On the Distribution of Pseudorandom Numbers and Vectors Generated by Inversive Methods

Sequences of pseudorandom numbers are needed for applications of stochastic simulation. Usually they are generated in the computer by a deterministic algorithm which produces a sequence of standard pseudorandom numbers, i.e. a sequence of numbers which “behave” as a realization of a sequence of independent identically distributed random variables having a uniform distribution on the unit interval. In a second step these standard pseudorandom numbers are often transformed in order to fit a given (non-uniform) distribution. For example, Devroy’s book [4] is devoted to this problem. This paper deals only with algorithms producing sequences of standard pseudorandom numbers. Such algorithms are called pseudorandom number generators. Many users of simulation techniques do not think about the pseudorandom number generators implemented in their computers and use the standard software at hand. This habitude is dangerous since the “stochastic quality” of the pseudorandom number sequences is fundamental for the results of stochastic simulation and many pseudorandom number generators in use have serious defects (see e.g. [2], [6], p. 10, and [26], p. 18). The present paper gives a survey on methods for generating deterministic sequences of real numbers which can be used as standard pseudorandom number sequences. Special emphasis is given to generation methods which have been studied by the author’s research group at Darmstadt Technical University.

A new Pseudo random number generator based on generalized Newton complex map with dynamic key

Most commonly-used pseudo-random number generators (PNGs) in computer systems are based on linear recurrence. These deterministic PNGs have fast and compact implementations, andean ensure very long periods. However, the points generated by linear PNGs in fact have a regular lattice structure and are thus not suit able for applications that rely on the assumption of uniformly distributed pseudo-random numbers (PNs). In this paper we propose and evaluate several fast and compact linear, non-linear, and hybrid PNGs for a field- programmable gate array (FPGA). The PNGs have excellent equidistribution properties and very small autocorrelations, and have very long repetition periods. The distribution and long-range correlation properties of the new generators are efficiently, and much more rapidly, estimated at hardware speeds using designed modules within the FPGA. The results of these statistical tests confirm that the combination of several linear PNGs or the combination of even one small non-linear PNG with a linear PNG significantly improves the statistical properties of the generated PNs.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

Many fields need random and pseudorandom numbers, especially in cryptographic applications. d-bit segment binary pseudorandom numbers can be more easily used for block encryption. This paper studies on the statistical test for binary d string pseudorandom number generator (PRNG). Three postulates on the randomness for ideal pseudorandom d-bit segment sequences have been proposed. Based on the FIPS140-2 tests, a statistical test suite for d-bit segment sequences generated by PRNG has been proposed. Using the test suite tests the 100 key streams generated by RC4 PRNG and Matlab PRNG, respectively. The test results show that in the two 100 key streams generated via the two PRNGs with different seeds, about 97% 8-bit strings have passed the test suite, respectively. Using the key streams generated via the two PRNGs encrypts an RGB image. The results have shown that the encrypted RGB images have significant stream encryption avalanche effect.KeywordsInformation TheoryPseudorandom Number StringRandomness PostulateFIPS 140-2 TestNew Test Standard

Chaos has been widely combined with cryptography in the field of information security, especially, a considerable amount of studies of generating pseudo-random numbers based on chaotic systems have been proposed in recent decades. However, many of them are easy to be attacked via utilizing the nonlinear prediction method based on phase space reconstruction and other analysis. Furthermore, under the finite precision environment of computer simulation, there does not exist a random sequence which is truly non-periodic. Unfortunately, few researches had made a related analysis on the above two discussions. This paper is devoted to designing a pseudo-random number generator based on coupled map lattice with time-varying delay, analyzing the random properties of the generated pseudo-random numbers and discussing the dynamical degradation of the system under finite precision of computer simulation. The proposed scheme merely depends on the determining equation; thus, the algorithm itself is not complex, which does not impose high demand on computer hardware and its efficiency is excellent. In order to meet the requirements of using the proposed pseudo-random number generator in cryptography and other practical engineering applications, the proposed pseudo-random number generator is subjected to statistical tests utilizing the well-known test suites, such as NIST SP800-22 and TestU01. Moreover, other related properties, such as permutation entropy, invariant distribution, degradation of dynamical characteristics and parameter test, are also investigated. All results illustrate that the new pseudo-random number generator can generate a high percentage of available pseudo-random numbers for scientific computer simulation and practical applications in the field of information security.

In this paper, after a short survey describing several known constructions recommended for generating sequences of pseudo-random numbers based on elliptic curves over finite fields of prime order, we propose a method of generating such sequences of points with algorithms driven by a chaotic map. Our construction improves randomness of the sequence generated since it combines good statistical properties of an ECPRNG (Elliptic Curve Pseudo-Random Number Generator) and a CPRNG (Chaotic Pseudo-Random Number Generator). Theoretical analysis shows that periods of the proposed constructions are longer than in the case of the ECPRNG without modulation by a chaotic map. In the second part of the paper we present numerical analysis of the proposed construction to obtain optimal parameters of the generator. We also use some tests from the NIST’s SP 800-22 Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications to analyze statistical properties of the proposed constructions for different values of parameters.

The generalized feedback shift register pseudorandom number algorithm has several advantages over all other pseudorandom number generators. These advantages are: (1) it produces multidimensional pseudorandom numbers; (2) it has an arbitrarily long period independent of the word size of the computer on which it is implemented; (3) it is faster than other pseudorandom number generators; (4) the “same” floating-point pseudorandom number sequence is obtained on any machine, that is, the high order mantissa bits of each pseudorandom number agree on all machines— examples are given for IBM 360, Sperry-Rand-Univac 1108, Control Data 6000, and Hewlett-Packard 2100 series computers; (5) it can be coded in compiler languages (it is portable); (6) the algorithm is easily implemented in microcode and has been programmed for an Interdata computer.

A new statistical test for uniform pseudo-random number generators (PRNGs) is presented. The idea is that a sequence of pseudo-random numbers should have numbers reappear with a certain probability. The expectation time that a repetition occurs provides the metric for the test. For linear congruential generators (LCGs) failure can be shown theoretically. Empirical test results for a number of commonly used PRNGs are reported, showing that some PRNGs considered to have good statistical properties fail. A sample implementation of the test is provided over the Internet.

The inversive congruential method with modulus m = 2W for the generation of uniform pseudorandom numbers has recently been introduced.The discrepancy Z)' L of A:-tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m/2 is the crucial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test.It is proved that for a positive proportion of the inversive congruential generators with maximal period length, the discrepancy £>¡J2 is at least of the order of magnitude m~xl2 for all k > 2 .This shows that the bound D{2)/2 = 0(m_1/2(logw)2) established by the second author is essentially best possible.G m.

This article explores randomness and how general purpose computers generate it. It also discusses the security requirements that a pseudo-random number generator must satisfy for use in cryptographic applications. And also special attention is paid to the known and new methods of testing random bit sequences. Analyzing the effectiveness of pseudo-random sequence generators is a pressing problem for smart cards when using more advanced methods of encryption and information protection. The available methods show low flexibility and versatility in the means of finding hidden patterns in data. To solve this problem, it is proposed to use algorithms based on multivariate statistics. These algorithms combine all the advantages of statistical methods and are the only alternative for analyzing short and medium-length sequences. The paper considers the scheme of operation of pseudo-random number generators in limited devices. The main requirements for modern smart cards are highlighted. A criterion for checking the randomness of bit sequences of small length (up to 100 bits) is proposed. This approach is appropriate for testing a lightweight pseudo-random number generator in devices with certain resource constraints. The paper presents the compatible distributions of the number of 2-strings and the number of 3-strings of a fixed form of a random bit sequence, which make it possible to carry out a statistical analysis of local sections of this sequence. A possible application of the obtained formulas can be to test the hypothesis of the randomness of the arrangement of zeros and ones in a (0, 1) -sequence of finite length. Research has shown that even with limited resources and a limited entropy environment like a smart card, good quality pseudo-random sequences can be created that can satisfy all the requirements for pseudo-random number generators, even those used for general purpose computers. In the work, the set of statistical tests was expanded to include other tests that are not included in the statistical set of NIST tests, and to analyze the work of the proposed algorithms. The paper presents algorithms for testing a pseudo-random sequence using multivariate statistics to illustrate their possible application in a smart card environment.

The article is dedicated to a detailed review of pseudorandom number generators (PRNGs), their operating principles, features, advantages, limitations, and various implementation methods. Pseudorandom number generators have become an integral part of modern information security, leveraging advanced cryptographic algorithms to ensure reliability and resistance to attacks. Pseudorandom numbers are widely used in cryptography, simulation, computer games, and many other fields. The main goal of the article is to analyze various methods of generating pseudorandom numbers, their features, advantages, and disadvantages, as well as to study their impact on cryptographic strength. The article begins with the basic principles of PRNG operation, focusing on linear congruential generators (LCGs) as the simplest example. Although LCGs are easy to implement, they have significant limitations in the context of cryptographic security, such as short periods and relative predictability. For higher security requirements, cryptographically secure pseudorandom number generators (CSPRNGs) based on block ciphers, hash functions, or other complex mathematical algorithms are recommended. Additional attention is given to generators such as ChaCha20 and NORX, which demonstrate high speed and resistance to cryptanalytic attacks. Theoretical cryptographic strength is also analyzed for generators based on modified algorithms, such as BBS and BM, supported by the complexity of computational problems. In conclusion, the article emphasizes the critical importance of understanding the operation of various PRNGs for information security professionals, as the choice of generator can significantly impact the overall security level of a system. A comparison of the efficiency and cryptographic strength of different pseudorandom number generation methods is presented in a table, supplemented by program algorithms, their characteristics, and examples of their use in cryptography.

Background: Pseudo random numbers have indispensable role in designing cryptography systems such as key stream in stream ciphers. Efficiency of most crypto systems are in depend on the quality of secret key generated by a pseudo random number generator. Improvements/Methods: In the present paper, an efficient pseudo random number generator is presented for cryptographic applications. The algorithm is based on controlling distribution of generated random numbers with the chaotic henon congruential generator. Statistical Analyses: Statistical tests and histograms are performed over the proposed generator and the results confirm the improvements over the proposed algorithms. According to the results of statistical tests, proposed algorithms generate pseudo random numbers with acceptable independency and uniformity and random sequences with long enough period. Applications: Key streams in stream cipher system can be considered as the most important applications of pseudo random numbers. With this regard proposed generators are statistically proved as proper key stream generator for designing stream cipher systems.

Monte Carlo methods rely on sequences of random numbers to obtain solutions to many problems in science and engineering. In this work, we evaluate the performance of different pseudo-random number generators (PRNGs) of the Curand library on a number of modern Nvidia GPU cards. As a numerical test, we generate pseudo-random number (PRN) sequences and obtain non-uniform distributions using the acceptance-rejection method. We consider GPU, CPU, and hybrid CPU/GPU implementations. For the GPU, we additionally consider two different implementations using the host and device application programming interfaces (API). We study how the performance depends on implementation parameters, including the number of threads per block and the number of blocks per streaming multiprocessor. To achieve the fastest performance, one has to minimize the time consumed by PRNG seed setup and state update. The duration of seed setup time increases with the number of threads, while PRNG state update decreases. Hence, the fastest performance is achieved by the optimal balance of these opposing effects.

As a critical technology of 5G air interface waveform, filtered orthogonal frequency division multiplexing (F-OFDM) not only inherits the technical advantages of OFDM, but also has outstanding advantages in system flexibility and spectrum efficiency. However, as a multi-carrier technology, it is still extremely sensitive to sample timing offset (STO) and carrier frequency offset (CFO). In this letter, an improved Park frequency domain training sequence (FS-Park) is proposed to complete STO and CFO estimation of F-OFDM system. Firstly, a real-value pseudorandom number (PN) sequence is sent to each subcarrier as training sequence in frequency domain, the corresponding time domain training symbol has a conjugate symmetry structure. Secondly, the training symbol is utilized for timing synchronization, then the fractional frequency offset is estimated based on the cyclic prefix in time domain. Finally, the integer frequency offset is estimated in frequency domain based on the auto-correlation of PN sequence. The simulation results illustrate that the FS-Park algorithm not only has a single pulse timing metric curve and great STO estimation accuracy, but also has better performance of CFO estimation than classical Park algorithm and Liang Xiao’s method.