Parameters Of Chaotic System Research Articles

Recurrence microstates for machine learning classification.

Recurrence microstates are obtained from the cross recurrence of two sequences of values embedded in a time series, being the generalization of the concept of recurrence of a given state in phase space. The probability of occurrence of each microstate constitutes a recurrence quantifier. The set of probabilities of all microstates are capable of detecting even small changes in the data pattern. This creates an ideal tool for generating features in machine learning algorithms. Thanks to the sensitivity of the set of probabilities of occurrence of microstates, it can be used to feed a deep neural network, namely, a microstate multi-layer perceptron (MMLP) to classify parameters of chaotic systems. Additionally, we show that with more microstates, the accuracy of the MMLP increases, showing that the increasing size and number of microstates insert new and independent information into the analysis. We also explore potential applications of the proposed method when adapted to different contexts.

A metaheuristic image cryptosystem using improved parallel model and many‐objective optimization

AbstractMetaheuristic is one of the techniques to improve the security of image encryption. However, existing metaheuristic image cryptosystems based on metaheuristic may have convergence difficulties during the optimization process, which cause insecurity and slow convergence. Besides, the time cost of the parallel execution model applied to metaheuristic image cryptosystems is not low enough. Therefore, a parallel many‐objective optimized key generation framework is proposed. Firstly, if four or more security indicators of cryptosystem, which are the results of security test, need to be optimized, the many‐objective optimization algorithm is employed to the proposed framework. With method adjusts the chaotic system parameters as the optimization key, which effectively avoid the convergence difficulty of the encryption key. Secondly, a master‐slave parallel model is improved to metaheuristic encryption. The model allocates the most time‐consuming fitness calculation work to slave nodes, which makes the modified model more reasonable and thus reduces the encryption time. To evaluate the performance of the proposed framework, a specific encryption scheme is constructed, that utilizes a 2D quadric map and many‐objective optimization algorithm based on dominance and decomposition (MOEA/DD) to optimize five security indicators. Experimental results reveal that this scheme has good security performances and less parallel encryption time.

Adaptive internal synchronization control of chaotic systems based on DNA strand displacement

AbstractIn the application of DNA‐based chaotic synchronization, the precise values of the system parameters are often complicated to attain due to the interference of external factors, thus weakening the performance of synchronization. In this paper, considering the case of unknown parameters in chaotic systems, a method of adaptive synchronization based on DNA strand displacement (DSD) is proposed. Firstly, the chain replacement reaction modules of emergence, activation and degradation are given through the design of DNA strand, which enable the realization of Yang chaotic system using the mass action dynamics law. Further, in order to achieve the estimation of the unknown parameters and the stability of the error system, the rules for identification of unknown parameters are derived and adaptive controllers are designed under the adaptive control principle. Finally, an internal synchronization approach is recommended to decompose the chaotic system into subsystems of different orders, which can freely choose subsystems to complete synchronization. Subsequently, numerical simulations show that DNA‐based adaptive controllers can fulfill the internal synchronization of Yang system with unknown parameters and are robust to parameter deviations. In addition, the strategy presented in this study can be extended to the synchronization control of other DSD‐based multivariable chaotic systems.

Secure Communication via Chaotic Synchronization Based on Reservoir Computing.

Information security occupies a very important part of national security. Chaos communication can provide high-level physical layer security, but its harsh claims on the chaotic system parameters of the transmitter and the receiver resulting in reduced synchronization coefficient and more difficult consistent synchronization of point to multipoint networking. In this article, a chaotic synchronization and communication system based on reservoir computing (RC) has been proposed. In this scheme, the trained RC highly synchronized with the emitter acts as the receiver with simplified structure under the premise of ensuring safety. Simultaneously, the cross-prediction algorithm has been proposed to weaken the accumulation effect of prediction synchronization error of RC and facilitate the realization of long-term communication. Furthermore, the tolerance of the system performance to the signal-to-noise ratio with the variations of the mask coefficients has been investigated, and the optimal operation point under the condition of the adjustable number of nodes and leakage rate of RC has been numerically analyzed. The simulation results show that the normalized mean-square error of synchronization of 10⁻⁶ magnitude and the bit error rate of decryption at 10⁻⁸ level can be obtained. Finally, from the operational perspective, a 100-m short-distance experiment confirms that its communication performance is consistent with the simulation results. We strongly believe that the proposed system offers the opportunity of a new research direction in chaotic secure communications.

Chaotic Complex Hashing: A simple chaotic keyed hash function based on complex quadratic map

In the last two decades, the use of chaotic iterative maps in cryptographic systems, especially chaotic hash functions, has been expanding. Sensitivity to small changes in initial conditions and control parameters in chaotic systems is the best feature for designing a chaotic hash function. This paper introduces a new generalized chaotic map based on a complex quadratic map, which in addition to the chaotic behavior in the real and imaginary parts, also has a high key length, which can guarantee the security of a cryptographic system. Various analyses of dynamical systems, such as the bifurcation diagram and the Lyapunov exponent, show that the proposed chaotic map has the characteristics of a chaotic dynamical system. The foundation of a secure and efficient algorithm for designing and developing a new hash function lies in how to use this new chaotic map. Also, statistical analysis, collision analysis, key space analysis, and speed analysis use to prove the security and efficiency of the proposed method. The results of these analyses and their comparison with similar methods show that the proposed method can be a reliable and efficient method for practical applications in information technology.

Employing Quantum Fruit Fly Optimization Algorithm for Solving Three-Dimensional Chaotic Equations

In a chaotic system, deterministic, nonlinear, irregular, and initial-condition-sensitive features are desired. Due to its chaotic nature, it is difficult to quantify a chaotic system’s parameters. Parameter estimation is a major issue because it depends on the stability analysis of a chaotic system, and communication systems that are based on chaos make it difficult to give accurate estimates or a fast rate of convergence. Several nature-inspired metaheuristic algorithms have been used to estimate chaotic system parameters; however, many are unable to balance exploration and exploitation. The fruit fly optimization algorithm (FOA) is not only efficient in solving difficult optimization problems, but also simpler and easier to construct than other currently available population-based algorithms. In this study, the quantum fruit fly optimization algorithm (QFOA) was suggested to find the optimum values for chaotic parameters that would help algorithms converge faster and avoid the local optimum. The recommended technique used quantum theory probability and uncertainty to overcome the classic FA’s premature convergence and local optimum trapping. QFOA modifies the basic Newtonian-based search technique of FA by including a quantum behavior-based searching mechanism used to pinpoint the position of the fruit fly swarm. The suggested model has been assessed using a well-known Lorenz system with a specified set of parameter values and benchmarked signals. The results showed a considerable improvement in the accuracy of parameter estimates and better estimation power than state-of-the art parameter estimation approaches.

Hybrid mapping algorithm based on 1‐DCM and Lorenz

A hybrid chaotic mapping algorithm(HCM) based on 1-DCM (one-dimensional chaotic map) and Lorenz is proposed to address the security issues of low-dimensional chaotic systems and the complexity of high-dimensional chaotic systems. Firstly, the color images are divided into three high 4-bit planes (BP) and low 4-bit planes. By optimizing the parameters and structure of trigonometric function, the new arcsine-sine (AS) function is designedto generate the 1-DCM and permutation matrix, which is constructed to change the plane position and pixel position of the high 12BP to obtain the scrambling plane, which is encrypted by Lorenz chaotic system to produce the selfencryption matrix, with which the low 12BP is encrypted to generate the low bit encryption planes. Then, the ciphertext image is obtained by combining the high bit scrambling plane and low bit encryption plane. The SHA function is also used as the chaotic system's parameter generator, which improves the correlation between plaintext images and the key and resists selective plaintext attacks. The simulation results show that by modularizing the image's high and low bit planes, the encryption process is simplified and improved. The security and reliability of color image transmission are also improved.

A Connection between Lyapunov Exponents and Sensitive Dependence on Parameters of Chaotic Systems

A Connection between Lyapunov Exponents and Sensitive Dependence on Parameters of Chaotic Systems

Fast synchronization of symmetric Hénon maps using adaptive symmetry control

The article discusses the possibility of synchronizing adaptive discrete chaotic maps through the control of the symmetry coefficient. Since a change in the symmetry coefficient in symmetric chaotic maps possesses a much less influence on the system oscillation mode than a change in nonlinearity parameters, we assume that the synchronization of such systems can be achieved in a small number of iterations. To experimentally examine this hypothesis, we studied three cases of adaptive synchronization feedback controllers for the conventional Hénon map and adaptive Hénon map. We found that synchronization occurs faster while the symmetry coefficient is controlled by comparing the synchronization times in all considered cases. Averagely, an accurate estimate of the adaptive coefficient is achieved after 3–5 iterations. Applying this approach to the communication systems based on modulation through switching parameters of chaotic systems can significantly reduce the transient processes inherent in this method. The numerical experiments show that it is possible to decrease the transmission time by more than 25%, even in the case of short messages.

Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are critically used in science and engineering. In this paper, we propose a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems. We numerically show that the existence of the derivative depends on the Lebesgue-integrability of a certain density gradient function, which we define as the derivative of logarithmic SRB density along the unstable manifold. We develop a recursive formula for the density gradient that can be efficiently computed along trajectories, and demonstrate its use in determining the differentiability of statistics. Our numerical procedure is illustrated on low-dimensional chaotic systems whose statistics exhibit both smooth and rough regions in parameter space.

A novel color image encryption algorithm based on image hashing, 6D hyperchaotic and DNA coding

In order to improve the key space of color image encryption algorithm, the sensitivity to the contents of plain images, the robustness against various types of known attacks, and to achieve the tamper location analysis, a novel color image encryption algorithm based on image hashing, six-dimensional (6D) hyperchaotic and dynamic DNA coding is proposed. Firstly, the color image is pre-processed and the hash sequence is extracted by image hashing algorithm which is used as the initial value and control parameter of chaotic system. Secondly, three color channels of the color image RGB are synthesized into a two-dimensional matrix and the pixels replacement are performed by using the improved two-dimensional chaotic map. Finally, a 6D hyperchaotic system is used to generate random sequences for DNA dynamic coding and arithmetic operations of color images, the encrypted image is obtained. The experimental results show that, compared with the existing methods, the proposed algorithm has a large enough key space, better plain-image sensitivity, better statistical and differential characteristics, as well as can resistant various forms of attacks such as noise and cropping, and the tampering image can be tamper-located analyzed, which has good security and strong robustness.

Designing a secure round function based on chaos

A round function based on chaos is designed combining Feistel structure's pseudo-randomness, chaotic system's parameter sensitivity and image data characteristics. The round function composes of two parts—data transformation based on Feistel (abbreviated as FST) and sampling output based on chaos (abbreviated as SMP). FST bases on Feistel structure and several efficient operations including bitwise xor, permutation and circulating shift. SMP is a chaos based pseudo-random sampling algorithm. It is from theoretical analysis that the round function is a pseudo-random function. The upper bounds of the average maximum differential probability and average maximum linear probability are p2 and q2 respectively. Finally, the good pseudo-randomness of the round function is examined with the NIST random test. The design of this round function provides an important cryptographic component for the design of chaotic image encryption algorithm.

An Improved Return Maps Method for Parameter Estimation of Chaotic Systems

Recently, an effective method called return maps is proposed for the parameter estimation of chaotic systems. However, high time-consumption limits practical applications. In this paper, we focus on this problem, and an improved return maps method is proposed. It combines the differential evolution algorithm with the return maps method, and simplifies the calculation process of Euclidean distance. Numerical simulations are carried out on two fractional-order chaotic systems, and the other five methods are used as the comparison. Results show that the improved method can accurately estimate the parameters of chaotic systems, and it saves much time than does the classical return maps method. Furthermore, the proposed method also exhibits good anti-noise performance.

Hardware realization of a secure and enhanced s-box based speech encryption engine

This paper presents a secure and efficient substitution box (s-box) for speech encryption applications. The proposed s-box data changes every clock cycle to swap the input signal with different data, where it generated based on a new algorithm and a memristor chaotic system. Bifurcation diagrams for all memristor chaotic system parameters are introduced to stand for the chaotic range of each parameter. Moreover, the effect of each component inside the proposed encryption system is studied, and the security of the system is validated through perceptual and statistical tests. The size of the encryption key is 175 bits to meet the global standards for the optimum encryption key width (> 128). MATLAB software is used to calculate entropy, MSE, and correlation coefficient. Both chaotic circuit and encryption/decryption schemes are designed using Verilog HDL and simulated by Xilinx ISE 14.7. Xilinx Virtex 5 FPGA kit is used to realize the proposed algorithm with a throughput 0.536 of Gbit/s. The cryptosystem is tested using two different speech files to examine its efficiency.

Estimation of Parameters in Fractional order Financial Chaotic system with Nature Inspired Algorithms

Abstract Chaotic system parameter modeling is one of the significant topics in the field of non-linear science and therefore has attracted the attention of many researchers in this field. This paper discusses parameter recognition of Fractional Order Financial Chaotic System by employing different metaheuristic algorithms. The implemented algorithms include Artificial bee colony, Grey wolf optimizer, Whale optimization algorithm and Ant colony optimizer. Mean Square Error is employed as objective function to estimate parameters of system under consideration. Results of optimized parameters using the above-mentioned optimization are compared with each other and discussed in the paper. The overall outcome shows that Whale optimization algorithm gives more accurate and robust results with higher convergence rate as compared with other algorithms.

Pseudo-Random Number Generator Based on Logistic Chaotic System

In recent years, a chaotic system is considered as an important pseudo-random source to pseudo-random number generators (PRNGs). This paper proposes a PRNG based on a modified logistic chaotic system. This chaotic system with fixed system parameters is convergent and its chaotic behavior is analyzed and proved. In order to improve the complexity and randomness of modified PRNGs, the chaotic system parameter denoted by floating point numbers generated by the chaotic system is confused and rearranged to increase its key space and reduce the possibility of an exhaustive attack. It is hard to speculate on the pseudo-random number by chaotic behavior because there is no statistical characteristics and infer the pseudo-random number generated by chaotic behavior. The system parameters of the next chaotic system are related to the chaotic values generated by the previous ones, which makes the PRNG generate enough results. By confusing and rearranging the output sequence, the system parameters of the previous time cannot be gotten from the next time which ensures the security. The analysis shows that the pseudo-random sequence generated by this method has perfect randomness, cryptographic properties and can pass the statistical tests.

Parameter estimation of chaotic systems based on extreme value points

Parameter estimation and synchronisation of chaotic systems are one of the hottest topics in the field of nonlinear science. In this paper, we addressed how to utilise the obtained experimental time series to estimate multiple parameters in chaotic systems. On the basis of relations of critical points and extreme value points, as well as the least squares estimation, we deduced a novel statistical parameter estimation corollary method to evaluate the unknown parameters in chaotic systems. In order to illustrate the feasibility and effectiveness of the proposed method, three numerical simulation results are presented, where the validity of the proposed method is verified in detail. Furthermore, we also investigated the effects of time-series noise and system disturbances for the proposed method, and the results showed that the proposed method is robust to uncertainties.

A Novel Method to Identify Initial Values of Chaotic Maps in Cybersecurity

Chaos theory has applications in several disciplines and is focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. Chaotic dynamics are the impromptu behavior displayed by some nonlinear dynamical frameworks and have been used as a source of diffusion in cybersecurity for more than two decades. With the addition of chaos, the overall strength of communication security systems can be increased, as seen in recent proposals. However, there is a major drawback of using chaos in communication security systems. Chaotic communication security systems rely on private keys, which are the initial values and parameters of chaotic systems. This paper shows that these chaotic communication security systems can be broken by identifying those initial values through the statistical analysis of standard deviation and variance. The proposed analyses are done on the chaotic sequences of Lorenz chaotic system and Logistic chaotic map and show that the initial values and parameters, which serve as security keys, can be retrieved and broken in short computer times. Furthermore, the proposed model of identifying the initial values can also be applied on other chaotic maps as well.

Parallel non-dominated sorting genetic algorithm-II-based image encryption technique

ABSTRACTChaotic system requires parameters to generate random sequences. Recent studies show that the improper selection of parameter values make secret keys generated from chaotic system vulnerable. Meta-heuristic techniques have been introduced in the area of image encryption to improve the selection of chaotic system parameters. But, these techniques suffer from poor computational speed. To overcome this issue, in this paper, a parallel Non-Dominated Sorting Genetic Algorithm (NSGA-II)-based intertwining logistic map is proposed to encrypt the images. To implement NSGA-II in parallel fashion, master–slave environment is designed. Initially, the execution time analysis of NSGA-II is done to determine the computationally expensive operations. Thereafter, NSGA-II operators are divided into master and slave jobs. The Message Passing Interface (MPI) is used for intercommunication between master and slave nodes. The simulation results show that the parallel proposed technique provides a significant improvement in computational speed as compared to the existing techniques.

A color image encryption technique using exclusive-OR with DNA complementary rules based on chaos theory and SHA-2

A novel color image encryption algorithm is proposed in this paper which utilizes SHA-256 hash function to modify the initial conditions and control parameters of chaotic system. Three channels of a color image called red, green and blue is arranged into one dimensional vector and sort according to chaotic sequence generated by Piecewise Linear Chaotic Map. Then this permuted array split into three parts where each represents a color channel and permuted again using Lorenz’s chaotic system independently. After dual permutation, each pixel of every channel is independently encoded into Deoxyribonucleic Acid (DNA) bases chaotically. The novelty of algorithm is that the substitution of each pixel of a channel is achieved by Exclusive-OR operation with DNA complementary rules. This operation is repeated with multiple DNA rules in a sequence to some random number of times. This repetition of operation continues in a cycle. The selection of DNA rule in the beginning of this cyclic operation and continuation of operation is made reliant on Chen’s chaotic sequence. The extensive simulated experimental results such as Key Sensitivity, Correlation, entropy, Number of Pixel Chang Rate (NPCR), Unified Average Cipher Intensity (UACI) scores prove that proposed algorithm has excellent encryption results obtained in just one round. This scheme has advantage over other encryption techniques that it has nice feature to resist transmission noise and loss of a part of an encrypted image.