A Class of Quadratic Polynomial Chaotic Maps and Their Fixed Points Analysis
TL;DR: It is proved that such class quadratic polynomial chaotic maps cannot have hidden chaotic attractors and can accurately control the amplitude of chaotic time series through the existence and stability analysis of fixed points.
Abstract: When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors.
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TL;DR: The experimental results show that the proposed encryption scheme has good encryption effect and image compression capability.
Abstract: In this paper, a new three-dimensional chaotic system is proposed for image encryption. The core of the encryption algorithm is the combination of chaotic system and compressed sensing, which can complete image encryption and compression at the same time. The Lyapunov exponent, bifurcation diagram and complexity of the new three-dimensional chaotic system are analyzed. The performance analysis shows that the chaotic system has two positive Lyapunov exponents and high complexity. In the encryption scheme, a new chaotic system is used as the measurement matrix for compressed sensing, and Arnold is used to scrambling the image further. The proposed method has better reconfiguration ability in the compressible range of the algorithm compared with other methods. The experimental results show that the proposed encryption scheme has good encryption effect and image compression capability.
44 citations
TL;DR: The security analyses and experimental results are confirmed that 2D Correlation Coefficient, Information Entropy, Number of Pixels Change Rate (NPCR), Unified Average Changing Intensity (UACI), Mean Absolute Error (MAE), and decryption quality are able to meet the encryption security demands.
Abstract: Chaotic maps that can provide highly secure key sequences and ease of structure implementation are predominant requirements in image encryption systems. One Dimensional (1D) chaotic maps have the advantage of a simple structure and can be easily implemented by software and hardware. However, key sequences produced by 1D chaotic maps are not adequately secure. Therefore, to improve the 1D chaotic maps sequence security, we propose two chaotic maps: 1D Improved Logistic Map (1D-ILM) and 1D Improved Quadratic Map (1D-IQM). The proposed maps have shown higher efficiency than existing maps in terms of Lyapunov exponent, complexity, wider chaotic range, and higher sensitivity. Additionally, we present an efficient and fast encryption method based on 1D-ILM and 1D-IQM to enhance image encryption system performance. This paper also introduces a key expansion method to reduce the number of chaotic map iteration needs, thereby decreasing encryption time. The security analyses and experimental results are confirmed that 2D Correlation Coefficient (CC) Information Entropy (IE), Number of Pixels Change Rate (NPCR), Unified Average Changing Intensity (UACI), Mean Absolute Error (MAE), and decryption quality are able to meet the encryption security demands (CC = −0.00139, IE = 7.9990, NPCR = 99.6114%, UACI = 33.46952% and MAE = 85.3473). Furthermore, the proposed keyspace reaches 10240, and the encryption time is 0.025s for an image with a size of 256 × 256. The proposed system can yield efficacious security results compared to obtained results from other encryption systems.
15 citations
TL;DR: Wang et al. as mentioned in this paper proposed a six-dimensional non-degenerate discrete hyperchaotic system with six positive Lyapunov exponents to construct the measurement matrix. But, the chaotic system has low complexity and generate sequences with poor randomness.
Abstract: Digital images can be large in size and contain sensitive information that needs protection. Compression using compressed sensing performs well, but the measurement matrix directly affects the signal compression and reconstruction performance. The good cryptographic characteristics of chaotic systems mean that using one to construct the measurement matrix has obvious advantages. However, existing low-dimensional chaotic systems have low complexity and generate sequences with poor randomness. Hence, a new six-dimensional non-degenerate discrete hyperchaotic system with six positive Lyapunov exponents is proposed in this paper. Using this chaotic system to design the measurement matrix can improve the performance of image compression and reconstruction. Because image encryption using compressed sensing cannot resist known- and chosen-plaintext attacks, the chaotic system proposed in this paper is introduced into the compressed sensing encryption framework. A scrambling algorithm and two-way diffusion algorithm for the plaintext are used to encrypt the measured value matrix. The security of the encryption system is further improved by generating the SHA-256 value of the original image to calculate the initial conditions of the chaotic map. A simulation and performance analysis shows that the proposed image compression-encryption scheme has high compression and reconstruction performance and the ability to resist known- and chosen-plaintext attacks.
12 citations
TL;DR: In this article , a summary of how chaotic maps can be used to generate pseudo-random numbers and perform multimedia encryption is given, where the lowest correlation coefficient was 0.00006 and the highest entropy was 7.999995 bits per byte using the quantum chaotic map.
Abstract: Because of the COVID-19 pandemic, most of the tasks have shifted to an online platform. Sectors such as e-commerce, sensitive multi-media transfer, online banking have skyrocketed. Because of this, there is an urgent need to develop highly secure algorithms which can not be hacked into by unauthorized users. The method which is the backbone for building encryption algorithms is the pseudo-random number generator based on chaotic maps. Chaotic maps are mathematical functions that generate a highly arbitrary pattern based on the initial seed value. This manuscript gives a summary of how the chaotic maps are used to generate pseudo-random numbers and perform multimedia encryption. After carefully analyzing all the recent literature, we found that the lowest correlation coefficient was 0.00006, which was achieved by Ikeda chaotic map. The highest entropy was 7.999995 bits per byte using the quantum chaotic map. The lowest execution time observed was 0.23 seconds with the Zaslavsky chaotic map and the highest data rate was 15.367 Mbits per second using a hyperchaotic map. Chaotic map-based pseudo-random number generation can be utilized in multi-media encryption, video-game animations, digital marketing, chaotic system simulation, chaotic missile systems, and other applications.
10 citations
TL;DR: In the last few years, entropy has been a fundamental and essential concept in information theory and has become an important concept in computer science.
Abstract: In the last few years, entropy has been a fundamental and essential concept in information theory [...].
10 citations
References
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TL;DR: This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
Abstract: First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.
6,118 citations
"A Class of Quadratic Polynomial Cha..." refers background in this paper
...For example, classical quadratic logistic chaotic maps [18] are defined as...
[...]
TL;DR: In this article, a generalized logistic equation was used to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if this distribution is helpful in predicting uneven wear of the bit.
Abstract: The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the simplest mathematical situations occurs when the phenomenon can be described by a single number as, for example, when the number of children susceptible to some disease at the beginning of a school year can be estimated purely as a function of the number for the previous year. That is, when the number x n+1, at the beginning of the n + 1st year (or time period) can be written
$${x_{n + 1}} = F({x_n}),$$
(1.1)
where F maps an interval J into itself. Of course such a model for the year by year progress of the disease would be very simplistic and would contain only a shadow of the more complicated phenomena. For other phenomena this model might be more accurate. This equation has been used successfully to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if [8, 11] knowing this distribution is helpful in predicting uneven wear of the bit. For another example, if a population of insects has discrete generations, the size of the n + 1st generation will be a function of the nth. A reasonable model would then be a generalized logistic equation
$${x_{n + 1}} = r{x_n}[1 - {x_n}/K].$$
(1.2)
3,278 citations
"A Class of Quadratic Polynomial Cha..." refers background in this paper
...The period-three points play a vital role in the chaotic map because Li and Yorke have shown that period-three implies chaos, and proposed the famous period-three theorem [16]....
[...]
TL;DR: In this paper, it was shown that for any continuous mapping of the real line into itself, the existence of a cycle of order n2 follows from the existence (1 − n 2 ) of cycle n 1.
Abstract: The basic result of this investigation may be formulated as follows. Consider the set of natural numbers in which the following relationship is introduced: n1 precedes n2 (n1 ≼ n2) if for any continuous mapping of the real line into itself the existence of a cycle of order n2 follows from the existence of a cycle of order n1. The following theorem holds.
567 citations
"A Class of Quadratic Polynomial Cha..." refers background in this paper
...In the first 11 years, Sharkovskii also proposed the Sharkovskii theorem on period-three points [17]....
[...]
TL;DR: The detailed dynamical behaviors of this hyperchaotic system are further investigated, including Lyapunov exponents spectrum, bifurcation, and Poincare mapping.
Abstract: This paper constructs a new hyperchaotic system based on Lu system by using a state feedback controller. The detailed dynamical behaviors of this hyperchaotic system are further investigated, including Lyapunov exponents spectrum, bifurcation, and Poincare mapping. Moreover, a novel circuit diagram is designed for verifying the hyperchaotic behaviors and some experimental observations are also given.
402 citations
TL;DR: In this article, a four-dimensional hyperchaotic Lorenz system was obtained by adding a nonlinear controller to the Lorenz chaotic system, which is studied by bifurcation diagram, Lyapunov exponents spectrum and phase diagram.
Abstract: This paper presents a four-dimension hyperchaotic Lorenz system, obtained by adding a nonlinear controller to Lorenz chaotic system. The hyperchaotic Lorenz system is studied by bifurcation diagram, Lyapunov exponents spectrum and phase diagram. Numerical simulations show that the new system’s behavior can be convergent, divergent, periodic, chaotic and hyperchaotic when the parameter varies.
219 citations