Chaos-based secure communications in a large community

Lorenz-like systems and Lorenz-like attractors: Definition, examples, and equivalences

Since the early 1970s, numerous systems exhibiting an algebraic structure resembling that of the 1963 Lorenz system have been proposed. These systems have occasionally yielded the same attractor as the Lorenz system, while in other cases, they have not. Conversely, some systems that are evidently distinct from the Lorenz system, particularly in terms of symmetry, have resulted in attractors that bear a resemblance to the Lorenz attractor. In this paper, we put forward a definition for Lorenz-like systems and Lorenz-like attractors. The former definition is based on the algebraic structure of the governing equations, while the latter relies on topological characterization. Our analysis encompasses over 20 explicitly examined chaotic systems.

A 2D Hyperchaotic Map: Amplitude Control, Coexisting Symmetrical Attractors and Circuit Implementation

An absolute value function was introduced for chaos construction, where hyperchaotic oscillation was found with amplitude rescaling. The nonlinear absolute term brings the convenience for amplitude control. Two regimes of amplitude control including total and partial amplitude control are discussed, where the attractor can be rescaled separately by two independent coefficients. Symmetrical pairs of coexisting attractors are captured by corresponding initial conditions. Circuit implementation by the platform STM32 is consistent with the numerical exploration and the theoretical observation. This finding is helpful for promoting discrete map application, where amplitude control is realized in an easy way and coexisting symmetrical sequences with opposite polarity are obtained.

Doubling the coexisting attractors

When the offset boosting technique is introduced into a chaotic system for attractor shifting, the number of coexisting attractors in the system can be doubled under the application of the employed absolute-value function. Consequently, the offset booster becomes a doubling parameter determining the distance between the two coexisting attractors, and therefore can polymerize these attractors to become a pseudo-multi-scroll attractor. This paper demonstrates that the attractor doubling operation can be applied to any dimension of the system and can also be nested at any time leading to the geometric growth of the coexisting attractors. Furthermore, various regimes of coexistence can be merged and composed together to reproduce an integrated attractor in the system.

A new chaotic oscillator with free control

A novel chaotic system is explored in which all terms are quadratic except for a linear function. The slope of the linear function rescales the amplitude and frequency of the variables linearly while its zero intercept allows offset boosting for one of the variables. Therefore, a free-controlled chaotic oscillation can be obtained with any desired amplitude, frequency, and offset by an easy modification of the linear function. When implemented as an electronic circuit, the corresponding chaotic signal can be controlled by two independent potentiometers, which is convenient for constructing a chaos-based application system. To the best of our knowledge, this class of chaotic oscillators has never been reported.

Breaking a chaotic direct sequence spread spectrum communication system using interacting multiple model-unscented Kalman filter

In this paper, a new method to break chaotic direct sequence spread spectrum (CD3S) communication systems is proposed. Here, the CD3S communication system transmitting different information symbols is considered as a combination of two subsystems which are driven by two different chaotic dynamic models, respectively. At every single time moment, the CD3S signal can be regarded as generated by the subsystem corresponding to the information symbol transmitted. Then, based on the multiple model form of CD3S signals, an interacting multiple model unscented Kalman filter with model switching detection mechanism is exploited to track the CD3S signals. The l(2)-norm of tracking errors is used to choose the model which best matches the intercepted signals. Thus, the information symbols are recovered indirectly. Compared with the existing methods, the proposed algorithm can: (1) reduce the influence of a low spreading factor; (2) calculate the spreading factor using the length of time intervals between model switching; and (3) be more effective under scenarios of low signal-to-noise ratio or multipath fading. Simulation results verify the superiority of the proposed method.

Spatial chaos of traveling waves has a given velocity

We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy. These arguments provide a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.

Cluster synchronization in community networks with nonidentical nodes

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

On the security of a spatiotemporal chaotic cryptosystem

Recently, Tang et al. [Phys Lett. A 318, 388 (2003)] proposed a spatiotemporal cryptosystem based on one-way coupled map lattices. They claim that it has high security, fast encryption speed, and short synchronization transient. This work proves that this scheme has some security weaknesses. Indeed, we propose an attack which the system cannot withstand.

Hash function based on chaotic map lattices

A new hash function system, based on coupled chaotic map dynamics, is suggested. By combining floating point computation of chaos and some simple algebraic operations, the system reaches very high bit confusion and diffusion rates, and this enables the system to have desired statistical properties and strong collision resistance. The chaos-based hash function has its advantages for high security and fast performance, and it serves as one of the most highly competitive candidates for practical applications of hash function for software realization and secure information communications in computer networks.

Selective image encryption using a spatiotemporal chaotic system

A universal selective image encryption algorithm, in which the spatiotemporal chaotic system is utilized, is proposed to encrypt gray-level images. In order to resolve the tradeoff between security and performance, the effectiveness of selective encryption is discussed based on simulation results. The scheme is then extended to encrypt RGB color images. Security analyses for both scenarios show that the proposed schemes achieve high security and efficiency.

Encryption and decryption of images with chaotic map lattices

We propose a secure algorithm for direct encryption and decryption of digital images with chaotic map lattices. The basic idea is to convert, pixel by pixel, the image color to chaotic logistic maps one-way coupled by initial conditions. After small numbers of iterations and cycles, the image becomes indistinguishable due to inherent properties of chaotic systems. Since the maps are coupled, the image can be completely recovered by the decryption algorithm if map parameters, number of iterations, number of cycles, and the image size are exactly known.

Chaotic synchronization through coupling strategies

Usually, complete synchronization (CS) is regarded as the form of synchronization proper of identical chaotic systems, while generalized synchronization (GS) extends CS in nonidentical systems. However, this generally accepted view ignores the role that the coupling plays in determining the type of synchronization. In this work, we show that by choosing appropriate coupling strategies, CS can be observed in coupled chaotic systems with parameter mismatch, and GS can also be achieved in coupled identical systems. Numerical examples are provided to demonstrate these findings. Moreover, experimental verification based on electronic circuits has been carried out to support the numerical results. Our work provides a method to obtain robust CS in synchronization-based chaos communications.

Public-key encryption based on generalized synchronization of coupled map lattices

Currently used public-key cryptosystems are based on difficulties in solving certain numeric theoretic problems, in which the way to predict the private key from the knowledge of the public key is computationally infeasible. Here we propose a method of constructing public-key cryptosystems by generalized synchronization of coupled map lattices, in which the difficulty in predicting the synchronous function is used as the trap-door function to deduce the private key from the public key. In specific, we implement this idea on the method of "Merkle's puzzles," and find that, incorporated with the chaotic dynamics, this traditional method is equipped with some new features and can be practical in certain situations.

Encryption with synchronized time-delayed systems

We propose a new communication scheme that uses time-delayed chaotic systems with delay time modulation. In this method, the transmitter encodes a message as an additional modulation of the delay time and then the receiver decodes the message by tracking the delay time. We demonstrate our communication scheme in a system of coupled logistic maps. Also we discuss the error of the transferred message due to an external noise and present its correction method.

Synchronized pseudorandom systems and their application to speech communication

An approach to the synchronization of pseudorandom systems is proposed and applied to secure speech communication. The encoding signal produced by the pseudorandom synchronization scheme passes the random test, and shows much more complex dynamics, better random properties, and greater sensitivity to parameter mismatches than that produced by the active-passive decomposition scheme. Also, two coupled pseudorandom systems can be exactly synchronized despite their different initial states or seeds. Pseudorandom encoding and synchronization may yield great security in communication.

Public-key encryption with chaos

We propose public-key encryption algorithms based on chaotic maps, which are generalization of well-known and commercially used algorithms: Rivest-Shamir-Adleman (RSA), ElGamal, and Rabin. For the case of generalized RSA algorithm we discuss in detail its software implementation and properties. We show that our algorithm is as secure as RSA algorithm.

A new spatiotemporally chaotic cryptosystem and its security and performance analyses

A one-way-coupled chaotic map lattice is proposed for cryptography of self-synchronizing stream cipher. The system performs basic floating-point analytical computation on real numbers, incorporating auxiliarily with few simple algebraic operations on integer numbers. Parallel encryption (decryption) operations of multiple chaotic sites are conducted. It is observed that the system has high practical security, fast encryption (decryption) speed with software realization, and excellent reliability against strong channel noise, and its overall cryptographic properties are considerably better than both known chaotic cryptosystems and currently used conventional cryptosystems, including the advanced encryption standard.

Propagation of desynchronous disturbances in synchronized chaotic one-way coupled map lattices

Propagations of desynchronous perturbations in synchronization processes of spatiotemporal chaos are investigated by considering chaotic one-way coupled map lattices. Under large coupling approximation the desynchronous area in time space is analytically calculated, based on the concepts of comoving Lyapunov exponent and absolute largest Lyapunov exponent. The ideas used in this paper are expected to be applicable to synchronizations of spatiotemporally chaotic systems of coupled maps and coupled oscillators with convective instability.

Error function attack of chaos synchronization based encryption schemes

Different chaos synchronization based encryption schemes are reviewed and compared from the practical point of view. As an efficient cryptanalysis tool for chaos encryption, a proposal based on the error function attack is presented systematically and used to evaluate system security. We define a quantitative measure (quality factor) of the effective applicability of a chaos encryption scheme, which takes into account the security, the encryption speed, and the robustness against channel noise. A comparison is made of several encryption schemes and it is found that a scheme based on one-way coupled chaotic map lattices performs outstandingly well, as judged from quality factor.

Complete synchronization and generalized synchronization of one-way coupled time-delay systems

The complete synchronization and generalized synchronization (GS) of one-way coupled time-delay systems are studied. We find that GS can be achieved by a single scalar signal, and its synchronization threshold for different delay times shows the parameter resonance effect, i.e., we can obtain stable synchronization at a smaller coupling if the delay time of the driven system is chosen such that it is in resonance with the driving system. Near chaos synchronization, the desynchronization dynamics displays periodic bursts with the period equal to the delay time of the driven system. These features can be easily applied to the recovery of time-delay systems.