Low-dimensional Chaotic Maps Research Articles

A highly secure stream cipher based on analog-digital hybrid chaotic system

In recent decades, an avalanche of chaos-based cryptosystems have been proposed for information security. Most of these systems are not immune to the dynamical degradation of digital chaos and many have been shown to suffer from a lack of security. In this paper, a stream cipher system based on an analog–digital hybrid chaotic system is presented. The hybrid model can construct digital chaotic maps without degeneration and guarantee synchronization of analog chaotic systems for successful decryption. Moreover, focusing on the characteristics of low-dimensional chaotic maps, a modified three-dimensional Logistic map is proposed to improve the weaknesses of uneven distribution, low complexity and limited parameter space. Combining the three-dimensional Logistic map and the hybrid model, the proposed stream cipher has advantages of huge key space, virtually infinite cycle length and tight security. In particular, it is not affected by the dynamical degradation. Performance and security analyses indicate that the proposed stream cipher is highly resistant to various chaos-based attacks and cryptanalytic attacks.

Medical Image Encryption Based on 2D Zigzag Confusion and Dynamic Diffusion

For the security of medical image, a new algorithm of medical image encryption is designed. The novel algorithm is based on a chaotic system composed of the two-dimensional Sine Logistic modulation map (2D-SLMM) and the two-dimensional Hénon-Sine map (2D-HSM). The main encryption procedure includes zigzag scan scramble, pixel grey value transformation, and dynamic diffusion. On the pixel grey value transformation stage, a password feedback is added. This makes the relationship between password and key more complicated. The proposed scheme is lossless for medical image encryption and decryption. It avoids the problems of low-dimensional chaotic map such as narrow interval and few parameters, as well as the problem of the special texture and contour of medical images. The key space of the novel algorithm is big enough, and the encryption and decryption processing are sensitive to the key. Simulation and experiments validate the effectiveness and efficiency of the novel algorithm. Security analysis proves the algorithm is resistant to common attacks.

Zooming into chaos as a pathway for the creation of a fast, light and reliable cryptosystem

Motivated by today’s huge volume of data that needs to be handled in secrecy, there is a wish to develop not only fast and light but also reliably secure cryptosystems. Chaos allows for the creation of pseudo-random numbers (PRNs) by low-dimensional transformations that need to be applied only a small number of times. These two properties may translate into a chaos-based cryptosystem that is both fast (short running time) and light (little computational effort). What we propose here is an approach to generate PRNs—and consequently digital secret keys—that can serve as a seed for an enhanced chaos-based cryptosystem. We use low-dimensional chaotic maps to quickly generate PRNs that have little correlation, and then, we quickly (“fast”) enhance secrecy by several orders (“reliability”) with very little computational cost (“light”) by simply looking at the less significant digits of the initial chaotic trajectory. This paper demonstrates this idea with rigor, by showing that a transformation applied a small number of times to chaotic trajectories significantly increases its entropy and Lyapunov exponents, as a consequence of the smoothing out of the probability density towards a uniform distribution.

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.

Mode transition in a memristive dynamical system and its application in image encryption

Chaotic systems can be used for secure communication and image encryption by applying a variety of encryption algorithms. While most of the low-dimensional chaotic systems and maps can be estimated by using phase reconstruction and thus the safety in signal processing and propagation is attacked. In this paper, an initial-dependent dynamical system, which is developed from the Rössler system by adding memristive function and disturbance function on the memristive variable [Formula: see text], is presented for realizing image encryption and bifurcation analysis is supplied in detail. Time-varying disturbance from sampled variables is applied to control the memristive variable and the dependence of mode oscillation on initial values is enhanced. As a result, the dynamics of this memristive system is switched between different oscillation modes (e.g., periodical to chaotic, chaotic to chaotic) by activating the initial value, memristive gain and disturbance gain, respectively. From a dynamical viewpoint, the involvement of stochastic adjustment on the memristive variable can reset the initial value and then induce time-varying parameter regulation or switch on certain parameter embedded in the memristive nonlinearity and function, and thus the dynamics dependence on the initial setting is enhanced. Standard bifurcation analysis is carried out on this memristive system and then the sampled time series are used for image encryption, furthermore, the reliability for this scheme is discussed and suggestions for further study are supplied in the end.

A new one-dimensional chaotic map and its application in a novel permutation-less image encryption scheme

In this paper, we propose a new real one-dimensional cosine fractional (1-DCF) chaotic map. Several chaos-theory analysis tests demonstrate that the proposed map has many good cryptography properties, such as a highly chaotic behavior, a large chaotic range, an infinite number of unstable fixed points, and a widely superior sensitivity to the initial conditions than most of the low-dimensional chaotic maps. Regarding these attractive features, we use the 1-DCF map to design a novel fast image encryption scheme for real-time image processing. Unlike most of the existing encryption schemes, we adopt a permutation-less architecture to increase the encryption speed. Regardless of the permutation phase absence, a high-security level is obtained by using a substitution process with a high sensitivity to the plain image. Moreover, we replace the natural row-order encryption with a more secure random-like encryption order generated from the secret key. Experimentation and simulations show that the new scheme is better than many recently proposed encryption schemes in both security and rapidity.

Hyperchaotic image encryption algorithm based on bit-level permutation and DNA encoding

Many chaos-based image encryption algorithms using low-dimensional chaotic mapping and permutation diffusion structures have been proposed recently. However, low-dimensional chaotic maps are less secure than high-dimensional chaotic systems. Furthermore, the permutation process is independent of the plaintext and diffusion process. As a result, they are not very resistant to chosen plaintext attacks and chosen ciphertext attacks. In this paper, we propose a hyper-chaos-based image encryption algorithm that uses a 6-dimensional hyperchaotic system; the key stream generated by hyperchaotic system is related to the plaintext image. Then, bit-level permutation is employed to strengthen the security of the cryptosystem. Finally, DNA coding and operations are employed to change pixels. Theoretical analysis and numerical simulations demonstrate that the proposed algorithm is safe and reliable for image encryption.

A novel method of constructing high-dimensional digital chaotic systems on finite-state automata**Project supported by the National Key R&D Program of China (Grant No. 2017YFB0802000) and the Cryptography Theoretical Research of National Cryptography Development Fund, China (Grant No. MMJJ20170109).

When chaotic systems are implemented on finite precision machines, it will lead to the problem of dynamical degradation. Aiming at this problem, most previous related works have been proposed to improve the dynamical degradation of low-dimensional chaotic maps. This paper presents a novel method to construct high-dimensional digital chaotic systems in the domain of finite computing precision. The model is proposed by coupling a high-dimensional digital system with a continuous chaotic system. A rigorous proof is given that the controlled digital system is chaotic in the sense of Devaney’s definition of chaos. Numerical experimental results for different high-dimensional digital systems indicate that the proposed method can overcome the degradation problem and construct high-dimensional digital chaos with complicated dynamical properties. Based on the construction method, a kind of pseudorandom number generator (PRNG) is also proposed as an application.

Study on an efficient hyper-chaos-based image encryption scheme using global bit permutation.

BACKGROUND: The chaotic system with low dimensions has a low security compared to the high-dimensional chaotic system. Furthermore, major pixel-level permutations merely transform the pixel position and cannot change the intensity distribution of the original image. Bit-level permutation could change the intensity distribution, as it devotes more time to conduct bit-level computation.OBJECTIVE: In this study, we present a more efficient image encryption approach based on hyper-chaos and a global bit cycle shift (HC-GBCS).METHODS: According to the input image we adopted the SHA-256 secure hash algorithm to obtain the initial key, which served as the premier parameter of the chaotic system. Then we employed a 4D hyper-chaotic system for generating the chaotic series, on which we utilized global bit permutation to enhance the security of the encryption system. Finally, the diffusion process was conducted by using the generated chaotic series extended with a logistic map.RESULTS: Experimental results and analysis reveal that the presented approach encrypts plain images effectively and achieves high security and stability.CONCLUSION: The proposed method can deal with the problems inherently existing in encryption methods utilizing low-dimensional chaotic map. Furthermore, global bit permutation can transform the pixel distribution of plain images and enhance the cryptosystem security.

Performance of Finite Precision on Discrete Chaotic Map Based on a Feedback Shift Register

The scintillating technological improvements have changed the process of communication in all parts of the world. Shopping, banking, instant communication, and so on can be operated online and do not care about the linkage of those Internet communications. Because of simple chaotic structure, discrete nature, less arithmetic computation, and high complexity, low-dimensional chaotic systems such as a logistic map and tent map are more attractive than a high-dimensional chaotic system. To overcome the disadvantages of a low-dimensional chaotic map with a finite precision in chaos-based application, the chaotic map with a feedback shift register (CMFSR) is proposed. The related properties and performance associated with CMFSR are analysed. The relationship among the precision of the system, the architecture, and the performance are discussed. The experiments about new pseudorandom number generator (PRNG) based on CMFSR show that our scheme is simple, secure, and easy to accomplish. Experiments show that the proposed architecture of CMFSR is secure for random number generation.

Color image encryption via compressive sensing and chaotic systems

In this paper, we design a color image encryption algorithm based on chaotic system and block compressive sensing. Firstly, the sparse representation of the plain-image is obtained by an adaptive learning dictionary. Secondly, the key streams are produced from two excellent low-dimensional chaotic maps, where updating the initial values and parameters rely on the SHA-384 and the input image. Thirdly, three measurement matrices of R, G, B components are constructed from the chaotic sequences, respectively. Finally, a random rows and columns diffusion method is performed on the encrypted image. Experimental results and safety analysis prove that the proposed scheme has excellent performance.

Symbolic dynamics techniques for complex systems: Application to share price dynamics

The symbolic dynamics technique is well known for low-dimensional dynamical systems and chaotic maps, and lies at the roots of the thermodynamic formalism of dynamical systems. Here we show that this technique can also be successfully applied to time series generated by complex systems of much higher dimensionality. Our main example is the investigation of share price returns in a coarse-grained way. A nontrivial spectrum of Rényi entropies is found. We study how the spectrum depends on the time scale of returns, the sector of stocks considered, as well as the number of symbols used for the symbolic description. Overall our analysis confirms that in the symbol space transition probabilities of observed share price returns depend on the entire history of previous symbols, thus emphasizing the need for a modelling based on non-Markovian stochastic processes. Our method allows for quantitative comparisons of entirely different complex systems, for example the statistics of symbol sequences generated by share price returns using 4 symbols can be compared with that of genomic sequences.

An efficient image encryption algorithm based on a novel chaotic map

In this paper, we propose a novel low dimensional chaotic map, namely, compounded piecewise linear map (CPLM) in order to balance between security and computational complexity. Then we prove the chaotic properties of CPLM theoretically and numerically. Based on the CPLM, an efficient chaotic image encryption algorithm is presented, in which the self-adaptive model and feedback mechanism are initially mingled to enhance the security. Besides, initial values of iteration are generated in a new way to ensure the sensitivity for the changes of plain-image. Simulation results show that our scheme possesses the high key sensitivity, large key space, fast encryption speed and resists the common attacks, especially differential attack. In addition, our algorithm satisfies the applicabilities of all-zero image and binary image which are seldom considered in the existing algorithms. The proposed algorithm justifies the superior security and good efficiency, which can be regarded as an excellent candidate for practical applications of image encryption.

A hyper-chaos-based image encryption algorithm using pixel-level permutation and bit-level permutation

Recently, a number of chaos-based image encryption algorithms that use low-dimensional chaotic map and permutation-diffusion architecture have been proposed. However, low-dimensional chaotic map is less safe than high-dimensional chaotic system. And permutation process is independent of plaintext and diffusion process. Therefore, they cannot resist efficiently the chosen-plaintext attack and chosen-ciphertext attack. In this paper, we propose a hyper-chaos-based image encryption algorithm. The algorithm adopts a 5-D multi-wing hyper-chaotic system, and the key stream generated by hyper-chaotic system is related to the original image. Then, pixel-level permutation and bit-level permutation are employed to strengthen security of the cryptosystem. Finally, a diffusion operation is employed to change pixels. Theoretical analysis and numerical simulations demonstrate that the proposed algorithm is secure and reliable for image encryption.

A novel pseudorandom number generator based on pseudorandomly enhanced logistic map

In last years, low-dimensional and high-dimensional chaotic systems have been implemented in cryptography. The efficiency and performance of these nonlinear systems play an important role in limited hardware implementations. In this context, low-dimensional chaotic systems are more attractive than high-dimensional chaotic systems to produce the pseudorandom key stream used for encryption purposes. Although low-dimensional chaotic maps present some security disadvantages when they are used in cryptography, they are highly attractive due its simple structure, discrete nature, less arithmetic operations, high output processing, and relatively easy to implement in a digital system. In this paper, we proposed both a pseudorandomly enhanced logistic map (PELM) and its application in a novel pseudorandom number generator (PRNG) algorithm, which produces pseudorandom stream with excellent statistical properties. The proposed PELM is compared with logistic map by using histograms and Lyapunov exponents to show its higher benefits in pseudorandom number generator. In contrast to recent schemes in the literature, we present a comprehensive security analysis over the proposed pseudorandom number generator based on pseudorandomly enhanced logistic map (PRNG–PELM) from a cryptographic point of view to show its potential use in secure communications. In addition, the randomness of the PRNG–PELM is verified with the most complete random test suit of National Institute of Standards and Technology (NIST 800-22) and with TestU01. Based on security results, few arithmetic operations required, and high output rate, the proposed PRNG–PELM scheme can be implemented in secure encryption applications, even in embedded systems with limited hardware resources.

A Symmetric Image Encryption Scheme Base on Hyperchaotic System

In recent years, chaos-based image encryption technologies have been widely studied to meet the increasing demand for real-time secure image transmission applications. To overcome the drawbacks of small key space and weak security in many existing schemes based on low-dimensional chaotic maps, this paper suggests a security improved scheme with a permutation-diffusion architecture. In the permutation stage, baker map is employed to shuffle the pixel positions. In the diffusion stage, the value of each pixel is altered by using a key stream derived from hyperchaotic system. Compared with ordinary chaotic systems, hyperchaotic systems, with more complex dynamical behaviors and number of system variables, offer greater potential for secure cryptosystem construction. Extensive security analysis has been performed on the proposed scheme, including the most important ones like key space analysis, statistical analysis and key sensitivity, which has demonstrated the satisfactory security of the proposed scheme.

The linearisation of maps in data assimilation

For the purpose of linearising maps in data assimilation, the tangent-linear approximation is often used. We compare this with the use of the ‘best linear’ approximation, which is the linear map that minimises the mean square error. As a benchmark, we use minimum variance filters and smoothers which are non-linear generalisations of Kalman filters and smoothers. We show that use of the best linear approximation leads to a filter whose prior has first moment unapproximated compared with the benchmark, and second moment whose departure from the benchmark is bounded independently of the derivative of the map, with similar results for smoothers. This is particularly advantageous where the maps in question are strongly non-linear on the scale of the increments. Furthermore, the best linear approximation works equally well for maps which are non-differentiable. We illustrate the results with examples using low-dimensional chaotic maps.

A Digital Image Encryption Algorithm Based on Hyper-chaotic Cellular Neural Network

Using Chaotic characteristics of dynamic system is a promising direction to design cryptosystems that play a pivotal role in a very important engineering application of cognitive informatics, i.e., information assurance and security. However, encryption algorithms based on the lowdimensional chaotic maps face a potential risk of the keystream being reconstructed via return map technique or neural network method. In this paper, we propose a new digital image encryption algorithm that employs a hyper-chaotic cellular neural network. To substantiate its security characteristics, we conduct the following security analyses of the proposed algorithm: key space analysis, sensitivity analysis, information entropy analysis and correlation coefficients analysis of adjacent pixels. The results demonstrate that the proposed encryption algorithm has desirable security properties and can be deployed as a cornerstone in a sound security cryptosystem. The comparison of the proposed algorithm with five other chaos-based image encryption algorithms indicates that our algorithm has a better security performance.

OCML-based colour image encryption

The chaos-based cryptographic algorithms have suggested some new ways to develop efficient image-encryption schemes. While most of these schemes are based on low-dimensional chaotic maps, it has been proposed recently to use high-dimensional chaos namely spatiotemporal chaos, which is modelled by one-way coupled-map lattices (OCML). Owing to their hyperchaotic behaviour, such systems are assumed to enhance the cryptosystem security. In this paper, we propose an OCML-based colour image encryption scheme with a stream cipher structure. We use a 192-bit-long external key to generate the initial conditions and the parameters of the OCML. We have made several tests to check the security of the proposed cryptosystem namely, statistical tests including histogram analysis, calculus of the correlation coefficients of adjacent pixels, security test against differential attack including calculus of the number of pixel change rate (NPCR) and unified average changing intensity (UACI), and entropy calculus. The cryptosystem speed is analyzed and tested as well.

Stochastic cooperativity in non-linear dynamics of genetic regulatory networks

Two major approaches are known in the field of stochastic dynamics of genetic regulatory networks (GRN). The first one, referred here to as the Markov Process Paradigm (MPP), places the focus of attention on the fact that many biochemical constituents vitally important for the network functionality are present only in small quantities within the cell, and therefore the regulatory process is essentially discrete and prone to relatively big fluctuations. The Master Equation of Markov Processes is an appropriate tool for the description of this kind of stochasticity. The second approach, the Non-linear Dynamics Paradigm (NDP), treats the regulatory process as essentially continuous. A natural tool for the description of such processes are deterministic differential equations. According to NDP, stochasticity in such systems occurs due to possible bistability and oscillatory motion within the limit cycles. The goal of this paper is to outline a third scenario of stochasticity in the regulatory process. This scenario is only conceivable in high-dimensional, highly non-linear systems, and thus represents an adequate framework for conceptually modeling the GRN. We refer to this framework as the Stochastic Cooperativity Paradigm (SCP). In this approach, the focus of attention is placed on the fact that in systems with the size and link density of GRN (∼25 000 and ∼100, respectively), the confluence of all the factors which are necessary for gene expression is a comparatively rare event, and only massive redundancy makes such events sufficiently frequent. An immediate consequence of this rareness is ‘burstiness’ in mRNA and protein concentrations, a well known effect in intracellular dynamics. We demonstrate that a high-dimensional non-linear system, despite the absence of explicit mechanisms for suppressing inherent instability, may nevertheless reside in a state of stationary pseudo-random fluctuations which for all practical purposes may be regarded as a stochastic process. This type of stochastic behavior is an inherent property of such systems and requires neither an external random force as in the Langevin approach, nor the discreteness of the process as in MPP, nor highly specialized conditions of bistability as in NDP, nor bifurcations with transition to chaos as in low-dimensional chaotic maps.