Discrete Chaotic Maps Research Articles

Analogue circuit implementation of a new logistic‐like map

Discrete chaotic system has complex dynamics and becomes to be a present hot topic, but its analogue circuit implementation is rarely discussed. In this paper, by simplifying the logistic map, a new Logistic-like map is proposed. Its cobweb plots, bifurcations, Lyapunov exponents, and dynamic map are investigated theoretically and numerically. The results show that the Logistic-like map has complex dynamical behaviours. And then Psim simulation model is proposed to verify the feasibility of analogue circuit implementation. Finally, an analogue circuit experiment is performed and its results are consistent with the simulation results. The circuit designed is very simple and has less components, which can further broaden the applications of discrete chaotic map in applications.

Design and analysis of a three-dimensional discrete memristive chaotic map with infinitely wide parameter range

In this paper, a new discrete memristive chaotic map with infinitely wide parameter range is designed. Firstly, a discrete memristor based on a triangular wave function is constructed. The memristor conforms to the definition of generalized memristor, and a new three-dimensional memristive chaotic map is designed based on it. Numerical simulations show that the map has complex dynamic behavior. An improved perturbation method is proposed to estimate the output sequence of the chaotic system. At the same time, it is proved mathematically that the new map can always be in chaotic or hyperchaotic state with infinitely wide parameter range under certain conditions. By observing the Lyapunov exponent spectrum and the phase diagram, it is found as the absolute value of the parameter increases, the output range and ergodicity of the new map are also enhanced. We demonstrate that the new map has an initial-boosting behavior that depends on the initial conditions of the memristor. By changing the initial values of the memristor, we can control the appearance of attractor at different locations without loss. At the same time, this paper analyzes the mechanism of the discrete memristive chaotic map generating initial-boosting behavior, puts forward a method to make ordinary chaotic maps easier to obtain this behavior. Finally, the DSP hardware platform is used to implement the new map, which proves the physical existence and realizability of the map.

Dynamics, synchronization control of a class of discrete quantum game chaotic map

In this work, A four-dimensional (4D) discrete chaotic map model based on quantum-Cournot duopoly game is proposed, and the dynamics of this map using bounded rationality are studied. It is observed that the stable region of mapping equilibrium increases with the increase of quantum entanglement γ. and the occurrence time of bifurcation and chaos delays with the increase of mapped Nash equilibrium. In addition, A novel synchronization control approach which adopts both linear and nonlinear feedback strategies is proposed by constructing a dynamic controller. The simulation results verify the effectiveness of the proposed method.

Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor

The application of a memristor in chaotic circuits is increasingly becoming a popular research topic. The influence of a memristor on the dynamics of chaotic systems is worthy of further exploration. In this paper, a multi-dimensional closed-loop coupling model based on a Logistic map and Sine map (CLS) is proposed. The new chaotic model is constructed by cascade operation in which the output of the Logistic map is used as the input of the Sine map. Additionally, the one-dimensional map is extended to any dimension through the coupling modulation. In order to further increase the complexity and stability of CLS, the discrete memristor model is introduced to construct a discrete memristor-based coupling model with a Logistic map and a Sine map (MCLS). By analyzing the Lyapunov exponents, bifurcation diagram, complexity, and the 0–1 test result, the comparison result between CLS and MCLS is obtained. The dynamics performance analysis shows that the Lyapunov exponents and bifurcation diagrams present symmetrical distribution with variations of some parameters. The MCLS has parameters whose values can be set in a wider range and can generate more complex and more stable chaotic sequences. It proves that the proposed discrete memristor-based closed-loop coupling model can produce any higher dimension hyperchaotic system and the discrete memristor model can effectively improve the performance of discrete chaotic map and make this hyperchaotic system more stable.

Enhanced design and hardware implementation of a chaos-based block cipher for image protection

Nowadays chaos-based cryptography systems are widely used to protect sensitive data. A variety of chaos-based cryptosystems with software implementation has been published in the literature. However, due to the rapid growth of connected devices, the development of hardware secure and low-resource cryptographic systems is necessary. In this paper, we enhance the design of one of our software cryptosystems and then we address its FPGA-based implementation. The improvement in the design consists essentially to introduce an Initial Vector (IV) entry to the Chaotic Pseudo Random Number Generator (CPRNG) thus allowing to generate a new dynamic keys for the confusion layer (using a modified 2-D cat map) and diffusion layer (using a logistic map) with each new execution of the system. The CPRNG is formed by XORred a discrete Skew Tent Map (STM) with a discrete Piecewise Linear Chaotic Map (PWLCM). The proposed cryptosystem, which operates in Cipher Block Chaining (CBC) mode, is controlled by a Finite State Machine (FSM) to ensure that both the encryption and decryption are performed by the same reconfigurable architecture. The system is implemented on a Xilinx XC7Z020 ZYNQ FPGA platform. The results obtained in terms of hardware implementation costs (logic resources, throughput, and efficiency) and in terms of security against cryptographic attacks demonstrate the effectiveness of the proposed chaos-based cryptosystem.

A Novel Dynamic Mathematical Model Applied in Hash Function Based on DNA Algorithm and Chaotic Maps

This paper aims to improve SHA-512 security without increasing complexity; therefore, we focused on hash functions depending on DNA sequences and chaotic maps. After analysis of 45 various chaotic map types, only 5 types are selected in this proposal—namely, improved logistic, cosine logistic map, logistic sine system, tent sine system, and hybrid. Using DNA features and binary coding technology with complementary rules to hide information is a key challenge. This article proposes improving SHA-512 in two aspects: the modification of original hash buffer values, and the modification of additive constants Kt. This proposal is to make hash buffer values (a, b, c, d, e, f, g, and h) and Kt dependent on one-dimensional discrete chaotic maps and DNA sequences instead of constant. This modification complicates the relationship between the original message and hash value, making it unexpected. The performance of the proposed hash function is tested and analyzed the confusion, diffusion, and distributive and compared with the original SHA-512. The performance of security is analyzed by collision analysis, for which the maximum number of hits is only three, showing that the proposed hash function enhances the security and robustness of SHA-512. The statistical data and experimental analysis indicate that the proposed scheme has good properties and satisfies high-performance requirements for secure hash functions.

A novel discrete memristive chaotic map

A novel discrete memristive chaotic map

A new discrete chaotic map application in image encryption algorithm

In this work, a new discrete chaotic map is developed from the 1D ICMIC (Iterative Map with Infinite Collapses) to control the 2D Hénon map. The performances of the proposed discrete map are evaluated by using the phase diagram, Lyapunov exponent spectrum and complexity. It is found that the new chaotic map has larger value of complexity, and chaotic states are distributed in large parameter ranges. Thus the new chaotic map has better chaotic features and randomness, it is especially suitable for the chaotic secure communication. Then an image encryption algorithm is proposed based on the new chaotic map. In this algorithm, the sequences generated by the new chaotic map are used for pixel position scrambling and pixel value diffusion operations. The security features of the designed encryption scheme are evaluated by common attacks methods. The results confirmed that the proposed algorithm is effective to improve the security in image encryption.

A stable meaningful image encryption scheme using the newly-designed 2D discrete fractional-order chaotic map and Bayesian compressive sensing

In order to safely and covertly protect the digital image information during public channel transmission and localized storage, a stable image visually secure encryption algorithm by using 2D discrete fractional-order chaotic map (FOCM), Bayesian compressive sensing (BCS), and discrete V transform (DVT) is presented in this paper. First, the key-controlled measurement matrix constructed by the infinite collapse-Chebyshev-coupling chaotic map is employed to measure the scrambled wavelet packet coefficients of plain image in parallel. Then, the fractional-order chaotic map designed through piece-wise constant arguments method is employed to generate the secret code streams with assistance of the counter mode, and under its control, the compressed image is re-encrypted into the intermediate secret image. Next, the DVT-based embedding technology is proposed to stochastically embed the secret data into the non-secret-involved host image. Eventually, experimental results show that under the premise of ensuring security, the proposed algorithm improves the visual security around 1∼9 dB, and stabilizes its compression performance within 1 dB, compared with other existing related encryption algorithms.

Cryptanalysis and Improvement of Novel Image Encryption Technique Using Hybrid Method of Discrete Dynamical Chaotic Maps and Brownian Motion

Cryptanalysis and Improvement of Novel Image Encryption Technique Using Hybrid Method of Discrete Dynamical Chaotic Maps and Brownian Motion

A Strong Hybrid S-Box Scheme Based on Chaos, 2D Cellular Automata and Algebraic Structure

Substitution-boxes are the main deciding components in symmetric-key cryptosystems for resisting many cryptanalytic attacks. It has been a challenging task for the designers to construct strong S-box which satisfies multiple cryptographic properties simultaneously. A number of S-box studies have been investigated in literature; but, the generated S-box found to exhibit one single property with good score. This paper proposes a novel creation of S-boxes which possess excellent scores of multiple cryptographic properties instead of only one property. The suggested hybrid S-box method explores the science of two-dimensional cellular automata theory, discrete chaotic maps, and algebraic group structure. The proposed anticipated 8×8 S-box holds excellent security performance features such as: minimum nonlinearity as high as 110, no fixed points, satisfaction of strict avalanche and bits independence criterions, differential uniformity as low as 6, linear approximation probability as low as 0.0703, and auto-correlation function (absolute indicator) of 40. The performance comparison indicates the proposed S-box has superior features, greater inherent security and robustness strength than many available state of the art S-box methods.

Atmospheric Convection Model Based Digital Confidentiality Scheme

Nonlinear dynamics is a fascinating area that is intensely affecting a wide range of different disciplines of science and technology globally. The combination of different innovative topics of information security and high-speed computing has added new visions into the behavior of complex nonlinear dynamical systems which uncovered amazing results even in the least difficult nonlinear models. The generation of complex actions from a very simple dynamical method has a strong relation with information security. The protection of digital content is one of the inescapable concerns of the digitally advanced world. Today, information plays an important role in everyday life and affects the surroundings rapidly. These digital contents consist of text, images, audio, and videos, respectively. Due to the vast usage of digital images in the number of social and web applications, its security is one of the biggest issues. In this work, we have offered an innovative image encryption technique based on present criteria of confusion and diffusion. The designed scheme comprises two major nonlinear dynamical systems. We have employed discrete fractional chaotic iterative maps to add confusion capability in our suggested algorithm and continuous chaotic Lorenz system. We have verified our offered scheme by using statistical analysis. The investigations under the statistical tests suggested that our proposed technique is quite reasonable for the security of digital data.

Design, Analysis, and Application of Flipped Product Chaotic System

In this paper, a novel method is proposed to build an improved 1-D discrete chaotic map called flipped product chaotic system (FPCS) by multiplying the output of one map with the output of a vertically flipped second map. Two variants, each with nine combinations, are shown with trade-off between computational cost and performance. The chaotic properties are explored using the bifurcation diagram, Lyapunov exponent, Kolmogorov entropy, and correlation coefficient. The proposed schemes offer a wider chaotic range and improved chaotic performance compared to the constituent maps and several prior works of similar nature. Wide chaotic window and improved chaotic complexity are two desired characteristics for several security applications as these two characteristics ensure enhanced design space with elevated entropic properties. We present a general Field-Programmable Gate Array (FPGA) design framework for the hardware implementation of the proposed flipped-product schemes and the results show good qualitative agreement with the numerical results from MATLAB simulation. Finally, we present a new Pseudo Random Number Generator (PRNG) using the two variants of the proposed chaotic map and validate their excellent randomness property using four standard statistical tests, namely NIST, FIPS, TestU01, and Diehard.

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.

A new fractional-order 2D discrete chaotic map and its DSP implement

A new fractional-order 2D discrete chaotic map and its DSP implement

An improved LogNNet classifier for IoT applications

In the age of neural networks and Internet of Things (IoT), the search for new neural network architectures capable of operating on devices with limited computing power and small memory size is becoming an urgent agenda. Designing suitable algorithms for IoT applications is an important task. The paper proposes a feed forward LogNNet neural network, which uses a semi-linear Henon type discrete chaotic map to classify MNIST-10 dataset. The model is composed of reservoir part and trainable classifier. The aim of the reservoir part is transforming the inputs to maximize the classification accuracy using a special matrix filing method and a time series generated by the chaotic map. The parameters of the chaotic map are optimized using particle swarm optimization with random immigrants. As a result, the proposed LogNNet/Henon classifier has higher accuracy and the same RAM usage, compared to the original version of LogNNet, and offers promising opportunities for implementation in IoT devices. In addition, a direct relation between the value of entropy and accuracy of the classification is demonstrated.

Parameter identification for discrete memristive chaotic map using adaptive differential evolution algorithm

Since the concept of discrete memristor was proposed, more and more scholars began to study this topic. At present, most of works on the discrete memristor are devoted to the mathematical modeling and digital circuit implementation, but the research on its synchronization control has not received much attention. This paper focuses on the parameter identification for the discrete memristive chaotic map, and a modified intelligent optimization algorithm named adaptive differential evolution algorithm is proposed. To deal with the complex behaviors of hyperchaos and coexisting attractors of the considered discrete memristive chaotic maps, the identification objective function adopts two special parts: time sequences and return maps. Numerical simulations demonstrate that the proposed algorithm has the best performance among six existing algorithms, and it can still accurately identify the parameters of the original system under noise interference.

IMAGE ENCRYPTION BASED ON TWO-DIMENSIONAL FRACTIONAL QUADRIC POLYNOMIAL MAP

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.

IMAGE ENCRYPTION TECHNOLOGY BASED ON FRACTIONAL TWO-DIMENSIONAL DISCRETE CHAOTIC MAP ACCOMPANIED WITH MENEZES-VANSTONE ELLIPTIC CURVE CRYPTOSYSTEM

A new fractional two-dimensional triangle function combination discrete chaotic map (2DTFCDM) with the discrete fractional difference is proposed in this paper. The chaos behaviors are observed through the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits. The proposed map is applied in color image encryption with the secret keys generated by Menezes–Vanstone Elliptic Curve Cryptosystem. The image encryption system is analyzed using six aspects indicating the superiority of the proposed algorithm compared to the other algorithms.

Image encryption using adaptive multiband signal decomposition

Due to the rapid growth of multimedia transmission over the internet, the challenges of image security have become an important research topic. In this paper, an Adaptive Multiband Signal Decomposition (AMSD) is proposed and its application for image encryption is explored. Like the conventional multiband wavelet transform, the AMSD can decompose the original image into multiband subimages. The perfect reconstruction of the original image from the decomposed multibands is achieved. In addition, a novel image encryption algorithm based on the adaptive multiband image decomposition with three dimensional discrete chaotic maps is developed and its performance is evaluated using common security analysis methods. Simulation results show that the proposed encryption algorithm has great degree of security and can resist various typical attacks.