Statistical Randomness Tests Research Articles

Security Analyses of Random Number Generation with Image Encryption Using Improved Chaotic Map

A chaotic system can have a series of values for the developing variable that completely repeat themselves, giving periodic behavior beginning from any point in that series. It is very sensitive to initial conditions and control parameters which make them suitable for image encryption. The two maps Duffing and Logistic maps are combined to generate a random sequence. The Logistic map is 1 Dimensional and contains one control parameter (r). Duffing map contains 2 dimensions X and Y, and two control parameters A and B. Both are XOR'ed to get the random numbers. The control parameters of Duffing map are controlled by the control parameter r. X and Y are looped to get repetitive iterations. The random sequences are analyzed for statistical tests for randomness, keyspace, key sensitivity (or) cross-correlation, periodicity, auto-correlation, information entropy, equilibrium, linear complexity, and speed are the analyses to verify their conformity for use in cryptographic applications. Image encryption is the process of encoding a grayscale image with the help of a series of random numbers created by using a combination of chaotic maps, in such a way that uncertified users can't access it. NPCR, UACI, Histogram, and Information Entropy are the analyses performed. Image encryption and secret key generation are presented based on the Duffing map and logistic chaotic map. The series of random numbers and the greyscale image pixels are XOR'ed to shuffle the pixels and it turned the original image into a cipher image.

A New Energy-Efficient and High Throughput Two-Phase Multi-Bit per Cycle Ring Oscillator-Based True Random Number Generator

Oscillator-based elementary true random number generator (TRNG) uses a slow jittery ring oscillator (RO) to sample a fast RO. The ROs are always on but most of the oscillatory cycles of the fast RO are not sampled into random bits. In this paper, a new lightweight TRNG design is proposed to minimize the power wasted by the superfluous oscillations. Random bits are extracted from both phases of the slow ROs to increase the throughput and the fast RO is activated only during the narrow transition time difference between two symmetrically designed slow ROs. The slow jittery ROs are implemented using current starved inverters biased in the weak inversion region to reduce their power consumption. Their jitter amplitudes are increased by lowering the oscillation frequency and reducing the drain current of the transistors. The narrow jittery pulse generated by the differential pair of slow ROs is quantized by the fastest three-stage RO. Two random bits from each phase of the jittery ROs can be extracted by using a gigahertz dynamic toggled D flip-flop counter to count the number of oscillatory cycles of the fast RO. The proposed TRNG is fabricated in a standard 65 nm 1.2 V CMOS process. Measurement results of the fabricated chips show that the proposed TRNG consumes merely <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$260~\mu \text{W}$ </tex-math></inline-formula> at a bit rate of 52 Mbps. It outperforms the state-of-art on-chip jitter-based TRNGs with the best figure-of-merit of 5 pJ/bit and the smallest footprint of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$366~\mu \text{m}^{2}$ </tex-math></inline-formula> . Its generated bit sequence passes the statistical randomness tests including National Institute of Standards and Technology (NIST) test, Auto Correlation Factor (ACF) test and bias. The mean redundancy of the ten tested chips is measured to be less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> bit/symbol.

Parallel and independent true random bitstreams from optical emission spectra of atmospheric microplasma arc discharge

AbstractIn this study, we propose the possibility of generating several parallel and independent random bitstreams from the time‐varying optical emission spectra of an atmospheric pressure air microplasma system. This is achieved by splitting the plasma arc emission into discrete wavelengths using an optical spectrometer and then monitoring the fluctuating intensities of each wavelength as an independent time series. As a proof of concept, we considered eight wavelengths centered at 377.8, 389.1, 425.8, 591.4, 630.5, 673.0, 714.2, and 776.4 nm corresponding to atomic emissions lines from species either from the surrounding atmospheric air gap or from the electrodes' materials. NIST SP 800‐22 statistical randomness tests and other statistical estimates (auto‐ and cross‐correlation analysis and binary vector similarity measures) are subsequently applied to the binarized data, and the obtained results confirm the possibility of generating several parallel and independent random bitstreams from the microplasma system. The data throughput is, however, relatively low with the optical setup we used, which can be improved using faster spectrometry.

Fast Software Implementation of Serial Test and Approximate Entropy Test of Binary Sequence

In many cryptographic applications, random numbers and pseudorandom numbers are required. Many cryptographic protocols require using random or pseudorandom numbers at various points, e.g., for auxiliary data in digital signatures or challenges in authentication protocols. In NIST SP800-22, the focus is on the need for randomness for encryption purposes and describes how to apply a set of statistical randomness tests. These tests can be used to evaluate the data generated by cryptographic algorithms. This paper will study the fast software implementation of the serial test and the approximate entropy test and propose two types of fast implementations of these tests. The first method is to follow the basic steps of these tests and replace bit operations with byte operations. Through this method, compared with the implementation of Fast NIST STS, the efficiency of the serial test and approximate entropy test is increased by 2.164 and 2.100 times, respectively. The second method is based on the first method, combining the statistical characteristics of subsequences of different lengths and further combining the two detections with different detection parameters. In this way, compared to the individual implementation of these tests, the efficiency has been significantly improved. Compared with the implementation of Fast NIST STS, the efficiency of this paper is increased by 4.078 times.

Private Weakly-Random Sequences from Human Heart Rate for Quantum Amplification.

We investigate whether the heart rate can be treated as a semi-random source with the aim of amplification by quantum devices. We use a semi-random source model called -Santha–Vazirani source, which can be amplified via quantum protocols to obtain a fully private random sequence. We analyze time intervals between consecutive heartbeats obtained from Holter electrocardiogram (ECG) recordings of people of different sex and age. We propose several transformations of the original time series into binary sequences. We have performed different statistical randomness tests and estimated quality parameters. We find that the heart can be treated as a good enough, and private by its nature, source of randomness that every human possesses. As such, in principle, it can be used as input to quantum device-independent randomness amplification protocols. The properly interpreted parameter can potentially serve as a new characteristic of the human heart from the perspective of medicine.

Recommendations on Statistical Randomness Test Batteries for Cryptographic Purposes

Security in different applications is closely related to the goodness of the sequences generated for such purposes. Not only in Cryptography but also in other areas, it is necessary to obtain long sequences of random numbers or that, at least, behave as such. To decide whether the generator used produces sequences that are random, unpredictable and independent, statistical checks are needed. Different batteries of hypothesis tests have been proposed for this purpose.In this work, a survey of the main test batteries is presented, indicating their pros and cons, giving some guidelines for their use and presenting some practical examples.

Magniber v2 Ransomware Decryption: Exploiting the Vulnerability of a Self-Developed Pseudo Random Number Generator

With the rapid increase in computer storage capabilities, user data has become increasingly important. Although user data can be maintained by various protection techniques, its safety has been threatened by the advent of ransomware, defined as malware that encrypts user data, such as documents, photographs and videos, and demands money to victims in exchange for data recovery. Ransomware-infected files can be recovered only by obtaining the encryption key used to encrypt the files. However, the encryption key is derived using a Pseudo Random Number Generator (PRNG) and is recoverable only by the attacker. For this reason, the encryption keys of malware are known to be difficult to obtain. In this paper, we analyzed Magniber v2, which has exerted a large impact in the Asian region. We revealed the operation process of Magniber v2 including PRNG and file encryption algorithms. In our analysis, we found a vulnerability in the PRNG of Magniber v2 developed by the attacker. We exploited this vulnerability to successfully recover the encryption keys, which was by verified the result in padding verification and statistical randomness tests. To our knowledge, we report the first recovery result of Magniber v2-infected files.

A fully CMOS true random number generator based on hidden attractor hyperchaotic system

Low-power devices used in Internet-of-things networks have been short of security due to the high power consumption of random number generators. This paper presents a low-power hyperchaos-based true random number generator, which is highly recommended for secure communications. The proposed system, which is based on a four-dimensional chaotic system with hidden attractors and oscillators, exhibits rich dynamics. Numerical analysis is provided to verify the dynamic characteristics of the proposed system. A fully customized circuit is deployed using 130 nm CMOS technology to enable integration into low-power devices. Four output signals are used to seed a SHIFT-XOR-based chaotic data post-processing to generate random bit output. The chip prototype was simulated and tested at 100 MHz sampling frequency. The hyperchaotic circuit consumes a maximum of 980 upmu W in generating chaotic signals while dissipates a static current of 623 upmu A. Moreover, the proposed system provides ready-to-use binary random bit sequences which have passed the well-known statistical randomness test suite NIST SP800-22. The proposed novel system design and its circuit implementation provide a best energy efficiency of 4.37 pJ/b at a maximum sampling frequency of 100 MHz.

Propose a new approach for key generation based on Chicken Swarm Optimization and HTML Parser

The random key plays a basic role in the design of any cryptographic algorithms. In this paper, a proposed model for generating a random key is presented, which will be used for security purposes. Chicken swarm optimization (CSO) and HTML parser are used for scattering the bits of the key. The statistical tests of randomness have given good and acceptable security results. The proposed key generation method was programmed by JavaScript language.

Unified Analog PUF and TRNG Based on Current-Steering DAC and VCO

This work presents a unified weak physical unclonable function (PUF) and a true random number generator (TRNG) based on the current-steering digital-to-analog converter (DAC) and ring voltage-controlled oscillator (VCO). Entropy source for the weak PUF is the mismatch between NMOS and PMOS transistors in a cascode current DAC as well as the mismatch between VCO quantizers, while entropy source for the TRNG is thermal noise in the DAC and VCO and clock jitter. Instead of using spatial entropy sources for the PUF, i.e., multiple unit PUF elements, the proposed architecture utilizes temporal entropy source by capturing the output of unit PUF element over multiple cycles, which reduces area significantly. A unified PUF/TRNG prototype is fabricated in 65-nm CMOS and consumes 0.36 pJ/bit at a throughput of 100 Mb/s. The PUF has a measured intra-HD of 0.0906 and inter-HD of 0.4859, while the raw TRNG bitstream has an entropy of 0.9991 and passes all the NIST statistical randomness tests.

Measuring Independence between Statistical Randomness Tests by Mutual Information.

The analysis of independence between statistical randomness tests has had great attention in the literature recently. Dependency detection between statistical randomness tests allows one to discriminate statistical randomness tests that measure similar characteristics, and thus minimize the amount of statistical randomness tests that need to be used. In this work, a method for detecting statistical dependency by using mutual information is proposed. The main advantage of using mutual information is its ability to detect nonlinear correlations, which cannot be detected by the linear correlation coefficient used in previous work. This method analyzes the correlation between the battery tests of the National Institute of Standards and Technology, used as a standard in the evaluation of randomness. The results of the experiments show the existence of statistical dependencies between the tests that have not been previously detected.

Parity-based, bias-free optical quantum random number generation with min-entropy estimation

We describe the generation of sequences of random bits from the parity of photon counts produced by polarization measurements on a polarization-entangled state. The resulting sequences are bias free, pass the applicable tests in the NIST battery of statistical randomness tests, and are shown to be Borel normal, without the need for experimental calibration stages or postprocessing of the output. Because the photon counts are produced in the course of a measurement of the violation of the Clauser–Horne–Shimony–Holt inequality, we are able to concurrently verify the nonclassical nature of the photon statistics and estimate a lower bound on the min-entropy of the bit-generating source. The rate of bit production in our experiment is around 13 bits/s.

Using nanoresonators with robust chaos as hardware random number generators.

In this paper, we investigate theoretically the potential of a nanoelectromechanical suspended beam resonator excited by two-external frequencies as a hardware random number generator. This system exhibits robust chaos, which is usually required for practical applications of chaos. Taking advantage of the robust chaotic oscillations, we consider the beam position as a possible random variable and perform tests to check its randomness. The beam position collected at fixed time intervals is used to create a set of values that is a candidate for a random sequence of numbers. To determine how close to a random sequence this set is, we perform several known statistical tests of randomness. The performance of the random sequence in the simulation of two relevant physical problems, the random walk and the Ising model, is also investigated. An excellent overall performance of the system as a random number generator is obtained.

Chaos Based Cryptographic Pseudo-Random Number Generator Template with Dynamic State Change

This article presents a configurable, high-throughput pseudo-random number generator template targeting cryptographic applications. The template is parameterized using a chaotic map that generates data, an entropy builder that is used to periodically change the parameters of the map and a parameter change interval, which is the number of iterations after which the entropy builder will change the generator’s parameters. The system is implemented in C++ and evaluated using the TestU01 and NIST RNG statistical tests. The same implementation is used for a stream cipher that can encrypt and decrypt PNG images. A Monte-Carlo analysis of the seed space was performed. Results show that for certain combinations of maps and entropy builders, more than 90% of initial states (seeds) tested pass all statistical randomness tests. Also, the throughput is large enough so that a 8 K color image can be encrypted in 2 s on a modern laptop CPU (exact specifications are given in the paper). The conclusion is that chaotic maps can be successfully used as a building block for cryptographic random number generators.

True Random Number Generator for Reliable Hardware Security Modules Based on a Neuromorphic Variation-Tolerant Spintronic Structure

The generation of true random numbers is one of the most important tasks in a hardware security module (HSM), particularly for cryptography applications. The stochastic behavior of electronic devices can be used to generate random numbers. In this paper, a reliable neuromorphic true random number generator (TRNG) relying on stochastic switching of the magnetic tunnel junction (MTJ) in the subcritical current regime is proposed. Thanks to the efficient structure of the proposed design as well as the fascinating features of the MTJs and carbon nanotube field-effect transistors (CNTFET), the proposed TRNG consumes low power. Moreover, the neuromorphic structure of the proposed design leads to variation tolerance and guarantees the truly random number generation even in the presence of the process variations. HSPICE simulations verify the functionality of the proposed TRNG. Furthermore, by considering the corners of the fabrication process, the randomness of the bitstream, generated by the proposed TRNG, is validated by the statistical randomness test provided by the U.S National Institute of Standards and Technology (NIST).

Random Number Generators Based on Irregular Sampling and Fibonacci–Galois Ring Oscillators

This brief presents a random number generator (RNG) based on irregular sampling of regular waveform method where the irregular signal is obtained by combining Fibonacci-Galois ring oscillators with an XOR gate. The RNG is implemented on a FPGA (field-programmable gate array). The regular waveform generated by the digital clock manager of the FPGA, is sampled at times corresponding to certain number of rising edges of the irregular signal, and the resulting bit stream is subjected to statistical tests of randomness. It is demonstrated that the resulting bit sequence from the proposed RNG satisfies NIST 800–22 test suit and Rabbit and SmallCrush batteries from TestU01 library without any need for post-processing such as Von Neumann or XOR. A comparison between the methods regular sampling of irregular waveform and irregular sampling of regular waveform is given in terms of robustness against external interference. The impact of selection of Fibonacci-Galois polynomials is discussed. Using digital design flow for TSMC 65nm process, an asic implementation of the proposed RNG is given having 1115 gates and 4.811 mW estimated power.

Periodic template tests: A family of statistical randomness tests for a collection of binary sequences

In this work, we classify all templates according to their periods and for each template we evaluate the exact probabilities using generating functions. Afterwards, we propose a new family of statistical randomness tests, that is periodic template tests, for a collection of binary sequences. We apply these tests to the outputs of AES, SHA-3, SHA-2 family, SHA-1 and MD5 and the binary expansion of π and 2, and biased non-random data to test the power of new tests. Moreover, we give the probabilities for all templates for the overlapping template matching test in the NIST test suite. Afterwards, we analyse the power of templates and compare the periodic template tests with NIST overlapping template test.

Nano-Intrinsic True Random Number Generation: A Device to Data Study

We present a circuit technique to extract true random numbers from carrier capture and emission in oxide traps in the emerging redox-based resistive memory (ReRAM). This phenomenon that appears as small changes in current magnitude passing through the device is known as random telegraph noise (RTN) and is increasingly becoming a source of reliability issues in nanometer-scale devices. We demonstrate a circuit that exploits TRN suitable for a true random number generator (TRNG) in security applications, where the system is secure from different adversarial attacks, including side-channel monitoring and machine learning analysis. We experimentally characterize RTN in ReRAMs and extract its dependency to temperature, voltage, and area. We introduce an RTN harvesting circuit to mitigate sensitivities to temperature fluctuations, injected supply noise, and power signal monitoring. We reduced bias and imbalance in data due to high-speed sampling via von Neumann whitening. The circuit is compared to conventional non-differential readout approach. Our approach shows a 7.26 times improvement in autocorrelation and significant resilience against the injected supply noise. We also demonstrate the TRNG’s quality and robustness using statistical tests and machine learning attacks. The output of the generator satisfies statistical tests for randomness and is immune to modeling attacks based on the machine learning methods.

Practical True Random Number Generator Using CMOS Image Sensor Dark Noise

We present a true random number generator (TRNG) using dark noise of a CMOS image sensor. Because the proposed TRNG is based on the dark characteristics of the CMOS image sensor, it does not require any additional hardware, such as light source and optics, for providing true randomness. Therefore, it can be a promising solution for compact and low-cost mobile application. By using NIST SP 800-90B entropy assessment suite, we evaluate the min-entropy for the raw outputs of our original noise source and the final random numbers including post-processing as well. We also adopt NIST SP 800-22 statistical randomness test suite for the evaluation of the random numbers. The test results demonstrate that the generated random numbers pass all the statistical tests and have high entropy.

Robust Digital Image Encryption Approach Based on Extended Large-Scale Randomization Key-Stream Generator

This paper presents a novel image encryption scheme based on extended large-scale randomization key-stream generator. The basic form of the key-stream generator is presented, and employed in digital image ciphering. The modification of the basic form also, presented, and gives encouraging results in image encryption as compared with classical non-linear stream cipher generators and the basic form. Pixel shuffling is performed via vertical and horizontal permutation. Shuffling is used to expand diffusion in the image and dissipate high correlation among image pixels the sequences generated from all presented generators are introduced to five well-known statistical tests of randomness to judge their randomness characteristic. The ciphered images are tested for their residual intelligibility subjectively. The measures applied to images ciphered by one of the classical key-stream cipher generators (Threshold generator) for the purpose of comparison with the presented key-stream algorithms. Experiments results show that the proposed algorithm achieves the image security. In order to evaluate performance, the proposed algorithm was measured through a series of tests. Experimental results illustrate that the proposed scheme shows a good resistance against brute-force and statistical attacks.