Chaos-based Pseudorandom Number Generator Research Articles

Fractal Tent Map with Application to Surrogate Testing

Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks.

Comparison of two new chaos-based pseudorandom number generators implemented in microcontroller

Chaos-based encryption algorithms uses a pseudorandom number generator (PRNG) to generate chaotic dynamics to carry out the processes to encrypt different types of information. One of the main problems is that some chaotic systems do not present high randomness and uniformity in time series affecting directly the security of the chaos-based cryptosystem. In this work, we proposed two new 2D hyperchaotic maps designed from 1D chaotic maps to use in the design of two new PRNGs for cryptographic applications. The proposed PRNGs were implemented in Renesas RA4M1 32-bit microcontroller for first time in the literature. The chaotic sequences with double-precision floating-point format are also validated with different randomness and security analyses such as the Lyapunov exponent, key space, histograms, autocorrelation, information entropy and the NIST SP800-22 test. In addition, a comparison of the results with similar schemes in the literature about design, security and performance is made. The results achieved in this work can help scholars and designers to improve the effectiveness in randomness of new chaos-based PRNGs for cryptographic applications.

An Improved Absolute-Cosine Chaotification Model and Its Simple Application in PRNG

Chaotic systems are widely used in various aspects such as information security, signal processing, and synchronous control. The structural complexity and the chaotic behavior of chaotic systems are two significant factors affecting their practical applications. In this paper, we propose a universal two-dimensional (2D) absolute-cosine chaotic model (ACCM). The 2D-ACCM is composed of a nonlinear bounded cosine function and an absolute value function. It can construct new chaotic maps with simple structures and complex chaotic behaviors on the basis of existing chaotic systems. To verify the effectiveness of the proposed system, we first choose two existing one-dimensional (1D) chaotic maps and one existing 2D chaotic map as the seed maps of the 2D-ACCM to generate two new maps, respectively. The results of chaotic behavior analysis show that these two new maps have more complex chaotic behavior and wider chaotic ranges than seed maps and some advanced chaotic maps. Then a hardware experiment platform based on a field-programmable gate array (FPGA) is used for the hardware implementation of the new maps. Finally, a simple chaos-based pseudo-random number generator (PRNG) is introduced to show the practical application. The experimental results show that the new maps can be easily implemented on the FPGA and the chaos-PRNGs can generate pseudo-random numbers with excellent randomness.

A New Chaos-Based PRNG Hardware Architecture Using the HUB Fixed-Point Format

Chaotic systems have been applied in many applications involving instrumentation and measurements, such as in sensors and control systems, due to their pseudorandom proprieties. However, reproducing chaos in digital systems is challenging because of the dynamical degradation of chaotic digital systems and, consequently, their intrinsic periodic orbits. Many techniques have been used to guarantee a sufficiently large period, making them suitable for such applications. Nevertheless, few articles pay attention to the effects of using different numerical representations on chaos degradation and, at the same time, keeping key design parameters, such as few logical resources and power consumption. Thus, this article aims to provide a new hardware architecture for a chaos-based pseudorandom number generator (PRNG) using the Half-Unit-Biased (HUB) format for fixed-point numbers by bi-coupling the tent map in conjunction with the Bernoulli map, causing a significant impact on its logical resources and performance. Results show that the HUB format is more effective than the standard fixed-point numerical representation and that the proposed approach is chaotic, with pseudorandomness.

Pseudorandom number generator based on novel 2D Hénon-Sine hyperchaotic map with microcontroller implementation.

Recently, chaotic maps have been considered to design pseudorandom number generator (PRNG). However, some chaotic maps present security disadvantages, such as low uniformity and low randomness properties. Nowadays, chaos-based PRNGs are used as the main source for the development of cryptographic algorithms. In this work, to overcome such weaknesses, a novel 2D hyperchaotic map is proposed based on discrete-time feedback by using Hénon map and Sine map. In addition, the dynamics of the hyperchaotic map are enhanced by using the remainder after division function (rem), where better random statistical properties are obtained. A comparison is made between the enhanced Hénon-Sine hyperchaotic map (EHSHM) and the Hénon-Sine hyperchaotic map through Lyapunov exponent analysis, attractor trajectory, histograms and sensitivity at initialization. Then, 8-bit pseudorandom number generator based on the proposed hyperchaotic map (PRNG-EHSHM) is designed and the initial seed of the PRNG is calculated by a secret key of 60 hexadecimal characters. It is implemented in both MATLAB and Arduino Mega microcontroller for experimental results. A complete security analysis is presented from a cryptographic point of view, such as key space, floating frequency, histograms and entropy of the information. Moreover, the randomness is verified with the tests of the National Institute of Standards and Technology (NIST 800-22). Based on the security results obtained, the proposed PRNG-EHSHM can be implemented in embedded cryptographic applications based on chaos.

PSDCE: Physiological signal-based double chaotic encryption for instantaneous E-healthcare services

As fundamental technical support to instantaneous E-healthcare services, Wireless Body Area Networks (WBANs) have attracted huge attention in the scientific and industrial community in recent years, which defines a unique wireless communication protocol for medical and healthcare circumstances. However, the concerns of the network and data security in WBANs have become an obstacle to the rapid development of it. In this paper, a double chaotic encryption method containing the natural randomness by adding physiological signal, electroencephalogram (EEG), is proposed. Initially, the three-dimensional eigenvalues contained deep multi-domain features are extracted as the initial factors of the chaos-based pseudo-random number generator to ensure the natural randomness of the initial values. Then, a hybrid pseudo-random number generator based on two chaotic maps, Piecewise linear Map (PLCM) and Coupled Logistic-Tent map (LTM), is proposed, which has the advantages of high randomness and low hardware complexity. Finally, the double chaotic encryption method based on the EEG eigenvalue for WBANs is presented. Various types of evaluations are performed to verify the efficiency and robustness of the proposed solution.

Two-Dimensional Parametric Polynomial Chaotic System

When used in engineering applications, most existing chaotic systems may have many disadvantages, including discontinuous chaotic parameter ranges, lack of robust chaos, and easy occurrence of chaos degradation. In this article, we propose a two-dimensional (2-D) parametric polynomial chaotic system (2D-PPCS) as a general system that can yield many 2-D chaotic maps with different exponent coefficient settings. The 2D-PPCS initializes two parametric polynomials and then applies modular chaotification to the polynomials. Setting different control parameters allows the 2D-PPCS to customize its Lyapunov exponents in order to obtain robust chaos and behaviors with desired complexity. Our theoretical analysis demonstrates the robust chaotic behavior of the 2D-PPCS. Two illustrative examples are provided and tested based on numeral experiments to verify the effectiveness of the 2D-PPCS. A chaos-based pseudorandom number generator is also developed to illustrate the applications of the 2D-PPCS. The experimental results demonstrate that these examples of the 2D-PPCS can achieve robust and desired chaos, have better performance, and generate higher randomness pseudorandom numbers than some representative 2-D chaotic maps.

Rating of Modern Color Image Cryptography: A Next-Generation Computing Perspective

Issues such as inefficient encryption architectures, nonstandard formats of image datasets, weak randomness of chaos-based Pseudorandom Number Generators (PRNGs), omitted S-boxes, and unconvincing security metrics leading to increased computational time and inadequate security level of chaos and Deoxyribonucleic Acid- (DNA-) based image encryption schemes need careful examination towards the development of more stable encryption schemes in terms of efficiency and reasonable security. A new taxonomy of image encryption based on chaotic systems, hyperchaotic systems, and DNA is propounded to assess the impact of these issues on the performance and security metrics. The primary emphasis of this research is to study various recent encryption architectures centered on a variety of confusion and diffusion methods. It is aimed at assessing the performance and security of various ciphers using a cipher rating criterion that categorizes ciphers into different classes. The parameters that are included in the rating criteria are information entropy, chi-squared goodness of fit test for histogram uniformity analysis, encryption efficiency, key space, differential attacks (Number of Pixels Change Rate and Universal Average Changing Intensity), key sensitivity analysis, encryption time, randomness tests such as NIST-R (a statistical suite for validating the randomness designed by the National Institute of Standards and Technology), correlation coefficient analysis, contrast analysis, energy analysis, homogeneity analysis, Mean Absolute Error, peak signal-to-noise ratio, and robustness to noise and occlusion attacks.

From Continuous-Time Chaotic Systems to Pseudo Random Number Generators: Analysis and Generalized Methodology.

The use of chaotic systems in electronics, such as Pseudo-Random Number Generators (PRNGs), is very appealing. Among them, continuous-time ones are used less because, in addition to having strong temporal correlations, they require further computations to obtain the discrete solutions. Here, the time step and discretization method selection are first studied by conducting a detailed analysis of their effect on the systems’ statistical and chaotic behavior. We employ an approach based on interpreting the time step as a parameter of the new “maps”. From our analysis, it follows that to use them as PRNGs, two actions should be achieved (i) to keep the chaotic oscillation and (ii) to destroy the inner and temporal correlations. We then propose a simple methodology to achieve chaos-based PRNGs with good statistical characteristics and high throughput, which can be applied to any continuous-time chaotic system. We analyze the generated sequences by means of quantifiers based on information theory (permutation entropy, permutation complexity, and causal entropy × complexity plane). We show that the proposed PRNG generates sequences that successfully pass Marsaglia Diehard and NIST (National Institute of Standards and Technology) tests. Finally, we show that its hardware implementation requires very few resources.

FPGA Implementation of Improved Security Approach for Medical Image Encryption and Decryption

Securing medical images is a great challenge to protect medical privacy. An image encryption model founded on a complex chaos-based Pseudorandom Number Generator (PRNG) and Modified Advanced Encryption Standard (MAES) is put forward in this paper. Our work consists of the following three main points. First, we propose the use of a complex PRNG based on two different chaotic systems which are the 2D Logistic map in a complex set and Henon’s system in the key generation procedure. Second, in the MAES 128 bits, the subbytes’ operation is performed using four different S-boxes for more complexity. Third, both shift-rows’ and mix-columns’ transformations are eliminated and replaced with a random permutation method which increases the complexity. More importantly, only four rounds of encryption are performed in a loop that reduces significantly the execution time. The overall system is implemented on the Altera Cyclone III board, which is completed with an SD card interface for medical image storage and a VGA interface for image display. The HPS software runs on μClinux and is used to control the FPGA encryption-decryption algorithm and image transmission. Experimental findings prove that the propounded map used has a keyspace sufficiently large and the proposed image encryption algorithm augments the entropy of the ciphered image compared to the AES standard and reduces the complexity time by 97%. The power consumption of the system is 136.87 mw and the throughput is 1.34 Gbit/s. The proposed technique is compared to recent image cryptosystems including hardware performances and different security analysis properties, such as randomness, sensitivity, and correlation of the encrypted images and results prove that our cryptographic algorithm is faster, more efficient, and can resist any kind of attacks.

A visual analysis method of randomness for classifying and ranking pseudo-random number generators

The development of new pseudo-random number generators (PRNGs) has steadily increased over the years. Commonly, PRNGs’ randomness is “measured” by using statistical pass/fail suite tests, but the question remains, which PRNG is the best when compared to others. Existing randomness tests lack means for comparisons between PRNGs, since they are not quantitatively analysing. It is, therefore, an important task to analyze the quality of randomness for each PRNG, or, in general, comparing the randomness property among PRNGs. In this paper, we propose a novel visual approach to analyze PRNGs randomness allowing for a ranking comparison concerning the PRNGs’ quality. Our analysis approach is applied to ensembles of time series which are outcomes of different PRNG runs. The ensembles are generated by using a single PRNG method with different parameter settings or by using different PRNG methods. We propose a similarity metric for PRNG time series for randomness and apply it within an interactive visual approach for analyzing similarities of PRNG time series and relating them to an optimal result of perfect randomness. The interactive analysis leads to an unsupervised classification, from which respective conclusions about the impact of the PRNGs’ parameters or rankings of PRNGs on randomness are derived. We report new findings using our approach in a study of randomness for state-of-the-art numerical PRNGs such as LCG, PCG, SplitMix, Mersenne Twister, and RANDU as well as chaos-based PRNG families such as K-Logistic map and K-Tent map with varying parameter K.

Coupling chaotic system based on unit transform and its applications in image encryption

Chaotic maps are very important for establishing chaos-based image encryption systems. This paper introduces a coupling chaotic system based on a certain unit transform, which can combine any two 1D chaotic maps to generate a new one with better performance in cryptography applications. The chaotic behavior analysis has verified this coupling system’s effectiveness and progress. In particular, we give a specific strategy about selecting an appropriate unit transform function to enhance the chaotic behaviors of generated maps. Besides, a new chaos-based pseudo-random number generator, shorted as CBPRNG, is designed to improve the randomness of chaotic sequences. Moreover, based on CBPRNG, a novel digital image encryption algorithm is introduced, where we design a two-way multi-round transformation network to encrypt the higher bits and lower bits of input images separately. Simulation results and security analysis indicate that the proposed image encryption scheme is competitive with some existing methods.

A novel secure chaos-based pseudo random number generator based on ANN-based chaotic and ring oscillator: design and its FPGA implementation

This paper presents a novel, real time, high speed and robust chaos-based pseudo random number generator (PRNG) design using the structures of artificial neural network (ANN)-based 2D chaotic oscillator and ring oscillator. In this study, four different robust PRNGs have been implemented using four different approaches (TS-55, Elliott-93, Elliott-2, Cordic-LUT) of TanSig activation functions (TSAF) that have been used in the design of ANN-based 2D chaotic oscillators. The designs have been coded in VHDL using IEEE-754–1985 number standard. The PRNGs have been synthesized for Virtex-6 FPGA chip using Xilinx ISE Design Tools. After Place&Route operation, FPGA chip statistics and maximum operating frequencies have been presented. The maximum operating frequencies of the proposed PRNGs range between 184 and 241 MHz. The 1 Mbit of bit streams generated by PRNGs have been subjected to NIST-800–22 randomness tests. Among 4 different proposed PRNGs, the proposed PRNGs that designed using the Elliott-93 and Cordic-LUT approaches have successfully passed all NIST-800–22 tests and have a bit production rate of 241 Mbps. The proposed secure hybrid chaos-based PRNG structures were compared with similar studies conducted in the literature in recent years. According to the results, the proposed FPGA-based secure new chaotic PRNG structures are useful in cryptographic applications.

Robust Chaos of Cubic Polynomial Discrete Maps with Application to Pseudorandom Number Generators

Based on the robust chaos theorem of S-unimodal maps, this paper studies a kind of cubic polynomial discrete maps (CPDMs) and sets up a novel theorem. This theorem gives general conditions for the occurrence of robust chaos in the CPDMs. By using the theorem, we construct a CPDM. The parameter regions of chaotic robustness of the CPDM are larger than these of Logistic map. By using a fixed point arithmetic, we investigate the cycle lengths of the CPDM and a Logistic map. The results show that the maximum cycle lengths of 1000 chaotic sequences with length 3×107 generated by different initial value conditions exponentially increase with the resolutions. When the resolutions reach 10-7~10-13, the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches 10-14, there is the situation without cycle for 1000 cubic polynomial chaotic sequences with length 3×107. By using the CPDM and Logistic map, we design four chaos-based pseudorandom number generators (CPRNGs): CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV. The randomness of two 1000 key streams consisting of 20000 bits is tested, respectively, generated by the four CPRNGs. The result suggests that CPRNGIII based on the cubic polynomial chaotic generalized synchronic system has better performance.

Dynamics and Optimization Control of a Robust Chaotic Map

Robust chaos in the discrete system is suggested to have practical as well as theoretical importance since it can obtain reliable operation in the chaotic mode. However, it receives only moderate attention and only focuses on a finite chaotic parameter space and small Lyapunov exponents. This paper introduces a two-dimensional smooth map and studies its robustness of chaos in the infinite parameter space. Then, a compound operation-based optimization control method is introduced to increase the map complexity in the measure of Lyapunov exponent. The introduced method is simple and provides a new pathway for exploring the robustness and complexity of discrete chaotic system. Finally, we design a chaos-based pseudo-random number generator (CPNG) based on the optimized robust chaotic map, and the careful analysis shows that the proposed CPNG has high quality of randomness and has passed the rigorous National Institute of Standards and Technology (NIST) test.

A selective bitplane image encryption scheme using chaotic maps

Partial encryption is one of the viable solutions for low power, high speed, real time secure multimedia communication. In this paper, a chaotic tent map based selective bitplane encryption technique is proposed for both gray scale and color images. After decomposing the original image into eight bitplanes, each bitplane is classified into either significant or non-significant category by defining a flexible threshold value of 0.3, deduced experimentally. Following this segregation, only the significant bitplanes are encrypted with the key stream sequences generated by a chaos-based pseudo-random binary number generator. The cipher image is then transmitted through public channel. The proposed scheme has three important contributions v.i.z. a) determination of significant bitplanes, b) encryption of only the significant bitplanes leading to reduction in computational complexity and c) elimination of the need for separate channel for transmitting the information about the significant bitplanes. It is shown that the proposed partial encryption scheme saves around 35% computation on the image database used here. Different types of attacks against this scheme are also analysed to show the robustness of this approach.

A Hardware and Secure Pseudorandom Generator for Constrained Devices

Hardware security for an Internet of Things or cyber physical system drives the need for ubiquitous cryptography to different sensing infrastructures in these fields. In particular, generating strong cryptographic keys on such resource-constrained device depends on a lightweight and cryptographically secure random number generator. In this research work, we have introduced a new hardware chaos-based pseudorandom number generator, which is mainly based on the deletion of an Hamilton cycle within the N-cube (or on the vectorial negation), plus one single permutation. We have rigorously proven the chaotic behavior and cryptographically secure property of the whole proposal: the mid-term effects of a slight modification of the seed (proven to be sensitive to the initial conditions) or of the inputted generator cannot be predicted. The proposal has been fully deployed on a FPGA and 65 nm ASIC, it runs completely in parallel while consuming as low resources as possible, and achieving: (a) 11.5 Gb/s for FPGA and 9.4 Gb/s for ASIC random bit throughput, (b) 3.3 μW (LF) to 7.8 mW (UHF) total power consumption with 5% leakage power, measured at 1.32 V, and (c) able to successfully pass the statistical tests of NIST and TestU01 (BigCrush).

Hyperchaotic system‐based pseudorandom number generator

Pseudorandom sequences are very important in the field of cryptography. The characteristics such as non-linearity and random-like behaviours make chaotic systems suited to generate pseudorandom sequences. However, most of chaos-based pseudorandom number generators have a typical shortcoming. That is, the finite precision in all processors may cause the chaotic systems to degenerate into a periodic function or a fixed point. To overcome this shortcoming, a hyperchaos-based generator is proposed. First, a new hyperchaotic system with bigger Lyapunov exponent is constructed. Then the self-shrinking generator, which is superior to many other linear feedback shift register-based generators, is used to perturb the hyperchaotic sequences to decrease the period degeneration and improve the performance of the sequences. The proposed generator is named as hyperchaos with self-shrinking perturbance generator (H-SSP generator). The analysis results show that the H-SSP generator has better performance.

A Stream Encryption Scheme with Both Key and Plaintext Avalanche Effects for Designing Chaos-Based Pseudorandom Number Generator with Application to Image Encryption

Based on a stream encryption scheme with avalanche effect (SESAE), a stream encryption scheme with both key avalanche effect and plaintext avalanche effect (SESKPAE) is introduced. Using this scheme and an ideal [Formula: see text]-word ([Formula: see text]-segment) pseudorandom number generator (PRNG), a plaintext can be encrypted such that each bit of the ciphertext block has a change with the probable probability of [Formula: see text] when any word of the key is changed or any bit of the plaintext is changed. To that end, a novel four-dimensional discrete chaotic system (4DDCS) is proposed. Combining the 4DDCS with a generalized synchronization (GS) theorem, a novel eight-dimensional discrete GS chaotic system (8DDGSCS) is constructed. Using the 8DDGSCS, a [Formula: see text]-word chaotic pseudorandom number generator (CPRNG) is designed. The keyspace of the [Formula: see text]-word CPRNG is larger than [Formula: see text]. Then, the FIPS 140-2 test suit/generalized FIPS 140-2 test suit is used to test the randomness of the 1000-key streams consisting of 20[Formula: see text]000 bits generated by the [Formula: see text]-word CPRNG, the RC4 algorithm PRNG and the ZUC algorithm PRNG, respectively. The test results show that for the three PRNGs, there are 100%/98.9%, 99.9%/98.8%, 100%/97.9% key streams passing the tests, respectively. Furthermore, the SP800-22 test suite is used to test the randomness of four 100-key streams consisting of 1000[Formula: see text]000 bits generated by four PRNGs, respectively. The numerical results show that the randomness performances of the [Formula: see text]-word CPRNG is promising, showing that there are no significant correlations between the key streams and the perturbed key streams generated via the [Formula: see text]-word CPRNG. Finally, using the [Formula: see text]-word CPRNG and the SESKPAE to encrypt two gray-scale images, test results demonstrate that the [Formula: see text]-word CPRNG is able to generate both key avalanche effect and plaintext avalanche effect, which are similar to those generated via an ideal CPRNG, and performs better than other comparable schemes.

A novel method to improve the performance of chaos based evolutionary algorithms

Many methods have been proposed to improve performance of evolutionary algorithms. One of the most common methods used in these studies is the chaos theory. Chaotic systems are especially widely used in creation of the initial populations and to express optimization variables. Although, chaos theory is a good application field in the design of evolutionary algorithms, many proposals become unsatisfactory. Since using many chaotic systems do not have a uniform distribution. In this study, we analyze the weaknesses of the evolutionary algorithms based on chaotic maps. By applying statistical distribution test, we show that using chaos based pseudorandom number generators are not appropriate for pseudorandom characteristics. In order to overcome this problem, a novel method has been proposed. Statistical tests and distribution test show that proposed new generator has good pseudorandom characteristics.