Two-dimensional Chaotic Map Research Articles

Constructing keyed strong S-box with optimized nonlinearity using nondegenerate 2D hyper chaotic map

Abstract In the realm of information security, the nonlinearity of an S-Box is a pivotal attribute for ensuring its effectiveness. S-Boxes randomly generated with chaotic maps, however, often exhibit insufficient nonlinearity. To construct a robust S-Box with enhanced nonlinearity, we have developed a nondegenerate two-dimensional hyper chaotic map (2D-NDHCM). We rigorously evaluated its dynamic properties through bifurcation and phase diagrams, Lyapunov exponents, Kolmogorov entropy, correlation dimension, and randomness tests to assess the map's resilience. Subsequently, we employed the 2D-NDHCM to devise a novel algorithm for constructing and optimizing a keyed strong S-Box, leveraging simulated annealing technique to achieve superior nonlinearity. Finally, We conducted performance analyses of the optimized S-Box using the classic six criteria, and the results confirmed the algorithm's practicality and effectiveness.

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

Dynamical analysis of a novel 2D Lyapunov exponent controllable memristive chaotic map.

The proposal of discrete memristors has made memristive chaotic maps based on them an important research topic. In this study, a new two-dimensional chaotic map without fixed points is constructed, and numerical simulation results display its rich dynamical behaviors. The analysis reveals the map's center inversion symmetry and Lyapunov exponent controller. The map exhibits complex dynamical behaviors, including memristor initial-boosting and single-parameter-offset boosting. Embedding the absolute value function within the memristor results in the emergence of localized boosting-free regions. Moreover, a class of multicavity transients is captured that greatly enhances the system's complexity. Ultimately, this map is implemented on the STM32 platform, demonstrating its practical applicability in potential practical application scenarios.

Visually secure traffic image encryption scheme using new two-dimensional Sigmoid-type memristive chaotic map and Laguerre transform embedding

In intelligent transportation system, unprotected bare data transmission faces serious security threats and challenges. To this end, this paper proposes a visually secure traffic image encryption scheme that combines a newly designed two-dimensional Sigmoid-type memristive chaotic map (2D-SMCM) with two-dimensional compressive sensing (2D-CS) and Laguerre transform (LT) embedding to provide services for secure transmission of private images. Specifically, first, the 2D-SMCM is used to generate pseudo-random sequences for subsequent compression, encryption and hiding operations. Second, the 2D-CS is utilized to compress the plain image to reduce the amount of data transmission. Then, encryption is completed by modifying the data values and their positions through index permutation and bidirectional diffusion. Finally, the encrypted data is embedded in the LT-processed public carrier medium for covert transmission. Experiments and performance analysis illustrate that the proposed scheme has good security, imperceptibility and reconstruction performance, with the average PSNRs of the cipher images and decrypted secret images up to 45.90 dB and 34.85 dB, respectively, using 500 grayscale images from the database BOWS2.

An n-Dimensional Chaotic Map with Application in Reversible Data Hiding for Medical Images.

The drawbacks of a one-dimensional chaotic map are its straightforward structure, abrupt intervals, and ease of signal prediction. Richer performance and a more complicated structure are required for multidimensional chaotic mapping. To address the shortcomings of current chaotic systems, an n-dimensional cosine-transform-based chaotic system (nD-CTBCS) with a chaotic coupling model is suggested in this study. To create chaotic maps of any desired dimension, nD-CTBCS can take advantage of already-existing 1D chaotic maps as seed chaotic maps. Three two-dimensional chaotic maps are provided as examples to illustrate the impact. The findings of the evaluation and experiments demonstrate that the newly created chaotic maps function better, have broader chaotic intervals, and display hyperchaotic behavior. To further demonstrate the practicability of nD-CTBCS, a reversible data hiding scheme is proposed for the secure communication of medical images. The experimental results show that the proposed method has higher security than the existing methods.

An Efficient Image Encryption Method Based on Enhanced Josephus Problem and a Non-Invertible Economic Map

As an increasing number of digital images are created and transmitted over the internet, there is growing concern over their unauthorized use, which has a big impact on both security and privacy concerns. In this research, we provide a fast and secure image encryption scheme by using an enhanced Josephus problem and two-dimensional non-invertible economic chaotic map (2D-ECM) to safely and covertly protect digital image information throughout public channel transmission. First, the initial values of the 2D-ECM map are generated based on Secure Hash Algorithm (SHA-256) and the input secret key. Then, the Josephus problem is enhanced by substituting the extract operation with location exchange and dynamic start location and step size is employed to scramble the image pixels. In order to integrate the confusion process and diffusion process, the enhanced Josephus problem is utilized indirectly to choose two random columns from the scrambled image and random image to XOR them with the current column. The evaluation results prove that the proposed image cryptosystem is more efficient compared to existing cryptosystems.

An image crypt model based on enhanced two-dimensional sine-cosine chaotic map

Sensitive images are being transmitted over insecure media as network and information technology evolve at a rapid pace. As a result, people are becoming more concerned about information security, specifically digital image protection. For securing digital images, numerous image encryption algorithms have been proposed. From that, a chaotic map has been proved a perfect image protection tool as it contains efficient random behaviour. This work proposes a novel two-dimensional sine-cosine map for developing an image cryptosystem. In the developed framework, at first, two random chaotic series are generated by the developed sine-cosine map, then the given image is divided into two blocks, and each block is confused by utilizing different chaotic key series. In the diffusion process, the chaotic series is first changed into integer form, which is applicable for executing some arithmetic operation between chaotic values and image pixels. In the diffusion phase, each confused image block pixel is converted into binary form, and a bit shifting operation is executed depending on the integer form of chaotic series. In the final process, a bit-xor operation is performed between the partial diffused image and chaotic series to obtain the final encrypted image. The proposed model's security strength will be evaluated using statistical and differential attack analysis.

Quantification of chaos in a time series generated from a traffic flow model using the extended entropic chaos degree

Several Traffic flow models have been proposed to address the underlying causes of traffic congestion and develop effective methods for its suppression. Typically, these models are represented using differential or difference equations. Traffic flow models can exhibit chaotic behaviors. Traffic situations become unstable if chaos prevails in the traffic flow. Therefore, the conditions under which chaos occurs in the traffic flow and the intensity of chaos at that time must be investigated. The Lyapunov exponent is used for quantifying the chaos in the dynamical systems. However, computing the Lyapunov exponent over time series without a dynamical map is challenging. Recently, the extended entropic chaos degree has been introduced as an extension of the entropic chaos degree. The extended entropic chaos degree can be computed directly for any time series. Analytically, the extended entropic chaos degree equals the sum of all the Lyapunov exponents for multidimensional non-periodic maps. Moreover, the extended entropic chaos degree is mathematically shown to equal the sum of one positive and one negative Lyapunov exponents for two-dimensional typical chaotic maps, such as a generalized Baker’s map, and a standard map.In this study, we utilize the extended entropic chaos degree to quantify chaos in a time series with less computing complexity compared with conventional methods. First, we demonstrate that chaos can be quantified in a time series generated from a traffic flow model using an extended entropic chaos degree. Lyapunov exponent for a time series of time traveling between two adjacent traffic lights in the traffic flow model is not directly computable because its dynamics are not given as an explicit differential map. Second, we demonstrate that the computational complexity of the extended entropic chaos degree only corresponds to O(M). However, the computational complexities of the estimated Lyapunov exponents using the Wolf method are O(M2), where M is the number of points on a one-dimensional time series. Thus, we show that the extended chaos degree can quantify the chaos in the traffic flow model, such as one vehicle moving through a sequence of traffic lights in one dimension, with less complexity, where the chaos occurs depending on the timing of signal switching.

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.

Two-dimensional memristive hyperchaotic maps with different coupling frames and its hardware implementation.

Continuous-time memristors have been used in numerous chaotic circuit systems. Similarly, the discrete memristor model applied to a discrete map is also worthy of further study. To this end, this paper first proposes a discrete memristor model and analyzes the voltage-current characteristics of the memristor. Also, the discrete memristor is coupled with a one-dimensional (1D) sine chaotic map through different coupling frameworks, and two different two-dimensional (2D) chaotic map models are generated. Due to the presence of linear fixed points, the stability of the 2D memristor-coupled chaotic map depends on the choice of control parameters and initial states. The dynamic behavior of the chaotic map under different coupled map frameworks is investigated by using various analytical methods, and the results show that different coupling frameworks can produce different complex dynamical behaviors for memristor chaotic maps. The dynamic behavior based on parameter control is also investigated. The numerical experimental results show that the change of parameters can not only enrich the dynamic behavior of a chaotic map, but also increase the complexity of the memristor-coupled sine map. In addition, a simple encryption algorithm is designed based on the memristor chaotic map under the new coupling framework, and the performance analysis shows that the algorithm has a strong ability of image encryption. Finally, the numerical results are verified by hardware experiments.

A Robust Chaotic Map and Its Application to Speech Encryption in Dual Frequency Domain

When chaotic systems are used for speech encryption, their chaotic performance largely determines the security of speech encryption. However, traditional chaotic systems have problems such as parameter discontinuity, easy occurrence of chaos degradation, low complexity, and the existence of periodic windows in chaotic intervals. In real applications, chaotic mappings may fall into periodic windows, which is extremely unfavorable for security. In this paper, a new chaotic mapping 2D-LMSM is proposed by improving the chaotic logistic and sine mappings, and applied to speech encryption. Performance evaluation shows that this map can effectively generate robust chaotic signals in a wide parameter range. The 2D-LMSM achieves better robustness and desired chaotic properties than several existing two-dimensional chaotic maps. We propose a novel speech encryption algorithm using this map. First, it performs Fast Fourier Transform (FFT) on the input speech signal to obtain real and imaginary values, which are encrypted by one-time scrambling encryption and XOR diffusion encryption with pseudorandom numbers generated by chaos; then, it performs secondary scrambling encryption by Discrete Wavelet Transform (DWT) and 2D-LMSM; finally, it obtains encrypted speech data by Discrete Wavelet Inverse Transform (IDWT) and Fast Fourier Inverse Transform (IFFT). Experimental results show that this algorithm has good encryption and decryption performances and ensures system security.

Image encryption using the new two-dimensional Beta chaotic map

Image encryption using the new two-dimensional Beta chaotic map

Proposed Biometric Security System Based on Deep Learning and Chaos燗lgorithms

Nowadays, there is tremendous growth in biometric authentication and cybersecurity applications. Thus, the efficient way of storing and securing personal biometric patterns is mandatory in most governmental and private sectors. Therefore, designing and implementing robust security algorithms for users’ biometrics is still a hot research area to be investigated. This work presents a powerful biometric security system (BSS) to protect different biometric modalities such as faces, iris, and fingerprints. The proposed BSS model is based on hybridizing auto-encoder (AE) network and a chaos-based ciphering algorithm to cipher the details of the stored biometric patterns and ensures their secrecy. The employed AE network is unsupervised deep learning (DL) structure used in the proposed BSS model to extract main biometric features. These obtained features are utilized to generate two random chaos matrices. The first random chaos matrix is used to permute the pixels of biometric images. In contrast, the second random matrix is used to further cipher and confuse the resulting permuted biometric pixels using a two-dimensional (2D) chaotic logistic map (CLM) algorithm. To assess the efficiency of the proposed BSS, (1) different standardized color and grayscale images of the examined fingerprint, faces, and iris biometrics were used (2) comprehensive security and recognition evaluation metrics were measured. The assessment results have proven the authentication and robustness superiority of the proposed BSS model compared to other existing BSS models. For example, the proposed BSS succeeds in getting a high area under the receiver operating characteristic (AROC) value that reached 99.97% and low rates of 0.00137, 0.00148, and 0.00157 for equal error rate (EER), false reject rate (FRR), and a false accept rate (FAR), respectively.

VLSI Implementation of Optimized 2D SIMM Chaotic Map for Image Encryption

The current research work proposed a novel optimization-based 2D-SIMM (Two-Dimensional Sine Iterative chaotic map with infinite collapse Modulation Map) model for image encryption. The proposed 2D-SIMM model is derived out of sine map and Iterative Chaotic Map with Infinite Collapse (ICMIC). In this technique, scrambling effect is achieved with the help of Chaotic Shift Transform (CST). Chaotic Shift Transform is used to change the value of pixels in the input image while the substituted value is cyclically shifted according to the chaotic sequence generated by 2D-SIMM model. These chaotic sequences, generated using 2D-SIMM model, are sensitive to initial conditions. In the proposed algorithm, these initial conditions are optimized using JAYA optimization algorithm. Correlation coefficient and entropy are considered as fitness functions in this study to evaluate the best solution for initial conditions. The simulation results clearly shows that the proposed algorithm achieved a better performance over existing algorithms. In addition, the VLSI implementation of the proposed algorithm was also carried out using Xilinx system generator. With optimization, the correlation coefficient was −0.014096 and without optimization, it was 0.002585.

Dual Image Cryptosystem Using Henon Map and Discrete Fourier Transform

This paper introduces an efficient image cryptography system. The proposed image cryptography system is based on employing the two-dimensional (2D) chaotic henon map (CHM) in the Discrete Fourier Transform (DFT). The proposed DFT-based CHM image cryptography has two procedures which are the encryption and decryption procedures. In the proposed DFT-based CHM image cryptography, the confusion is employed using the CHM while the diffusion is realized using the DFT. So, the proposed DFT-based CHM image cryptography achieves both confusion and diffusion characteristics. The encryption procedure starts by applying the DFT on the image then the DFT transformed image is scrambled using the CHM and the inverse DFT is applied to get the finally encrypted image. The decryption procedure follows the inverse procedure of encryption. The proposed DFT-based CHM image cryptography system is examined using a set of security tests like statistical tests, entropy tests, differential tests, and sensitivity tests. The obtained results confirm and ensure the superiority of the proposed DFT-based CHM image cryptography system. These outcomes encourage the employment of the proposed DFT-based CHM image cryptography system in real-time image and video applications.

A Henon-like chaotic map and its application in image encryption combined with compressed sensing

A new method of color image encryption is proposed in this paper, with the combination of a two-dimensional (2D) Henon-like chaotic map and compressed sensing for better performance. A sinusoidal function is introduced into the 2D Henon map for increasing randomness. Image encryption algorithm includes three procedures: compression of a color image through the Bernoulli observation matrix, encryption of the compressed image by 2D-Henon map with sinusoidal factor, and reconstruction of color image based on compressed sensing algorithm of orthogonal matching pursuit (OMP). Numerical simulation shows that the proposed encryption algorithm has some advantages including being hardware-friendly, sufficient keyspace, and higher security.

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.

Image encryption algorithm based on a new chaotic system with Rubik's cube transform and Brownian motion model

In this paper a new two-dimensional chaotic map based on Logistic map and Chebyshev map is proposed. The performance analysis and evaluation show the proposed map has a very wide range of chaos and good ergodicity. Further, a new image encryption algorithm is designed using the chaotic map with Rubik's Cube transform and Brownian motion model. The encryption process is ‘confusion - Rubik's cube transformation - diffusion’. In the confusion stage, a method combining Brownian motion theory and chaotic sequence is proposed. Then combining the advantages of chaotic system and Rubik's cube, a Rubik's Cube transformation algorithm is proposed. The scrambled pixel matrix and the chaotic matrix are spliced into a hexahedron structure, and the Rubik's Cube is dynamically scrambled through a pseudo-random sequence. Finally, the XOR operation is performed on the basis of the two-dimensional Hénon chaotic map. The experimental results show that the algorithm has the advantages of large key space, strong robustness and high security, and can resist common cryptanalysis attacks.

Learning chaotic systems from noisy data via multi-step optimization and adaptive training.

A data-driven sparse identification method is developed to discover the underlying governing equations from noisy measurement data through the minimization of Multi-Step-Accumulation (MSA) in error. The method focuses on the multi-step model, while conventional sparse regression methods, such as the Sparse Identification of Nonlinear Dynamics method (SINDy), are one-step models. We adopt sparse representation and assume that the underlying equations involve only a small number of functions among possible candidates in a library. The new development in MSA is to use a multi-step model, i.e., predictions from an approximate evolution scheme based on initial points. Accordingly, the loss function comprises the total error at all time steps between the measured series and predicted series with the same initial point. This enables MSA to capture the dynamics directly from the noisy measurements, resisting the corruption of noise. By use of several numerical examples, we demonstrate the robustness and accuracy of the proposed MSA method, including a two-dimensional chaotic map, the logistic map, a two-dimensional damped oscillator, the Lorenz system, and a reduced order model of a self-sustaining process in turbulent shear flows. We also perform further studies under challenging conditions, such as noisy measurements, missing data, and large time step sizes. Furthermore, in order to resolve the difficulty of the nonlinear optimization, we suggest an adaptive training strategy, namely, by gradually increasing the length of time series for training. Higher prediction accuracy is achieved in an illustrative example of the chaotic map by the adaptive strategy.

Color image encryption based on 2D enhanced hyperchaotic logistic-sine map and two-way Josephus traversing

To improve the chaotic behavior of the chaotic map, this paper proposes a new two-dimensional (2D) chaotic map, namely, 2D enhanced hyperchaotic Logistic-sine map (2D-EHLM), which is derived from the idea of the 2D Logistic map and the Sine map. The performance analyses demonstrate that the 2D-EHLM has significantly better hyperchaotic property, unpredictability characteristics in terms of their Lyapunov exponents, trajectories, and 0-1test in comparison to some existing 2D chaotic maps. Such as the 2D-EHLM performs chaotic behavior when their control parameter μ within (−3,4), which indicates a large chaotic range comparing with other 2D dynamical systems. To investigate its application, this paper proposes a new image confusion strategy based on the two-way Josephus traversing (TJT) to efficiently scramble the color image components and pixel positions. Combining 2D-EHLM and TJT operation, a new color image encryption scheme is proposed. The stimulation experiments and security analysis demonstrate that the proposed algorithm is able to protect the image information with a high security as well as to resist common types of attacks, which also verify the superior hyperchaotic behaviors of the 2D-EHLM from the aspect of encryption effect.