One-dimensional Chaotic Maps Research Articles

A non-trivial relation between some many-dimensional chaotic discrete dynamical systems and some one-dimensional chaotic discrete dynamical systems

Chaotic maps are very useful in practical applications. In this paper, we present a method for constructing the many-dimensional chaotic discrete dynamical systems using semiconjugacy property. The chaotic property in one dimension may be influenced the chaotic property in higher-dimensions. In fact, using the one-dimensional chaotic maps and semiconjugacy property, we construct some many-dimensional chaotic discrete dynamical systems. These systems may be used as random number generators in Monte Carlo simulations. Also, these systems may be used in practical applications such as chaotic cryptography and evolutionary algorithms.

The Schröder functional equation and its relation to the invariant measures of chaotic maps

The aim of this paper is to show that the invariant measure for a class of one-dimensional chaotic maps, T(x), is an extended solution of the Schröder functional equation, q(T(x)) = λq(x), induced by them. Hence, we give a unified treatment of a collection of exactly solved examples worked out in the current literature. In particular, we show that these examples belong to a class of functions introduced by Mira (see the text). Moreover, as a new example, we compute the invariant densities for a class of rational maps having the Weierstrass ℘ function as an invariant one. Also, we study the relation between that equation and the well-known Frobenius–Perron and Koopman's operators.

DESIGN OF CHAOTIC PSEUDO-RANDOM BIT GENERATOR AND ITS APPLICATIONS IN STREAM-CIPHER CRYPTOGRAPHY

Because of the sensitivity of chaotic systems on initial conditions/control parameters, when chaotic systems are realized in a discrete space with finite states, the dynamical properties will be far different from the ones described by the continuous chaos theory, and some degradation will arise. This problem will cause the chaotic trajectory eventually periodic. In order to solve the problem, a new binary stream-cipher algorithm based on one-dimensional piecewise linear chaotic map is proposed in this paper. In the process of encryption and decryption, we employ a secret variant to perturb the chaotic trajectory and the control parameter to lengthen the cycle-length of chaotic trajectory. In addition, we design a nonlinear principle to generate a pseudo-random chaotic bit sequence as key stream. Cryptanalysis shows that the cryptosystem is of high security.

Image encryption with compound chaotic sequence cipher shifting dynamically

We design a new two-dimensional chaotic function using two one-dimensional chaotic functions, and then prove the chaotic properties to a new function based on a strict Devaney definition. And we propose a new encrypting image scheme using the new compound chaotic function by choosing one of the two one-dimensional chaotic functions randomly. We give statistical analysis, sequence random analysis, and sensitivity analysis to plaintext and key on the proposed scheme. The experimental results show that the new scheme has a very fast encryption speed and the key space is expanded and it can resist all kinds of cryptanalytic, statistical and brute-force attacks, and especially, our new method can be also used to solve the problem that is easily exposed to chosen plaintext attack and low digitization of one-dimensional chaotic function.

Design and implementation of improved watermarking system in WT domain

This article presents a watermarking algorithm based on one-dimensional (1-D) chaotic map in wavelet transform (WT) domain. Traditional encryption and watermarking technologies in the WT domain are combined in this algorithm. First, the watermark is encoded by a chaotic sequence, which is generated by a key. On the basis of human visual system (HVS), it is then divided into two parts and embedded into the low- and intermediate-frequency bands of three-layer WT domain. The encryption of the watermark and the choice of embedded areas based on the perceptivity not only confer the watermark with good invisibility but also with good robustness. Meanwhile, it is difficult to decrypt the watermark extracted from the WT domain.

FAULT TOLERANCE AND DETECTION IN CHAOTIC COMPUTERS

We introduce a structural testing method for a dynamics based computing device. Our scheme detects different physical defects, manifesting themselves as parameter variations in the chaotic system at the core of the logic blocks. Since this testing method exploits the dynamical properties of chaotic systems to detect damaged logic blocks, the damaged elements can be detected by very few testing inputs, leading to very low testing time. Further the method does not entail dedicated or extra hardware for testing. Specifically, we demonstrate the method on one-dimensional unimodal chaotic maps. Some ideas for testing higher dimensional maps and flows are also presented.

A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps

In recent years, a growing number of discrete chaotic cryptographic algorithms have been proposed. However, most of them encounter some problems such as the lack of robustness and security. In this Letter, we introduce a new image encryption algorithm based on one-dimensional piecewise nonlinear chaotic maps. The system is a measurable dynamical system with an interesting property of being either ergodic or having stable period-one fixed point. They bifurcate from a stable single periodic state to chaotic one and vice versa without having usual period-doubling or period-n-tupling scenario. Also, we present the KS-entropy of this maps with respect to control parameter. This algorithm tries to improve the problem of failure of encryption such as small key space, encryption speed and level of security.

Adaptation to the edge of chaos in one-dimensional chaotic maps

We present a method that enables chaotic systems to change its dynamics to stable periodic dynamics by a feedback adjustment. The proposed method uses feedback of a largest value obtained from observations of a fixed interval of time series of the system variable and therefore does not require any a priori detailed information. We apply this method to several chaotic systems and confirm numerically that chaotic states are stabilized to stable periodic ones. Since the stabilized states in the system are formed around a boundary between regular states and chaotic ones, the method provides a kind of adaptation to the edge of chaos.

A novel symmetrical cryptosystem based on discretized two-dimensional chaotic map

A prevalent situation in existing digitized chaotic cryptosystems is that multi-dimensional chaotic maps are usually employed for image encryption as the elements of an image are managed in two-dimensional way. On the other hand, one-dimensional chaotic maps are widely adopted in document encryption. In this Letter, a novel cryptosystem employing two-dimensional chaotic maps for document encryption is proposed. Several widely used two-dimensional chaotic maps are considered and their performance is investigated. Both theoretical analysis and numerical simulation verify their feasibility and superiority for practical applications.

Cryptanalysis of a chaotic block cipher with external key and its improved version

Recently, Pareek et al. proposed a symmetric key block cipher using multiple one-dimensional chaotic maps. This paper reports some new findings on the security problems of this kind of chaotic cipher: (1) a number of weak keys exist; (2) some important intermediate data of the cipher are not sufficiently random; (3) the whole secret key can be broken by a known-plaintext attack with only 120 consecutive known plain-bytes in one known plaintext. In addition, it is pointed out that an improved version of the chaotic cipher proposed by Wei et al. still suffers from all the same security defects.

A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.

Cryptography with chaotic mixing

We propose a cryptosystem based on one-dimensional chaotic maps of the form H p ( x ) = r p - 1 ∘ G ∘ r p ( x ) defined in the interval [0, 10 p ) for a positive integer parameter p, where G ( x ) = 10 x ( mod 10 ) and r p ( x ) = x p , which is a topological conjugacy between G and the shift map σ on the space Σ of the sequences with 10 symbols. There are three advantages in comparison with the recently proposed cryptosystem based on chaotic logistic maps F μ ( x ) = μ x ( 1 - x ) with 3 < μ ⩽ 4: (a) H p is always chaotic for all parameters p, (b) the knowledge of an ergodic measure allows assignments of the alphabetic symbols to equiprobable sites of H p ’s domain and (c) for each p, the security of the cryptosystem is manageable against brute force attacks.

A novel algorithm for image encryption based on mixture of chaotic maps

Chaos-based encryption appeared recently in the early 1990s as an original application of nonlinear dynamics in the chaotic regime. In this paper, an implementation of digital image encryption scheme based on the mixture of chaotic systems is reported. The chaotic cryptography technique used in this paper is a symmetric key cryptography. In this algorithm, a typical coupled map was mixed with a one-dimensional chaotic map and used for high degree security image encryption while its speed is acceptable. The proposed algorithm is described in detail, along with its security analysis and implementation. The experimental results based on mixture of chaotic maps approves the effectiveness of the proposed method and the implementation of the algorithm. This mixture application of chaotic maps shows advantages of large key space and high-level security. The ciphertext generated by this method is the same size as the plaintext and is suitable for practical use in the secure transmission of confidential information over the Internet.

Low-hardware complexity PRBGs based on a piecewise-linear chaotic map

In this brief, a family of discretized one-dimensional chaotic maps derived from the Sawtooth map is analyzed to evaluate its suitability for the integrated implementation of low-complexity digital pseudorandom bit generators (PRBGs). The proposed PRBGs, classifiable as nonlinear congruential generators, are investigated in terms of period length, statistical properties of the generated sequences, hardware complexity, and are compared with traditional PRBGs.

A feedback strategy to improve the entropy of a chaos-based random bit generator

In this paper, the guidelines to design a true random bit generator (TRBG) circuit with a predefined minimum entropy are discussed. The approach is proposed for a TRBG based on a one-dimensional piecewise-linear chaotic map; it does not require bit throughput reduction, and it is suitable for the development of integrated TRBG circuits. In particular, the proposed design strategy is based on a feedback control procedure that allows to dynamically change the system parameters for the correction of the circuit nonidealities (e.g., the circuit offsets). The correction algorithm does not require a direct measurement of the system nonidealities or of the effective value of the map parameters, but only a dynamic estimation of these quantities based on the observation of the TRBG output. The design approach is validated by a hardware prototype implemented on a field-programmable analog array. The results of the NIST FIPS 140-2 test suite, the DIEHARD test suite, and the Coron's Universal test, applied to the TRBG output sequences before and after a simple post processing without throughput reduction, are reported and discussed.

Symbolic Dynamics of Coupled Map Lattices

We present a method to reduce the [FORMULA: SEE TEXT] dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences. An example utilizing coupled quadratic maps is given.

Evaluation of channel coding and decoding algorithms using discrete chaotic maps

In this paper we address the design of channel encoding algorithms using one-dimensional nonlinear chaotic maps starting from the desired invariant probability density function (pdf) of the data sent to the channel. We show that, with some simple changes, it is straightforward to make use of a known encoding framework based upon the Bernoulli shift map and adapt it readily to carry the information bit sequence produced by a binary source in a practical way. On the decoder side, we introduce four already known decoding algorithms and compare the resulting performance of the corresponding transmitters. The performance in terms of the bit error rate shows that the most important design clue is related not only to the pdf of the data produced by the chosen discrete map: the own dynamics of the maps is also of the highest importance and has to be taken into account when designing the whole transmitting and receiving system. We also show that a good performance in such systems needs the extensive use of all the evidence stored in the whole chaotic sequence.

Lyapunov spectrum of a lattice of chaotic systems with local and non-local couplings

We consider a one-dimensional chaotic piecewise linear map lattice with periodic boundary conditions and two types of interactions: (i) local couplings between nearest and next-to-the-nearest neighbors; and (ii) non-local couplings randomly chosen along the lattice according to a specified probability. The chaoticity of the lattice is described by means of its Lyapunov spectrum, which furnishes also information about the system global attractor in a high-dimensional phase space. We study in particular the dependence of this spectrum with the coupling parameters, as well as make comparisons with limiting cases, for which the Lyapunov spectrum is known. 2005 Elsevier Ltd. All rights reserved.

Statistical Properties of Modulo-p Added p-Ary Sequences

Some statistical analyses of modulo-2 added binary sequences are generalized to modulo-p added p-ary sequences. First, we theoretically evaluate statistics of sequences obtained by modulo-p addition of two general p-ary random variables. Next, we consider statistics of modulo-p added chaotic p-ary sequences generated by a class of one-dimensional chaotic maps.

Design of binary sequences with tunable exponential autocorrelations and run statistics based on one-dimensional chaotic maps

This paper describes simple design methods of chaotic binary sequences with prescribed exponential autocorrelation properties and run statistics. We employ one-dimensional piecewise monotonic onto maps and a simple threshold function for generating such sequences. Some examples of such designs are also given. Furthermore, bounds on run statistics are discussed and compared to the result for sequences of general binary random variables.