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We describe random processes (with binary alphabet) whose entropy is less than 1 (per letter), but they mimic true random process, i.e., by definition, generated sequence can be interpreted as the result of the flips of a fair coin with sides that are labeled 0 and 1. It gives a possibility to construct Random Number Generators which possess theoretical guarantees. This, in turn, is important for applications such as those in cryptography.

This paper discusses the testing of methods for generating uniform numbers in a computer--the commonly used multiplicative and mixed congruential generators as well as two methods. Tests proposed here are more stringent than those usually applied, because the usual tests for randomness have passed several of the commonly used pprocedures which subsequently gave poor results in actual Monte Carlo calculations. The principal difficulty seems to be that certain simple functions of n-tuples of uniform random numbers do not have the distribution that probability theory predicts. Two alternative generating methods are described, one of them using a table of uniform numbers, the other one combining two congruential generators. Both of these methods passed the tests, whereas the conventional multiplicative and mixed congruential methods did not.

The RANDOM computer program evaluates the usefulness of pseudo-uniform random number generators. RANDOM was written in FORTRAN IV for the CDC 6700 computer system at the Naval Surface Weapons Center, Dahlgren. This program can also be used to evaluate candidate pseudo-uniform random number generators designed for use on any other computer system. RANDOM subjects the candidate generator to 11 different statistical tests designed to detect departures from randomness. These tests serve as useful tolls for determining the adequacy of a candidate generator. (Author)

Quantum random-number generators (QRNGs) can offer a means to generate information-theoretically provable random numbers, in principle. In practice, unfortunately, the quantum randomness is inevitably mixed with classical randomness due to classical noises. To distill this quantum randomness, one needs to quantify the randomness of the source and apply a randomness extractor. Here, we propose a generic framework for evaluating quantum randomness of real-life QRNGs by min-entropy, and apply it to two different existing quantum random-number systems in the literature. Moreover, we provide a guideline of QRNG data postprocessing for which we implement two information-theoretically provable randomness extractors: Toeplitz-hashing extractor and Trevisan's extractor.

Quantum random number generators (QRNG) represent an advanced solution for randomness generation, essential in every cryptographic applications. In this context, integrated arrays of single photon detectors have promising applications as QRNGs based on the spatial detection of photons. For the employment of QRNGs in Cryptography, it is necessary to have efficient methods to evaluate the so called quantum min-entropy that corresponds to the amount of the true extractable quantum randomness from the QRNG. Here we present an efficient method that allow to estimate the quantum min-entropy for a multi-detector QRNG. In particular, we will consider a scenario in which an attacker can control the efficiency of the detectors and knows the emitted number of photons. Eventually, we apply the method to a QRNG with $10^3$ detectors.

Revisions are almost entirely in the introduction and conclusion. Results are unchanged, however the comments and recommendations on different generators were changed, and more references were added.

Revisions are almost entirely in the introduction and conclusion. Results are unchanged, however the comments and recommendations on different generators were changed, and more references were added.

Methods are presented for the generation of random numbers with uniform and normal distributions. Subprogram listings of Fortran generators for the Univac 1108, SDS 930, and CDC 3200 digital computers are also included. The generators are of the mixed multiplicative type, and the mathematical method employed is that of Marsaglia and Bray.

Experimented on heuristic, based on an ant system approach for traveling Salesman problem, are several quasi and pseudo-random number generators. This experiment is to explore if any particular generator is most desirable. Such an experiment on large samples has the potential to rank the performance of the generators for the foregoing heuristic. This is just to seek an answer to the controversial performance ranking of the generators in probabilistic/statically sense.

We consider pseudo-random number generators suitable for vector processors. In particular, we describe vectorised implementations of the Box-Muller and Polar methods, and show that they give good performance on the Fujitsu VP2200. We also consider some other popular methods, e.g. the Ratio method of Kinderman and Monahan (1977) (as improved by Leva (1992)), and the method of Von Neumann and Forsythe, and show why they are unlikely to be competitive with the Polar method on vector processors.

We outline some of Chris Wallace's contributions to pseudo-random number generation. In particular, we consider his idea for generating normally distributed variates without relying on a source of uniform random numbers, and compare it with more conventional methods for generating normal random numbers. Implementations of Wallace's idea can be very fast (approximately as fast as good uniform generators). We discuss the statistical quality of the output, and mention how certain pitfalls can be avoided.

In this paper, a new pseudo-random number generator (PRNG) based on chaotic iterations is proposed. This method also combines the digits of two XORshifts PRNGs. The statistical properties of this new generator are improved: the generated sequences can pass all the DieHARD statistical test suite. In addition, this generator behaves chaotically, as defined by Devaney. This makes our generator suitable for cryptographic applications. An illustration in the field of data hiding is presented and the robustness of the obtained data hiding algorithm against attacks is evaluated.

ADA027292

Three different pseudo-random number generators were examined for use in a sonar detection simulation using a whiteness test. The fluctuation of the whiteness measure and the probability that an acceptable sequence of numbers could be produced were studied. The ran1 generator, initially thought to be suitable, was found to be unacceptable for our use. Random was the recommended generator.

Three different pseudo-random number generators were examined for use in a sonar detection simulation using a whiteness test. The fluctuation of the whiteness measure and the probability that an acceptable sequence of numbers could be produced were studied. The ran1 generator, initially thought to be suitable, was found to be unacceptable for our use. Random was the recommended generator.

Several widely used uniform random number generators have been extensively subjected to three commonly used statistical tests of uniformity and randomness. The object was (1) to examine the power of these statistical tests to discriminate between good and bad random number generators, (2) to correlate these results with recently proposed mathematical characterizations of random number generators which might also be useful in such a discrimination, and (3) to examine the effect of shuffling on the random number generators. Briefly the results show that the commonly used runs test has virtually no power to discriminate between good and bad generators, while serial tests perform better. Also shuffling does help, although much more needs to be done in this area. And finally, there is some utility to the mathematical characterizations, but many unanswered questions.

The relatively slow speed and small word size of the current crop of micro-computers causes the efficient production of pseudo-random numbers on these machines to be considerably more difficult than on larger computers. As a consequence, some micro-computer-based algorithms are excessivly time consuming, while other algorithms trade off speed against 'randomness'. To alleviate this problem we present in this paper several families of pseudo random number generators explicitly designed for use on micro-computer environment, others are new or lesser known algorihtms designed to overcome some of the restrictions intrinsic to the micro-computer's 8 bit environment. For each generator the basic algorithm is discussed and a Pascal implementation is presented. Values of coefficients leading to pseudo random number streams with good statistical properties are recommended and an empirical evaluation of the computational efficiency of the Pascal procedures is offered.

There exist today many 'good' pseudo-random number generators; the problem is to retrieve them. This document discusses three commonly used pseudo- random number generators, the first being RANDU, a notoriously bad generator, but one which is still occasionally used. The next is the widely used prime modulus, multiplicative congruential generator used in LL-RANDOMII, the Naval Postgraduate School random number package, and the last is the random number generator provided for microcomputers with the DOS operating system. This latter pseudo-random number generator is completely defective. Simple graphical methods for initial screening of pseudo-random number generators are given, and the problems which arise with bad pseudo-random number generators are detailed with graphics. Finally, recent work on obtaining even better pseudo-random number generators is discussed.

We present a new computational scheme, GRIP (Geometric Random Inner Products), for testing the quality of random number generators. The GRIP formalism utilizes geometric probability techniques to calculate the average scalar products of random vectors generated in geometric objects, such as circles and spheres. We show that these average scalar products define a family of geometric constants which can be used to evaluate the quality of random number generators. We explicitly apply the GRIP tests to several random number generators frequently used in Monte Carlo simulations, and demonstrate a new statistical property for good random number generators.

The paper suggests that points in n-space produced by congruential random number generators are too regular for general Monte Carlo use. Regularity was established previously for multiplicative congruential generators by showing that all the points fall in sets of relatively few parallel hyperplanes. The existence of many containing sets of parallel hyperplanes was easily established, but proof that the number of hyperplanes was small required a result of Minkowski from the geometry of numbers--a symmetric, convex set of volume 2 to the nth power must contain at least two points with integral coordinates. The present paper takes a different approach to establishing the course lattice structure of congruential generators. It gives a simple, selfcontained proof that points in n-space produced by the general congruential generator r sub (i+1) is identically equal to a(r sub i) + b mod m must fall on a lattice with unit-cell volume at least m to the power (n-1). There is no restriction on a or b; this means that all congruential random number generators must be considered unsatisfactory in terms of lattices containing the points they produce, for a good generator of random integers should have an n-lattice with unit-cell volume 1.

We introduce several classes of pseudorandom sequences which represent a natural extension of classical methods in random number generation. The sequences are obtained from constructions on labeled binary trees, generalizing the well-known Stern-Brocot tree.

Basic uniform pseudo-random number generators are implemented on ATI Graphics Processing Units (GPU). The performance results of the realized generators (multiplicative linear congruential (GGL), XOR-shift (XOR128), RANECU, RANMAR, RANLUX and Mersenne Twister (MT19937)) on CPU and GPU are discussed. The obtained speed-up factor is hundreds of times in comparison with CPU. RANLUX generator is found to be the most appropriate for using on GPU in Monte Carlo simulations. The brief review of the pseudo-random number generators used in modern software packages for Monte Carlo simulations in high-energy physics is present.

In many publications, authors showed that chaotic pseudo random number generators (PRNGs) may improve performance of the evolutionary algorithms. In this paper, we use two chaotic maps Gingerbread man and Tinkerbell as the chaotic PRNGs instead of the classical PRNG in the differential evolution. Numbers generated by this maps are normalized to the unit interval by three different methods -- operation modulo, straightforward number normalization where we know minimal and maximal generated number and arctangent of the two variables $x$ and $y$, where numbers $x$ and $y$ are generated by the Gingerbread man map and Tinkerbell map. The first goal of this paper is to show whether the differential evolution convergence speed might be affected by the way how we normalize number generated by the chaotic map. The second goal is to find out the influence of the probability distribution function of the selected chaotic PRNGs. The results mentioned below showed that the selected normalization method may improve differential evolution convergence speed, especially in the case of arctangent and straightforward number normalization, where we know the minimal and maximal generated numbers.

Two structural tests for random number generators of the Lehmer congruential type are discussed. They are known now to be essentially equivalent but are formulated incorrectly and the computational algorithms to implement the tests are unnecessarily complicated. New algorithms for these tests will be sketched.

Two structural tests for random number generators of the Lehmer congruential type are discussed. They are known now to be essentially equivalent but are formulated incorrectly and the computational algorithms to implement the tests are unnecessarily complicated. New algorithms for these tests will be sketched.

It is well-known that the quality of random number generators can often be improved by combining several generators, e.g. by summing or subtracting their results. In this paper we investigate the ratio of two random number generators as an alternative approach: the smaller of two input random numbers is divided by the larger, resulting in a rational number from $[0,1]$. We investigate theoretical properties of this approach and show that it yields a good approximation to the ideal uniform distribution. To evaluate the empirical properties we use the well-known test suite \textsc{TestU01}. We apply the ratio transformation to moderately bad generators, i.e. those that failed up to 40\% of the tests from the test battery \textsc{Crush} of \textsc{TestU01}. We show that more than half of them turn into very good generators that pass all tests of \textsc{Crush} and \textsc{BigCrush} from \textsc{TestU01} when the ratio transformation is applied. In particular, generators based on linear operations seem to benefit from the ratio, as this breaks up some of the unwanted regularities in the input sequences. Thus the additional effort to produce a second random number and to calculate the ratio allows to increase the quality of available random number generators.

Marsaglia recently introduced a class of xorshift random number generators (RNGs) with periods 2n-1 for n = 32, 64, etc. Here we give a generalisation of Marsaglia's xorshift generators in order to obtain fast and high-quality RNGs with extremely long periods. RNGs based on primitive trinomials may be unsatisfactory because a trinomial has very small weight. In contrast, our generators can be chosen so that their minimal polynomials have large weight (number of nonzero terms). A computer search using Magma has found good generators for n a power of two up to 4096. These have been implemented in a free software package xorgens.

Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises the issue of generating large amounts of random numbers in a parallel application. In this lecture we will learn just enough of the theory of pseudo random number generation to make wise decisions on how to choose and how to use random number generators when it comes to large scale, parallel simulations.

A scalar random number generator, RANDUL, and a vector random number generator, VRANF, are available on the ASC at NRL. Several tests to which they have been subjected are described, and the results indicate that both are generally acceptable for most applications. (Author)

A scalar random number generator, RANDUL, and a vector random number generator, VRANF, are available on the ASC at NRL. Several tests to which they have been subjected are described, and the results indicate that both are generally acceptable for most applications. (Author)

This paper discusses some aspects of selecting and testing random and pseudorandom number generators. The outputs of such generators may he used in many cryptographic applications, such as the generation of key material. Generators suitable for use in cryptographic applications may need to meet stronger requirements than for other applications. In particular, their outputs must he unpredictable in the absence of knowledge of the inputs. Some criteria for characterizing and selecting appropriate generators are discussed in this document. The subject of statistical testing and its relation to cryptanalysis is also discussed, and some recommended statistical tests are provided. These tests may he useful as a first step in determining whether or not a generator is suitable for a particular cryptographic application. The design and cryptanalysis of generators is outside the scope of this paper.

30 p. 28 cm

We are developing further our earlier suggestion to use hyperbolic Anosov C-systems for the Monte-Carlo simulations in high energy particle physics. The hyperbolic dynamical systems have homogeneous instability of all trajectories and as such they have mixing of all orders, countable Lebesgue spectrum and positive Kolmogorov entropy. These extraordinary ergodic properties follow from the C-condition introduced by Anosov. The C-condition defines a rich class of dynamical systems which span an open set in the space of all dynamical systems. The important property of C-systems is that they have a countable set of everywhere dense periodic trajectories and that their density exponentially increases with entropy. Of special interest are C-systems that are defined on a high dimensional torus. The C-systems on a torus are perfect candidates to be used for Monte-Carlo simulations. Recently an efficient algorithm was found, which allows very fast generation of long trajectories of the C-systems. These trajectories have high quality statistical properties and we are suggesting to use them for the QCD lattice simulations and at high energy particle physics.

Chaos and its applications in the field of secure communications have attracted a lot of attention. Chaos-based pseudo-random number generators are critical to guarantee security over open networks as the Internet. We have previously demonstrated that it is possible to define such generators with good statistical properties by using a tool called "chaotic iterations", which depends on an iteration function. An approach to find update functions such that the associated generator presents a random-like and chaotic behavior is proposed in this research work. To do so, we use the vectorial Boolean negation as a prototype and explain how to modify this iteration function without deflating the good properties of the associated generator. Simulation results and basic security analysis are then presented to evaluate the randomness of this new family of generators.

Random numbers are a fundamental resource in science and engineering with important applications in simulation and cryptography. The inherent randomness at the core of quantum mechanics makes quantum systems a perfect source of entropy. Quantum random number generation is one of the most mature quantum technologies with many alternative generation methods. We discuss the different technologies in quantum random number generation from the early devices based on radioactive decay to the multiple ways to use the quantum states of light to gather entropy from a quantum origin. We also discuss randomness extraction and amplification and the notable possibility of generating trusted random numbers even with untrusted hardware using device independent generation protocols.

We report large systematic errors in Monte Carlo simulations of the tricritical Blume-Capel model using single spin Metropolis updating. The error, manifest as a $20\%$ asymmetry in the magnetisation distribution, is traced to the interplay between strong triplet correlations in the shift register random number generator and the large tricritical clusters. The effect of these correlations is visible only when the system volume is a multiple of the random number generator lag parameter. No such effects are observed in related models.

It is shown how correlations in the generalized feedback shift-register (GFSR) random-number generator are greatly diminished when the number of feedback taps is increased from two to four (or more) and the tap offsets are lengthened. Simple formulas for producing maximal-cycle four-tap rules from available primitive trinomials are given, and explicit three- and four-point correlations are found for some of those rules. A number of generators are also tested using a simple but sensitive random-walk simulation that relates to a problem in percolation theory. While virtually all two-tap generators fail this test, four-tap generators with offset greater than about 500 pass it, have passed tests carried out by others, and appear to be good multi-purpose high-quality random-number generators.

Presented at Computer Science and Statistics, Seventh Annual Symposium on the Interface held at Iowa State University, Ames, Iowa, October 18 - 19, 1973.

This paper presents the results of an exhaustive search to find optimal full period multipliers for the multiplicative congruential random number generator with prime modulus 2 to the 31st power -1. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for k=2,...,6. This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 414 multipliers among the more than 534 million candidate multipliers. Section 1 reviews the basic properties of linear congruential generators and Section 2 describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k-tuples, the lattice ratio and the discrepancy. Section 3 presents the five best multipliers and compares their performance with those of three commonly employed multipliers for all measures but the lattice test. Comparisons using packing measures in the space of k-tuples and in the dual space are also made. Section 4 presents the results of applying a battery of statistical tests to the best five to detect local departures from randomness. None were found. The Appendix contains a list of all optimal multipliers. (Author)

Three random number generators, which produce Gaussian white noise sequences, were compared to assess their suitability in aircraft dynamic modeling applications. The first generator considered was the MATLAB (registered) implementation of the Mersenne-Twister algorithm. The second generator was a website called Random.org, which processes atmospheric noise measured using radios to create the random numbers. The third generator was based on synthesis of the Fourier series, where the random number sequences are constructed from prescribed amplitude and phase spectra. A total of 200 sequences, each having 601 random numbers, for each generator were collected and analyzed in terms of the mean, variance, normality, autocorrelation, and power spectral density. These sequences were then applied to two problems in aircraft dynamic modeling, namely estimating stability and control derivatives from simulated onboard sensor data, and simulating flight in atmospheric turbulence. In general, each random number generator had good performance and is well-suited for aircraft dynamic modeling applications. Specific strengths and weaknesses of each generator are discussed. For Monte Carlo simulation, the Fourier synthesis method is recommended because it most accurately and consistently approximated Gaussian white noise and can be implemented with reasonable computational effort.

An exhaustive evaluation of state-of-the-art random number generators with several well-known suites of tests provides the basis for selection of suitable random number generators for use in stochastic simulations. Implementation details for the selected algorithms include the synthesis of a virtually unlimited number of extremely long-period statistically independent random streams in an efficient and transparent manner.

Title from cover

Two structural tests for random number generators of the Lehmer congruential type are discussed. They are known now to be essentially equivalent but are formulated incorrectly and the computational algorithms to implement the tests are unnecessarily complicated. New algorithms for these tests will be sketched.

Recent exact analytical results developed for the random number generators with taps are reported. These results are applicable to a wide class of algorithms, including random walks, cluster algorithms, Ising models. Practical considerations on the improvement of the quality of random numbers are discussed as well.

In this paper, we approximate the hidden Markov model of chaotic-map truly random number generators (TRNGs) and describe its fundamental limits based on the approximate entropy-rate of the underlying bit-generation process. We demonstrate that entropy-rate plays a key role in the performance and robustness of chaotic-map TRNGs, which must be taken into account in the circuit design optimization. We further derive optimality conditions for post-processing units that extract truly random bits from a raw-RNG.

We deal with randomness-quantifiers and concentrate on their ability do discern the hallmark of chaos in time-series used in connection with pseudo random number generators (PRNG). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely, i) its invariant measure and ii) the mixing constant. This is of help in answering two questions that arise in applications, that is, (1) which is the best PRNG among the available ones? and (2) If a given PRNG turns out not to be good enough and a randomization procedure must still be applied to it, which is the best applicable randomization procedure?. Our answer provides a comparative analysis of several quantifiers advanced in the extant literature.

We use a set of Schwinger-Dyson equations for the Ising Model to check several random number generators. For the model in two and three dimensions, it is shown that the equations are sensitive tests of bias originated by the random numbers. The method is almost costless in computer time when added to any simulation.

Presented at Computer Science and Statistics, Seventh Annual Symposium on the Interface held at Iowa State University, Ames, Iowa, October 18 - 19, 1973.

Presented at Computer Science and Statistics, Seventh Annual Symposium on the Interface held at Iowa State University, Ames, Iowa, October 18 - 19, 1973.

This report describes the use of three random number generators installed on the SSPS VAX 11/785. There are three random number generators: a uniform (URAND), normal (ANRMRN), and exponential (EXPRN).