Cat Map Research Articles

Semiclassical Measures for Higher-Dimensional Quantum Cat Maps

Consider a quantum cat map M associated with a matrix A∈Sp(2n,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in {{\\,\ extrm{Sp}\\,}}(2n,{\\mathbb {Z}})$$\\end{document}, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of A of largest absolute value and (2) the characteristic polynomial of A is irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with n=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=1$$\\end{document}, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.

Iterative approximation of common fixed points of two nonself asymptotically nonexpansive mappings in CAT(0) spaces with numerical examples

In this manuscript, we investigate and approximate common fixed points of two nonself asymptotically nonexpansive mappings in the setting of CAT(0) spaces. We provide three examples and conduct numerical experiments to show the implementation of the approximation schemes. Our results extend and improve the related results in the literature.

An 8-bit integer true periodic orbit PRNG based on delayed Arnold’s cat map

Designing a pseudo-random number generator (PRNG) exhibiting true periodic orbits with large periods is promising for embedded applications. Implementing chaotic behaviors on a finite precision digital platform using conventional unsigned integer or floating-point arithmetic yields short period limit-cycles. It leads the output to diverge from the true chaotic orbits. This paper presents an unsigned integer encoded PRNG based on the delayed quantized Arnold’s Cat Map (QACM) that exhibits true periodic orbits with large periods. The proposed PRNG is obtained by delaying one of the 2-D QACM outputs with a linear feedback shift register (LFSR) thereby increasing its period. By investigating its period, we show that the periodicity increases with an increase in the delay and bit-precision. We also present the proposed PRNG unit’s hardware architecture and its synthesis results targeting on a Xilinx Zynq 7000 Field Programmable Gate Array (FPGA) with 8-bit precision and delay δ=58. Further, we synthesize the design using 90 nm CMOS technology for ASIC realization. The synthesis results demonstrate the throughput as 2.208 Gbps and 5.46 Gbps at 138 MHz and 342 MHz operating frequency on FPGA and ASIC respectively. We experimentally evaluate the performance of random numbers using the NIST 800-22-1 tests along with the correlation and key sensitivity tests.

Non-blind Image Watermarking Algorithm based on Non-Separable Haar Wavelet Transform against Image Processing and Geometric Attacks

The protection of digital media become crucial and needs to be developed as technology continues to become more advanced. This is to cater to the problems in copyright protection and data security. This paper presents a new non-blind image watermarking algorithm applying modified non-separable Haar wavelet transform (NSHWT), singular value decomposition (SVD), Arnold’s cat map and Rabin-p cryptosystem. The traditional transform domain watermarking (DWT) is resource-consuming, especially when performed on large image data. In order to improve on this issue, the proposed algorithm applied the modified NSHWT, a much more efficient alternative to the DWT. Another concern of digital watermarking algorithms is it is often publicly known, so encryptions are important to keep the image secure. The embedded watermark image is protected by scrambling the image using Arnold’s cat map, and the scrambling parameters are encrypted with the Rabin-p cryptosystem to keep the algorithm secure. In general, the application of modified NSHWT and SVD makes the watermark highly robust against image processing and geometric processing attacks, and its security is ensured with the protection by Arnold’s cat map and Rabin-p cryptosystem. There are already many algorithms with high imperceptibility and robustness, but in finding the perfect balance, authors often forsake other metrics such as efficiency, security, and flexibility. These are also essential when the algorithm is considered in real applications. Thus, the proposed algorithm is evaluated in terms of imperceptibility, robustness, and efficiency. The algorithm achieved imperceptibility results of 51.5157 average PSNR and 0.9991 average SSIM, and robustness results from 0.9710 to 0.9966 NC values. The algorithm is also around 40% faster to execute with modified NSHWT compared to the traditional DWT. Finally, it has a high embedding capacity of 6 bits per pixel. Overall, results show that the algorithm is highly imperceptible and robust against image processing and geometric attacks, while additionally being secure and flexible.

Using chaotic dynamics to characterize the complexity of rough surfaces.

In this work, we use the basic ingredients of chaotic dynamics (stretching and folding of phase space points) for the characterization of the complexity of microscopy images of rough surfaces. The key idea is to use an image as the initial condition of a chaotic discrete dynamical system, such as the Arnold cat map, and track its transformations during the first iterations of the map. Since the basic effects of the Arnold map are the stretching and folding of image texture, the application of the map leads to an enhancement of the high frequency content of images along with an increase of discontinuities in pixel intensities. We exploit these effects to quantify the complexity of S type (lying between homogeneity and randomness) of the image texture since the first (enhancement of high frequencies) can be used to quantify the distance of texture from randomness and noise and the second (the proliferation of discontinuities) the distance from order and homogeneity. The method is validated in synthetic images which are generated from computer generated surfaces with controlled correlation length and fractal dimension and it is applied in real images of nanostructured surfaces obtained from a scanning electron microscope with very interesting results.

Best proximity pairs of asymptotically relatively nonexpansive mappings in CAT(0) spaces

Best proximity pairs of asymptotically relatively nonexpansive mappings in CAT(0) spaces

Bl-IEA: A Bit-Level Image Encryption Algorithm for Cognitive Services in Intelligent Transportation Systems

In Intelligent Transportation Systems, images are the main data sources to be analyzed for providing intelligent and precision cognitive services. Therefore, how to protect the privacy of sensitive images in the process of information transmission has become an important research issue, especially in future no non-private data era. In this article, we design the Rearrangement-Arnold Cat Map (R-ACM) to disturb the relationship between adjacent pixels and further propose an efficient Bit-level Image Encryption Algorithm ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{B}{l}$ </tex-math></inline-formula> -IEA) based on R-ACM. Experiments show that the correlation coefficients of two adjacent pixels are 0.0022 in the horizontal direction, -0.0105 in the vertical direction, and -0.0035 in the diagonal direction respectively, which are obviously weaker than that of the original image with high correlations of adjacent pixels. What’s more, the NPCR is 0.996120172, and the UACI is 0.334613406, which indicate that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{B}{l}$ </tex-math></inline-formula> -IEA has stronger ability to resist different attacks compared with other solutions. Especially, the lower time complexity and only one round permutation make it particularly suitable to be used in the time-limited intelligent transportation field.

Securing 3D Point and Mesh Fog Data Using Novel Chaotic Cat Map

With the rapid evolution of Internet technology, fog computing has taken a major role in managing large amounts of data. The major concerns in this domain are security and privacy. Therefore, attaining a reliable level of confidentiality in the fog computing environment is a pivotal task. Among different types of data stored in the fog, the 3D point and mesh fog data are increasingly popular in recent days, due to the growth of 3D modelling and 3D printing technologies. Hence, in this research, we propose a novel scheme for preserving the privacy of 3D point and mesh fog data. Chaotic Cat map-based data encryption is a recently trending research area due to its unique properties like pseudo-randomness, deterministic nature, sensitivity to initial conditions, ergodicity, etc. To boost encryption efficiency significantly, in this work, we propose a novel Chaotic Cat map. The sequence generated by this map is used to transform the coordinates of the fog data. The improved range of the proposed map is depicted using bifurcation analysis. The quality of the proposed Chaotic Cat map is also analyzed using metrics like Lyapunov exponent and approximate entropy. We also demonstrate the performance of the proposed encryption framework using attacks like brute-force attack and statistical attack. The experimental results clearly depict that the proposed framework produces the best results compared to the previous works in the literature.

Design Low Complexity SCMA Codebook Using Arnold’s Cat Map

In 5G wireless communications, sparse code multiple access (SCMA) – a multi-dimensional codebook based on a specific category of the non-orthogonal multiple access (NOMA) technique - enables many users to share non-orthogonal resource components with a low level of detection complexity. The multi-dimensional SCMA (MD-SCMA) codebook design presented in this study is based on the constellation rotation and interleaving method. Initially, a subset of the lattice Z 2 is used to form the mother constellation’s initial dimension. The first dimension is then rotated to produce other dimensions. Additionally, interleaving is employed for even dimensions to enhance fading channel performance. Arnold’s chaotic cat map is proposed as the interleaving method to reduce computational complexity. Performance of the SCMA codebook based on interleaving is evaluated by comparing it with selected codebooks for SCMA multiplexing. The metrics used for performance evaluation purposes include bit error rate (BER), peak to average power ratio (PAPR), and minimum Euclidean distance (MED), as well as complexity. The results demonstrate that the suggested codebook with chaotic interleaving offers performance that is equivalent to that of the conventional codebook based on interleaving. It is characterized by lower MED and higher BER compared to computer-generated and 16-star QAM codebook design approaches, but its complexity is lower than that of the conventional codebook based on interleaving.

Local correlations in coupled cat maps with space-time duality

We study quantum and classical correlations between local observables in perturbed coupled cat map model. In spite of fully chaotic dynamics, local correlations can be calculated explicitly due to the presence of spatiotemporal symmetry. This symmetry restricts correlations to the ‘light rays’ because the causality applies both in time and in space. We obtain detailed form of correlations of 2 and 3 sites’ observables demonstrating that exponential decay holds, generically. Furthermore, for unperturbed, pure linear, map, correlations between classical observables with a finite support exhibit superfast decay—the correlations disappear completely after a finite time.

Multi population-based chaotic differential evolution for multi-modal and multi-objective optimization problems

Differential evolution (DE) is a simple but powerful evolutionary algorithm used in multiple sciences and engineering disciplines to tackle optimization problems. DE has some disadvantages, such as premature convergence and the low convergence rate that prompts the worst DE execution structure in the constrained environment. The occurrence of these constraints split up the exploration area into viable and un-viable intervals. To overcome the abovementioned issues, we chose to take advantage of the vital characteristics of two mutation strategies: DE/rand/1 and DE/best/2. This research proposes a novel DE variant called Multi-population-based chaotic DE (MPC-DE) to solve multi-model and multi-objective optimization problems. The proposed MPC-DE is divided into two sub-populations with chaotic-based enhanced population initialization approaches, Sinusoidal and Tent map chaotic population initialization. Each sub-population follows the proposed improved mutation strategies based on two-dimensional chaotic maps, i.e., Baker’s map and Arnold’s Cat Map for DE/rand/1 in the first sub-population, and Zaslavskii Map for DE/best/2 in the second sub-population. Finally, the selection criteria are proposed to select the best offspring produced by each sub-population following the mutant vectors generated by the proposed mutation strategies. MPC-DE is evaluated on the dynamic multi-model and multi-objective optimization problems, i.e., benchmark problems for CEC 2017 and CEC 2020, respectively. To verify MPC-DE’s performance, we compare it with the latest DE variants, namely, EFADE, MPEDE, SHADE, EPSDE, L-SHADE, ESMDE, CoDE, and JADE. The proposed MPC-DE is also employed to solve the Economic Load Dispatch Problem (EDP) and reduce fuel costs. We used a 60-unit bus system and a 180-unit bus system to solve EDP and compared it to recent EDP solvers such as DPADE, JADE, EPSDE, SaDE, DE/BBO, DE, MIMO, TLBO, BPSO, CSO, ORCSA, CSA, ORCCRO, BBO, and ED-DE. The empirical results confirmed that MPC-DE outperformed other recent variants for multi-objective optimization problems and EDP.

Self-Super-Resolution of an MRI Image with Assistance of the DSTTD System.

Motivation. In the modern world of information technology, the need for ensuring the safety of wireless transmissions while transiting through a given network is growing rapidly. The process of transmitting images via a wireless network is fraught with difficulty. There is a possibility that data may be corrupted while being transmitted, which would result in an image with low resolution. Both of these issues were investigated head-on in this research methodology using the aiding double space-time block coding (DSTTD) system and the self-super-resolution (SSR) method. Description. In recent times, medical image transmission over a wireless network has received a significant amount of attention, as a result of the sharing of medical images between patients and doctors. They would want to make sure that the image was sent in a risk-free and protected manner. Arnold cat map, often known as ACM, is a well-known and widely implemented method of image transmission encryption that has been in use for quite some time. At the receiver end, SSR is now being employed in order to view the transmitted medical image in the finest possible resolution. It is anticipated that in the near future, image transmission through wireless DSTTD will be technically feasible. This is performed in order to maximize the benefits that the system has to offer in terms of both spatial diversity and multiplexing as much as is possible. Conclusion. The SSR approach is used in order to represent the image in a document pertaining to human resources. ACM is used so that the image may be sent in a risk-free and protected way. The adoption of a DSTTD-based architecture for wireless communication is suggested. A comparison of the results is provided, and PSNR and SSIM values are detailed towards the results and discussion of the article.

Large deviations in chaotic systems: Exact results and dynamical phase transition.

Large deviations in chaotic dynamics have potentially significant and dramatic consequences. We study large deviations of series of finite lengths N generated by chaotic maps. The distributions generally display an exponential decay with N, associated with large-deviation (rate) functions. We obtain the exact rate functions analytically for the doubling, tent, and logistic maps. For the latter two, the solution is given as a power series whose coefficients can be systematically calculated to any order. We also obtain the rate function for the cat map numerically, uncovering strong evidence for the existence of a remarkable singularity of it that we interpret as a second-order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps.

An Efficient Lightweight Image Encryption Scheme Using Multichaos

With an immense increase in Internet multimedia applications over the past few years, digital content such as digital images are stored and shared over global networks, the probability for information leakage and illegal modifications to the digital content is at high risk. These digital images are transferred using the network bandwidth; therefore, secure encryption schemes facilitate both information security and bandwidth issues. Hence, a state-of-the-art lightweight information security methodology is required to address this challenge. The main objective of this work is to develop a lightweight nonlinear mechanism for digital image security using chaos theory. The proposed scheme starts by changing a plain image into an encrypted image to improve its security. A block cipher, using lightweight chaos, has been added to achieve this objective for digital image security. We utilized multiple chaotic maps to generate random keys for each channel. Also, Arnold cat map and chaotic gingerbread map are used to add confusion and diffusion. During the permutation stage, image pixels are permuted, while in diffusion stage, pixels are distorted utilizing gingerbread map to add more security. The proposed scheme has been validated using different security parameter tests such as correlation coefficient tests (CC), whose results have been observed closer to zero and information entropy (IE) value is 7.99, respectively, which is almost equal to the ideal value of 8. Moreover, number of pixels changing rate (NPCR) obtained value is higher than 99.50%, while the unified average changing intensity (UACI) is 33.33. Other parameters such as mean absolute error (MAE), mean square error (MSE), lower value of peak to signal noise ratio (PSNR), structural content (SC), maximum difference (MD), average difference (AD), normalized cross-correlation (NCC), and histogram analysis (HA) is tested. The computed values of the proposed scheme are better. The achieved results after comparison with existing schemes highlight that the proposed scheme is highly secure, lightweight, and feasible for real-time communications.

Approximating a Common Solution of Monotone Inclusion Problems and Fixed Point of Quasi-Pseudocontractive Mappings in CAT(0) Spaces

In this paper, we aimed to introduce a new viscosity-type approximation method for finding the common fixed point of a class of quasi-pseudocontractive mapping and a system of monotone inclusion problems in CAT(0) spaces. We proved some fixed-point properties concerning the class of quasi-pseudocontractive mapping in CAT(0) spaces, which is more general than many other mappings such as nonexpansive, quasi-nonexpansive, pseudocontractive and demicontractive mappings which have been studied by other authors. A strong convergence result is proved under some mild conditions on the control sequences and some numerical examples were presented to illustrate the performance and efficiency of the proposed method.

Design, Implementation, and Analysis of a Block Cipher Based on a Secure Chaotic Generator

This work proposes a new secure chaos-based encryption/decryption system, operating in cipher block chaining (CBC) mode, and analyze its performance. The cryptosystem includes a robust pseudorandom number generator of chaotic sequences (PRNG-CS). A strong chaos-based S-box is proposed to perform a circular substitution operation (confusion process). This PRNG-CS consists of four discrete 1-D chaotic maps, weakly coupled by a predefined coupling matrix M, to avoid, on the one hand, the divide-and-conquer attack and, on the other hand, to improve the generated sequence’s randomness and lengths. The noun is also used in the construction of the S-box. Moreover, a 2-D modified cat map and a horizontal addition diffusion (HAD) preceded by a vertical addition diffusion (VAD) are introduced to perform the diffusion process. The security analysis and numerous simulation results of the main components (PRNG-CS and S-box) as well as the whole cryptosystem reveal that the proposed chaos-based cryptosystem holds up against various types of statistical and cryptographic attacks.

An extendable key space integer image-cipher using 4-bit piece-wise linear cat map

This paper presents a multiplierless image-cipher, with extendable 2048-bit key-space, based on a 4-dimensional (4D) quantized piece-wise linear cat map (PWLCM). The quantized PWLCM exhibits limit-cycles of 4-bit encoded integers with periods greater than 107. The synthesis of the PWLCM in a finite state space allows to eliminate the undesirable finite precision effect due to the hardware realization. The proposed image-cipher combines chaos, modular arithmetic, and lattice-based cryptography to encrypt a color image by performing pixel permutation and diffusion in a single operation. Further, an image-dependent confusion operation based on an 8-bit 2D-PWLCM is performed on the whole image to enhance security. In order to increase the key-space without key duplication, 16 × 16 sub-images are modified using sub-keys of different lattice length vectors generated from the external key. Both simulations and security analyses confirm that the proposed algorithm can resist common cipher attacks, in addition to its advantages such as simplicity, ease of implementation on low-end processors and extensibility of key-space that allows it to easily adapt even for future post-quantum computing attacks.

A New Block Based Non-Blind Hybrid Color Image Watermarking Approach Using Lifting Scheme and Chaotic Encryption Based on Arnold Cat Map

Online platforms became preferred mode of communication due to advancement in communication technology. Sharing of digital documents over online communication medium grown exponentially and thus demanded a secure, robust and transparent watermarking technique for authenticity of digital media and copyright protection. This research study proposes a robust and secure non-blind SVD-LWT watermarking technique. Color images are employed instead of gray scale images and Y channel of YCbCr color model is utilized to embed secret digital information. The selected color model is in accordance with the human visual system and Y channel is ideal for data hiding. Two level LWT, SVD is used and diagonal matrix of Y channel of host (cover) and watermark image along with scaling factor (α) is used to embed digital data. Block based and chaotic image encryption transform are used for image scrambling. The performances of presented watermarking scheme evaluated with the aid of fidelity parameters namely MSE, PSNR, SSIM and NCC.

Approximation of fixed points of enriched asymptotically nonexpansive mappings in CAT(0) spaces

Approximation of fixed points of enriched asymptotically nonexpansive mappings in CAT(0) spaces

Enhanced Gravitational Search Optimization with Hybrid Deep Learning Model for COVID-19 Diagnosis on Epidemiology Data.

Effective screening provides efficient and quick diagnoses of COVID-19 and could alleviate related problems in the health care system. A prediction model that combines multiple features to assess contamination risks was established in the hope of supporting healthcare workers worldwide in triaging patients, particularly in situations with limited health care resources. Furthermore, a lack of diagnosis kits and asymptomatic cases can lead to missed or delayed diagnoses, exposing visitors, medical staff, and patients to 2019-nCoV contamination. Non-clinical techniques including data mining, expert systems, machine learning, and other artificial intelligence technologies have a crucial role to play in containment and diagnosis in the COVID-19 outbreak. This study developed Enhanced Gravitational Search Optimization with a Hybrid Deep Learning Model (EGSO-HDLM) for COVID-19 diagnoses using epidemiology data. The major aim of designing the EGSO-HDLM model was the identification and classification of COVID-19 using epidemiology data. In order to examine the epidemiology data, the EGSO-HDLM model employed a hybrid convolutional neural network with a gated recurrent unit based fusion (HCNN-GRUF) model. In addition, the hyperparameter optimization of the HCNN-GRUF model was improved by the use of the EGSO algorithm, which was derived by including the concepts of cat map and the traditional GSO algorithm. The design of the EGSO algorithm helps in reducing the ergodic problem, avoiding premature convergence, and enhancing algorithm efficiency. To demonstrate the better performance of the EGSO-HDLM model, experimental validation on a benchmark dataset was performed. The simulation results ensured the enhanced performance of the EGSO-HDLM model over recent approaches.