1D Map Research Articles

Hybrid Chaotic-Based PRNG for Secure Cryptography Applications

This paper suggests a novel one-dimensional (1D) map to address the limitations of traditional chaotic 1D maps. In contrast to traditional 1D maps, the proposed map has three control parameters a, μ, and c, allowing it to exhibit chaotic behavior over a wide range of values. The dynamic behavior of the new 1D map was analyzed using well-known numerical methods, including the bifurcation diagram and Lyapunov exponent. Both tests showed their complex and diverse behavior. In addition, a novel image encryption scheme was devised using the new function as its pseudorandom number generator. Rigorous statistical testing was applied to the proposed encryption algorithm. The mean square error (MSE) and peak signal-to-noise ratio (PSNR) results, in addition to subjecting 28 images to number of pixels change rate (NPCR) and unified average changing intensity (UACI) tests demonstrated the robustness of the system. The results of this study demonstrate the effectiveness of the new 1D map for use in secure image cryptography applications, providing a more robust and secure alternative to traditional chaotic 1D maps.

In-Depth Analysis of Smooth and Nonsmooth Bifurcations for an Open-Loop Boost Converter Feeding Constant Power Loads in Discontinuous Conduction Mode

This paper presents a study of the existence conditions of limit cycles and mechanisms when losing their stability in an open-loop DC–DC boost converter loaded with a constant power load and operating in discontinuous conduction mode. First, the existence conditions are determined. Then, boundaries associated with different kinds of instabilities such as the classical smooth fold and period-doubling bifurcations and nonsmooth border-collision bifurcations such as fold border-collision bifurcations are elucidated. To perform the stability analysis, an implicit 1D map for the system is derived. From this model, it is obtained that smooth period-doubling and fold bifurcations may occur at certain values of the circuit parameters. Some results from numerical simulations performed on the circuit-based switched model are provided showing consistency with the theoretical analysis from the derived model.

Real-time medical image encryption for H-IoT applications using improved sequences from chaotic maps

A new medical cryptosystem based on four chaotic maps is presented as a case study. The message queuing telemetry transport (MQTT) protocol is used in this research to propose an end-to-end chaotic encryption technique that would enhance security and secrecy to the transmission of medical images from any Healthcare Internet of Things (H-IoT) device connected to the Internet. The maximal Lyapunov exponent (LE), the analysis of bifurcation diagrams, the Statistical Test Suit from the National Institute of Standard and Technology in the Special Publication 800–22 (NIST SP 800–22), and TestU01 statistical tests are used to evaluate the randomness of the sequences. The proposed medical image encryption technique is confirmed to be strong enough to withstand most current assaults, including differential attacks, the correlation between adjacent pixels, statistical histograms, entropy, key-space, and key-space attacks. The suggested cryptosystem is suitable for edge computing devices and has throughputs of up to 10.53 Mbit/s on a Raspberry Pi 4 (RPi4) and 47.44 Mbit/s utilizing a 2.9 GHz CPU on a desktop PC with improved sequences from the Logistic 1D map. Because of this, the proposed embedded medical cryptosystem may be used to strengthen the security of real-time medical image transmission across WiFi networks and the Internet.

Prospective oxygen-ion conductors LnaXbOz: Geometry and energy calculations

Using a combined approach based on the geometrical-topological (GT) analysis of crystal structures and bond valence site energy (BVSE) calculations, a high-throughput computer screening was carried out for 2073 compounds of the LnaXbOz composition (X is any chemical element of III-VI periods, excluding elements 1, 16, 17 and 18-groups, P, As, Ba, Sr), which taken from ICSD (version 2021/2). By means GT approach, 720 most promising oxygen-containing compounds with 1D, 2D or 3D migration map of O2−-ions were obtained. According to the BVSE results, 225 most stable compounds were divided to groups based on structural type, in each of which experimental data were established for known oxygen-ion conductors and the migration energy barriers of were calculated by the BVSE method. 46 compounds were found that do not belong to known structural types, while 41 of them had low barriers to migration of oxygen ions (0.10–1.20 eV), which made it possible to identify them as new potential oxygen-ion conductors.

3D Model Encryption Algorithm by Parallel Bidirectional Diffusion and 1D Map with Sin and Logistic Coupling

3D Model Encryption Algorithm by Parallel Bidirectional Diffusion and 1D Map with Sin and Logistic Coupling

Period adding with symmetry breaking/recovering in a power inverter with hysteresis control

In the present work, we investigate a period adding structure observed in a discontinuous 1D map modeling the behavior of a DC–AC H-bridge inverter with hysteresis control. Supported by experimental measurements from a laboratory prototype, we demonstrate that the structure is partially affected by bistability, so that for some parameter values the system has a single attracting cycle while for other parameter values a pair of attracting cycles of the same period exists. We show that the reason for possible bistability is related to the symmetry of the underlying model. Then, by means of symbolic dynamics, we explain which of the cycles are affected by bistability and which are not.

A cluster of 1D quadratic chaotic map and its applications in image encryption

Chaotic systems are widely used in designing encryption algorithms for their ideal dynamical performances. One-dimensional (1D) chaotic maps have the highest efficiency in implementation and have achieved great attention. However, 1D chaotic maps have a common security weakness, which is that their key space is relatively small. Therefore, in this paper, a cluster of 1D quadratic chaotic maps is proposed according to the topological conjugate theory. The 1D chaotic map has three tunable parameters which have a significant expansion in parameter space than the traditional 1D chaotic maps. The 1D map is proved to be chaotic theoretically since it is topologically conjugated with a logistic chaotic map. An example of a 1D quadratic chaotic map is provided in this paper, and several numerical simulation results indicate that this 1D quadratic map has ideal chaotic characteristics, which are consistent with the theoretical analysis. To verify the effectiveness of this 1D quadratic map, a novel image encryption algorithm is proposed. The security of this image encryption algorithm is completely dependent on the properties of the 1D quadratic map. Security experimental test results show that this image encryption algorithm has a high-security level, and is quite competitive with other chaos-based image encryption algorithms.

Design and Implementation of a New Hyperchaotic Memristive Map

The two-dimensional (2D) discrete maps with complicated dynamics have lately received a lot of interest. However, the discrete enhanced 1D map models and their combinations with memristors haven’t received much attention. In this brief, we first present an enhanced 1D map with more significant chaotic characteristics. By coupling a discrete memristor into the enhanced map, a novel 2D memristive map with infinite fixed points is further proposed. We also study the map’s parameter-relied dynamical behaviors and multi-stability phenomena using various analysis methods and demonstrate its hyperchaos and coexisting attractors. The numerical results show that the addition of a memristor boosts the map’s chaotic complexity. Digital experiments based on a microcomputer are also shown, with results that are consistent with the numerical analysis.

3D chaos graph deep learning method to encrypt and decrypt digital image

We live in technological age development’s where many important data transmitted electronically from one device to another and in every place. Deep learning algorithms have facilitated the process of encoding and decoding digital images. Chaotic graph systems, on the other hand, are one of the most recent techniques utilized to encode image data based on the methods of cryptography. The chaos maps are divided into two main aspects, first one deals with the 1D map which requires fewer features and can be developed easily, the second one is the high dimensional map which is more complex than the 1D graph and it requires more features, more parameters, and it is relatively hard to develop. In this paper, we present a method for image encoding and decoding electronically using deep learning, the proposed algorithm was developed by using the hybrid technique of 3D chaos map generation, the best case of the proposed technique gave the following results: The average entropy calculation was (7.4838) before image encryption and (7.9896) after image encryption with average number of pixels change rate (NPCR) of (99.7085%) and the unified average changing intensity (UACI) of (33.2030%) which are the best outcomes when compared to other similar works.

Real-time RGB image encryption for IoT applications using enhanced sequences from chaotic maps

Four chaotic maps are used herein as case study to design an embedded cryptosystem based on a pseudo-random number generator (PRNG). The randomness of the sequences is enhanced by applying the mod 1023 function and verified by analyzing bifurcation diagrams, the maximum Lyapunov exponent, and performing NIST SP 800-22 and TestU01 statistical tests. The PRNG is applied in a simple algorithm for real-time RGB images encryption on a machine-to-machine (M2M) scheme, using message queuing telemetry transport (MQTT) protocol over WiFi network and through Internet. The cryptanalysis confirms that the proposed image encryption scheme is robust to resist most of the existing attacks, such as statistical histograms, entropy, key-space, correlation of adjacent pixels, and differential attacks. The implementation of the proposed cryptosystem is done using enhanced sequences from the Logistic 1D map, and it reaches a throughput of up to 47.44 Mbit/s using a personal computer with a 2.9 GHz clock, and 10.53 Mbit/s using a Raspberry Pi 4. As a result, our proposed embedded cryptosystem is suitable to increase the security in the transmission of RGB images in real-time through WiFi networks and Internet.

Type III intermittency without characteristic relation.

Chaotic intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories from the chaotic region into the laminar one. The RPD classically was taken as a constant. This hypothesis is behind the classically reported characteristic relations, a tool describing how the mean value of the laminar length goes to infinity as the control parameter goes to zero. Recently, a generalized non-uniform RPD has been observed in a wide class of 1D maps; hence, the intermittency theory has been generalized. Consequently, the characteristic relations were also generalized. However, the RPD and the characteristic relations observed in some experimental Poincaré maps still cannot be well explained in the actual intermittency framework. We extend the previous analytical results to deal with the mentioned class of maps. We found that in the mentioned maps, there is not a well-defined RPD in the sense that its shape drastically changes depending on a small variation of the parameter of the map. Consequently, the characteristic relation classically associated to every type of intermittency is not well defined and, in general, cannot be determined experimentally. We illustrate the results with a 1D map and we develop the analytical expressions for every RPD and its characteristic relations. Moreover, we found a characteristic relation going to a constant value, instead of increasing to infinity. We found a good agreement with the numerical simulation.

Complex dynamics of a vibration machine caused by a relay feedback control

In this paper we describe some nonlinear phenomena observed in a vibration machine with an unbalanced rotor driven by a DC motor controlled by a relay system. We demonstrate that this type of control leads to the appearance of several complex kinds of behavior caused by interactions between slow and fast dynamics. Our numerical results are verified through comparison with an experimental vibration system. In order to understand the mechanism leading to these phenomena, we apply a modeling approach based on the reduction of a 5D autonomous non-smooth vector field to a 1D discontinuous map for the phase of the switching events. We demonstrate that there are two different reasons causing this map to be discontinuous, so that depending on parameters it may be either a gap map or a circle diffeomorphism. In both parameter domains the map exhibits a period adding structure. We explain also which features of the behavior of the modeled vibration machine are captured by the 1D map and which are not.

Probing dark energy with tomographic weak-lensing aperture mass statistics

We forecast and optimize the cosmological power of various weak-lensing aperture mass (Map) map statistics for future cosmic shear surveys, including peaks, voids, and the full distribution of pixels (1DMap). These alternative methods probe the non-Gaussian regime of the matter distribution, adding complementary cosmological information to the classical two-point estimators. Based on the SLICS and cosmo-SLICSN-body simulations, we buildEuclid-like mocks to explore theS8 − Ωm − w0parameter space. We develop a new tomographic formalism that exploits the cross-information between redshift slices (cross-Map) in addition to the information from individual slices (auto-Map) probed in the standard approach. Our auto-Mapforecast precision is in good agreement with the recent literature on weak-lensing peak statistics and is improved by ∼50% when including cross-Map. It is further boosted by the use of 1DMapthat outperforms all other estimators, including the shear two-point correlation function (γ-2PCF). When considering all tomographic terms, our uncertainty range on the structure growth parameterS8is enhanced by ∼45% (almost twice better) when combining 1DMapand theγ-2PCF compared to theγ-2PCF alone. We additionally measure the first combined forecasts on the dark energy equation of statew0, finding a factor of three reduction in the statistical error compared to theγ-2PCF alone. This demonstrates that the complementary cosmological information explored by non-GaussianMapmap statistics not only offers the potential to improve the constraints on the recentσ8–Ωmtension, but also constitutes an avenue to understanding the accelerated expansion of our Universe.

2D mixed pseudo-random coupling PS map lattice and its application in S-box generation

In this paper, firstly, we investigate a new 1D PWLCM-Sin (PS) map which derived from PWLCM and Sin map by modulo operation. Due to the stronger parameter space, bigger Lyapunov exponents and better ergodicity than simple 1D map, the PS map is more suitable for local map of spatiotemporal dynamics. Secondly, with the novel 2D pseudo-random mixed coupling method we present a spatiotemporal chaos which used PS map as local map f(x). This spatiotemporal chaos named 2D Mixed pseudo-random Coupling PS Map Lattice (2DMCPML). The experimental results of bifurcation diagrams, Kolmogorov–Sinai entropy density and spatiotemporal chaotic diagrams showed that 2DMCPML has advantages of larger parameter space, more complex chaotic behavior and more ergodic output sequence than CML. Therefore, 2DMCPML is more suitable in cryptography than CML. Subsequently, we proposed a chaos-based random S-box design algorithm employed the spatial chaotic character of 2DMCPML to generate a large number of S-boxes. The cryptographic performance indicated that generated S-boxes can resist cryptanalysis attack well. Finally, four criteria bounds are set. The numbers of S-boxes satisfying these bounds generated by 2DMCPML and several 1D chaotic maps is calculated, respectively. The result showed that spatiotemporal chaos can generate more S-boxes with high cryptographic quality than low-dimensional chaos. This new discovery is significant to the development of some cryptographic researches such as dynamic S-box algorithm.

Deterministic chaotic finite-state automata

Chaotic dynamics have been widely applied in various domains such as cryptography, watermarking and optimization algorithms. Enhancing chaotic complexity of simple one-dimensional (1D) chaotic maps has been a popular topic of research in recent years. However, most of the proposed methods have low complexity and are not suitable for practical applications. To overcome these issues, this paper introduces a novel approach known as deterministic chaotic finite-state automata (DCFSA). Existing 1D chaotic maps are associated with deterministic finite automata states. Then, a transition rule dynamically selects which 1D chaotic map to compute. DCFSA allows the creation of a large number of possible chaotic configurations with enhanced nonlinearity while retaining the computational complexity comparable to a 1D map. Theoretical and performance analyses show that DCFSA provides a larger chaotic parameter range, higher nonlinearity and chaotic complexity, as well as longer cycle length as compared to its underlying 1D chaotic maps. Moreover, performance comparison against other existing chaotification methods demonstrates DCFSA’s superiority.

Design of an analog chaos-generating circuit using piecewise-constant dynamics

e return map is constructed from a piecewise-linear 1D map. We analytically prove the generation of chaos with a positive Lyapunov exponent through a simple systematic procedure, and observe the attractor through laboratory measurements. We strongly believe that the explicit simplicity of the proposed circuit dynamics could significantly affectthearchitectureofchaos-generating circuitdesigninapplicationbecausetheprocedurefor proving chaos is systematic, and our circuit is expected to extend to higher-dimensional circuits in a systematic manner.

Scanning and resetting the phase of a pinned spiral wave using periodic far field pulses

Spiral waves in cardiac tissue can pin to tissue heterogeneities and form stable pinned waves. These waves can be unpinned by electric stimuli applied close to the pinning center during the vulnerable window of the spiral. Using a phase transition curve (PTC), we quantify the response of a pinned wave in a cardiac monolayer to secondary excitations generated electric field pulses. The PTC can be used to construct a one-dimensional map that faithfully predicts the pinned wave’s response to periodic field stimuli. Based on this 1D map, we predict that pacing at a frequency greater than the spiral frequency, over drive pacing, leads to phase locking of the spiral to the stimulus, which hinders unpinning. In contrast, under drive pacing can lead to scanning of the phase window of the spiral, which facilitates unpinning. The predicted mechanisms of phase scanning and phase locking are experimentally tested and confirmed in the same monolayers that were used to obtain the PTC. Our results have the potential to help choose optimal parameters for low energy antifibrillation pacing schemes.

Exponential-Algebraic Maps and Chaos in 3D Autonomous Quadratic Systems

For some 3D autonomous quadratic dynamical systems an explicit autonomous exponential-algebraic 1D map, generating chaos in mentioned systems, is designed. Examples of the systems, where chaos is generated by such discrete maps, are given. New results about an existence of chaotic dynamics in the quadratic 3D systems are also derived. Besides, for the Lanford system (it is 3D autonomous quadratic dynamical system) the value of some parameter at which the system shows increased chaotic behavior is indicated. This assertion is based on the construction for the Lanford system of 2D exponential-algebraic discrete map which possesses chaotic properties.

Noise and Chaos Contributions in Fast Random Bit Sequence Generated From Broadband Optoelectronic Entropy Sources

During the last 4 years, chaotic waveforms for random number generation found a deep interest within the community of analogue broadband chaotic optical systems. Earlier investigations on chaos-based RNG were proposed in the 90s and early 2000, however mainly based on piecewise linear (PL) 1D map, with bit rate determined by analog electronic processing capabilities to provide the PL nonlinear function of concern. Optical chaos came with promises for much higher bit rate, and entropy sources based on high complexity (high dimensional) continuous time (differential) dynamics. More specifically in 2009, Reidler published a paper entitled “An optical ultrafast random bit generator”, in which they presented a physical system for a random number generator based on a chaotic semiconductor laser. This generator is claimed to reach potentially the extremely high rate of 300 Gb/s. We report on analysis and experiments of their method, which leads to the discussion about the actual origin of the obtained randomness. Through standard signal theory arguments, we show that the actual binary randomness quality obtained from this method, can be interpreted as a complex mixing operated on the initial analogue entropy source. Our analysis suggests an explanation about the already reported issue that this method does not necessarily require any specific deterministic property (i.e., chaos) from the physical signal used as the source of entropy. The bit stream randomness quality is found to result from “aliasing” phenomena performed by the post-processing method, both for the sampling and the quantization operations. As an illustration, such random bit sequences extracted from different entropy sources are investigated. Optoelectronic noise is used as a non deterministic entropy source. Electro-optic phase chaotic signal, as well as simulations of its deterministic model, are used as deterministic entropy sources. In all cases, the extracted bit sequence reveals excellent randomness.

Studies of phase return map and symbolic dynamics in a periodically driven Hodgkin—Huxley neuron

How neuronal spike trains encode external information is a hot topic in neurodynamics studies. In this paper, we investigate the dynamical states of the Hodgkin—Huxley neuron under periodic forcing. Depending on the parameters of the stimulus, the neuron exhibits periodic, quasiperiodic and chaotic spike trains. In order to analyze these spike trains quantitatively, we use the phase return map to describe the dynamical behavior on a one-dimensional (1D) map. According to the monotonicity or discontinuous point of the 1D map, the spike trains are transformed into symbolic sequences by implementing a coarse-grained algorithm — symbolic dynamics. Based on the ordering rules of symbolic dynamics, the parameters of the external stimulus can be measured in high resolution with finite length symbolic sequences. A reasonable explanation for why the nervous system can discriminate or cognize the small change of the external signals in a short time is also presented.