years, with modern studies focusing on systems such as the fractional-order Chen, Lorenz, and Liu systems [42]. A novel switching fractional-order chaos system was proposed by Hou [42] and utilizes controlling switches to switch between its sub-systems and achieve a strong chaos source for applying the exclusive Or (XOR) operation against the plaintext image.

  Another algorithm utilizing fractional-order systems was proposed by Wei [43], which opts to use a more standard third-order fractional system, as well as a novel Josephus scrambling algorithm and circular diffusion to achieve desirable encryption properties and resilience against common attacks.

  –

  Arnold cat map

  Arnold mapping is a well-know transposition chaotic map that, in the context of cryptography, was used by Ranimol and Gopakumar [44], as well as Zhang et al. [45] to provide a method of permutating and de-correlating adjacent pixels in their proposed encryption algorithms. Both algorithms were proven to exhibit a large keyspace with high key sensitivity and be capable of resisting common attacks such as brute force, entropy, CPT, and KPT.

  –

  Coupled map lattice

  A Coupled Map Lattice (CML) is a form of spatiotemporal chaos map efficient for random number stream generation. In one use case, Wu [46] proposed a novel implementation of the CML to create encryption streams dependent not only on initial values, but also on intermediate cipher images by using said ciphertexts to modify the CML parameters. This adds a layer of plaintext dependency, which aids in the defense against several attacks.

  –

  Lorenz map

  A Lorenz system is a type of differential equation that is highly susceptible to initial conditions. Jiang and Fu [47] proposed an image encryption procedure in which the key is composed of the three inputs to a 3D Lorenz system and utilizes the chaotic nature of said system to provide strong security.

  –

  Logistic map

  A logistic map is a relativity simplistic mathematical mapping function, which when influenced by particular control values, acts chaotically. An algorithm proposed by Sharma and Bhargava [48] utilizes a two-step interactive logistic map, where the next input is dependent on the previous two outputs, as a source of chaos. Similar work was performed by Li-Hong et al. [49], where they used a more standard logistic map and paired it with a hyper-chaos system to improve key generation effectiveness. Likewise, Mu and Lui [50] also found success utilizing the logistic map for key generation.

  –

  Tent map

  Wu et al. [51] proposed an image encryption algorithm using the CTM, and the rectangular transform was later analyzed by Zhu et al. [52] and improved upon to better protect against plaintext attacks such as CPT and KPT. The Chaotic Tent Map (CTM) is a mapping function that, when configured with control values in a particular range, behaves chaotically.

  –

  Lotka–Volterra

  A Lotka–Volterra chaotic system is a third-order differential equation in a similar family to other systems such as Lorenz, Rossler, Shua, and Chen. In a particular case study by Zahir et al. [53], an encryption procedure was proposed that utilizes the Lotka–Volterra chaotic system to aid in the creation of Substitution boxes (S-boxes) with strong confusion properties. The resulting S-boxes were found to satisfy the five criteria (bijective, non-linearity, strict avalanche, bit independence, input/output XOR distribution) required for acceptable use in cryptographic algorithms.

  –

  Henon map

  The Henon map was first discovered in 1978 and can be described as a 2D mapping function with two control parameters, which, when chosen strategically, enable the map to behave chaotically. Tresor et al. [54] proposed an image encryption algorithm utilizing Henon maps for shuffling the pixels of the image and 4D Qi hyper-chaos to generate keys for encryption. Experimental analysis of the algorithm demonstrates strong cryptographic properties and resistance against common attacks.

  –

  Logistic-sine system

  A Logistic-Sine System (LSS) is a discrete combination of the logistic and sine maps, both of which exhibit chaotic behavior under particular initial conditions. Zeng and Chen [55] referred to such a combination of the two maps as a compound chaotic map and utilized it in a novel encryption algorithm using XOR and modulus operations.

  Zhao et al. [56] investigated the inefficiencies with single chaos systems and proposed a novel algorithm utilizing LSS and cascade chaos to improve upon said inefficiencies. Experimental analysis through simulation has proven the new algorithm to be highly resilient

  In another study, Lu et al. [57] conducted cryptanalysis on an existing algorithm based on multiple S-boxes, but were able to break it using CPT attacks. A new algorithm was proposed to improve upon the old one and involved only a single S-box constructed utilizing LSS. Further cryptanalysis of the new algorithm showed improvement over the original and was also quite fast.

  Variants of LSS have also been employed in encryption algorithms, such as a 2D Logistic Modulated Sine Coupling Logistic (LSMCL) map proposed by Zhu et al. [58], a Logistic Sine Modulation Map (LSIMM) proposed by Zhang et al. [59], and a 2D Logistic Adjusted Sine Chaotic Map (LASCM) proposed by Balakrishnan and Mubarak [60]. In all cases, theoretical analysis and simulations determined the algorithms to be both secure and efficient.

  –

  Baker map

  The baker map is a bijective permutation function that operates on an MxM matrix by randomizing its cells according to a secret key and is well respected in the image encryption community. Elshamy et al. [61] utilized the baker map in an image encryption algorithm to improve upon a classic technique known as Double Random Phase Encoding (DRPE). The proposed algorithm uses the map to pre-process the image before applying DRPE, and experimental analysis showed significant increases in security as opposed to using DRPE alone.

  Another algorithm utilizing the baker map was proposed by Tong et al. [62], where high-dimensional dynamical multiple chaos was paired with the baker map to achieve a larger avalanche effect. Experimental results again showed significant increases in security when

  –

  Tinkerbell map

  Krishna [63] proposed an encryption algorithm utilizing Tinkerbell maps, a pair of chaotic functions, to inject strong pseudo-random numbers in multiple points during the encryption and decryption process. Differential and correlational analysis of the algorithm showed the proposed method to be highly efficient.

  –

  Cubic map

  A cubic map is a single-dimensional chaotic function that produces values on the interval [0, 1] and can be controlled by a single mapping parameter. Kavinmozhi et al. [64] proposed an encryption technique that employs a hybrid chaos source composed of the cubic and tent maps, as well as the Iterative Chaotic Map with Infinite Collapses (ICMICs). The resulting hybrid map is used with the XOR operation to achieve encryption, and an analysis of the algorithm showed that it is suitable for repeated use and is resilient against attacks.

  –

  Gingerbreadman map

  Savitri et al. [65] used the Gingerbreadman map, a 2D chaotic map, to generate encrypted keys for use with the well-known Cipher Block Chain (CBC) encryption algorithm. Using the map in this algorithm greatly improves CBC’s performance when applied to images, and a visual comparison demonstrated massive improvements.

  –

  Tangent map

  Moysis et al. [66] proposed a Random Number Generation (RNG) algorithm based on the usage of the mathematical hyperbolic tangent function. When the RNG algorithm was applied to image encryption, the resulting procedure demonstrated strong cryptography

  –

  Multiple maps

  Mixing multiple mapping functions in image encryption algorithms can serve multiple purposes. For example, Bisht et al. [67] employed a verity of different maps to achieve tasks such as more chaotic permutation, diffusion, and RNG. A similar technique employing various maps in different stages of the encryption procedure was also proposed by Wang et al. [68].

  Fu et al. [69] proposed a novel keystream generation technique utilizing multiple chaotic maps that incorporates the plaintext itself into the stream. The algorithm was motivated by the need to defend against CPT and KPT attacks, and an analysis of the algorithm showed it is effective in achieving its goal.

  In terms of areas of application, stronger algorithms enforced by the use of multiple chaotic maps are important in numerous fields. For example, Choi et al. [70] proposed an algorithm using multiple maps for encrypting colored medical images, which can be seen as unique in their size and sensitivity. Experimental and statistical analysis of the resulting procedure showed it is secure for use with healthcare images.

• Other mathematical sources
  In addition to chaotic systems and maps, some researchers have used the following mathematical designs, which have not been originally defined for chaos creation:

  –

  Space-filling curves

  Fractal geometry has several intriguing properties, such as self-similarity, composition by iterative methods, and a complex structure. Zhang et al. [71] utilized Hilbert curves and H-fractals, types of self-filling curves, in a novel image encryption algorithm. This algorithm alternates the use of both curves to efficiently scramble the pixels of the image.

  –

  Memory cellular automata

  Cellular Automata (CA) can best be described as a grid of cells with a finite set of states and a transition function that governs how cells change state over time. Whereas a standard CA only depends on the generation t-1, Memory Cellular Automata (MCA) depend on more parameters. When the MCA’s rules are defined by chaotic maps, the structure becomes a powerful tool for image encryption. Several algorithms using various-order MCAs have been proposed, for example a 4D MCA by Aslam et al. [72], a 2D MCA by Hibibipour et al. [73], and an indefinite CA by Hibibipour et al. [74].

  –

  Transcendental numbers

  In mathematics, a transcendental number has the characteristic that digits to the right of the decimal have no pattern [75]. Garcia et al. [75] proposed an image encryption algorithm that uses chaos and the transcendental number Pi, dubbed Chaotic Pi Ciphering (CPC). The algorithm uses Pi and a chaos source created using differential equations to generate cipher keys and substitution boxes.

Physical Sources

In addition to mathematical sources, chaos can be created using physical phenomena and used in chaotic image encryption:

• Optical Chaos
  Our physical world can provide many forms of chaos, with just one example being light. In studies by Xie et al. [76] and Lui et al. [77], they found success in producing a chaotic base for image encryption algorithms using lasers. Extensive security testing of both algorithms showed them to be highly secure and feasible for practical use.
  Other studies have also been carried out, such as those by Li et al. [78] and Liu et al. [79], where optical chaos is utilized for encrypting and then transmitting images for storage in the cloud. Experimental results showed both procedures to be secure and safe for production use.
• Chaotic circuits

  –

  Chua circuit

  Some physical electronic circuits such as the Chua circuit can produce chaotic behavior. AlMutairi et al. [80] utilized the circuit as a key generator in their proposed image encryption algorithm. By contrast, Lin et al. [81] proposed a similar encryption model, but instead utilized a variant of the classic Chua circuit with a PWL memristor. In both cases, analysis showed the algorithms to exhibit strong security properties.

  –

  Memristive circuits

  A memristor is a form of electrical component that is capable of exhibiting chaotic behavior. Liu et al. [82] proposed an image encryption algorithm that utilizes 4D memristive hyper-chaos to create chaos matrices. Security analysis showed strong security and cryptographic properties.

  Another image encryption algorithm was proposed by Sun et al. [83] using a memristive chaotic system. The presented system demonstrates a unique property known as multistability, which further improves the chaoticness of the system. Again, security analysis showed the algorithm to possess strong cryptographic properties.

  –

  Physically Unclonable Functions (PUFs)

  True Random Number Generators (TRNGs), although very important in cryptography, are impossible to achieve in software. To counter this fact, Muhammad et al. [84] proposed an encryption algorithm using a hardware device, a form of physically unclonable function, to generate true random numbers. Through extensive experiments and analysis, the TRNG was successful in passing all tests required for safe use in cryptographic algorithms.

3.1.3. Chaos Dimension

The dimension of a chaos map refers to the number of functions (x(t), y(t), etc.) it is composed of. Many image encryption algorithms utilize chaotic functions of varying dimensions. Chaotic functions used in chaotic image encryption can be categorized as follows:

• One-dimensional
  Work with one-dimensional chaos includes that by Wang and Lui [85], where the novel 1D Sine Chaotic System (1DSCS) was proposed. This system exhibits a large parameter interval as compared to the standard sine map it was built upon.
  Elghandour et al. [86] proposed an image encryption algorithm utilizing the 1D tent map. A similar algorithm also using the tent map was proposed by Tiwari et al. [87]. Extensive testing proved both algorithms to be effective at resisting common cryptographic attacks. The former paper also elaborated on the low chaotic range for the tent map and suggested that future work use a variant with a larger range such as the tent-sine map.
• Two-dimensional
  An image encryption algorithm based on two-dimensional chaos was proposed by Yang and Tong [88]. This algorithm uses the 2D logistic chaotic system and a novel block image encryption procedure. Experimental results demonstrated the algorithm to have strong randomness, low pixel correlations, and high key sensitivity.
• Three-dimensional
  Many image encryption algorithms utilize three-dimensional chaos. One such algorithm was proposed by Qian et al. [89], where they utilized the 3D logistic and cat maps. The novel usage of image reconstruction techniques also improved the effectiveness of the algorithm.
  In an algorithm proposed by Asl et al. [90], the 2D image was converted into three-dimensional space by creating three streams from the red, green, and blue channels of the image. The 3D modular chaotic map was used as the chaos source for encryption.
  Two other algorithms using three-dimensional chaos systems were proposed by Cao and Fu [91] and Xiu-chun and E-Nuo [92], respectively. In the former, the Rossler chaos system was used, whereas the latter study opted to use the Lorenz system.
• Four-dimensional
  Huang et al. [93] proposed a novel four-dimensional chaos system based on concepts known as “shape synchronization” and “driver-response”. The complex mathematical underpinnings make the algorithm very difficult to break, and experimental tests in the application of image encryption showed promising results for its effectiveness.
• Five-dimensional
  Zhu and Zhu [94] proposed a novel five-dimensional chaotic map composed of the 2D logistic map and 3D discrete Lorenz map. Experimental simulations of the system when applied to image encryption resulted in high scores in many common encryption strength tests.
• Multiple dimensional
  Work related to mixing maps of varying dimensions in image encryption has also been performed. For example, Qui and Yan [95] proposed an image encryption algorithm using both the 1D logistic map and 3D Lorenz system. Experimental results demonstrated that the algorithm has strong security.
  Parida et al. [96] proposed a novel image encryption and transmission procedure based on Elliptic Curve Cryptography (ECC). Encryption is achieved using 3D and 4D Arnold cat maps as chaos sources, and the Elliptic Curve Diffie–Hellman (EDCH) key exchange algorithm is utilized to establish a shared key between parties. Digital signatures allow the algorithm to authenticate the encrypted message before decryption, and experimental results showed the method to be effective.

3.2. State-of-the-Art in the Image Aspect

The image aspect of chaotic image encryption is about 2D versus 3D, color versus gray-scale, and image transforms. In the following, we discuss each of these topics and show how they are dealt with in research works focusing on chaotic image encryption.

3.2.1. Two-Dimensional versus Three-Dimensional Image

Although 2D images are much more common, 3D images, which can be visualized as 3D meshes, do exist and possess the same encryption requirements as their two-dimensional counterparts. Due to this fact, 3D image encryption algorithms are much less common. However, one such algorithm was proposed by Xu et al. [97].

3.2.2. Gray-Scale versus Color Image

Several algorithms that focus specifically on gray-scale image encryption have been proposed such as one that interestingly utilizes the concept of water waves [98] and another that uses the integer wavelet transform [99]. If color image encryption is required, then gray-scale-specific algorithms will typically not suffice. Algorithms that encrypt color images employ a wide range of techniques such as matrix convolution [100] and 4D memristive hyper-chaos [82]. An approach for encrypting multiple colored images has also been proposed [101], as well as a unique procedure for encrypting and transmitting color images using audio signals [102].

3.2.3. Transforms

Image transforms are of critical importance in chaotic image encryption. Some of them are studied below.

Wavelet

The wavelet transform is a popular method of permutating the cells of a 2D matrix and can yield a significant increase in encryption effectiveness [103]. To fulfill the chaos requirement of good encryption, several chaos sources have been paired with the wavelet transform including an improved 3D cat map [104], a 1D logistic map [105], a 3D logistic map [106], the Arnold map [107,108], and a logistic sequence [108]. Other algorithms utilizing variations of the standard wavelet transform such as the Integer Wavelet Transform (IWT) have also been proposed [109].

Zigzag Transform

The zigzag transform is capable of rearranging the cells of a 2D matrix to heavily decrease the correlation between adjacent pixels, an important property when considering image encryption. Gao et al. [110] proposed an algorithm for image encryption utilizing a more complicated implementation of the transform that yields better security.

Cosine Transform

Zhang et al. [111] proposed an image encryption algorithm utilizing the Discrete Fraction Cosine Transform (DFrCT), which has additional benefits over the standard Discrete Cosine Transform (DCT) that make it more suitable for image encryption.

Contourlet Transform

The contourlet transform provides a method of decorrelating the cells of a 2D matrix and was designed to improve upon the shortcomings of the wavelet transform when dealing with natural images. Jiang et al. [112] proposed an image encryption algorithm utilizing the transform, which has some desirable attack resistances, for example against JPEG compression.

Linear Canonical Transform (LCT)

Li et al. [113] proposed an image encryption algorithm utilizing LCT that is both speedy in execution and also boasts a large keyspace to protect against brute-force attacks.

3.3. State-of-the-Art in the Encryption Aspect

The last aspect of chaotic image encryption is the encryption aspect, which is about symmetric versus asymmetric cryptography, block versus stream ciphers, and cryptographic primitives. The state-of-the-art in this aspect is studied below.

3.3.1. Symmetric versus Asymmetric Cryptography

Symmetric key encryption involves the use of the same key in both encryption and decryption and is common in many algorithms such as the Advanced Encryption Standard (AES). Ashtiyani et al. [114] proposed an image encryption algorithm for encrypting medical images using a chaotic variant of the Simplified Advanced Encryption Standard (S-AES). Asymmetric key encryption involves the use of different (but related) keys for encryption and decryption. Wu et al. [115] proposed an algorithm that utilizes a complex and irreversible function that causes the algorithm to exhibit asymmetric properties.

3.3.2. Block Cipher vs. Stream Cipher

Block and stream ciphers, although common with arbitrary binary encryption, typically fall short when encrypting images. However, when paired with sufficient chaos, they can be used effectively. Some block ciphers used in chaotic image encryption are studied below:

• Blowflish
  Bora et al. [116] proposed a block cipher using the Blowfish algorithm and cross-chaos map. Cryptanalysis results showed strong security.
• Ecliptic Curve Cryptography (ECC)
  Abbas et al. [117] proposed an Ecliptic-Curve (EC)-based algorithm that utilizes pixel-level parallelism for faster encryption speeds. A different proposal by Benssalah and Rhaskali et al. [118] uses ECC combined with the Hill cipher and Arnold cat map to achieve a strong encryption algorithm targeted at medical images.
• El-Gamal
  El-Gamal is a type of EC commonly utilized in cryptography. For example, Luo et al. [119] proposed a public-key-based image encryption algorithm utilizing the El-Gamal EC to address common issues with key management. In another proposal by Yousif et al. [118], El-Gamal was used to encrypt images that were first permutated using zigzag and spiral scanning.
• Rijindal
  Dsouza and Sonawane [120] proposed a novel technique of using images as the key to encrypt/decrypt a directory in a file system. This technique employs both AES and Rijideal ciphers, and evaluation results demonstrated its effectiveness.
• Rivest–Shamir–Adleman Cryptosystem (RSA)
  Nkapkop et al. [121] developed a novel asymmetric image encryption algorithm using RSA to solve the issue of key management. This algorithm uses the RSA key pairs to encrypt the initial values and parameters of the chaotic function so that the public key can encrypt images and only the private key can decrypt.
• Data Encryption Standard (DES)
  Zhang et al. [122] proposed an image encryption algorithm utilizing the logistic chaos sequence for chaotic sequence generation and an improved DES algorithm for encryption. Simulation results demonstrated good security and speed, making it suitable for real-time use.
• Novel block ciphers
  Gupta et al. [123] proposed a novel block cipher using two keys where the image is split into four blocks; each is encrypted n times, and finally, the keys are inverted and the blocks further encrypted m more times. Evaluation through standard tests demonstrated strong cryptographic properties, making the algorithm usable for real-time connections.
  Rani and Kumar [124] proposed a novel stream cipher using a modified RC4 algorithm. The algorithm converts the image into three vectors for each color channel and uses the modified RC4 stream algorithm to encrypt them. Another algorithm utilizing the RC4 stream cipher was proposed by Ginting and Dillak [125]. This algorithm uses the logistic map to generate a keystream for encryption. The algorithm is lossless, which was verified by comparing the hash values of the image before encryption and the image after encrypting, then decrypting.
• Hybrid ciphers
  A hybrid approach utilizing both block and stream ciphers was proposed by Goumidi and Hachouf [126]. This algorithm splits the image into two sub-images and encrypts one using the block cipher and the other using the stream ciphers. The encrypted sub-images are then merged back together to form the final image. The use of two different types of ciphers greatly increases the complexity of the algorithm, leading to stronger cryptographic properties.

3.3.3. Primitives

In the following, we examine the role of cryptographic primitives such as scrambling, bit shuffling, hashing, secret sharing, one-time key, permutation, substitution, confusion, and diffusion in chaotic image encryption.

Scrambling

Scrambling is the process of permutating the pixels (or even bits in a pixel) so that the new ordering is unrecognizable from the original. Various methods of scrambling have been employed including Latin rectangle [127], logistic chaotic [128], and spiral [129].

Bit Shuffling

Bit shuffling is another method of permutating the pixels of an image, specifically at the bit level. Krishnamoorthi et al. [130] proposed a method of bit shuffling in the spatial domain using a tent map.

Hashing

Hashing algorithms are special types of functions that take an input of any length and produce an output that is always the same length. The SHA algorithm specifically also has the added bonus of being highly input sensitive, that is to say, small changes in the input create a very different output.

In the context of image encryption, one common use of hash algorithms is to generate the keystream. For example, Bhadke et al. [131] utilized SHA-256 and the Lorenz chaos attractor to generate strong key streams. Slimane et al. [132] also proposed an algorithm using the Lorenz chaos attractor and a hash algorithm, although they opted to use SHA-1 instead.

In a paper by Lui [133], a novel encryption algorithm using SHA-3 and stenography was proposed. This algorithm embeds the hash of the plaintext image into the cipher image using stenography. This makes the algorithm very sensitive to the plain image, which in turn yields stronger security.

Permutation, Substitution, Confusion, and Diffusion

• Permutation and diffusion
  Permutation is the process of rearranging elements in a structure, which, in the context of images, refers to scrambling the pixels. Abd-El-Hafiz et al. [134] performed an evaluation on three different permutation methods (discrete chaos, vectors, and Arnold cat map) and found that discrete performed the best.
  Diffusion is the process of ensuring there is no statistical significance to the resulting structure. In the context of images, this refers to scrambling the pixels of the image to eliminate the correlation between adjacent pixels. Ping et al. [135] proposed a novel digit-level permeation algorithm that additionally employs a high-speed diffusion algorithm. Evaluation results demonstrated high security and efficiency.
  Combining permutation with diffusion into the same stage of encryption aims to combat hackers who try to break each stage separately [136]. Liu et al. [137] proposed an algorithm to perform permutation and diffusion simultaneously. The algorithm additionally uses a Hopfield chaotic neural network to perform further diffusion, which gives the algorithm greater key sensitivity.
• Confusion
  Confusion in encryption refers to the level of dependency elements of the cipher-text have with the key. As seen with permutation, confusion is often integrated with diffusion for the same reasons. For example, Run-he et al. [138] proposed an image encryption algorithm that achieves an integration of confusion and diffusion by XORing the plain image with chaotically generated offset matrices.
• Substitution
  Substitution involves replacing an element with something else in a predictable and invertible manner. The substitution requirement is commonly implemented using S-boxes, which are matrices that define how each input maps to its substituted value.
  For image encryption, chaos-based S-boxes include those generated from the chaotic sine map [139] and the logistic map [140]. Wang and Zhang [141] also proposed an algorithm with multiple S-box substitutions, where the order of the boxes is determined by a random chaos sequence. Another algorithm proposed by Khan et al. [142] splits the image into four blocks and applies a different S-box to each block. These S-boxes each originate from a different encryption algorithm (AES, PQL, APL, and Shipjack). Another paper by Lidong et al. [143] proposes a dynamic encryption algorithm so that the cipher image is always different even if the same key and plain image are used.

One-Time Key

Rehman et al. [144] proposed an image encryption algorithm that uses a one-time-key to generate chaotic maps using the hash of the plaintext image. The algorithm employs a novel concept known as a rotor machine, and through simulation, the results showed that the algorithm possesses strong cryptographic properties.

Secret Sharing

Multiple Secret Sharing (MSS) in the context of image encryption is where k plaintext images are required to create k cipher-text images, and those same k cipher-text images are required to obtain even just one plaintext image [145]. Guo et al. [145] proposed an MSS algorithm for images that addresses common shortcomings.

4. Ecosystem

In this section, we try to establish an ecosystem for chaotic image encryption. To this end, we try to highlight challenges, application areas, and enabling technologies.

4.1. Challenges

The main challenges faced by researchers focusing on chaotic image encryption include encrypted image processing, attack protection, and evaluation. These challenges are studied below.

4.1.1. Encrypted Image Processing

Processing encrypted images is a highly challenging job. The related challenges are discussed in the following.

Data Hiding

Data hiding refers to the technique of hiding information in an encrypted image without knowledge of the original contents [146]. One common method of Reversible Data Hiding in Encrypted Images (RDHEIs) is Block Permutation and Co-Modulation (BPCM); however, Wang et al. [147], through cryptanalysis, determined that the method cannot protect against KPAs.

Another big challenge with current data hiding methods is capacity limits. A high-capacity alternative utilizing a blockwise multi-predictor and Huffman coding was proposed by Zhang et al. [146]. Other capacities improving techniques based on pixel correlation preservation [148] have also been proposed.

Other novel contributions to data hiding have been proposed. Xiong et al. [149] proposed improvements to common techniques to increase the resistance against common attacks such as JPEG compression and noise addition. Wang et al. [150] proposed a novel method of decryption using separate keys for both the decryption of the image and the reversion of the data hiding.

Image Retrieval

The image retrieval problem refers to the issue of maintaining privacy when outsourcing image search services in the cloud. Work by Huang et al. [151] aimed to fix issues with low accuracy and high involvement from image owners using the process of extracting image features using neural networks and encrypting using the k-Nearest Neighbors (kNN) algorithm. Other progress in the field includes work involving multi-owner access by Tong et al. [152].

Image Compression

Techniques of mixing image encryption with lossy compression have been proposed. One such proposal was made by Zhang [153], where an iterative reconstruction technique was utilized for decompression. However, the encryption technique used was relatively weak compared to exiting algorithms.

Another algorithm was proposed by Qin et al. [154], where a selective compression technique is paired with inpainting, a method of restoring missing pixels, to reconstruct the image.

Image Folding

A novel procedure known as encrypted image folding was proposed by Bowley and Rebollo-Neira [155]. This technique not only encrypts the image in question, but also decreases its dimensionality, allowing for more efficient storage.

4.1.2. Attack Protection

There are several attacks that can be conducted against chaotic image encryption. Research works focusing on these attacks are studied below.

Plaintext-Related Image Encryption (Protection against Plaintext Attacks)

Good encryption algorithms should possess resistance to common types of attacks including plaintext attacks. Many resilient algorithms have been proposed such as the one by Khan et al. [156], where they used a random DNA sequence to initialize chaos maps. Niu and Zhang [157] proposed another resilient algorithm utilizing pixel permutation and Josephus traversing. Yet another algorithm based on Chen’s chaotic system was proposed by Fu et al. [158], aiming at improving resilience against known/chosen plaintext attacks. In the diffusion stage of this algorithm, the key stream elements created by Chen’s chaotic system are rotated in such a way that the rotation is dependent on the value of the plain pixel. This way, the key stream is a function not only of the key, but also of the plain image. This notably strengthens the cryptosystem against known/chosen plaintext attack. Critiques based on the attack resilience of existing proposed algorithms have also been made. One such critique was made by Liu et al. [159], where vulnerabilities related to diffusion and permutation in a hyper-chaos algorithm were discovered and improved upon.

4.1.3. Evaluation

Many researchers are interested on the evaluation of chaotic image encryption methods. The evaluation process may include attack, cryptanalysis, or benchmarking. These evaluation approaches are discussed in the following.

Attack

Yan-Qing and Zhuo-Min [160] performed an analysis of the HYPER-HIE image encryption algorithm based on hyper-chaos. Their analysis attempted to exploit the algorithm’s weaknesses in its permutation and diffusion techniques, and their research concluded that the algorithm could not resist chosen plaintext attacks.

Cryptanalysis

The cryptanalysis of many existing algorithms has been performed. One such analysis was performed by Feng and He [161], where an algorithm based on hyper-chaos and DNA failed to account for chosen plaintext attacks. To this extent, the authors of the paper were able to design an attack that reveals the plain image with no knowledge of the key. Another analysis was performed by Liu et al. [162], where an image encryption scheme combining bit-plane extraction with multiple chaotic maps (IESBC) was found to be vulnerable to chosen and known plaintext attacks. The algorithm was then improved upon to score highly with various cryptographic metrics.

Some algorithms also claim to have secure properties, but may fall short. One example is a chaotic Image Encryption Algorithm based on Information Entropy (IEAIE), which was analyzed by Li et al. [163]. This algorithm was found to possess many pitfalls and questionable security metrics. Another example is a color image encryption scheme based on a hybrid hyper-chaotic system and cellular automata analyzed by Li et al. [164]. This algorithm possesses several security drawbacks despite claiming to resist applicable attacks. A more general study of issues with algorithms based on cryptanalysis-driven design were outlined by Muhammed et al. [165].

Benchmarking

Hraoui et al. [166] performed benchmarking on the AES encryption algorithm and another algorithm based on the logistic map. Their results demonstrated that the AES exhibits better security performance, but is computationally slower. The logistic map, on the other hand, is less secure due to a few specific vulnerabilities, but is faster and simpler to implement, making it more ideal for real-time communications.

4.2. Application Areas

As suggested by existing research works, chaotic image encryption is mainly applied to IoT systems, medical systems, and satellite systems. In the following, we study these application areas.

4.2.1. IoT

Image encryption is a necessary requirement in many modern IoT systems, yet the requirements of said algorithms are very steep and their efficiency must be proven. One such algorithm proposed by Nath et al. [167] was simulated in MATLAB and contrasted against existing algorithms to prove its efficiency.

Another novel algorithm was proposed by Boutros et al. [168], where hardware acceleration was used to meet the speed requirements of real-time IoT applications. For a 512 × 512 image, the algorithm achieved a maximum frequency of 135 MHz and 256 fps.

4.2.2. Medical Systems

Protecting medical records (including images) is not just a privacy concern, but a legal requirement [169]. This is especially true when transferring said medical images over open or public networks [170,171].

Many chaos-based approaches for encryption have been proposed. These approaches often involve chaos maps [169], especially the Arnold cat map [172] and logistic map [171]. Approaches using DNA-based algorithms have also been proposed [173], including a rather novel one by Kumari and Nagaraju [171], where chaotic maps were used to decrease the computational complexity of DNA-rule-based sequences.

The use of key streams is also a common technique. Vaseghi et al. [174] proposed a method for synchronizing chaotic systems at both the transmission and receiver ends so that keys can be obtained from the streams and no keys need to be directly shared. Another novel proposal was given by Han et al. [175], where a logistic map was used to generate a sequence from which a Hermite neural network was trained. This trained network was then used to generate key streams for use in encryption.

4.2.3. Satellite Systems

Transmission of image data over satellite connection poses a unique problem of a non-negligible time delay between the involved parties. A novel solution to this issue was proposed by Vaseghi et al. [176], where Lyapunov stability theory was applied to achieve time synchronization in finite time.

Another algorithm dealing with satellite transmission was proposed by Ibrahim et al. [177], where a new Cascade Modular Tent Logistic Map (CMTLM) was used. Simulation results demonstrated high effectiveness for the algorithm to perform in the satellite cryptosystem.

4.3. Enabling Technologies

In this subsection, we examine the technologies that support chaotic image encryption.

4.3.1. DNA Computing

DNA, formally known as Deoxyribonucleic Acid, is the biological substance that forms our genetic code. DNA can be broken down into four pieces known as ATCG, or formally Adenine, Thymine, Cytosine, and Guanine [178]. This concept can be used in encryption by encoding the data using this structure and leveraging existing DNA technology and sequences to manipulate it [179]. This encoding scheme has many advantages such as immense keyspace [180], high storage space for data [181], and high parallelism for algorithms [182].

The most common way to implement DNA concepts in image encryption is to pair it with a chaos source that aids in choosing random DNA coding rules or sequences [179]. Many algorithms with varying sources of chaos have been proposed. These include the logistic map [183,184,185,186], sine chaotic map [182], sine-Henon alteration map [185], cellular automata [187], optical chaos [188], etc.

The encryption algorithms themselves also employ a wide range techniques. These include encryption then transmission [188], using DNA sequences as a one-time pad [184], the Vigenere cipher [189], matrix subdivision [186], splitting color streams [190], the use of the Chinese remainder theorem [191], and the use of reconfigurable hardware called FPGA.

Several more unique approaches have also been proposed. One such proposal was by Hao et al. [178], where DNA mutations were implemented to improve the randomness of DNA algorithms. A different approach by Gasimov and Mammadov [192] utilizes DNA pseudo-symbols where the standard DNA sequence of four characters (ATCG) is expanded to use eight pseudo-symbols.

A particularly interesting proposal was also made by Iqbal et al. [193], where the concept of the castle piece from the game Chess (also called the rook) was used. These authors argued that simply mixing chaos and DNA is not enough, and to more effectively scramble and confuse the pixels of the image, random movements of the castle piece were simulated on a chessboard with dimensions equivalent to the image.

Statistical and security analyses of several algorithms based on DNA encoding and chaos have been performed; one examples is Özkaynak et al. [194]. In this study, it was found that existing algorithms are vulnerable, and choosing the plaintext strategically can reveal some or all secret parameters. Another research proposal by Ahgue et al. [181] takes a graphical approach to analyzing the security of algorithms. The provided GUI allows for configuring the secret key, encrypting a chosen image, and viewing the statistical analysis.

4.3.2. Quantum Computing

Xu et al. [195] proposed a novel image encryption algorithm involving a quantum chaos map by applying quantum correlation theory to the classic logistic map. The resulting map exhibits strong chaotic behavior and is used to generate chaotic streams for permuting the pixels of the image in question.

4.3.3. Signal Processing

Some signal processing methods such as compressive sensing and differential evolution have played roles in research on chaotic image encryption. These methods and their roles are investigated below.

Compressive (Compression) Sensing

Image encryption algorithms based on compressive sensing theory have been proposed. One such proposal is by Shao et al. [196], where a unique source of chaos from hybrid analog–digital electro-optic sources was used. Another proposal was by Zhu et al. [197], where Gauss random matrices were used for compression. Both algorithms exhibited strong cryptographic strength through simulations and experimental analysis.

Differential Evolution

An image encryption algorithm utilizing differential evolution was proposed by Toktas et al. [198]. The differential equation utilizes an algorithm based on natural evolution to find optimal decision variables to seed the [198] chaotic map optimization for image encryption using the triple objective differential evolution algorithm.

4.3.4. Hardware Technology

Many chaotic systems can be very expensive and slow to compute, making them infeasible for real-time scenarios. By comparison, hardware-based approaches are much faster and still maintain the high level of security required [199].

In a proposal by Paliwal et al. [200], chaos generators were implemented in part by the Coordinate Rotational Digital Computer (CORDIC), a hardware-efficient algorithm for calculating various functions such as tangents and hyperbolics. The types of chaos generators used in hardware algorithms are much the same as standard software approaches, such as iterative maps [199], cat maps [201], and the double-humped logistic map [202].

These algorithms are often tested on physical hardware to prove their claimed effectiveness. Examples of the hardware used are the Virtex-IV [203] and the Virtex-5 FPGA [202]. These tests resulted in high throughput with a small utilized area. Zhang et al. [201] also proposed a more generic hardware structure for an algorithm that exhibits good reusability for use in different systems.

4.3.5. Parallel Processing

Parallel processing sees much use in image encryption. In one such proposal by Gu et al. [204], the problem of implementing image encryption in green IoT environments was tackled. The paper discussed the primary concerns, those being limited computing power and long lifetime. These issues were addressed by proposing a novel parallel encryption technique using various chaos maps implemented in a parallel structure.

In another proposal by Fu et al. [205], a novel encryption algorithm was proposed. Said algorithm decomposed the image into eight subplanes and processed each in parallel. Experimental analysis demonstrated that the algorithm possesses high security and runs five-times faster than an equivalent serial method.

4.3.6. Fuzzy Systems

A fuzzy system is based on fuzzy logic, that is values are real numbers between 0 and 1 as opposed to exactly 0 or exactly 1. In a proposal by Mohamed et al. [206], the Choquet Fuzzy Integral (CFI) was utilized to construct better substitution boxes than standard cryptographic methods.

Figure 3 illustrates the established ecosystem.

In Figure 3, columns represent enabling technologies. The blocks in the roof denote application areas. Lastly, the rain drops and the thunder represent challenges.

5. Future Roadmap: The Promise of AI

We anticipate that research on chaotic image encryption will move towards quantum-inspired and bio-inspired neural chaotic image encryption in the near future. Our reason for such an anticipation is the existence of the trends discussed in Section 5.2, Section 5.3, Section 5.4, Section 5.5, Section 5.6, Section 5.7, Section 5.8, Section 5.9, Section 5.10 and Section 5.11.

5.1. Neural Networks in Chaos: Chaotic Neural Networks and Their Applications

The use of chaotic neural networks is wide in application. For example, to solve the problem of degrading performance in Code Division Multiple Access (CDMA) systems due to system capacity and multiple access interferences, Wang et al. [207,208] used a CNN to detect multiple users, a task that is NP-complete. Experimental results demonstrated superior performance to the existing Hopfield-Neural-Network (HNN)-based detector. CDMA is a communication technique where multiple transmitters can send signals simultaneously over a single channel.

CNNs can even be applied on a hardware level. For example, Cao et al. [209] used a novel Chaotic Hopfield Neural Network (CHNN) to optimize the SF6 circuit breaker. Work by Zhang et al. [210] used CNNs to solve a hysteresis bottleneck for the usage of the Magnetic Shape Memory Alloy (MSMA) smart material with promising properties for use in micro-positioning applications.

5.2. Neural Networks in Image Processing

Convolutional Neural Networks (CNNs) see much use in the realm of image processing. Many complex image problems are solved using CNNs, including facade parsing (determining distinct faces of buildings) from street-level images [211], crowd stability calculations (count and density) from video surveillance [212], Hyperspectral Image (HSI) classification [213,214], Polarimetric Synthetic Aperture Radar (PolSAR) image classification [215], medical image diagnostics [216], and tree species identification from aerial images [217].

Another effective use for image processing is to convert data into images so that existing image processing techniques can be used to solve other problems. For example, a technique proposed by Song et al. [218] converts EGG data into images to convert the problem of EGG emotion reading to the problem of image recognition. In another proposal by Lu et al. [219], graph classification using CNNs was mixed with image segmentation to yield better structural information.

CNNs also present several shortcomings when used with image processing. A common issue is the complexity of the networks, which leads to high computational costs and infeasibility for real-time applications [214,215]. The lack of a labeled training table can also be a common issue and leads to over-fitting and loss of detail [213]. To overcome these issues, Fernandes and Yen [216] proposed an algorithm that uses a deepening then pruning approach to shrink the size of a network while maintaining its effeteness. Another proposal by Mei et al. [214] trims the input to the network to use integer values instead of floating point values, which greatly speeds up the computation time.

5.3. Neural Networks in Encryption: Neural Cryptography

Neural cryptography is a recent research trend. Different kinds of neural networks have been used for this purpose [9,220]. Moreover, different cryptographic algorithms have been examined in this area [221]. More importantly, different enabling technologies are used to support neural cryptography. For example, hardware technology can be utilized to embed cryptographic algorithms into neural network accelerators [220].

5.4. Neural Networks in Image Security

Deep-Neural-Network (DNN) and CNN-based image processing have many applications in security, one of the most common being computer vision [222]. Examples include facial and body detection [222], detecting threats (specifically explosives) in X-ray images of passenger baggage [223], and detecting man-made items in millimeter wave images.

Neural networks also see much use in improving the security of image encryption algorithms. In a proposal by Seethalakshmi et al. [224], steganography techniques were improved by splitting the source image into n shares and using neural networks to determine where in the destination image to hide the shares to maximize its security. In another proposal by Sirichotedumrong et al. [225], a privacy-preserving DNN was developed. This DNN takes specially encrypted images as the input so that it can still extract the required features without being able to see the plain image. Furthermore, the testing and training phases for the model can use different encryption keys for further privacy.

In another proposal by Ito et al. [226], a DNN was trained on plain images to transform them into visually secure images. Through analysis, the network was shown to be resistant to inverse transformation attacks where DNNs were trained to invert the transformation. In a different study by Sirichotedumrong et al. [227], this same inverse transformation attack was performed against many existing DNN-based encryption algorithms. It was found that if the images were encrypted using use different keys, then the encryption was secure. If the same key was used for different images, then the data were able to be retrieved.

CNNs can even be used to determine the security of already encrypted images. In a paper by Fezza et al. [228], a CNN was developed to rate an encrypted image based on visual security and visual quality. Su et al. [31] also examined existing algorithms to determine their security. In their study, 30 state-of-the-art image encryption algorithms using neural networks were examined to determine the optimal algorithm.

5.5. Neural Networks in Image Encryption

Neural networks are widely used in image encryption. Different aspects of this trend are explained below.

5.5.1. Network Types

• Back propagation:
  Yang et al. [229] proposed a Backpropagation (BP)-based neural network utilizing fractional-order memristive hyper-chaos to encrypt images. Another proposal by Ismail et al. [230] uses a similar network to encrypt large satellite images. This algorithm is unique in that it is not affected by geometrical image distortions such as translation, size, and rotation.
• Bidirectional:
  Memristive bidirectional associative memory neural networks have been applied to image encryption in proposals by Wang et al. [231] and Xiao et al. [232]. Simulations and experimental analysis demonstrated both algorithms to be highly secure and resist common attacks.
  Several image encryption schemes utilizing cellular neural networks have been proposed. These include one by Zhou [233] using parallel computing, one by Lin et al. [234] using compressive sensing to achieve encryption and compression, and one by Liu et al. [235] using a fractional-order quantum variant of the network. Lin et al. [236] also proposed an algorithm that uses the network to generate a chaotic stream, which is further used to power Latin squares.
  Another proposal by Hu et al. [237] implemented an encryption/decryption scheme using cellular neural networks that can decrypt the image even if parts of it were corrupted in transmission. A similar technique was used by Liu et al. [238], where noise removal was performed on a transmitted image using the special Cohen–Grossberg neural network. Noise removal is key, otherwise the decryption will not work correctly due to the avalanche property of encryption algorithms.

5.5.2. Encryption

Neural networks see much use in image encryption. Kumar and Rohit [239] proposed a novel Wavelet-based Chaotic Neural Network (WCNN) used for image encryption. A different proposal by Joshi et al. [240] also uses neural networks for encryption, but additionally adds impurities to confuse analysts. Other utilized techniques include multi-layer [241] and multi-image [242].

5.5.3. Steganography

Steganography is very similar to encryption, although the goal of steganography is to hide data so that it is not clear any data are even hidden. Preethi and Asokan [243] proposed a technique using neural networks to determine the Regions Of Non-Interest (RONI) in an image that are optimal for implanting a watermark. In a similar proposal by Duan et al. [221], the discrete cosine transform and ECC were used to encrypt the image first and then the well-known SegNet neural network to improve hiding.

5.5.4. Privacy

Maintaining the privacy of images that are processed by AI algorithms is a novel, but important concept. In a proposal by Sirichotedumrong et al. [244], a special privacy-preserving encryption algorithm was used that encrypts the image, but maintains its features, so that a DNN can still read them. In a different paper by Sirichotedumrong et al. [244], this algorithm was tested on the well-known ResNet-18 network.

5.5.5. Synchronization

Work on the synchronization of neural networks is plentiful. Researched techniques include Periodic Self-Triggered Impulsive (PSTI) control [245], neural activation function [246], and unbounded delay.

The types of neural networks involved in synchronization research are also various. These include arrays of coupled jumping delayed neural networks [247], reaction–diffusion neural networks with mixed delays [248], and chaotic memristive multidirectional associative memory neural networks [249].

5.6. Neural Networks in Chaotic Encryption

Thoms et al. [250] proposed an encryption algorithm using key-controlled neural networks, where key generation is based on the chaotic Lorenz system. The algorithm is designed for digital traffic encryption, and cryptographic analysis demonstrated that it performs on par with or better than existing algorithms.

5.7. Neural Networks in Chaotic Image Encryption

A common use for neural networks in chaotic image encryption is to generate the key streams. This can be seen in a proposal by Han et al. [175], where a Hermite chaotic network was trained from a logistic map, and in a proposal by Fang et al. [251].

In a different proposal by Qingmei and Guodong [252], the hyperchaotic properties of cellular neural networks were leveraged to achieve chaotic image encryption. Simulation results in MATLAB demonstrated strong cryptographic properties.

5.8. Neural-like Image Encryption

Zhang et al. [253] proposed an image encryption algorithm using a neuron-like scheme. The image information is used to adjust neuron weightings to achieve effective pixel diffusion. Simulation results demonstrated strong cryptographic properties.

Combinatorial Optimization Problems

Many important problems in computing such as the shortest path, the Traveling Salesman Problem (TSP), Cellular Channel Assignment (CCA), or more generally, sequencing and resource allocation problems have very inefficient exhaustive search solutions [254]. To find the optimal solution, neural networks have been utilized. Some examples are given below.

In a proposal by Shiyu and Jianying [255], a CNN was used to optimize the shortest path problem. In this proposal, a novel post-processing technique was utilized to achieve the optimal solution with high probability. In a different proposal, Zhao et al. [256] used a CNN with the novel ability to characterize local features to solve the TSP with a higher-than-average success rate for obtaining the optimal solution.

Work on optimizing CCA was conducted by He et al. [257] and Zhao and Gan [258], where noisy CNNs and transient CNNs were utilized, respectively. A related problem to CCA is the more general broadcast scheduling problem. Work by Sun et al. [259,260] proposed a novel noise-tuning-based Hysteretic Noisy Chaotic Neural Network (NHNCNN) that optimizes the problem by obtaining the minimum frame length and maximal channel usage. Their work was completed for both wireless multihop networks and pocket radio networks.

The same CNNs can also be used to solve multiple types of optimization problems. For instance, Wang et al. [261] combined the best parts of the existing Chaotic Simulated Annealing (CSA) and Stochastic Simulated Annealing (SSA) techniques to optimize multiple combinatorial problems such as the TSP and CCA. To compare and contrast the many different CNN approaches for solving these problems, Kwok and Smith [262] developed a framework.

5.9. Chaotic Neural Networks in Image Encryption

The application of chaotic neural networks has been of interest to the research community in recent years [263,264]. Different types of neural networks including the Once Forward-Long Short-Term Memory Structure (OF-LSTMS) [265], Hopefield neural networks [266], and cellular neural networks [267] have been used in this area. Moreover different chaotic maps [267] and sequences [265] have been examined in research in this area.

Wang et al. [249] proposed a novel chaotic neural network called a Multidirectional Associative Memory Neural Network (MAMNN). An image encryption algorithm utilizing the network was also designed to demonstrate its chaotic capabilities. In a different paper by Han et al. [175], they proposed an image encryption algorithm based on the Hermite chaotic neural network. This algorithm is targeted for use on medical images, and statistical analysis demonstrated strong cryptographic properties and resilience against common attacks.

5.10. Quantum-Inspired AI

Quantum-inspired Reinforcement Learning (QiRL) is a heavily researched application of quantum theory. It is inspired by the collapse phenomenon and amplitude amplification properties of quantum computing [268]. Li et al. [268] used QiRL to optimize the uplink transmission rate to an Unmanned Aerial Vehicle (UAV) with little context about the clients’ geographical location. Another proposal by Dong et al. [269] used QiRL to control the navigation of autonomous mobile robots. Simulation results of the system demonstrated good results as compared to other models.

Another proposal for use of quantum concepts in reinforcement learning comes from Wei et al. [270]. In this proposal, quantum concepts were used to adaptively choose experiences from the replay buffer in reinforcement learning models according to the complexity and the replayed times of each experience. This effectively achieves a balance between exploration and exploitation.

Quantum concepts also see much use in neural networks. Masuyama et al. [271] proposed Quantum-inspired Multidirectional Associative Memory (QMAM). This model allows the neural network to progressively develop a resonance state, from inputs to outputs. In another proposal by Patel et al. [272], a Quantum-inspired Fuzzy-based Neural Network (Q-FNN) was introduced. Quantum computing concepts were used to adjust the fuzzy parameter to optimize the network. Testing on 15 benchmark datasets demonstrated superior results.

5.11. Bio-Inspired AI

Bio-inspired learning is implemented in a wide range of fields, with a common one being computer vision. Yuan et al. [273] proposed a bio-inspired system for Visual Attention Prediction (VAP) by leveraging both low-level contrast and high-level semantic features. Another proposal in the computer vision realm was made by Xu et al. [274], where a combination of neural networks and collected eye-tracking data was utilized to determine if and why a face is perceived as beautiful. Extensive tests on this method using popular datasets demonstrated superior performance.

Reinforcement learning can also see great improvements by pairing it with bio-inspired techniques. For example, Lehnert et al. [275] proposed an autonomous navigation algorithm using visual sensors and auxiliary tasks to balance out the common problem of sparse rewards in standard reinforcement learning. Another proposal by Tu et al. [276] used reinforcement learning to control a mobile hummingbird robot to perform biological-based maneuvers such as rapid escapes and tight body flips.

Bio-inspired learning can even see use in the IoT space. Yang et al. [277] proposed a system for optimizing IoT services to reduce service time and energy consumption inspired by the cooperative nature of the various systems in a biological organism. Gibson et al. [278] also performed work with spam email detection with standard classification models.

Since AI techniques can often be very technical and confusing to newcomers, Duan et al. [279] developed an interactive learning environment for bio-inspired AI called BOLE. This system works with unmanned aerial vehicle paths.

Figure 4 demonstrates how current trends predict future trends. In the left column, icons are joined to form new icons for current trends. In the right column, different icons are joined in a similar vein to create predicted future trends. Arrows join the two columns to show how current trends predict future ones.

The trends in Figure 4 form the following two main lines:

(1)

The significant role of neural networks in the current state of chaotic image encryption.

This role is highlighted by the role of neural networks in the three aspects of chaotic image encryption, namely chaos, image, and encryption.

The role of neural networks in the creation and exploitation of chaos is studied in Section 5.1. Section 5.2 discusses the role of neural networks in image processing. Furthermore, the role of neural computing in encryption is highlighted in Section 5.3. The next few subsections investigate how neural networks appear in the intersections of these aspects.

(2)

The significant impact of quantum technology and biological inspirations on the future of neural networks.

Section 5.10 and Section 5.11 discuss the impact of quantum technology and biotechnology on the future of AI, and especially neural networks.

The above two lines clearly will converge at bio-inspired/quantum-inspired neural chaotic image encryption.

6. Conclusions

This paper presented of a comprehensive survey of existing work on chaotic image encryption and established an ecosystem, as well as a future roadmap for the field. We identified the challenges of the chaos aspect, the image aspect, and the encryption aspect of this research area. The choices among different chaos domains, sources, and dimensions, as well as the choice between block and stream ciphers or symmetric and asymmetric cryptography are some of these challenges. The work of this survey will help build a strong foundation from which further research can be built. While current research trends indicate a focus on AI and neural networks (chaotic neural networks, chaotic neural image encryption, etc.), predicted trends lean towards quantum and bio-inspired AI (bio-inspired and quantum-inspired neural chaotic encryption). Further research can be performed to expand the ecosystem of chaotic image encryption as these new trends produce more novel results.