Bernoulli Shift Map Research Articles

Exploring a family of Bernoulli-like shift chaotic maps and its amplitude control

Chaos theory is a branch of mathematics that studies systems highly sensitive to initial conditions. Various methods for analyzing and controlling these systems, such as bifurcation theory and amplitude control, have been developed and applied in diverse fields, including cryptography, physics, and engineering.In this study, we introduce a new class of Bernoulli-Like Shift (BLS) maps. These maps are generated by using a specific class of Piecewise Linear (PWL) maps combined with a modulo operation. We conduct a fixed point analysis by establishing propositions and corollaries, which outline the conditions that guarantee the presence or absence of fixed points. Additionally, we provide numerical examples to validate these theoretical results.Utilizing bifurcation diagrams and Lyapunov exponent calculations, we demonstrate that the range of chaotic behavior in the BLS maps is broader than those of the logistic and Bernoulli Shift maps. Moreover, we show that the BLS maps possess higher values of Lyapunov exponents and a better distribution than both the Bernoulli shift and logistic maps.Lastly, we introduce a transformation that generates a modified BLS map, allowing for amplitude control. This modification enables the regulation of amplitude while maintaining constant Lyapunov exponents.

SaCHBA_PDN: Modified honey badger algorithm with multi-strategy for UAV path planning

The honey badger algorithm (HBA) is a meta-heuristic optimization algorithm that simulates the foraging behavior of honey badgers. Since the algorithm is prone to premature convergence when solving complex optimization problems. To improve the overall optimization performance of the basic HBA, this paper develops a modified HBA named SaCHBA_PDN based on the Bernoulli shift map, piecewise optimal decreasing neighborhood, and horizontal crossing with strategy adaptation and applies it to solve the unmanned aerial vehicle (UAV) path planning problem. Firstly, the Bernoulli shift map is invoked to the HBA algorithm to change its initialization process, thus increasing the diversity of the population and speeding up the convergence speed. Secondly, a new piecewise optimal decreasing neighborhood strategy (PODNS) is proposed to address the shortcomings of unbalanced convergence of the traditional optimal neighborhood strategy. The proposed PODNS increases the optimization efficiency of HBA and enhances the local search ability to avoid falling into the local optimum. Finally, a novel horizontal crossing with strategy adaptation is introduced to balance exploration and exploitation and enhance the global optimization ability. These strategies collaborate to enhance HBA in accelerating overall performance. The superiority of SaCHBA_PDN is comprehensively verified by comparing it with the original HBA and numerous celebrated and newly developed algorithms on the well-known 23 classical benchmark functions and IEEE CEC2017 test suite, respectively. Experimental results show that SaCHBA_PDN has a better performance than other optimization algorithms. Furthermore, SaCHBA_PDN is used to solve a UAV path planning problem based on the threat source model and applied to circular and irregular obstacle scenarios as well as two-dimensional grid maps. Simulation results show that SaCHBA_PDN can obtain more feasible and efficient paths in different obstacle environments.

How to perturb Bernoulli shift map

The Bernoulli shift map is a fundamental example of a chaotic map with applications in algorithm design, data analysis, and numerical simulation. When implementing the Bernoulli shift map in the binary system, some sorts of perturbation methods are employed to make its outputs have long periods for practical reasons. We here look at one of such methods that perturbs underlying state space, and apply modular arithmetic to analyze the behavior of periods attained by this method, which reveals a close relationship with Artin’s conjecture on primitive roots. As a consequence, we obtain an exhaustive list of values for a dominant parameter of this method that are best possible in a theoretical sense.

Efficient JPEG Encoding Using Bernoulli Shift Map for Secure Communication

To ensure confidentiality and efficient network bandwidth, digital data must be compressed and encrypted. In most communication systems, these two factors are critical for information processing. Image compression and encryption may result in lower restoration quality and performance. Secure-JPEG is an effort to create a compression and encryption technique for digital data. This approach is based on the JPEG compression standard, which is the most extensively used lossy compression scheme. It enhances the usual JPEG compression algorithm to encrypt data during compression. The Secure-JPEG approach encrypts the data while it is compressed, and it may be easily modified to offer near lossless compression. Lossless compression, on the other hand, has a lower compression ratio and is only useful in certain situations. The paper addresses the issue of insufficient security as a result of the usage of a simple random number generator that is not cryptographically safe. The enhanced security characteristics are provided via the Generalized Bernoulli Shift Map, which has a chaotic system with proven security. Several cryptographic tests are used to validate the algorithm's security, and the chaotic system's behavior is also examined.

Probing bacterial cell wall growth by tracing wall-anchored protein complexes

The dynamic assembly of the cell wall is key to the maintenance of cell shape during bacterial growth. Here, we present a method for the analysis of Escherichia coli cell wall growth at high spatial and temporal resolution, which is achieved by tracing the movement of fluorescently labeled cell wall-anchored flagellar motors. Using this method, we clearly identify the active and inert zones of cell wall growth during bacterial elongation. Within the active zone, the insertion of newly synthesized peptidoglycan occurs homogeneously in the axial direction without twisting of the cell body. Based on the measured parameters, we formulate a Bernoulli shift map model to predict the partitioning of cell wall-anchored proteins following cell division.

A Novel Image Encryption Algorithm Based on Bernoulli Shift Map

This paper puts forward a chaotic encryption algorithm based on Bernoulli Shift Map. This algorithm adopts dynamic encryption key and uses plaintext image parameters to control generation of chaotic key, and enhance connection between chaotic key and plaintext data; the iterative operation precision of chaotic map during encryption is high, avoiding cycle of chaotic system iteration process. The NPCR and UACI analysis, adjacent pixels correlation analysis and image information entropy analysis on encrypted image show this encryption algorithm has favorable encryption characteristics such as larger space for key, sensitivity to plaintext and random distribution of ciphertext.

A TRNG using chaotic entropy pool as a post-processing technique: analysis, design and FPGA implementation

In this study, the hardware implementation of a TRNG using an alternative post-processing technique is presented. ROs (ROs) are used as the noise source of TRNG and true randomness is obtained from the jitter. The use of an entropy pool composed of discrete-time chaotic systems (quadratic map, cubic map, Bernoulli shift map and tent map) is proposed as the post-processing technique for TRNG. In the system, the statistical quality of true random numbers with low statistical quality, obtained from jitter, is improved to meet cryptographic qualifications with the proposed post-processing technique. Unlike other known post-processing techniques in the literature, the post-processing technique based on the effective use of chaos is user-controlled, and is dynamically adaptable to the chosen chaotic systems. Thus, optional post-processing inputs fed from a single chaotic system or different combinations of those systems can be obtained for the TRNG. The performance of the proposed TRNG with this post-processing is high. In addition to this, the fact that any random number sequence can be generated with the contribution of more than one chaotic input in the post-processing stage makes both the post-processing and TRNG safer and more secure. In addition to the chaos-based comprehensive security analysis of the post-processing, the statistical validation of the TRNG is made by using test-based tools. As a result, it has been found that the proposed post-processing technique can be used for cryptographic purposes.

Implementation and Performance Analysis of True Random Number Generator on FPGA Environment by Using Non-periodic Chaotic Signals Obtained from Chaotic Maps

In this study, FPGA implementation of a hybrid random number generator (HRNG) based on digital design techniques is given. The ring oscillators (ROs) are used as the noise source of HRNG, and true randomness is obtained by sampling jitter signals forming on the oscillators. The statistical quality and reliability of random number generators that used jitter as source of true randomness alone are often cryptographically insufficient. For this reason, one-dimensional discrete-time chaotic maps such as quadratic map, logistic map and Bernoulli shift map are benefited in order for HRNG to meet these cryptographic requirements. In contrast to many studies in the literature, non-periodic signals derived from chaotic systems of a powerful source of entropy are used instead of periodic signals for the sampling of jitter signals in the system. Depending on the usage of chaotic systems, output bit rate and reliability of high generator model that does not need post-processing techniques and is easily applicable to digital devices are obtained. The hybrid system is tested in total six different scenarios for two separate ring oscillator (RO) architectures of 25 and 114 pieces consisting of three different chaotic maps and equal-length inverters. The statistical qualifications of the random numbers obtained from HRNG for each scenario are verified by NIST 800-22 tests. Also, for each scenario, the design parameters of the generator are examined and the hardware performances and non-periodicity analyses of the chaotic maps are performed. Based on the obtained results, it is demonstrated that the HRNG based on non-periodic sampling can be used for cryptographic purposes.

Medical Image Encryption Based on Hybrid Chaotic DNA Diffusion

We explore the use of two chaotic systems (Bernoulli shift map and Zizag map) coupled with deoxyribonucleic acid coding in an encryption scheme for medical images in this paper. The scheme consists of two main phases: Chaotic key generation and DNA diffusion. Firstly, the message digest algorithm 5 hash function is performed on the plain medical image and the hash value used in combination with the value of an input ASCII string to generate initial conditions and control parameters for two chaotic systems (Bernoulli shift map and Zigzag map). These chaotic systems are subsequently used to produce two separate key matrices. Secondly, a row-by-row diffusion operation between the plain image matrix and the two chaotic key matrices, using the DNA XOR algebraic operation is performed in an alternating pattern to produce the cipher image. The logistic map is used to select the DNA encoding and decoding rules for each row. Experimental results of statistical, differential and key analyses demonstrate that the proposed scheme is robust and provides resistance to various forms of attacks.

BER performance analysis of compound chaotic sequence (CCS)-based NR-DCSK system under multipath fading channel

This paper presents the performance analysis of compound chaotic sequence (CCS)-based noise reduction differential chaos shift keying (NR-DCSK) system under multipath Rayleigh fading channel conditions. The special characteristics of chaotic sequences are their deterministic randomness behaviour that adds security and multipath immunity to the data when used as a carrier in communication systems. In this paper, the chaotic sequences are generated by combining the outputs of chaotic maps, such as logistic map, Chebyshev map, Bernoulli shift map, tent map, etc., leading to new complex sequences known as CCSs. This sequence possesses more randomness, overcomes severe interference levels encountered during transmission and provides higher multipath immunity compared with those of pseudo-noise (PN) codes. Since NR-DCSK is a spread spectrum technique, its performance in wireless multipath fading channels has important considerations. The CCS is used as a carrier in NR-DCSK systems, which leads to improved bit error rate (BER) performance. Comparisons of simulation results to theoretical BER expressions of additive white Gaussian noise (AWGN) and Rayleigh fading channels have been carried out to test the efficiency of the proposed CCS-based NR-DCSK system.

Chaos for implicit difference equations with snap-back repellers

This paper is a study of chaos for generalized dynamical systems derived from implicit difference equations. We define a snap-back repeller for an implicit difference equation and show that its existence implies chaotic dynamics for all small -perturbed systems. By chaotic dynamics, we mean that the solution set of an implicit difference equation contains a compact subset on which the Bernoulli shift map is invariant and has positive topological entropy.

Hardware implementation of pseudo-random number generators based on chaotic maps

We show the usefulness of bifurcation diagrams to implement a pseudo-random number generator (PRNG) based on chaotic maps. We provide details on the selection of the best parameter values to obtain high entropy and positive Lyapunov exponent from the bifurcation diagram of four chaotic maps, namely: Bernoulli shift map, tent, zigzag, and Borujeni maps. The binary sequences obtained from these maps are analyzed to implement a PRNG both in software and in hardware. The software implementation is realized using 32 and 64 bits microprocessor architectures, and with floating point and fixed point computer arithmetic. The hardware implementation is done by using a field-programmable gate array (FPGA) architecture. We developed a serial communication interface between the PRNG on the FPGA and a personal computer to obtain the generated sequences. We validate the randomness of the generated binary sequences with the NIST test suite 800-22-a both in floating point and fixed point arithmetic. At the end, we show that those chaotic maps are suitable to implement a PRNG but according to the hardware resources, the one based on the Bernoulli shift map is better. In addition, another advantage is that the required initial value for the sequences can be within the whole interval $$[-1,1]$$ , including its bounds.

Analysis of chaotic behavior in pipelined analog to digital converters

This paper presents a new approach to analysis of chaotic behavior in pipelined analog to digital converters. To this end, the fundamental theorem for function of a random variable is exploited and the propagation of the output pdf of chaotic maps in successive stages of pipelined converter is shown. It is illustrated that 1-bit stage of this converter can be used to implement the Bernoulli map, and also the 2k-way Bernoulli shift map can be implemented using k×1-bit stages or one k-bit stage. It is revealed that in the half-bit redundant structure, after a sufficiently large number of cycles, residue density becomes concentrated in the center half of the stage full-scale range. In this way, the 1.5-bit stage characteristic will be fully equivalent to the common chaotic map that is employed to generate random number sequences. By applying the proposed approach to the joint density of residue stages, it is shown that residues become independent and uniformly distributed. This feature is used to design multi bit chaotic random number generator using full-bit and half-bit redundant pipelined ADC structure. The validity of the proposed random number generator is confirmed by passing all NIST statistical tests even in the presence of nonidealities.

A Chaos-based Pseudorandom Permutation and Bilateral Diffusion Scheme for Image Encryption

A great many chaos-based image encryption schemes have been proposed in the past decades. Most of them use the permutation-diffusion architecture in pixel level, which has been proved insecure enough as they are not dependent on plain-images and so cannot resist chosen/known plain-image attack usually. In this paper, we propose a novel image encryption scheme comprising of one permutation process and one diffusion process. In the permutation process, the image sized M N  is expanded to one sized 2 M N  by dividing the plainimage into two parts: one consisting of the higher 4bits and one consisting of the lower 4bits. The permutation operations are done row-by-row and column-by-column to increase the speed of permutation process. The chaotic cat map is utilized to generate chaotic sequences, which are quantized to shuffle the expanded image. The chaotic sequence for permutation process is dependent on plainimage and cipher keys, resulting in good key sensitivity and plain-image sensitivity. To achieve more avalanche effect and larger key space, a chaotic Bernoulli shift map based bilateral (i.e., horizontal and vertical) diffusion function is applied as well. The security and performance of the proposed image encryption have been analyzed, including histograms, correlation coefficients, information entropy, key sensitivity analysis, key space analysis, differential analysis, encryption rate analysis etc. All the experimental results suggest that the proposed image encryption scheme is robust and secure and can be used for secure image and video communication applications.

Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulliστ-shift rule, which is members of Wolfram’s class II, and it was said to be simple as periodic before. It is worthwhile to study dynamical behaviors of rule 58 and whether it possesses chaotic attractors or not. It is shown that there exist two Bernoulli-measure attractors of rule 58. The dynamical properties of topological entropy and topological mixing of rule 58 are exploited on these two subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems, we divide determinative block systems into two subsets. In addition, it is shown that rule 58 possesses rich and complicated dynamical behaviors in the space of bi-infinite sequences. Furthermore, we prove that four rules of global equivalence classε43of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations, and we find that these attractors are not global attractors. The Lameray diagram is used to show clearly the iterative process of an attractor.

Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 24, which is Bernoulliστ-shift rule and is member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 24 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 24 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Furthermore, we prove that four rules of global equivalenceε52of cellular automata are topologically conjugate. Then, we use diagrams to explain the attractor of rule 24, where characteristic function is used to describe the fact that all points fall into Bernoulli-shift map after two iterations under rule 24.

An Image Encryption Scheme Based on Bit Circular Shift and Bi-directional Diffusion

A novel image encryption scheme based on chaotic system is proposed. The proposed encryption scheme utilizes one tent map to generate a pseudo- random sequence and then shift the bits of the expanding 0-1 image circularly so as to shuffle the image gray values. To make the encryption scheme resist differential attack efficiently, generalized Arnold maps and Bernoulli shift maps are applied to produce two pseudo-random gray value sequences and then diffuse the gray values bi-directionally. The bit circular shift process and diffusion processes greatly confuse the statistical nature between plain-images and cipher- images. Security analyses including key sensitivity analysis, key space analysis, statistical analysis, differential attack analysis and information entropy analysis are performed. All the experimental results demonstrate that the proposed image encryption scheme possesses large key space to frustrate brute-force attack efficiently and can resist statistical attack, differential attack, etc.

CDSK 방식의 디지털 카오스 통신 시스템

최근 정보 보안이 중요시됨에 따라서 향상된 보안성과 낮은 도청 확률을 가지는 무선 통신 기술에 대한 관심이 급격히 증가하고 있다. 카오스 신호는 불규칙한 현상으로 인하여 정보를 효율적으로 부호화 하는데 적용할 수 있다. 카오스 신호는 초기 조건에 매우 민감하며 초기 조건을 모른다면 신호의 감지가 어렵다는 단점을 가지고 있지만 다중 경로 간섭에 강한 장점을 가진다. 본 논문에서는 디지털 카오스 변조 방식인 CDSK 방식에서 여러 가지 카오스 맵을 적용하여 수신 성능을 평가한다. 또한, CDSK 방식에서 확산인자 (Spreading factor)의 변화에 따른 수신 BER 성능을 분석한다. 시뮬레이션 결과, Hennon map을 사용할 경우 다른 Chaotic map들 보다 훨씬 좋은 수신 성능을 얻을 수 있었으며, 확산인자가 70에서 가장 좋은 수신 BER 성능을 얻는 것을 확인하였다. Recently, interest for wireless communication technology with improved security and low eavesdropping probability is increasing rapidly recognizing that information security is an important. Chaos signal can be used encode information efficiently due to irregular phenomena. Chaotic signal is very sensitive to the initial condition. Chaos signal is difficult to detect the signal if you do not know the initial conditions. Also, chaotic signal has robustness to multipath interference. In this paper, we evaluate the performance of correlation delay shift keying (CDSK) modulation with different chaotic map such as Tent map, Logistic map, Henon map, and Bernoulli shift map. Also, we analyze the BER performance depending on the selection of spreading factor (SF) in CDSK. Through the theoretical analyses and simulations, it is confirmed that Henon map has better BER performance than the other three chaotic maps when spreading factor is 70.

A Secure and Robust Image Encryption Scheme Based on Mixture of Multiple Generalized Bernoulli Shift Maps and Arnold Maps

In this paper, a chaotic image encryption scheme with an efficient permutation-diffusion mechanism is constructed, where six generalized Bernoulli shift maps and one six-dimensional Arnold map are utilized to generate one hybrid chaotic orbit applied to disorder the pixel positions in the permutation process while four generalized Bernoulli shift maps and one Arnold map are employed to yield two random gray value sequences to change the gray values by a two-way diffusion process. Several merits of the proposed image encryption scheme are achieved, including a huge key space, good statistical properties resisting statistical attack and differential attack, desirable robustness against malicious attacks on cipher-images, such as cropping, noising, JPEG compression, etc. Experimental results have been carried out with detailed analysis to show that the proposed scheme can be a potential candidate for practical image encryption.

Complex Dynamic Behaviors in Cellular Automata Rule 14

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulli στ‐shift rule and is a member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 14 of one‐dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 14 is chaotic in the sense of both Li‐Yorke and Devaney on its attractor. Then, we prove that there exist fixed points in rule 14. Finally, we use diagrams to explain the attractor of rule 14, where characteristic function is used to describe that all points fall into Bernoulli‐shift map after two iterations under rule 14.