Piecewise Linear Chaotic Map Research Articles

An evolutionary robust solar radiation prediction model based on WT-CEEMDAN and IASO-optimized outlier robust extreme learning machine

As a kind of clean energy, solar energy occupies a pivotal position in energy applications. Accurate and reliable solar radiation prediction is critical to the application of solar energy. In particular, a novel solar radiation prediction based on wavelet transform (WT), complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), improved atom search optimization (IASO) and outlier-robust extreme learning machine (ORELM) is proposed for solar radiation prediction. First, WT is used to denoise the original solar radiation time series, and CEEMDAN method is used to decompose the denoised sequence into intrinsic mode function (IMF) components with different distributions according to the fluctuation scale. Then the IASO algorithm is used to optimize the weights and thresholds of the ORELM to improve the performance of the ORELM model. Levy flight is added to the ASO to enhance the local and global search capability while the chaos population initialization based on piecewise linear chaotic map (PWLCM) is taken to improve the randomness and ergodicity of the initial population within the feasible region. Finally, the comparison with other benchmark models verifies the robustness and accuracy of the proposed solar radiation prediction model.

A novel systematic byte substitution method to design strong bijective substitution box (S-box) using piece-wise-linear chaotic map

Cryptography deals with designing practical mathematical algorithms having the two primitive elements of confusion and diffusion. The security of encrypted data is highly dependent on these two primitive elements and a key. S-box is the nonlinear component present in a symmetric encryption algorithm that provides confusion. A cryptographically strong bijective S-box structure in cryptosystem ensures near-optimal resistance against cryptanalytic attacks. It provides uncertainty and nonlinearity that ensures high confidentiality and security against cryptanalysis attacks. The nonlinearity of an S-box is highly dependent on the dispersal of input data using an S-box. Cryptographic performance criteria of chaos-based S-boxes are worse than algebraic S-box design methods, especially differential probability. This article reports a novel approach to design an 8 × 8 S-box using chaos and randomization using dispersion property to S-box cryptographic properties, especially differential probability. The randomization using dispersion property is introduced within the design loop to achieve low differential uniformity possibly. Two steps are involved in generating the proposed S-box. In the first step, a piecewise linear chaotic map (PWLCM) is utilized to generate initial S-box positions. Generally, the dispersion property is a post-processing technique that measures maximum nonlinearity in a given random sequence. However, in the second step, the concept is carefully reverse engineered, and the dispersion property is used within the design loop for systematic dispersal of input substituting sequence. The proposed controlled randomization changes the probability distribution statistics of S-box’s differentials. The proposed methodology systematically substitutes the S-box positions that cause output differences to recur for a given input difference. The proposed S-box is analyzed using well-established and well-known statistical cryptographic criteria of nonlinearity, strict avalanche criteria (SAC), bit independence criteria (BIC), differential probability, and linear probability. Further, the S-box’s boomerang connectivity table (BCT) is generated to analyze its strength against boomerang attack. Boomerang is a relatively new attacking framework for cryptosystem. The proposed S-box is compared with the state-of-the-art latest related publications. Results show that the proposed S-box achieves an upper bound of cryptographic properties, especially differential probability. This work hypothesizes that highly dispersive hamming distances at output difference, generated a systematic S-box. The mixing property of chaos generated trajectories utilized for decimal mapping. To test the randomness of generated chaotic trajectories, a cryptographically secure pseudo-random sequence was generated using a chaotic map that was tested using the National Institute of Standards and Technology (NIST) NIST-800-22 test suit.

Enhanced design and hardware implementation of a chaos-based block cipher for image protection

Nowadays chaos-based cryptography systems are widely used to protect sensitive data. A variety of chaos-based cryptosystems with software implementation has been published in the literature. However, due to the rapid growth of connected devices, the development of hardware secure and low-resource cryptographic systems is necessary. In this paper, we enhance the design of one of our software cryptosystems and then we address its FPGA-based implementation. The improvement in the design consists essentially to introduce an Initial Vector (IV) entry to the Chaotic Pseudo Random Number Generator (CPRNG) thus allowing to generate a new dynamic keys for the confusion layer (using a modified 2-D cat map) and diffusion layer (using a logistic map) with each new execution of the system. The CPRNG is formed by XORred a discrete Skew Tent Map (STM) with a discrete Piecewise Linear Chaotic Map (PWLCM). The proposed cryptosystem, which operates in Cipher Block Chaining (CBC) mode, is controlled by a Finite State Machine (FSM) to ensure that both the encryption and decryption are performed by the same reconfigurable architecture. The system is implemented on a Xilinx XC7Z020 ZYNQ FPGA platform. The results obtained in terms of hardware implementation costs (logic resources, throughput, and efficiency) and in terms of security against cryptographic attacks demonstrate the effectiveness of the proposed chaos-based cryptosystem.

Efficient Hyperchaotic Image Encryption Algorithm Based on a Fast Key Generation Method and Simultaneous Permutation-Diffusion Structure

To provide faster encryption speed for protecting image data privacy, a highly efficient hyperchaotic image encryption algorithm using a fast key generation method and simultaneous permutation-diffusion structure is proposed in this paper. Firstly, we design a novel fast key generation method to obtain four key matrixes with the size of W × H (W and H are the width and height of an image) with the hyperchaotic Lorenz system only iterating 2W times. And within the key generation process, the piecewise linear chaotic map serves as a random selector to further enhance the randomness and unpredictability. Secondly, we introduce vectorization technique to parallelize the cipher blockchain-based encryption process. Hence, an image can be fast and sufficiently encrypted row by row and column by column through a four-round and interlinked diffusion process. Furthermore, during the diffusion process, the pixels are simultaneously scrambled by the Chebyshev map to enhance the antiattack capability of the proposed algorithm. Thirdly, we utilize a secure hash function to generate a plaintext-related secret key. Finally, cipher images can successfully pass common security tests with the correlation coefficients on the verge of 0, the entropies are greater than 7.99, and NPCR and UACI values are close to the expected values. Combined with other experimental results and analysis, we can safely conclude that the proposed algorithm has strong robustness against statistical, brute-force, differential, and some other common attacks. Moreover, only 0.009 s of encrypting a 256 × 256 image indicates that our algorithm has extensive application prosperity for real-time communications of massive digital images.

Image Watermarking Scheme Using LSB and Image Gradient

In the modern age, watermarking techniques are mandatory to secure digital communication over the internet. For an optimal technique, a high signal-to-noise ratio and normalized correctional is required. In this paper, a digital watermarking technique is proposed on the basis of the least significant bit through an image gradient and chaotic map. The image is segmented into noncorrelated blocks, and the gradient of each block is calculated. The gradient of the image expresses the rapid changes in an image. A chaotic substitution box (S-Box) is used to scramble the watermark according to a piecewise linear chaotic map (PWLCM). PWLCM has a positive Lyapunov exponent and better balance property as compared to other chaotic maps. This S-Box technique is capable of producing a disperse sequence with high nonlinearity in the generated sequence. Least significant bit is a simple technique for embedding but it has a high payload capacity and direct pixel manipulation. The embedding payload introduces a tradeoff between robustness and imperceptibility; hence, the image gradient is a technique to identify the best-suited place to embed a watermark and avoid image degradation. By modifying the least significant bits of the original image, the watermark signal is embedded according to the image gradient. In the image gradient, the direction and magnitude decide how much embeding can be done. In comparison with other methods, the experimental results show satisfactory progress in robustness against several image processing and geometrical attacks while maintaining the imperceptibility of the watermark signal.

Image encryption using modified perfect shuffle-based bit level permutation and learning with errors based diffusion for IoT devices

Image encryption is important for IoT devices which handle military, medical and other confidential images. In this paper, stream cipher based image diffusion is performed using tent chaotic map and block based encryption is done by taking four bytes every time to multiply the matrix formed using the secret key and error matrix. Pixel permutation is performed in two stages: block level pixel permutation and whole image permutation to resist the statistical attacks. Bit level permutation is performed with piecewise linear chaotic map using modified perfect shuffle technique to overcome differential attacks. Various statistical and differential analyses are performed on the encrypted image and the results obtained using these analyses convey its resistance to statistical and differential attacks.

Hardware architecture of a digital piecewise linear chaotic map with perturbation for pseudorandom number generation

Piecewise linear chaotic map (PWLCM) is a simple chaotic system popularly used for generating pseudorandom numbers (PRN). Digital implementation of PWLCM degrades the randomness of the pseudorandom number due to the finite precision effect. The current paper presents a digital 1-dimensional PWLCM that mitigates the degradation of digital chaos by cascading the chaotic map with a three-stage XORed shift register that perturbs the output of the PWLCM. We further perturb the output of the shift register using a one-stage XOR post-processing. We also present the hardware architecture of the proposed system to generate a high throughput PRN sequence. The designed architecture is synthesized on Xilinx Zynq 7000 Field Programmable Gate Array (FPGA) and Application Specific Integrated Circuit (ASIC) using TSMC (Taiwan Semiconductor Manufacturing Company) 90 nm CMOS technology. We experimentally validate the randomness of the output sequence using NIST test suite 800–22. In addition, we also perform security analyses such as phase space, key sensitivity, correlation, and information entropy. The statistical and security analysis confirms the efficiency of the proposed design in terms of randomness. The synthesis result demonstrates the achieved throughput as 1.296 Gbps at 81 MHz operating frequency on Zynq 7000 FPGA, whereas a throughput of 1.696 Gbps at 106 MHz operating frequency is achieved on ASIC.

A novel color image encryption scheme based on a new dynamic compound chaotic map and S-box.

In modern technological era image encryption has become an attractive and interesting field for researchers. They work for improving the security of image data from unauthorized sources. Chaos theory, due to its randomness and unpredictable behaviors, is considered favorite for the purpose of image encryption. This paper proposes a diffusion based image encryption algorithm by using chaotic maps. Firstly a chaotic map (piecewise linear chaotic map) is used for the generation of S-box, then it is used for the pixel values modification to generate element of non-linearity. After this these modified values are further diffused with another random sequence, generated by tent logistic chaotic map. Finally the color components of pre-encrypted image are mixed with each other so that the developed randomness uniformly distributed in them. For image data we develop non-linearity and diffusion by using S-box and then more randomness is added in the pre-encrypted image with the help of Boolean operation XOR. The use of this combination of chaotic maps along with S-box and Boolean operation XOR is a different technique, that provides satisfactory results for security aspects and also works efficiently.

IEPSBP: A Cost-Efficient Image Encryption Algorithm Based on Parallel Chaotic System for Green IoT

With the fast development of the Internet of Things (IoT) technologies, more IoT devices are currently connected with the Internet, resulting in more exchange of information. However, data privacy and security threats have become emerging challenges of the IoT. In this paper, we are concerned about the security of image transmission in green IoT. Image encryption algorithms for green IoT are faced with two challenges: 1) To save the cost, devices in green Internet of Things (IoT) always have very low computing power, so they cannot support high-precision computing and 2) The algorithm deployed on the device should be efficient to guarantee the device to run for a long time. To solve the problem of precision limited, we propose a parallel chaotic system named PSBP. <xref ref-type="fn" rid="fn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¹</xref> The PSBP is composed of Piecewise Linear Chaotic Map (PWLCM), Skew Tent Map (STM) and Bernoulli map in parallel model with the limited precision of 16-bit, which can generate good-performance key matrix for encryption. Then, we introduce a cost-efficient image encryption algorithm based on PSBP (named IEPSBP) which addresses the second challenge. Instead of bits or bytes, we use one row or one column as the basic unit to realize permutation and diffusion. The security analysis and performance experiments indicate that IEPSBP is secure and cost-efficient for green IoT.

Dynamic S-Box and PWLCM-Based Robust Watermarking Scheme

Due to the increased number of cyberattacks, numerous researchers are motivated towards the design of such schemes that can hide digital information in a signal. Watermarking is one of the promising technologies that can protect digital information. However, traditional watermarking schemes are either slow or less secure. In this paper, a dynamic S-Box based efficient watermarking scheme is presented. The original image was extracted at the receiver’s end without any loss of sensitive information. Firstly, the Secure Hash Algorithm is applied to the original image for the generation of the initial condition. Piece Wise Linear Chaotic Map is then used to generate 16 times 16 dynamic Substitution Box (S-Box). As an additional security feature, the watermark is substituted through dynamic S-Box. Hence, it is hard for the eavesdroppers to attack the proposed scheme due to the dynamic nature of S-Box. Lastly, lifting wavelet transform is applied to the host image and the High Low and High High blocks of host image are replaced with least significant bits and most significant bits of the substituted watermark, respectively. Robustness, efficiency and security of the proposed scheme is verified using Structure Similarity Index, Structure Dissimilarity Index, Structure Content, Mutual Information, energy, entropy, correlation tests and classical attacks analysis.

An efficient image encryption scheme for healthcare applications.

In recent years, there has been an enormous demand for the security of image multimedia in healthcare organizations. Many schemes have been developed for the security preservation of data in e-health systems however the schemes are not adaptive and cannot resist chosen and known-plaintext attacks. In this contribution, we present an adaptive framework aimed at preserving the security and confidentiality of images transmitted through an e-healthcare system. Our scheme utilizes the 3D-chaotic system to generate a keystream which is used to perform 8-bit and 2-bit permutations of the image. We perform pixel diffusion by a key-image generated using the Piecewise Linear Chaotic Map (PWLCM). We calculate an image parameter using the pixels of the image and perform criss-cross diffusion to enhance security. We evaluate the scheme’s performance in terms of histogram analysis, information entropy analysis, statistical analysis, and differential analysis. Using the scheme, we obtain the average Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) values for an image of size 256 × 256 equal to 99.5996 and 33.499 respectively. Furthermore, the average entropy is 7.9971 and the average Peak Signal to Noise Ratio (PSNR) is 7.4756. We further test the scheme on 50 chest X-Ray images of patients having COVID-19 and viral pneumonia and found the average values of variance, PSNR, entropy, and Structural Similarity Index (SSIM) to be 257.6268, 7.7389, 7.9971, and 0.0089 respectively. Furthermore, the scheme generates completely uniform histograms for medical images which reveals that the scheme can resist statistical attacks and can be applied as a security framework in AI-based healthcare.

A NOVEL METHOD FOR GENERATING DIGITAL IMAGE OF ENCRYPTED SPEECH SIGNAL BASED ON FWHT AND WPLCM

This paper suggests a novel algorithm for encrypting speech signals in common image formats and retrieve them from these image files. The speech signal is encrypted in three levels. In the first level,the sample positions are permuted based on keys generated using Game of Life matrix and Piecewise linear Chaotic Map (PWLCM) in order to reduce the correlation between adjust samples. In the second level, the resulting samples are then converted to Fast Walsh Hadamard Transform (FWHT) and their transactions are encrypted by using circular transformations in the row and column depending on the generated key. At the third level, the values of encrypted samples are converted to color pixels, which are then arranged in a puzzled manner and put in a 2-D matrix to achieve the secured data transfer across networks, with the image file contains the encrypted speech signal. Several objective measures have been used to evaluate the performance of the suggested method. The experimental results and numerical analyses show that the algorithm gives a high degree of security and robust against brute force attackers, statistical attack, strong diffusion and ambiguity so that the encrypted message has been saved in a different format from the original signal, and finally give the good quality of the reconstructed speech signal from image files.

Periodic Points of Asymmetric Bernoulli Shifts

It is well-known that Sharkovskii’s theorem gives a complete structure of periodic order for a continuous self-map on a closed bounded interval. As a further study, a natural problem is how to determine the location and number of periodic points for a specific map. This paper considers the periodic points of asymmetric Bernoulli shift, which is a piecewise linear chaotic map.

Image compression–encryption scheme using SPIHT and chaotic systems

The world is moving towards digitization for all forms of data. Images are one of the most used data that contains a lot of information. With the increasing network technology, the sharing of data across the internet has also manifold. Certain images are confidential and require security for storing or transmitting on an insecure communication channel. Digital images usually have highly correlated data where compression can be performed to reduce the load on the transmission channel or storage memory. The paper proposes an algorithm to compress the digital images and encryption is performed on the compressed data to provide security. Compression of the image is performed using the Set partitioning in hierarchical trees algorithm. The compressed image is encrypted using a chaotic sequence generated using the Henon map and the cipher image is scrambled using a permutation table obtained from Piecewise Linear chaotic map. To reduce the computation while generating the chaotic sequence, a base conversion operation is used. The initial parameters of the chaotic systems are generated from the elliptic curve point are shared securely using ElGamal based elliptic curve cryptosystem. Security analyses are conducted along with a comparison with other existing systems. The proposed algorithm is secured against various attacks and outperforms existing techniques concerning various statistical and security aspects of an enciphering algorithm.

A Multistage Blockchain-Based Secure and Trustworthy Smart Healthcare System Using ECG Characteristic

With the rapid development of advanced biomedical sensors, the Internet of Things, and modern wireless communication technologies, smart healthcare systems provide feasible solutions to the problems of population aging and telemedicine services. However, physiological information involves personal privacy, and the data security concern during the transmission of information on public channels has become a critical issue that has restricted the wider acceptance of smart healthcare systems. In this article, a secure and trustworthy smart healthcare system interoperating with wireless body area networks based on multistage blockchain is proposed. The security scheme provides a completely secure and trustworthy environment covering the entire data flow from the front-end to the back-end. The evaluation results show that the encryption scheme based on piecewise linear chaotic map can resist common attack methods and reduce encryption time. The system has the advantages of low complexity for encryption scheme, and larger capacity and efficiency for the blockchain-based data transmission and storage system.

Color image encryption system using combination of robust chaos and chaotic order fractional Hartley transformation

The paper proposes a robust color image encryption scheme based on a real fractional Hartley transform with chaotic transform orders. The multilayer scheme comprises of a PWLCM (piecewise linear chaotic map) based circular blending of color components, PWNCA (piecewise nonlinear chaotic map) based nonlinear processing in spatial domain followed by a multiple chaotic order fractional Hartley transform to obtain encrypted image. The input parameters for chaotic mapping used in each encryption stage serve as secret keys. A reality preserving transform enables to eliminate the complexity issues of transform.The proposed schemenot onlyovercomes the limitation of linearity in transform orders of multiparameter scheme but also thevulnerability of transformto information leakage. Besides this, the use of multilayer security has led to larger key space. Results and analysis substantiate the fact that proposed scheme is robust,highly sensitiveand showsconsiderably betterperformance than other similar-state-of-artschemes.

Robust medical image encryption based on DNA-chaos cryptosystem for secure telemedicine and healthcare applications

The security of healthcare and telemedicine systems is a critical issue that must be significantly investigated. Several smart telemedicine applications are expected to be adopted in the medical sector in the incoming years. Healthcare smart products that are connected through Internet to be accessible anytime and anywhere are expected to deal with critical and confidential information such as personal medical images. Therefore, medical image encryption is an important task in telemedicine and healthcare applications. This paper presents an efficient cryptosystem for medical image security based on exploiting the advantages of the de-oxyribo nucleic acid (DNA) rules and chaos maps. In the proposed medical image cryptosystem, logistic chaos map, piecewise linear chaotic map (PWLCM), and DNA encoding are employed. The PWLCM is employed to generate a secret key image. Then, the DNA rules are utilized for encoding the secret key image and the input plain image by rows that are encoded with the logistic chaos map. After that, the proposed logistic map is employed to obtain an intermediate image as another secret key image to set DNA functions row-by-row on the coded original image. Moreover, the intermediate image is decoded in the following stage. Finally, the previous actions are iterated through image columns once again to obtain the best ciphered image. The experimental results reveal that the suggested cryptosystem has a high security with an acceptable processing time. In addition, it can resist various kinds of attacks, such as known-plaintext and chosen-plaintext attacks.

A Robust and Fast Image Encryption Scheme Based on a Mixing Technique

This paper introduces a new image encryption scheme using a mixing technique as a way to encrypt one or multiple images of different types and sizes. The mixing model follows a nonlinear mathematical expression based on Cramer’s rule. Two 1D systems already developed in the literature, namely, the May-Gompertz map and the piecewise linear chaotic map, were used in the mixing process as pseudo-random number generators for their good chaotic properties. The image to be encrypted was first of all partitioned into N subimages of the same size. The subimages underwent a block permutation using the May-Gompertz map. This was followed by a pixel-based permutation using the piecewise linear chaotic map. The result of the two previous permutations was divided into 4 subimages, which were then mixed using pseudo-random matrices generated from the two maps mentioned above. Tests carried out on the cryptosystem designed proved that it was fast due to the 1D maps used, robust in terms of noise and data loss, exhibited a large key space, and resisted all common attacks. A very interesting feature of the proposed cryptosystem is that it works well for simultaneous multiple-image encryption.

Color image encryption scheme based on fractional Hartley transform and chaotic substitution–permutation

We propose a novel opto-digital method of color image encryption which utilizes compound chaotic mappings, the reality preserving fractional Hartley transformation and piecewise linear chaotic map for substitution, optical processing and permutation of image pixels, respectively. The image to be encrypted initially undergoes a chaos-based substitution in the spatial domain through the compound chaotic maps followed by a transformation to the combined time–frequency domain using the fractional Hartley transform. A reality preserving version of the fractional Hartley transform is used to eliminate the complexity associated with transform coefficients. Optical transformation of the image, in the fractional Hartley domain, is followed by a permutation through piecewise linear chaotic maps. Due to the intertwined application of optical transformation and chaos-based substitution and permutation processes, the proposed image encryption scheme possesses higher security. The input parameters (initial conditions, control parameters, and number of iterations) of chaotic maps along with fractional orders of the fractional Hartley transform collectively form the secret keys for encryption/decryption. The proposed scheme is a lossless and symmetric encryption scheme. The level of security provided in terms of high sensitivity to keys, resistivity to brute-force attack, classical attacks, differential attacks, entropy attack, noise and occlusion attack along with the elimination of complex coefficients proves its better efficacy as compared to other similar state-of-the-art schemes.

Bit-level image encryption algorithm based on BP neural network and gray code

In recent years, people have put forward various image encryption algorithms based on pixel level. In fact, bit level encryption has better effect than pixel level encryption. Therefore, this paper proposes a new bit-level image encryption algorithm based on Back Propagation (BP) neural network and Gray code. Firstly, the plaintext image is conversioned into binary image, then, the hyperchaotic Lorentz system is used to generate two sets of chaotic sequences for the Gray code bit-level permutation operation to generate the permutation matrix. Secondly, the permutation matrix is converted into a bit matrix reverse order output to generate a diffusion matrix. Finally, the algorithm uses a BP neural network composed of Logistic map and Piece-Wise Linear Chaotic (PWLCM) map to generate a key stream. The key stream is xored with the diffusion matrix to generate a ciphertext matrix. The experimental results show that the algorithm improves the encryption efficiency, has good security and can resist common attack methods.