Discrete Chaotic Maps Research Articles

Design and analysis of discrete fractional-order chaotic map with offset-boosting behavior

Fractional calculus, as a more accurate tool for depicting the dynamics of complex systems, has been introduced into discrete chaotic maps. To further describe the offset-boosting behavior in discrete systems, a discrete fractional-order chaotic map (DFOCM) based on the Caputo difference operator is constructed. The mapping order of this fractional-order model controls the stability of the fixed point, thereby affecting the dynamic behavior of the map. The dynamics of DFOCM is studied using numerical simulation methods such as bifurcation diagrams and maximum Lyapunov exponents, revealing the presence of multistability. By comparing with integer-order map, it is found that DFOCM exhibit a larger chaotic region. Based on this, the difference between fractional order and integer order offset-boosting behavior is theoretically derived. Specifically, the offset-boosting behavior of fractional-order maps concerning mapping parameters is related to the initial state, which was further demonstrated through numerical simulations. SE complexity proves that the chaotic sequences generated by DFOCM have high unpredictability and pseudo-randomness. Finally, the proposed DFOCM is implemented on the DSP hardware platform, and the physical feasibility of numerical simulation is verified.

Design of a discrete memristive chaotic map: fractional-order memory, dynamics and application

In this paper, a discrete fracmemristor (DFM) model is derived based on the Caputo difference, and a new fractional-order chaotic map is designed. Dynamics of the proposed map is investigated in detail by means of Lyapunov exponent spectra, bifurcation diagrams, PE complexity and multistability analyses. Compared with the coupled discrete integer-order memristor (DIM), the map coupled with the DFM products richer dynamics, including larger attractor distribution, fewer numerically periodic windows, and higher complexity. Besides, the order becomes additional bifurcation parameter. Finally, the proposed map is implemented on Field-Programmable Gate Array (FPGA) platform, and applied in a pseudorandom number generator (PRNG), which further demonstrates its application value.

A new image encryption algorithm with feedback key mechanism using two-dimensional dual discrete quadratic chaotic map

A new image encryption algorithm with feedback key mechanism using two-dimensional dual discrete quadratic chaotic map

A visual security multi-key selection image encryption algorithm based on a new four-dimensional chaos and compressed sensing

In this article, a visual security image encryption algorithm based on compressed sensing is proposed. The algorithm consists of two stages: the compression and encryption stage and the embedding stage. The key streams in the compression and encryption stage are generated by a newly constructed four-dimensional discrete chaotic map, while the Gaussian measurement matrix is generated by a Chebyshev map, and both of their generations are related to the feature code of the carrier image, which enhances the security of the ciphertext. In the compression and encryption stage, a scrambling-cyclic shift-diffusion encryption structure is adopted for the compressed image in which the shift number in the cyclic shift stage and the diffusion key streams are dynamically changed according to each pixel value, so the algorithm can resist chosen plaintext attack. In the embedding stage, the carrier image is first subjected to integer wavelet transform to obtain the high-frequency and low-frequency components of the image, and then the intermediate ciphertext information is embedded into its high-frequency components. Finally, the carrier image is subjected to inverse integer wavelet transform to obtain a visually secure ciphertext image. The experimental results and security analysis indicate that the encryption scheme has a large key space, high decryption key sensitivity, similar histogram distribution between the carrier image and the visual security ciphertext image, and good robustness to noise attacks.

Fractal Tent Map with Application to Surrogate Testing

Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks.

Sliced chaotic encrypted transmission scheme based on key masked distribution in a W-band millimeter-wave system

In order to guarantee the information of the W-band wireless communication system from the physical layer, this paper proposes the sliced chaotic encrypted (SCE) transmission scheme based on key masked distribution (KMD). The scheme improves the security of free space communication in the W-band millimeter-wave wireless data transmission system. In this scheme, the key information is embedded into the random position of the ciphertext information, and then the ciphertext carrying the key information is encrypted by multi-dimensional chaos. Chaotic system 1 constructs a three-dimensional discrete chaotic map for implementing KMD. Chaotic system 2 constructs complex nonlinear dynamic behavior through the coupling of two neurons, and the masking factor generated is used to realize SCE. In this paper, the transmission of 16QAM signals in a 4.5 m W-band millimeter-wave wireless communication system with a rate of 40 Gb/s is proved by experiments, and the performance of the system is analyzed. When the input optical power is 5 dBm, the bit error rate (BER) of the legitimate encrypted receiver is 1.23 × 10−3. When the offset of chaotic sequence x and chaotic sequence y is 100, their BERs are more than 0.21. The key space of the chaotic system reaches 10192, which can effectively prevent illegal attacks and improve the security performance of the system. The experimental results show that the scheme can effectively distribute the keys and improve the security of the system. It has great application potential in the future of W-band millimeter-wave wireless secure communication.

Fractional-order quantum game chaotic map and its synchronization with application

In this paper, according to the knowledge of discrete fractional calculus and quantum game chaotic map, we first get the fractional quantum game chaotic map, which describes the dynamics of a quantum duopoly game. Then, the phase space, maximum Lyapunov exponent, and bifurcation diagram of the proposed map are obtained for the purpose of observing dynamic behaviors compared with the original quantum game chaotic map. After that, the synchronous controller is designed and the chaos synchronization of the fractional quantum game chaotic map is observed. Finally, we develop the fractional quantum game chaotic map into the chaotic sequence generator and image encryption scheme.

An efficient image encryption algorithm using a discrete memory-based logistic map with deep neural network

In the last few years, multimedia technology has made tremendous strides. These days, the Web is frequently used to transfer multimedia content, including audio, video, and photos. However, the Internet is a very vulnerable medium with many security holes. To ensure that multimedia content carried across unprotected channels, like the Internet, is secure and private, several encryption techniques have been proposed. New encryption strategies must be developed because multimedia data streams cannot be encrypted using traditional methods. Therefore, the main goal of the recommended system is to present an analytical research approach for introducing a sophisticated framework wherein the suggested encryption technologies' efficacy is increased through the use of deep neural networks (DNNs). The robustness of the DNN principle is coupled with a discrete memristor-based logistic chaotic map notion for enhanced security performance. In this paper, three distinct encryption algorithms—Arnie cat with an artificial neural network (ANN), Henon map with an ANN, and logistic map with a DNN—are compared for security and performance with the suggested algorithm. Correlation coefficients, information entropy, number of pixels changing rate (NPCR), encryption quality, and encryption duration are the cryptographic analysis parameters examined here. The results show that the recommended implementation enhances security performance without degrading image quality. The proposed algorithm achieves 35.9% of UACI, 99.95% of NPCR, and 7.997231 of entropy.

Stability analysis of a discrete chaotic map in superior orbit

Stability analysis of a discrete chaotic map in superior orbit

Dynamical analysis and implementation of novel discrete memristive chaotic maps with hidden attractors

Dynamical analysis and implementation of novel discrete memristive chaotic maps with hidden attractors

The New Four-Dimensional Fractional Chaotic Map with Constant and Variable-Order: Chaos, Control and Synchronization

Using fractional difference equations to describe fractional and variable-order maps, this manuscript discusses the dynamics of the discrete 4D sinusoidal feedback sine iterative chaotic map with infinite collapse (ICMIC) modulation map (SF-SIMM) with fractional-order. Also, it presents a novel variable-order version of SF-SIMM and discusses their chaotic dynamic behavior by employing a distinct function for the variable fractional-order. To establish the existence of chaos in the suggested discrete SF-SIMM, some numerical methods such as phase plots, bifurcation and largest Lyapunov exponent diagrams, C0 complexity and 0–1 test are utilized. After that, two different control schemes are used for the conceived discrete system. The states are stabilized and asymptotically forced towards zero by the first controller. The second controller is used to synchronize a pair of maps with non–identical parameters. Finally, MATLAB simulations will be executed to confirm the results provided.

A Double Embedding of an Anti-Removable Visible Logo Using Discrete Wavelet Transform and Chaotic Map

A Double Embedding of an Anti-Removable Visible Logo Using Discrete Wavelet Transform and Chaotic Map

An improved key expansion algorithm based on a 2D discrete hyper chaotic map with memristor

An improved key expansion algorithm based on a 2D discrete hyper chaotic map with memristor

WAM 3D Discrete Chaotic Map for Secure Communication Applications

Chaotic systems have become widely adopted as an effective way for secure data communications, because of its simple mathematical complexity and good security. The relationship between encryption algorithms and chaos systems has gained a lot of attention in the past few years, since it avoids the data spreading as well as lower the transmission delay and costs. In this paper a novel 3D discrete chaotic map is proposed for data encryption and secure communication and named as WAM. For secure communication, the Pecora and Carroll (P-C) method was utilized to achieve synchronization between the master system and the slave system. The simulation results of WAM 3D discrete chaotic map showed that the system has a chaotic behavior and a characteristic randomness and can pass 0-1, Lyapunov exponent (LE) and NIST tests which are usually used to check chaotic behavior. The statistical outcomes of the LE test were 0.0193, the frequency test (FT) was 0.4237, and the run test (RT) yielded a value of 0.0607. As a result, it enrich the theoretical basis of the equations and implementation of chaos, and it is superior for encryption algorithms and communication security applications.

A time-efficient and noise-resistant cryptosystem based on discrete wavelet transform and chaos theory: An application in image encryption

Due to the exponential rise of multimedia technologies, digital information privacy is now more crucial than ever. The process of transmitting data across a channel such as the internet has become a challenging task. In recent years, a variety of techniques for encrypting digital data have been proposed. However, due to shortcomings such as a high computational cost and weak security, it is important to build an encryption scheme that can provide a suitable level of security while also being time-effective. This research introduces a Discrete Wavelet Transform (DWT), chaotic map, and substitution box-based encryption scheme that is both time efficient and provides high levels of security for digital images. DWT is utilized to reduce the computation complexity of the encryption scheme proposed in this research. This is done by transforming the input image into its four frequency sub-bands (ΓΓ, Γħ, ħΓ, and ħħ) and selecting only the low-frequency sub-band (ΓΓ) for encryption, while the substitution boxes (S-boxes) and a chaotic map are used to increase the level of security. Using chaotic maps, random sequences are also generated to introduce confusion and diffusion. The final encrypted image is created by combining all of the non-modified and encrypted frequency bands. Finally, the results show that proposed encryption system nears the optimum values with a key-space of 2265 to avoid a brute-force attack.

A visually meaningful image encryption algorithm based on P-tensor product compressive sensing and newly-designed 2D memristive chaotic map

The noise-like visual feature of cipher images produced by using the traditional image encryption technology explicitly reflects the presence of secret information. To overcome this issue, a visually meaningful image encryption algorithm is proposed based on a newly designed 2D memristive chaotic map, P-tensor product compressive sensing (PTP-CS) and discrete Hartley transform (DHT). For concreteness, a new two-dimensional discrete memristive chaotic map is first designed to provide highly unpredictable secret code streams while maintaining low time consumption. Second, the threshold processing and zigzag confusion operations are performed on the discrete wavelet coefficients of the plain image to meet the prerequisites for effective compression. Third, the intermediate secret information is obtained by utilizing PTP-CS in the compression layer. Information entropy and edge entropy are employed to adaptively identify the complex regions that are suitable for embedding due to inconspicuous visual degradation in the carrier image. Finally, the embedding of the secret information in the DHT domain of these regions is accomplished. Security test and performance analysis confirm that our algorithm has the advantage of a high balance between the encryption security and the decryption recovery, and exhibits excellent performance in important indicators such as visual quality, robustness and timeliness.

A Class of Discrete Memristor Chaotic Maps Based on the Internal Perturbation

Further exploration into the influence of a memristor on the behavior of chaotic systems deserves attention. When constructing memristor chaotic systems, it is commonly believed that increasing the number of memristors will lead to better system performance. This paper proposes a class of chaotic maps with different discrete memristors, achieved through internal perturbation based on the Sine map. The I-V curve of the discrete memristor has a symmetrical structure. The dynamic characteristics of the designed system are analyzed using the chaotic attractor phase diagram, Lyapunov exponent (LE) spectrum, and bifurcation diagram. Numerical simulations demonstrate that internal perturbations of discrete memristors enhance the Sine map’s chaotic characteristics, expand the chaos range, and improve the ergodicity and LE value. Moreover, the type of discrete memristors has a significant impact on the dynamic characteristics of the system, while the number of discrete memristors has little influence. Therefore, in this paper, a direction for the design of a discrete memristor chaotic system is provided. Finally, a discrete memristor chaotic map with a simple structure and better performance is selected. Based on this, a pseudo-random sequence generator is designed, and the generated sequence passes the National Institute of Standards and Technology (NIST) test.

Fast adaptive synchronization of discrete quantum chaotic maps

By using linear entropy, a discrete quantum logistic chaotic map is introduced and the dynamical evolution of the quantum radiation field interacting with the matter is studied by changing the independent parameters. Its typical effect is the chaotic property of the quantum optics and a quantum classical correspondence is obtained, and the chaos and the regular structure of phase space can be better revealed by using the coherence dynamics of phase space. In addition, an approach for fast adaptive synchronization of the discrete quantum maps is proposed. Results validate the effectiveness of the proposed scheme.

A class of novel discrete memristive chaotic map

Since Chua introduced the memristor as the fourth basic circuit component, memristors and memristive chaotic maps have received great attention. In this paper, we present a class of novel memristive chaotic map by coupling the discrete memristor with the internal item in an original boundary function. Several examples are presented based on classical one dimensional (1D) and high dimensional (HD) chaotic maps, and numerical simulation results display their improved dynamical performance and chaos complexity. We analyze the influences of introducing the memristor on the chaotic map and the generality for designing discrete chaotic maps of the proposed memristive model. Particularly, with the newly emerging features of no fixed points and multistability, we can robustly control the output sequence by setting their initial-states, which accommodates to many arisen applications of chaotic systems. Furthermore, based on the proposed memristive chaotic map, we construct the hardware platform to acquire its attractors, and design the Pseudorandom Number Generator (PRNG) to explore its application prospects.

A Novel Fractional Sine Chaotic Map and Its Application to Image Encryption and Watermarking

This article addresses the telecommunications industry’s priority of ensuring information security during the transition to next-generation networks. It proposes an image encryption system that combines watermarking techniques and a discrete fractional sine chaotic map. The authors also incorporate the principles of blockchain to enhance the security of transmitted and received image data. The proposed system utilizes a newly developed sine chaotic map with a fractional difference operator, exhibiting long-term chaotic dynamics. The complexity of this map is demonstrated by comparing it with three other fractional chaotic maps from existing literature, using bifurcation diagrams and the largest Lyapunov exponent. The authors also show the map’s sensitivity to changes in initial conditions through time-series diagrams. To encrypt images, the authors suggest a method involving watermarking of two secret images and encryption based on blockchain technology. The cover image is watermarked with the two hidden images using discrete wavelet transformations. Then, the image pixels undergo diffusion using a chaotic matrix generated from the discrete fractional sine chaotic map. This encryption process aims to protect the image data and make it resistant to unauthorized access. To evaluate the algorithm, the authors perform statistical analysis and critical sensitivity analysis to examine its characteristics. They also analyse different attacks to assess the algorithm’s ability to resist such threats and maintain image quality after decryption. The results demonstrate that the proposed algorithm effectively defends against attacks and ensures image security.