High-dimensional Chaotic Maps Research Articles

Mechanism of multistability in chaotic maps.

This research aims to investigate the mechanisms of multistability in chaotic maps. The study commences by examining the fundamental principles governing the development of homogeneous multistability using a basic one-dimensional chain-climbing map. Findings suggest that the phase space can be segmented into distinct uniform mediums where particles exhibit consistent movement. As critical parameter values are reached, channels emerge between these mediums, resulting in deterministic chaotic diffusion. Additionally, the study delves into the topic of introducing heterogeneous factors on the formation of heterogeneous multistability in the one-dimensional map. A thorough examination of phenomena such as multistate intermittency highlights the intimate connection between specific phase transition occurrences and channel formation. Finally, by analyzing two instances-a memristive chaotic map and a hyperchaotic map-the underlying factors contributing to the emergence of multistability are scrutinized. This study offers an alternative perspective for verifying the fundamental principles of homogenous and heterogeneous multistability in complex high-dimensional chaotic maps.

The hyperbolic sine chaotification model and its applications

Some existing chaotic systems suffer from issues such as period windows, discontinuous parameter ranges, and dynamical degradation, which seriously limit their application. Therefore, designing high-performance anti-degradation chaotic systems is of great significance. In this paper, a novel hyperbolic sine chaotification model (HSCM) is proposed. It allows for the use of any chaotic maps or linear functions as the seed maps, and employs a closed-loop modulation coupling (CMC) method to extend it to high-dimensional (HD) chaotic maps. Theoretical and experimental results show that this model can effectively improve the Lyapunov exponent (LE) of the seed chaotic map and expand the parameter ranges. In addition, it can also resist the dynamical degradation under finite computational precision. Based on the HSCM, a novel eight-dimensional (8D) HSCM is designed, and implemented through field-programmable gate array (FPGA) in both serial and parallel modes, respectively. Furthermore, the novel chaotic maps are applied to pseudo-random sequence generator (PRNG) and image compression under finite computing precision. Experimental results indicate that the novel chaotification model has greatly broad application prospects.

[formula omitted]D-CS: A circularly shifting chaotic map generation method

Due to more complex behavior and larger number of control parameters, high-dimensional chaotic map generation methods may provide more satisfactory performance in various practical applications compared to low-dimensional counterparts. Designing n-dimensional (nD) chaotic map generation methods for generating chaotic maps in arbitrary dimensions with desired dynamics is an interesting but challenging task. In this paper, we propose a universal framework called the nD circularly shifting chaotic map generation method (nD-CS), which utilizes existing one-dimensional chaotic maps as seed maps to generate nD chaotic maps with complex and robust behaviors. Theoretical analysis proves that when specific criteria are met by its control parameters, our nD-CS can exhibit complex and robust hyperchaotic behavior in the sense of Lyapunov exponent. To demonstrate the effectiveness of our nD-CS, we first employ it to generate three examples of three-dimensional hyperchaotic maps. The results indicate high-performance indicators of these new chaotic maps. Furthermore, we compare the nD chaotic maps generated by our nD-CS to those produced by existing methods. The results demonstrate that our chaotic maps have a better overall performance.

A class of novel discrete memristive chaotic map

Since Chua introduced the memristor as the fourth basic circuit component, memristors and memristive chaotic maps have received great attention. In this paper, we present a class of novel memristive chaotic map by coupling the discrete memristor with the internal item in an original boundary function. Several examples are presented based on classical one dimensional (1D) and high dimensional (HD) chaotic maps, and numerical simulation results display their improved dynamical performance and chaos complexity. We analyze the influences of introducing the memristor on the chaotic map and the generality for designing discrete chaotic maps of the proposed memristive model. Particularly, with the newly emerging features of no fixed points and multistability, we can robustly control the output sequence by setting their initial-states, which accommodates to many arisen applications of chaotic systems. Furthermore, based on the proposed memristive chaotic map, we construct the hardware platform to acquire its attractors, and design the Pseudorandom Number Generator (PRNG) to explore its application prospects.

Advanced 5D logistic and DNA encoding for medical images

ABSTRACT Since patient diagnostic data such as X-rays, MRIs, CT scans, etc. contain very sensitive information about a patient's health, it must be encrypted to prevent unauthorized access. In chaotic-based image encryption methods, the lower-dimensional chaotic maps are insufficient because they can be estimated through some signal estimation technologies; therefore, a higher-order novel 5D logistic chaotic map along with DNA encoding and the Arnold map for medical image encryption is presented to overcome the vital medical breaching that could cost a life. In this work, the advantages of DNA principles and high-dimensional chaotic maps are used to develop an efficient encryption method for the protection of medical images. The strategy is useful due to the fact that the higher order makes the system more complicated and tough to penetrate. The large keyspace up to, anti attack capabilities, and comparison of various standard parameters with different literatures illustrate the algorithm's superiority.

A novel hyperchaotic map with sine chaotification and discrete memristor

Discrete memristor has become a hotspot since it was proposed recently. However, the design of chaotic maps based on discrete memristor is in its early research stage. In this paper, a memristive seed chaotic map is proposed by combining a quadratic discrete memristor with the sine function. Furthermore, by applying the chaotification method, we obtain a high-dimensional chaotic map. Numerical analysis shows that it can generate hyperchaos. With the increase of cascade times, the generated map has more positive Lyapunov exponents and larger hyperchaotic range. The National Institute of Standards and Technology (NIST) test results show that the chaotic pseudo-random sequence generated by cascading two seed maps has good unpredictability, and it indicates the potential in practical application.

5D Gauss Map Perspective to Image Encryption with Transfer Learning Validation

Encryption of visual data is a requirement of the modern day. This is obvious and greatly required due to widespread use of digital communication mediums, their wide range of applications, and phishing activities. Chaos approaches have been shown to be extremely effective among many encryption methods. However, low-dimensional chaotic schemes are characterized by restricted system components and fundamental structures. As a result, chaotic signal estimation algorithms may be utilized to anticipate system properties and their initial values to breach the security. High-dimensional chaotic maps on the other hand, have exceptional chaotic behavior and complex structure because of increased number of system parameters. Therefore, to overcome the shortcomings of the lower order chaotic map, this paper proposes a 5D Gauss Map for image encryption for the first time. The work presented here is an expansion of the Gauss Map’s current 1D form. The performance of the stated work is evaluated using some of the most important metrics as well as the different attacks in the field. In addition to traditional and well-established metrics such as PSNR, MSE, SSIM, Information Entropy, NPCR, UACI, and Correlation Coefficient that have been used to validate encryption schemes, classification accuracy is also verified using transfer learning. The simulation was done on the MATLAB platform, and the classification accuracy after the encryption-decryption process is compared.

Improved Sine-Tangent chaotic map with application in medical images encryption

The attractive properties of ergodicity, unpredictability, and initial state sensitivity have made chaotic maps the go-to tools in many applications, including cryptography and cyber–physical systems. Despite this, two challenges arise when using chaos systems in cryptography: (i) some one-dimensional (1D) chaotic maps do not satisfy the unpredictability property, and (ii) although they exhibit a complex and chaotic behavior, high-dimensional (HD) chaotic maps incur higher computational complexity. To address these issues, this paper proposes a new 1D chaotic map dubbed the improved Sine-Tangent map (IST map) that is derived from of the Sine map, a tangent function, and a Chebyshev polynomial of the first kind. Relative to chaotic maps, the proposed IST map provides better unpredictability and ergodicity, a vast chaotic range, enhanced complex behavior, and competitive computational complexity. Based on the IST map, we also introduced an encryption scheme for securing medical images in telemedicine. It consists of two diffusion phases, i.e., a bitwise XOR operation and a bitwise expanded XOR (eXOR) operation, with an intermediated confusion one, i.e., random circular-shift. An overriding step is foremost first completed before performing these cryptography phases, i.e., key generation. The secret key of the IST map is updated using the sine and cosine values of the sum of pixels of the input image. This leads to a unique secret key for each image. That is, one-time chaotic sequences are produced for each input image. The cyclic pattern of the sine and cosine values of the sum of pixels provides a prominent sensibility to small changes in the input image. Thus, the proposed algorithm is capable of resisting any chosen/known plaintext attacks. A performance analysis shows that the proposed algorithm outperforms a set of state-of-the-art comparison algorithms and its variants based on Sine, SE, and ST maps since it allows the best performance/complexity trade-off.

N-Dimensional Polynomial Chaotic System With Applications

Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional polynomial chaotic system (<inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS) that can generate <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula> chaotic maps with any desired LEs. The <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS is constructed from <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS, we developed a chaos-based secure communication scheme. Simulation results show that <inline-formula> <tex-math notation="LaTeX">$n\text{D}$ </tex-math></inline-formula>-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.

A New Hyperchaotic Image Encryption Algorithm Based on Stochastic Signals

Image encryption is often used to protect private images during transmission on a public channel. A high dimensional chaotic map has a greater secret key space, better ergodicity and dynamic property than a low-dimensional chaotic system. A seven-dimensional (7D) hyperchaotic map is used to produce chaotic sequences. Given secret keys and SHA-512 function are employed to generate initial values for iteration. Many stochastic signals are injected into one of the variables during iteration to transfer trajectory and increase dynamic behavior of a chaotic system. Three matrices are constructed with generated chaotic sequences. Permutation is performed on the basis of a control matrix. It keeps the pixel far from its neighboring pixels. Cycle shift is executed during bit-level permutation. Characteristic values of scrambling image are calculated and temporary values are achieved successfully. Two dynamic values are also applied during the diffusion process. Experimental results display the effects of the proposed algorithm. Security analysis reveals that the proposed method has some special advantages.

A new fractional one dimensional chaotic map and its application in high-speed image encryption

Chaos theory has been widely used in the design of image encryption schemes. Some low-dimensional chaotic maps have been proved to be easily predictable because of their small chaotic space. On the other hand, high-dimensional chaotic maps have a larger chaotic space. However, their structures are too complicated, and consequently, they are not suitable for real-time image encryption. Motivated by this, we propose a new fractional one-dimensional chaotic map with a large chaotic space. The proposed map has a simple structure and a high chaotic behavior in an extensive range of its control parameters values. Several chaos theoretical tools and tests have been carried out to analyze and prove the proposed map’s high chaotic behavior. Moreover, we use the proposed map in the design of a novel real-time image encryption scheme. In this new scheme, we combine the substitution and permutation stages to simultaneously modify both of the pixels’ positions and values. The merge of these two stages and the use of the new simple one-dimensional chaotic map significantly increase the proposed scheme’s security and speed. Besides, the simulation and experimental analysis prove that the proposed scheme has high performances.

Research on cascading high-dimensional isomorphic chaotic maps

In order to overcome the security weakness of the discrete chaotic sequence caused by small Lyapunov exponent and keyspace, a general chaotic construction method by cascading multiple high-dimensional isomorphic maps is presented in this paper. Compared with the original map, the parameter space of the resulting chaotic map is enlarged many times. Moreover, the cascaded system has larger chaotic domain and bigger Lyapunov exponents with proper parameters. In order to evaluate the effectiveness of the presented method, the generalized 3-D Hénon map is utilized as an example to analyze the dynamical behaviors under various cascade modes. Diverse maps are obtained by cascading 3-D Hénon maps with different parameters or different permutations. It is worth noting that some new dynamical behaviors, such as coexisting attractors and hyperchaotic attractors are also discovered in cascaded systems. Finally, an application of image encryption is delivered to demonstrate the excellent performance of the obtained chaotic sequences.

Design new chaotic maps based on dimension expansion**Project supported by the National Natural Science Foundation of China (Grant Nos. 61161006 and 61573383).

Based on the high-dimensional (HD) chaotic maps and the sine function, a new methodology of designing new chaotic maps using dimension expansion is proposed. This method accepts N dimensions of any existing HD chaotic map as inputs to generate new dimensions based on the combined results of those inputs. The main principle of the proposed method is to combine the results of the input dimensions, and then performs a sine-transformation on them to generate new dimensions. The characteristics of the generated dimensions are totally different compared to the input dimensions. Thus, both of the generated dimensions and the input dimensions are used to create a new HD chaotic map. An example is illustrated using one of the existing HD chaotic maps. Results show that the generated dimensions have better chaotic performance and higher complexity compared to the input dimensions. Results also show that, in the most cases, the generated dimensions can obtain robust chaos which makes them attractive to usage in a different practical application.

An image encryption algorithm based on the perturbed high-dimensional chaotic map

Due to the secure problems of low-dimensional chaos in image encryption and cycle degradation caused by computers’ finite precision, a perturbed high-dimensional chaos is designed for image encryption according to Devaney and topological conjugate definition. Perturbation is added to the new high-dimensional chaos to enlarge cycle. The characteristics of the high-dimensional chaotic map are proved by Devaney definition, and the security features are analyzed. Owe to the periodic defects of Cat map, a new separated Cat map is designed for the confusion process. The “confusion–diffusion” structure is used in the image encryption. The security test results demonstrate the proposed methods have high security, and the speed of our algorithm is faster than classical solutions.

A new image encryption algorithm based on the high-dimensional chaotic map

Because precision degradation of low-dimensional chaos has seriously affected the security of encryption, a high-dimensional chaos is designed for image encryption according to Devaney and topological conjugate definition. The characteristics of the high-dimensional chaotic map are proved by Devaney definition and the security features are analysed. Owing to the periodic defects of Cat map, a new separated Cat map is designed for the confusion process. The bilateral-diffusion scheme is used in the novel image encryption. The experiment analysis shows that the algorithm passes the random and pseudo-random sequence of test standard SP800-22 released by the National Institute of Standards and Technology Commission and the DIEHARD pseudo-random sequence test. And the experiment results demonstrate that the encryption scheme has a high security.

A novel parallel hash function based on 3D chaotic map

As the core of cryptography, hash function is one of the basic techniques for information security. During the last few years, considerable effort has been devoted to research on chaos-based hash functions. Nevertheless, the corresponding analysis of them lag far behind. In this paper, a new efficient scheme for parallel hash function based on high-dimensional chaotic map is proposed. In the proposed scheme, the confusion as well as the diffusion effect is enhanced significantly by utilizing two nonlinear coupling parameters. Theoretical and experimental results indicate that the proposed scheme can satisfy all performance requirements of hash function such as desired statistical properties and strong collision resistance. At the same time, the proposed scheme can keep the parallel merit and message sensitivity with high potential to be adopted for network security.

A CHAOS-BASED RANDOM NUMBER GENERATOR FOR EIGHT-BIT MICRO-CONTROLLER SYSTEM

Random number generators are widely used in different applications. However, it is difficult to obtain a good random number generator in low precision and resource constrained system, such as an eight-bit micro-controller system which is still commonly used in industrial and consumer markets. This paper provides a practical solution for this problem based on chaotic systems. By the use of a modified Chua's circuit, it is demonstrated that the sampled state, after post-processing by a high-dimensional chaotic map, can be used as a random source even in an eight-bit environment. The randomness of the generated sequence is testified and confirmed by different statistical tests and the up-to-date statistical suite.

Image encryption using high-dimension chaotic system

In recent years, the chaos based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. This paper proposes a new approach for image encryption based on a high-dimensional chaotic map. The new scheme employs the Cat map to shuffle the positions, then to confuse the relationship between the cipher-image and the plain-image using the high-dimensional Lorenz chaotic map preprocessed. The results of experimental, statistical analysis and key space analysis show that the proposed image encryption scheme provides an efficient and secure way for real-time image encryption and transmission.

A CHAOS-BASED CRYPTOGRAPHIC HASH FUNCTION FOR MESSAGE AUTHENTICATION

In this paper, a chaos-based cryptographic hash function is proposed for message authentication where a fixed length message digest can be generated from any messages with arbitrary length. The compression function in-use is developed according to the diffusion and confusion properties of a novel high-dimensional discrete chaotic map. A symmetric key structure can also be easily embedded based on the high sensitivity to initial conditions and system parameters of the map. As demonstrated in experiments, the throughput of the proposed scheme is about 1.5 times of that with MD5, and the hash value appears randomly without any correlation with the message or the key.

Synchronization and parameter identification of high-dimensional discrete chaotic systems via parametric adaptive control

This paper is concerned with the synchronization of high-dimensional chaotic maps with multiple unknown parameters by parametric adaptive control, and a sufficient condition is drawn when the system parameters are considered to be in linear form. Moreover, the parameters of drive system can be identified when the two systems achieve synchronization. Finally, numerical experiments are presented to show the effectiveness and efficiency of the proposed method.