Discrete Chaotic Maps Research Articles

Application of Reservoir Computing Based on a 2D Hyperchaotic Discrete Memristive Map in Efficient Temporal Signal Processing

The analysis of time series is essential in many fields, and reservoir computing (RC) can provide effective temporal processing that makes it well-suited for time series analysis and prediction tasks. In this study, we introduce a new discrete memristor model and a corresponding two-dimensional hyperchaotic map with complex dynamic properties that are well-suited for reservoir computing. By applying this map to the RC, we enhance the state richness of the reservoir, resulting in improved performance. The paper evaluates the performance of the proposed RC approach using time series data for sunspot, exchange rate, and solar-E forecasting tasks. Our experimental results demonstrate that this approach is highly effective in handling temporal data with both accuracy and efficiency. And comparing with other discrete memristive chaotic maps, the proposed map is the best for improving the RC performance. Furthermore, the proposed RC model is characterized by a simple structure that enables it to fully exploit the time-dependence of the state values of the hyperchaotic map.

A Nondegenerate n-Dimensional Hyperchaotic Map Model with Application in a Keyed Parallel Hash Function

The construction of multidimensional discrete hyperchaotic maps with ergodicity and larger Lyapunov exponents is desired in cryptography. Here, we have designed a general [Formula: see text]D ([Formula: see text]) discrete hyperchaotic map ([Formula: see text]D-DHCM) model that can generate any nondegenerate [Formula: see text]D chaotic map with Lyapunov exponents of desired size through setting the control matrix. To verify the effectiveness of the [Formula: see text]D-DHCM, we have provided two illustrative examples and analyzed their dynamic behavior, and the results demonstrated that their state points have ergodicity within a sufficiently large interval. Furthermore, to address the finite precision effect of the simulation platform, we analyzed the relationship between the size of Lyapunov exponent and the randomness of the corresponding state time sequence of the [Formula: see text]D-DHCM. Finally, we designed a keyed parallel hash function based on a 6D-DHCM to evaluate the practicability of the [Formula: see text]D-DHCM. Experimental results have demonstrated that [Formula: see text]D discrete chaotic maps constructed using [Formula: see text]D-DHCM have desirable Lyapunov exponents, and can be applied to practical applications.

Chaos in popular metaheuristic optimizers – a bibliographic analysis

This paper presents an overview of the history and recent efforts in combining chaos theory and evolutionary computation techniques. Various algorithms from the evolutionary computation domain, also known as metaheuristic algorithms, have been successfully enhanced with chaotic components in the past. Numerous ways to incorporate chaos have been examined, and many impressive results have been reported. Implementations of discrete chaotic maps such as Lozi, Hénon, and logistic map as generators of chaotic pseudo-random sequences for controlling evolution operators in metaheuristics have achieved significant popularity. In this survey, we focus on the research field itself and perform a bibliographical analysis to show how broad and active is nowadays the research field of chaos-enhanced metaheuristics and what are some of the most recent works published.

Discrete fracmemristor-based chaotic map by Grunwald–Letnikov difference and its circuit implementation

The topic of discrete memristor has attracted more and more attention in recent years, but there is not enough discussion about discrete memristor under the fractional-order difference definition, especially for the system model defined by Grunwald–Letnikov. In this paper, a discrete fractional-order memristor model based on Grunwald–Letnikov definition is derived, and two new chaotic maps are designed based on this fractional-order model. Dynamic behaviors are investigated by the volt–ampere curve, bifurcation diagram, Lyapunov exponent spectrum, attractor phase diagram and fuzzy entropy complexity algorithm. Simulation results show that the two fractional-order systems can produce rich dynamic behaviors with the change of fractional order value, and both of them can present hyperchaos. Finally, the proposed systems are implemented in FPGA digital circuit and analog circuit respectively, and the phenomena in the numerical simulation are verified.

Design, Hardware Implementation on FPGA and Performance Analysis of Three Chaos-Based Stream Ciphers

In this paper, we come up with three secure chaos-based stream ciphers, implemented on an FPGA board, for data confidentiality and integrity. To do so, first, we performed the statistical security and hardware metrics of certain discrete chaotic map models, such as the Logistic, Skew-Tent, PWLCM, 3D-Chebyshev map, and 32-bit LFSR, which are the main components of the proposed chaotic generators. Based on the performance analysis collected from the discrete chaotic maps, we then designed, implemented, and analyzed the performance of three proposed robust pseudo-random number generators of chaotic sequences (PRNGs-CS) and their corresponding stream ciphers. The proposed PRNGs-CS are based on the predefined coupling matrix M. The latter achieves a weak mixing of the chaotic maps and a chaotic multiplexing technique or XOR operator for the output function. Therefore, the randomness of the sequences generated is expanded as well as their lengths, and divide-and-conquer attacks on chaotic systems are avoided. In addition, the proposed PRNGs-CS contain polynomial mappings of at least degree 2 or 3 to make algebraic attacks very difficult. Various experimental results obtained and analysis of performance in opposition to different kinds of numerical and cryptographic attacks determine the high level of security and good hardware metrics achieved by the proposed chaos system. The proposed system outperformed the state-of-the-art works in terms of high-security level and a high throughput which can be considered an alternative to the standard methods.

Response to: Commentary: Multidimensional discrete chaotic maps

Response to: Commentary: Multidimensional discrete chaotic maps

A DISCRETE MEMRISTOR COUPLED TWO-DIMENSIONAL GENERALIZED SQUARE HYPERCHAOTIC MAPS

In this paper, a new discrete chaotic map is constructed by introducing a discrete memristor in two-dimensional generalized square maps to enhance its chaotic performance. First, the fixed points of the new maps are analyzed, and the effects of different parameters on the system performance are investigated by bifurcation diagrams, Lyapunov exponential spectra and phase diagrams. Second, the fixed points of the new maps are analyzed, and the effects of different parameters on the system performance are investigated by bifurcation diagrams, Lyapunov exponential spectra and phase diagrams. The distinctive characteristic of a discrete system is the coexistence of various types of attractors, and there is coexistence of hyperchaos and cycles in the present maps. It is worth mentioning that symmetric chaotic attractors with different positive and negative parameters are found during the study. In addition, the phenomenon of state transition between chaos and cycles is also found. Finally, the discrete maps are designed and implemented using a DSP platform. The results of the study provide a reference for the application of discrete amnesic chaotic maps.

Grey Wolf Optimizer and Discrete Chaotic Map for Substitution Boxes Design and Optimization

A metaheuristic approach based on the nature-inspired and well-known Grey Wolf Optimization algorithm (GWO) was employed in this study to design an approach for retrieving strong designs of 8×8 substitution boxes (S-boxes). The GWO was developed as a novel metaheuristic based on inspiration from grey wolves and how they hunt. The ability of the GWO to quickly explore the search space for the near/optimal feature subsets that maximize any given fitness function (in consideration of its distinctive hierarchical structure) aids in the construction of strong S-boxes that can satisfy the required criteria. However, when tackling optimization problems, GWO may experience the problem of premature convergence. Therefore, a variant of GWO called Crossover Grey Wolf Optimizer (XGWO) has been proposed in this study. The performance of the proposed novel approach was evaluated using numerous cryptographic performance metrics, including bijective property, bit independence, strict avalanche, linear probability, and I/O XOR distribution and the result was contrasted with a couple of existing S-box creation techniques. Overall, the results of the experiment showed that the suggested S-box design had adequate cryptographic features.

Study of the dynamical behavior of an Ikeda-based map with a discrete memristor

In 1971, Chua suggested that there should be a fourth electrical component in addition to resistor, capacitor and inductor. This new component was proposed to be called “memristor”. Due to lack of commercially available memristors many emulators are mainly used in nonlinear circuits. Also, the memristance functions are used in many dynamical systems in order to enhance their complexity. Furthermore, the use of memristors in discrete chaotic maps is an interesting research topic, due to the applications of such systems. In this work, a memristor-based Ikeda mapping model is presented by coupling a discrete memristance function with Ikeda map. To investigate system’s dynamical behavior a host of nonlinear tools has been used, such as bifurcation and continuation diagrams, maximal Lyapunov exponent diagrams, and Kaplan–Yorke conjecture. Interesting phenomena related to chaos has been observed. More specifically, regular (periodic and quasiperiodic) and chaotic orbits, route to chaos through the mechanism of period doubling and crisis phenomena, have been found. Moreover, higher value of the internal state of the memristor, revealed chaotic behavior in a bigger area. Also, from the comparison of the bifurcation diagrams with the respective continuation diagrams coexisting attractors have been found. Finally, two-parameter bifurcation-like diagrams revealed the system’s rich dynamical behavior.

Novel memristor and memristor-based applications

This special issue presents 25 papers regarding memristor systems and their applications. Part of the manuscripts are about memristor-based neural systems, and the systems show rich dynamics. Multistability in memristive systems has aroused much interest, and nonlinear behaviors, such as hidden attractors, extreme multistability, and transient chaotic transition, are observed. There are also two works investigating controllers of fractional-order memristor chaotic systems and several for circuit implementation of memristor chaotic systems. As new research, discrete memristor chaotic maps are reported and show potential application value in the future. Finally, there are two papers about the application of memristive chaotic systems to image encryption fields, indicating that memristive chaotic systems can be used in the information security field.

Dynamic analysis of new two‐dimensional fractional‐order discrete chaotic map and its application in cryptosystem

A new fractional difference equation two‐dimensional triangle function combination discrete chaotic map (2D‐TFCDM) based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent, and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect.

A novel image encryption mechanism based on umbrella map and Yang-Gu algorithm

A novel image encryption algorithm based on nonlinear discrete chaotic umbrella map and Yang-Gu mixture amplitude-phase retrieval algorithm in fractional Fourier domain is proposed in this paper. Here in the proposed mechanism the input image is encrypted with a deterministic phase mask and later subjected to fractional Fourier transform. Thereafter, for scrambling the pixels of transformed image the umbrella map is utilized. The umbrella map along with providing a broader range of control parameters also enhances the key space. Scrambled image is further encoded with additional deterministic phase mask before subjecting it to another level of fractional Fourier transform. The phase truncated portion of the resultant image is used as an input to the Yang-Gu mixture amplitude-phase retrieval algorithm in the Fourier domain. The encrypted image attained by one-way binary modulation is real-valued, which makes it more appropriate for storage and transmission. Comprehensive statistical and attack analysis of the proposed scheme has been carried out. Sensitivity analysis of the scheme to various keys and crucial parameters has also been done. The results demonstrate the robustness and efficacy of the proposed mechanism. The exceptional resistance to noise attacks makes it suitable for use in extremely noisy channels and real valued cipher image is more convenient to store and transmit.

URUK 4D DISCRETE CHAOTIC MAP FOR SECURE COMMUNICATION APPLICATIONS

In this paper, URUK, a discrete four-dimensional chaotic map, is proposed for secure communication. The 0-1 tests, the Lyapunov Exponent (LE) tests, and the National Institute of Standard and Technology (NIST) tests, which are typically used to verify the randomness of bits, are employed to evaluate the dynamic behavior of the system based on a variety of criteria. Based on the results of tests demonstrating the system's characteristic chaotic and random behavior, it is reliable for use in covert communications and image encryption.

The synchronization of discrete quantum chaotic maps with uncertainty via compensated control technique

In this work, a two-dimensional discrete map model based on quantum game is introduced, and its dynamics are investigated. Moreover, the stability condition of Nash equilibrium is also analyzed. In addition, the synchronization control of discrete chaotic maps with uncertainties is studied, and a new dynamical compensation control (DCC) technique is presented by constructing a dynamical compensator. At a result, the influence of the uncertain term can be suppressed effectively. Compared with the previous synchronous control approaches, the proposed method avoids the complex derivation and amplification operation, because it does not use Lyapunov function, and the controller design is simple and the proposed scheme is well verified.

An image layered scrambling encryption algorithm based on a novel discrete chaotic map

At present, the image encryption algorithm based on the bit-plane has the problem that the encryption speed and the encryption effect cannot be well balanced. To address the problem, an image layered scrambling encryption algorithm is proposed in this paper. First, a new one-dimensional Cosine Logistic compound chaotic map (1DCLC) is proposed. Several characteristics of the new chaotic map are analyzed by using various measures, such as bifurcation diagram, Lyapunov exponent, sample entropy etc. And the results show that the novel chaotic map has a better chaotic performance. Second, based on the chaotic map, an image encryption algorithm is proposed. The algorithm first performs an XOR encryption operation on the image. Second, the layered scrambling is applied to the XOR encrypted image. Then the layered and scrambled image is recombined for the second XOR encryption operation. In particular, in the layered scrambling, not only can the position of the pixel be changed but also the value of the pixel can be modified, which improves the security of the algorithm. In addition, in order to defend against various plaintext attacks and increase the security of the overall algorithm, the Hash function is introduced to update the key of the proposed algorithm. Finally, the original algorithm is improved and the improved algorithm's ability to resist differential attacks is analyzed. The experimental simulation and security test results show that the proposed encryption algorithm has a comparable encryption effect and a faster encryption speed.

A novel image encryption scheme based on memristive chaotic system and combining bidirectional bit-level cyclic shift and dynamic DNA-level diffusion

In recent years, many researchers have leveraged various memristors to design many novel memristive chaotic systems with complex dynamics. Compared with other chaotic systems, applying these memristive chaotic systems to image encryption is expected to solve some key problems in this field. Therefore, exploiting a recently reported memristive chaotic system, this paper proposes a novel image encryption scheme based on the memristive chaotic system and combining bidirectional bit-level cyclic shift and dynamic DNA-level diffusion (IES-M-BD). First, a discrete memristive chaotic map is employed to generate chaotic sequences. Then, the plaintext image is shifted circularly on bit-level according to chaotic sequences and the hash value of the plaintext image. After that, the shifted matrix is recombined on the bit plane and encoded dynamically by DNA encoding rules. Next, dynamic DNA-level diffusion and DNA-level permutation are carried out in two rounds. Finally, the encrypted image is obtained after dynamic DNA decoding. Simulation tests and performance analyses are also carried out in this paper. The simulation results and the security analyses demonstrate that this encryption scheme has a high security level and can resist various attacks.

Constructing a discrete memristor chaotic map and application to hash function with dynamic S-Box

Constructing a discrete memristor chaotic map and application to hash function with dynamic S-Box

Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory network model

Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory network model

Chaotical PRNG based on composition of logistic and tent maps using deep-zoom

We propose the deep-zoom analysis of the composition of the logistic map and the tent map, which are well-known discrete one-dimensional chaotic maps. The deep-zoom technique transforms each point of a given chaotic orbit by removing the first k-digits after decimal separator. We found that the pseudo-random qualities of the composition map as a pseudo-random number generator (PRNG) improve as the k parameter increases. This was evidenced by the fact that it successfully passed the randomness tests and even outperformed the k-logistic map and the k-tent map PRNG. These dynamic properties show that the application of deep-zoom to the composition of chaotic maps, at least to these two well-known maps, is suitable for better randomization for PRNG purposes as well as for cryptographic systems.

A visually secure image encryption scheme using adaptive-thresholding sparsification compression sensing model and newly-designed memristive chaotic map

Aiming at the deficiencies of existing image compression-encryption algorithms, such as unstable reconstruction performance and weak crypticity, a novel visually secure image encryption scheme based on the adaptive-thresholding sparsification compression sensing model and the newly designed memristive chaotic map is proposed in this paper. First, the plain image is sparsely represented in the wavelet packet domain. Then, the coefficient matrix is scrambled and compressed using parallel compressive sensing with assistance from the newly developed adaptive-thresholding sparsification model. Next, the compressed image is re-encrypted by the FAN transform and diffusion to generate a noise-like secret image. Finally, the reversible matrix encoding-based embedding method is employed to create a meaningful cipher image. To improve the execution efficiency of the encryption algorithm without affecting the security performance, a low-dimensional discrete memristive chaotic map is designed and applied in the compression, encryption and embedding stages. Additionally, the Lp matrix norm is utilized to establish an adaptive threshold model to effectively enhance the compression performance. Simulation results and analyses are presented to demonstrate the effectiveness, the compressibility, and the secrecy of the proposed scheme.