1D Chaotic Maps Research Articles

A new pseudorandom bits generator based on a 2D-chaotic system and diffusion property

A remarkable correlation between chaotic systems and cryptography has been established with sensitivity to initial states, unpredictability, and complex behaviors. In one development, stages of a chaotic stream cipher are applied to a discrete chaotic dynamic system for the generation of pseudorandom bits. Some of these generators are based on 1D chaotic map and others on 2D ones. In the current study, a pseudorandom bit generator (PRBG) based on a new 2D chaotic logistic map is proposed that runs side-by-side and commences from random independent initial states. The structure of the proposed model consists of the three components of a mouse input device, the proposed 2D chaotic system, and an initial permutation (IP) table. Statistical tests of the generated sequence of bits are investigated by applying five evaluations as well as the ACF and NIST. The results of five standard tests of randomness have been illustrated and overcome a value of 0.160 in frequency test. While the run test presents the pass value t0=4.769 and t1=2.929. Likewise, poker test and serial test the outcomes was passed with 3.520 for poker test, and 4.720 for serial test. Finally, autocorrelation test passed in all shift numbers from 1 to 10.

A Novel S-Box Dynamic Design Based on Nonlinear-Transform of 1D Chaotic Maps

In this paper, a method to enhance the dynamic characteristics of one-dimension (1D) chaotic maps is first presented. Linear combinations and nonlinear transform based on existing chaotic systems (LNECS) are introduced. Then, a numerical chaotic map (LCLS), based on Logistic map and Sine map, is given. Through the analysis of a bifurcation diagram, Lyapunov exponent (LE), and Sample entropy (SE), we can see that CLS has overcome the shortcomings of a low-dimensional chaotic system and can be used in the field of cryptology. In addition, the construction of eight functions is designed to obtain an S-box. Finally, five security criteria of the S-box are shown, which indicate the S-box based on the proposed in this paper has strong encryption characteristics. The research of this paper is helpful for the development of cryptography study such as dynamic construction methods based on chaotic systems.

Image encryption algorithm based on 2D hyperchaotic map

It is well known that chaos is unpredictability and sensitivity to initial values and parameters. It is a best choice for cryptography application. However, the existing chaotic cryptography systems have many defects because of the chaotic maps that without complexity dynamic performances. To improve these weaknesses, we introduced a 2D hyperchaotic map through two 1D-chaotic maps, linear function and represents multiplier. The dynamic performances of the 2D hyperchaotic map are analyzed by attractor trajectory, 0–1 test, bifurcation diagram, Lyapunov exponents and Permutation entropy. The results illustrate that 2D hyperchaotic map has more complexity dynamic characteristics and randomness, which indicates that 2D hyperchaotic map is more suitable for image encryption algorithm. Therefore, an image encryption algorithm is introduced based on 2D hyperchaotic map. In encryption algorithm, based on 2D hyperchaotic map, the image is scrambled by row shift and column shift, then pixel values are diffused through forward and backward diffusion, finally, the ciphertext image is obtained. The simulation results show that the proposed image encryption algorithm is reliable. Moreover, the security analysis indicate that the introduced algorithm has a better security performances than several image encryption algorithms. This algorithm is not absolutely safe and efficient, and does not limit the improvement and optimization of the algorithm. However, this work provide a theoretical basis in chaotic theory and chaotic image encryption algorithm.

Designing an authenticated Hash function with a 2D chaotic map

This paper shows the use of four sets of coefficient values of the 2D chaotic map to generate pseudo-random number generators. We demonstrate that the generated sequences are random by applying NIST suite 800-22-a and TestU01 tests. The generated random sequences are used to implement a stream cipher and are applied to encrypt images. To detect if the images have been modified, we propose to use the random sequences as keys for a hash function based on the pseudo-dot product. This hash can be used as a message authentication code in the images to detect if the stored information has been compromised. The proposed schemes can be used to encrypt and authenticate any digital data not only images. The random sequences generator is probed also in a high-performance microcontroller STM32F746ZG obtaining a throughput of 173.35 Kbit/s.

A novel hybrid permutation substitution base colored image encryption scheme for multimedia data

A novel hybrid permutation substitution-based color image encryption scheme is presented. The permutation is desirable property to have which make any security system diffusing. In present article, permutation is applied through p-box. The substitution process is also vital because it create confusion. In presented work substitution is applied through S-p-box. 3D mixed chaotic map is utilized to make our algorithm more diffusing and confusing. The permutation box, permutation substitution box and initial conditions of 3D mixed chaotic map are also utilized as a key. The experimental outcomes of proposed technique have been critically analyzed and a comprehensive comparison with existing schemes is presented.

A new color image encryption using 2D improved logistic coupling map

This paper proposes a new 2D-ILCM (Two Dimension Improved Logistic Coupling Map) based on preceding works and performed the performance evaluation. Experimental results show that the proposed 2D-ILCM has better chaotic characteristics than the existing 2D chaotic maps. In order to confirm the application of the proposed 2D-ILCM to image encryptions, we propose a new image encryption algorithm which is simple in structure and combines bit level permutation, and pixel level diffusion and confirm the performance through experiments. Experimental results show that the proposed algorithm is robust to attacks and gives a good performance.

An image encryption algorithm based on a novel 1D chaotic map and compressive sensing

A new image encryption algorithm is presented based on a novel chaotic map and compressive sensing with excellent performance. At first, the sparse coefficient matrix is acquired by discrete wavelet transform (DWT) of the original image. Secondly, the SHA-512 hash value of the original image are regarded as the initial values of two novel 1D chaotic maps to generate two chaotic sequences. Furthermore, the measurement matrix is generated by one of the two chaotic maps. Next, the measurement result is scrambled based on bit-plane operations by another chaotic sequence. At last, the diffusion and rotation operations are carried out on the shuffled matrix to improve the security index of the proposed algorithm. It is indicated through the analyses of simulation experiment that the presented encryption algorithm is effective to resist statistical attacks, plaintext attacks and brute-force attacks and has good compression effect and robustness.

A Novel Image Encryption Algorithm Based on the Delayed Maps and Permutation-Confusion-Diffusion Architecture

To improve the dynamical behaviors of 1D chaotic maps, a new linear-delay-modulation method (LDM) is proposed. Derived from the Sine map, a delayed Sine map (DSM) is proposed based on the LDM. Then, we substitute the Sine map in the SIMM system with DSM and obtained a delayed SIMM system (DSIMM). Its chaotic performance is analyzed through the phase diagram, Lyapunov exponent spectrum, and complexity. The results show that the delayed chaotic map can generate more complex dynamical behaviors and more random sequences. Hence, we apply the two delayed systems to a novel image encryption algorithm with the permutation-confusion-diffusion architecture. Firstly, to permutate the pixel of the image efficiently, the plain-image is scrambled by using a multilayer of the nonlinear index. Secondly, the image is confused by using the chaotic matrix generated with two chaotic sequences, and then, the ciphertext is transformed into a 1D sequence. Finally, to improve the plaintext sensitivity and facilitate key management, we enhance the sensitivity by applying a novel diffusion algorithm instead of using plaintext-related keystream. The diffusion equation contains the sum of undiffused pixels and the operation of cyclic bit-shift. Simulation results for the gray image illustrate the effectiveness of the proposed encryption algorithm.

Spatial and temporal Taylor’s law in 1D chaotic maps

By using low-dimensional chaotic maps, the power-law relationship established between the sample mean and variance called Taylor's Law (TL) is studied. In particular, we aim to clarify the relationship between TL from the spatial ensemble (STL) and the temporal ensemble (TTL). Since the spatial ensemble corresponds to independent sampling from a stationary distribution, we confirm that STL is explained by the skewness of the distribution. The difference between TTL and STL is shown to be originated in the temporal correlation of a dynamics. In case of logistic and tent maps, the quadratic relationship in the sample mean and variance, called Bartlett's law, is found analytically. On the other hand, TTL in the Hassell model can be well explained by the chunk structure of the trajectory, whereas the TTL of the Ricker model has a different mechanism originated from the specific form of the map.

Securing Digital Images through Simple Permutation-Substitution Mechanism in Cloud-Based Smart City Environment

Data security plays a significant role in data transfer in cloud-based smart cities. Chaotic maps are commonly used in designing modern cryptographic applications, in which one-dimensional (1D) chaotic systems are widely used due to their simple design and low computational complexity. However, 1D chaotic maps suffer from different kinds of attacks because of their chaotic discontinuous ranges and small key-space. To own the benefits of 1D chaotic maps and avoid their drawbacks, the cascading of two integrated 1D chaotic systems has been utilized. In this paper, we report an image cryptosystem for data transfer in cloud-based smart cities using the cascading of Logistic-Chebyshev and Logistic-Sine maps. Logistic-Sine map has been utilized to permute the plain image, and Logistic-Chebyshev map has been used to substitute the permuted image, while the cascading of both integrated maps has been utilized in performing XOR procedure on the substituted image. The security analyses of the suggested approach prove that the encryption mechanism has good efficiency as well as lower encryption time compared with other related algorithms.

Improved El Gamal public key cryptosystem using 3D chaotic maps

Digital information is any type of data that is stored electronically. These data need to protect its assets and unauthorized access to it and therefore need measures to protect digital privacy. This process is done in different ways, one of which is encryption. Encryption provides secure transfer of unauthorized data over insecure channels. In this paper a method is proposed to create the keys to the El Gamal PKC algorithm based on chaos theory. The proposed algorithm uses 3D chaos maps to create keys used in the encryption and decryption process using an El Gamal algorithm. The time spent encoding and decoding in milliseconds was calculated and compared with methods that used 1D and 2D chaotic maps. Also, the results obtained exceeded most of the statistical and NIST tests. The generated results are tested their reactions against many types of attacks. Then, the results showed that the proposed method had an excellent randomness efficiency for creating public and private keys for El Gamal's algorithm.

True Random Number Generator Based on nondeterministic Device and Multi dimension Chaos Theory

There have been strong links between chaotic theory and cryptographic theory for the last three decades. The characterization of behaviors of the Chaotic system, such as: highly responsive to initial states, mixing attributes, deterministic existence, and often unable to predict long-term returns, allows researchers improve the security of cryptography systems. As a result, there are growing numbers of experiments that have been proposed using chaotic-based random number generators. Furthermore, These proposed generators suffer from minimal key space and those centered on a 1D chaotic map have limited capacity to produce entropy due to their limited number of exponents of Lyapunov (s). Random binary sequence generator producing sequence of bits has been proposed in this paper. The proposed system model consists of two parts that use mouse device data constructively as the basis for the theory of non-determinism and chaos. The chaos theory includes three forms with the mouse cursor coordinates while moving as the initial seeds in this proposed system (1D logistic chaotic system, 2D Hénon system and 3D chaotic system) and combines the values produced in the algorithm. With 1D, 2D Hénon, 3D chaotic maps, the mouse cursor coordinates are treated as an initial random number with post processing to increase the randomness and security of the keys. In the suggested research, there is high key space and a very long time. It is also evident that the developed keys have successful statistical features that require purely random binary sequences that are optimal for use in essential cryptography systems provided by evaluating the results of 16 NIST hardness tests (National Institute of Standards and Technology).

A lossless digital watermarking scheme based on a new 3D-hyper chaotic map

To protect owner's visual multimedia data, several watermarking techniques are proposed in the last few years. A 2D discrete cosine transform mathematical tool is used for host colour image copyright protection with new 3D discrete hyper chaotic map for creating watermark which is used for identification. In this research paper, a 3D discrete hyper chaotic map was proposed, which was used in the creation of secret keys and randomness in the image. It makes the system more robust and adds security to the system. The chaotic pattern generated by a chaotic map is applied to improve the protection of the watermark image. The experiments were conducted to check various common image processing attacks and watermark extraction process generates high quality image after different attacks.

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.

2D-SCMCI Hyperchaotic Map for Image Encryption Algorithm

Chaos has unpredictability and initial condition sensitivity, which known as a best candidate for cryptography application. However, there are many defects for the existing chaotic cryptography systems due to the use of chaotic maps that without complexity dynamic properties. To overcome these weaknesses, this work designed a 2D-SCMCI hyperchaotic map based on Cascade Modulation Couple (CMC) and two 1D-chaotic map. The dynamic characteristics of the 2D-SCMCI hyperchaotic map are analyzed through attractor trajectory, 0–1 test, bifurcation diagram, Lyapunov exponents and spectrum entropy (SE) complexity. The results of analysis indicate that 2D-SCMCI hyperchaotic map has rich dynamic performance and randomness, which illustrates that it is more suitable for image encryption algorithm. Therefore, an image encryption algorithm is proposed by 2D-SCMCI hyperchaotic map. In encryption algorithm, the image is scrambled by row and column, forward and backward diffusion are used to diffuse image pixel values. The security performances analysis results indicate that the introduced algorithm has better security characteristics.

A Finite Precision Implementation of an Image Encryption Scheme Based on DNA Encoding and Binarized Chaotic Cores

Under finite precision implementation, one dimensional (1D) chaotic maps suffer from limited number of control parameters and converged periodicity, making them unsuitable for hardware based ciphering systems despite their simple implementation and low hardware cost. This paper first discusses the limited periodicity of 1D maps under fixed point precision representation, then, presents an image encryption algorithm based on DNA encoding and two specially configured binarized chaotic cores. The function of both cores is to perform the confusion and diffusion stages of the image by generating pseudo random numbers with excellent cryptographic properties. DNA encoding adds an extra layer of security to the algorithm by converting both the image and the chaotic stream to DNA sequences using specific DNA encoding rule. Initial values of both chaotic cores are image dependent based on a calculated hamming distance. These initial condition and the utilized DNA rules composes the overall secret key of the system with a total length of 336 bits. On the condition that all calculations involved in the scheme are based on binary integer arithmetic, all performed security analysis subjected to the scheme proved that the system could withstand known attacks with excellent encryption properties.

Color image encryption using linear feedback shift registers by three dimensional permutation and substitution operations

This study proposes a scale-invariant digital color image encryption method that includes three main steps the pre-substitution, the 3D scale-invariant modular chaotic map, and the post-substitution. 1) The pre-substitution: At the first stage, pixels of plain sub-images are XOR with different key patterns. By starting from one of the plain sub-images, the pixels of the selected plain sub-image is XOR with the initial key, and the result is used as the cipher sub-image and as the next key pattern for performing XOR operations on the next plain sub-image. In other words, the XOR result of each step is used as the next step key pattern, 2) the 3D permutation: At the second stage, first the red, green, and blue components of a $M times N$ color image is divided to m sub-images with size $n times n$. Then m sub-images are partitioned into $k = leftlceil {frac{m}{n}} rightrceil $ windows $W_1$ to $W_k$ with size $n times n$ sub-images. The last two i؟¼$W_k-1$ and i؟¼$W_k$ windows may be overlap in several sub-images. Finally, the three-dimensional modular chaotic maps are performed on the $W_1$i؟¼ to $W_k$i؟¼ windows with MIPF keys and selected by LFSR, 3) the post-substitution: At the final stage, the $M times N$ color image is initially divided into a set of color sub-images. Then, the 24-bit pixels of each sub-images are circularly shifted with several bits specified in the secret key. Modular arithmetic is used in the 3D scale-invariant chaotic maps to increase keyspace and enhance security parameters. With repeat at least one round of main steps, the proposed encryption scheme reaches optimum parameter values and it is highly sensitive to minor differences in both secret key and plain image. The proposed encryption method for images improves the standard parameters of evaluation such as entropy, adjacent pixel correlations, histogram, and expanded keyspace.

Scale Invariant Digital Color Image Encryption Using a 3D Modular Chaotic Map

Cryptography is one of the most important security mechanisms for the transmission of digital media on the Internet. Most proposed cryptographic image methods based on chaotic maps are dependent upon image sizes and most of them worked on square images. In this paper for tackling this problem, a scale-invariant color image encryption method in three-dimensional space is presented. At first, the two-dimensional color image is converted into three-dimensional space, in this case, the red, green, and blue color spectrums are divided into a set of gray-level square sub-images. Then, to have confusion and diffusion properties, the 3D substitution and 3D permutation operation are performed on the sub-images. In substitution operations, the pixel values of the sub-images are changed with the help of XOR and circular shift operators with appropriate keys. In permutation operation, the position of the pixels is changed using modular three-dimensional chaos mappings. To have scale-invariant three-dimensional permutation, the sub-images are divided into one or more windows with equal size, and then 3D modular chaotic map operations are performed on each window with separate keys. Depending on the number of sub-images, there may be two last windows that have overlapping. To increase the speed of color image encryption, the steps of a 3D modular chaotic map on the windows can be implemented in parallel. The proposed approach relative to the similar color image encryption algorithms increases the key space and improves standard parameters, such as entropy, sensitivity, adjacent pixel correlations, and histogram uniformity.

Multi-Images Encryption Scheme Based on 3D Chaotic Map and Substitution Box

The protection of digital content is increasingly becoming a significant issue for researchers and engineers. In this context, nonlinear dynamic systems play a vital role in information security through their chaotic behavior and susceptibility to initial conditions. This research presents a 3D chaotic map-based symmetric algorithm for multiple images to improve encryption efficiency and encourage secure transmission. The proposed scheme comprises the following four modules: the combination (the images are combined into a single image by merging their color channels); the permutation (using the suggested 3D chaotic map); the S-box generation; and the substitution through the AES substitution method. The proposed algorithm’s encryption strength was determined through Entropy, Correlation coefficient, NPCR, and UACI analyses, which were then compared to the past techniques. Furthermore, the proposed method is assessed in terms of its computation time. Results demonstrate that it is highly efficient and secure for real-time communication.

A sensitive dynamic mutual encryption system based on a new 1D chaotic map

In this research paper, we propose a novel one-dimensional chaotic system based on the fraction of cosine over sine (1-DFCS). Evaluation of 1-DFCS indicates the existence of a complex chaotic behavior, an infinite chaotic range, and a high initial state sensitivity. We further propose a new sensitive dynamic mutual image encryption scheme (SDME) using 1-DFCS. SDME is designed with a dynamic diffusion and confusion, which gives a unique encryption process for each image. To ensure the dynamicity, SDME is enriched with a new proposed plain image sensitivity function (PISF). PISF uses the 1-DFCS, plain image, and secret key, which makes it unpredictable and sensitive to a tiny change. PISF is recommended to many image encryption schemes to raise their ability toward the differential attacks. Besides, we propose a mutual part encryption technique. This technique refers to two parties encrypting each other, where a small change in one part reflects the other part. These three characteristics make our scheme able to achieve a high-security level in a single round, which provides a combination of security and time efficiency. The experiment results prove that SDME provides better performance when compared with several state-of-the-art image encryption systems.