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This chapter discusses the structure of linear congruential sequences and their suitability as a source of random integers in a computer. Every congruential sequence is made up of a block of t < m residues, the effective period of the sequence, followed by translates of that block. A formula is given for the period, effective period, and translating constant for every sequence. Points in n-space, produced by a congruential generator, fall on a lattice with unit-cell volume m n-1. A congruential random number generator uses a linear transformation on the ring of reduced residues of some modulus to produce a sequence of integers. These integers are converted to fractions of the modulus and serve as independent uniform random variables in Monte Carlo calculations. Given a sequence of integers produced by a random number generator, points are formed in n-space, whose coordinates are successive n-tuples produced by the generator. The lattice spanned by that set of points is called the n-lattice of the generator.

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... As pointed out by Marsaglia [Mar72], in spite of a profuse number of articles having touted specific choices of increment c (c = 0), it is a simple known fact that the choice of c and initial seed x 0 is of no great consequence, since any linear congruential sequence {x i } can be obtained by an affine transformation of the fundamental sequence 0, 1, a + 1, a 2 + a + 1, . . . accordingly: ...

... , the structure and period length of linear congruential sequences can be derived by simply considering fundamental sequences. Marsaglia's paper [Mar72] provides an in-depth discussion of the structure of linear congruential sequences in terms of fundamental sequences. ...

... It is well known (see such articles as[Mar68,Mar70,BRW71,Bey72,Mar72]) that n-tuple vectors (points) of n consecutive terms of the normalized linear congruential pseudorandom number sequence {u i } form a lattice in the n-dimensional unit cube [0, 1] n , and the n-dimensional volume of a unit cell of the lattice is 1/m if {u i } has full period length m. However, according to Marsaglia[Mar68], this property should be regarded as defect, stemming from the simple nature of the underlying linear recursion, since such a coarse lattice structure implies several undesirable regularities. ...

... Theoretical tests examine the intrinsic structure of a given generator, the sequence does not necessarily need to be generated. Two classical examples are the lattice test [1] and the spectral test described in [2] (Section 3.3.4). See also [3] for a description of some standard tests from this class. ...

... Hence, if each number is generated independently with uniform distribution on V , then each combination of k bits is equally likely and therefore each bit of the output sequence is independent and equal to 0 or 1 with probability 1 2 . ...

... (1) The functions f , g (in Definition 2.1) now need to be efficiently computable i.e., need to be computed in polynomial time. 1 (2) Only generators for which there are no known efficient distinguishers (statistical tests) can be still called PRNGs. To introduce the definition of a cryptographic PRNG, we need a notion of negligible functions. ...

Testing the quality of pseudorandom number generators is an important issue. Security requirements become more and more demanding, weaknesses in this matter are simply not acceptable. There is a need for an in-depth analysis of statistical tests – one has to be sure that rejecting/accepting a generator as good is not a result of errors in computations or approximations. In this paper we propose a second level statistical test based on the arcsine law for random walks. We provide upper bounds for the approximation of the arcsine distribution, what allows us to perform a detailed error analysis of the proposed test.

... A consideration is the quality of random numbers obtained based on some function, such as the Cliff or cosine generators, in comparison to other published algorithms. Generators chosen for comparison are Schrage (1979), Wikramaratna (1989); as implemented in Deutsch and Journel (1992), and Marsaglia (1972); as implemented in (Deutsch and Journel, 1992). Scatterplots (Fig. 2) and quantiles (Table 1) are similar for all algorithms. ...

... The Schrage (1979) and Acorn algorithms perform the best. The Marsaglia (1972) and Power (1.5) algorithms perform Schrage (1979) a Tests: (1) birthday spacings test; (2) overlapping 5-permutation test; (3) binary rank test for 31 Â 31; 32 Â 32; and 6 Â 8 matrices; (4) bitstream test; (5) overlapping-pairs-sparse-occupancy, overlapping-quadruples-sparse-occupancy, and DNA tests; (6) count-the-ones test on a stream of bytes; (7) count-theones test for specific bytes; (8) parking lot test; (9) minimum distance test; (10) 3D-spheres test; (11) SQEEZE test; (12) overlapping sums test; (13) Runs test; (14) CRAPS test. Most of these tests yield a p-value as final outcome. ...

... Tests have been conducted in which up to 2 36 random values have been generated without any values repeating. In comparison, the Marsaglia (1972) algorithm repeats at the 41,744th value. ...

A wide variety of random number generators is discussed based on truncating functional outcomes and considering the fractional remainders as random digits in the interval, ]0,1[. These generators do not require seeding in the traditional sense, moreover offer an infinite number of outcomes, apparently without periodicity. These generators are trivial in their software implementation. Those that are based on logarithms perform best in tests of randomness. When applied for spatial simulation, though, quality of the random number generator seems unimportant to the outcome.

... We examine the set Gm of all generated pairs of a linear congruential generator with period length m. As is well known, this set has a lattice structure (see [2], [5], [8], [9] and [1], [12] for the sublattice structure). A change of the increment b of the generator yields only a shift of the lattice (that is the periodic continuation of the generated pairs). ...

... This is a very small number of exceptions, and the discrepancy is relatively large in these cases so that the exceptions are not interesting in applications. Example 3. We examine the generator (2) with modulus m = 232 and Marsaglia's multiplier a = 69069 and increment 6 = 1 (see [9]). The estimation mDm > 15545 given in [11] was sharpened to mDm > 15546. ...

Up to now, the rectangle discrepancy of linear congruential pseudorandom number generators could be exactly calculated only in some simple cases for a small number of generated points. Here an algorithm for the exact determination of the twodimensional rectangle discrepancy is presented which is practicable for large generators and requires less computation time. The algorithm is based on special properties of linear congruential generators.

... Theoretical tests examine the intrinsic structure of a given generator, the sequence does not necessarily need to be generated. Two classical examples are the lattice test [1] and the spectral test described in [2] (Section 3.3.4). See also [3] for a description of some standard tests from this class. ...

... Then (z 17 , . . . , z 32 ) = DP ρ(1) DP ρ(2) DP ρ(3) DP ρ(4) = (0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0). 1 The definitions of l i and r i imply that |O i | is even, i = 1, . . . , w DP i = sampleDyckPath(|O i |/2, rng, h(seed, D ri , i), D ri ) 9: end for 10: ρ ← RandPerm(w, rng(seed)) 11: (z 2 N −1 +1 , . . . ...

Testing the quality of pseudorandom number generators is an important issue. Security requirements become more and more demanding, weaknesses in this matter are simply not acceptable. There is a need for an in-depth analysis of statistical tests -- one has to be sure that rejecting/accepting a generator as good is not a result of errors in computations or approximations. In this paper we propose a second level statistical test based on the arcsine law for random walks. We provide a Berry-Essen type inequality for approximating the arcsine distribution, what allows us to perform a detailed error analysis of the proposed test.

... We examine the set Gm of all generated pairs of a linear congruential generator with period length m. As is well known, this set has a lattice structure (see [2], [5], [8], [9] and [1], [12] for the sublattice structure). A change of the increment b of the generator yields only a shift of the lattice (that is the periodic continuation of the generated pairs). ...

... This is a very small number of exceptions, and the discrepancy is relatively large in these cases so that the exceptions are not interesting in applications. Example 3. We examine the generator (2) with modulus m = 232 and Marsaglia's multiplier a = 69069 and increment 6 = 1 (see [9]). The estimation mDm > 15545 given in [11] was sharpened to mDm > 15546. ...

Up to now, the rectangle discrepancy of linear congruential pseudorandom number generators could be exactly calculated only in some simple cases for a small number of generated points. Here an algorithm for the exact determination of the two-dimensional rectangle discrepancy is presented which is practicable for large generators and requires less computation time. The algorithm is based on special properties of linear congruential generators.

... • Super-Duper [14]: x n = x n−1 * 69069 + 1 (mod 2 32 ), ...

Conventional random number generators provide the speed but not necessarily
the high quality output streams needed for large-scale stochastic simulations.
Cryptographically-based generators, on the other hand, provide superior quality
output but are often deemed too slow to be practical for use in large
simulations. We combine these two approaches to construct a family of hybrid
generators that permit users to choose the desired tradeoff between quality and
speed for a given application. We demonstrate the effectiveness, performance,
and practicality of this approach using a standard battery of tests, which show
that high quality streams of random numbers can be obtained at a cost
comparable to that of fast conventional generators.

... LCG(2 31 , 65539, 0) is the infamous RANDU [IBM 1968]. LCG(2 32 , 69069, 1) is from Marsaglia [1972] and has been much used in the past, alone and in combination with other RNGs. LCG(2 32 , 1099087573, 0) is an LCG with "optimal multiplier" found by Fishman [1990]. ...

We introduce TestU01, a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generator's period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01, the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.

... This type of generator goes back to Lehmer (1951), who originally introduced it. Marsaglia (1972) and Knuth (1998) then investigated desirable relationships between a, c and m to achieve that the sequence of random numbers has full period. 123 Reasonable choices of the parameters are, for instance, discussed by 122 Besides this class of estimators one might also use inverse congruential estimators or feedback shift register (see Matsumoto & Nishimura (1998) and Eichenauer-Herrmann, Herrmann & Wegenkittl (1998) respectively for more detailed information and assessment of the sampling accuracy and quality). ...

This dissertation analyzes Open-End Turbo Certificates (OETCs), a popular class of retail derivatives. OETCs can be exercised at any time at the investor’s discretion. In order to explain the existence of the certificates jump risk must be considered. We propose and implement an optimal stopping approach to price these securities, which further allows for determining optimal exercise thresholds. They result from the trade-off between benefits from downward jump protection and financing costs. We show that early exercise right has a significant impact on their values. In an empirical analysis pertaining to the years 2007 through 2009 it turns out that certificates which could be rationally held are very rare, although the degree by which the underlying exceeds the optimal exercise thresholds continually declines over the considered period. We suggest three lines of explanation: general market movement, jump risk perception by the market, and increased competition among issuers.

... Among the works devoted to the analysis of the structure of the space of LCG states, one of the main ones is the thorough scientific work of G. Marsaglia [23]. In addition, the work [24] is devoted to the development of the theory of PRS construction based on the LCG and LFSR. ...

The results of the study of the graph of states of a linear congruential generator (LCG) are considered and theoretically substantiated. A model of a generalized graph of LCG states has been developed. It represents each connected component of the graph in the form of cycles equipped with tree products, allows classifying the types of connectivity components of the graph of LCG states and investigating the influence of parameters on its topology. A method for generating a pseudorandom sequence (PRS) of numbers based on the linear congruential method is presented. This method allows generating uniformly distributed numbers regardless of the topology of the graph of LCG states and, consequently, minimizing the time spent on choosing its parameters, and increasing the size of the space of their allowable values to achieve the maximum period. Computer implementation of the algorithm for generating PRS of permutations based on LCG with any type of graph of its states has allowed increasing the speed of the generator compared to the permutation generator using the modern Fisher-Yates algorithm.

... If additionally F q is a finite prime field, i.e., q = p, this special lattice test for N = T is the one which was proposed by Marsaglia in [16]. ...

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter andWinterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions.We also use a result of Brandstätter andWinterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG.

... Fast generator of good pesudorandom numbers is crucial in Monte Carlo simulations. The linear congruential generator (LCG) [16] is most traditional and wellknown, however, due to the inherent linearity, the distribution of a sequence from the LCG has a unwanted regularity, lattice structure [17] [20,Chapter 8]. The inversive linear congruential pseudorandom generator (ICG) [8], which passes s-dimensional lattice test for all s (p + 1)/2 [20,Theorem 8.5], is an attractive alternative. ...

SUMMARY As pseudorandom number generators for Monte Carlo simulations, inversive linear congruential generators (ICG) have some advantages compared with traditional linear congruen- tial generators. It has been shown that a sequence generated by an ICG has a low discrepancy even if the length of the sequence is far shorter than its period. In this paper, we formulate fractional linear congruential generators (FCG), a generalized concept of the inversive linear congruential generators. It is shown that the sequence generated by an FCG is a geometrical shift of a sequence from an ICG and satisfies the same upper bounds of discrepancy. As an application of the general formulation, we show that un- der certain condition, "Leap-Frog technique," a way of splitting a random number sequence to parallel sequences, can be applied

... 216-220). More technical and detailed evaluations, including discussion of the choice of c, may be found in Coveyou and McPherson (1969), Marsaglia (1972), Knuth (1981), and Moore (1982, 1986). There are many elaborations on pseudorandom number generation that build on the primitive of the linear or multiplicative congruential generator. ...

This chapter discusses simulation methods that are both important and useful in the solution of integration problems, and discusses the principles for the practical application of simulation in economics with a focus on integration problems. The simulation methods are generally straightforward for the investigator to implement, relying on an understanding of a few principles of simulation and the structure of the problem at hand. By contrast, deterministic methods require much larger problem-specific investments in numerical methods. Simulation methods can provide solutions for two related integration problems. One integration problem arises in model solution for agents whose expected utilities cannot be expressed as a closed function of state and decision variables. The other occurs, when the investigator combines sources of uncertainty about models to draw conclusions about policy. Markov chain Monte Carlo methods, which make use of samples that are neither independently nor identically distributed, have greatly expanded the scope of integration problems with convenient practical solutions.

... .in this case. See [1], [3, Chapter 3], and [5]. For our purposes, the following characterization is convenient. ...

The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method is estimated for parts of the period which are somewhat larger than the square root of the modulus. Applications to numerical integration are mentioned.

... Park and Miller (1988) only mentioned it as a possible minimal standard, arguing that poorer generators (at the time) should be eliminated; it is not their generator. Marsaglia (1972) proposed a = 69069 as a "candidate for the best possible multiplier" for m = 2 32 , based on its good lattice structure in up to 5 dimensions and because "it is easy to remember". But this LCG turns out to have a bad lattice structure in 6 dimensions (Knuth 1998). ...

Random number generators were invented before there were symbols for writing numbers, and long before mechanical and electronic computers. All major civilizations through the ages found the urge to make random selections, for various reasons. Today, random number generators, particularly on computers, are an important (although often hidden) ingredient in human activity. In this article, we give a historical account on the design, implementation, and testing of uniform random number generators used for simulation.

... 216-220). More technical and detailed evaluations, including discussion of the choice of c, may be found in Coveyou and McPherson (1967), Marsaglia (1972), Knuth (1981), and Moore (1982, 1986). ...

Nowadays, RFID systems have been widely deployed for applications such as supply chain management and inventory control. One of their most essential operations is to swiftly identify individual tags to distinguish their associated objects. Most existing solutions identify tags sequentially in the temporal dimension to avoid signal collisions, whose performance degrades significantly as the system scale increases. In this paper, we propose a Parallel Identification Protocol (PIP) for RFID systems, which achieves the parallel identification paradigm and is compatible with current RFID devices. Uniquely, PIP encodes the tag ID into a specially designed pattern and thus greatly facilitates the reader to correctly and effectively recover them from collisions. Furthermore, we analytically investigate its performance and provide guidance on determining its optimal settings. Extensive simulations show that PIP reduces the identification delay by about 25%-50% when compared with the standard method in EPC C1G2 and the state-of-the-art solutions.

This paper presents the results of an exhaustive search to find optimal multipliers A for the multiplicative congruential random number generator Zi @@@@ A Zi-1 (mod M) with prime modulus M &equil; 231-1. Since Marsaglia (1968) has shown that k-tuples from this and the more general class of linear congruential generators lie on sets of parallel hyperplanes, it has become common practice to evaluate multipliers in terms of their induced hyperplane structures. This study continues the practice and regards a multiplier as optimal if for k &equil; 2,...,6 and each set of parallel hyperplanes the Euclidean distance between adjacent hyperplanes does not exceed the minimal achievable distance by more than a prespecified amount. The concept of using this distance measure to evaluate multipliers orginated in the spectral test of Coveyou and MacPherson (1967) and has been used notably by Knuth (1981). However, the criterion of optimality defined here is considerably more stringent than the criteria that these writers proposed. First proposed by Lehmer (1951), the multiplicative congruential random number generator has come to be the most commonly employed mechanism for generating random numbers. Jannson (1966) collected the then known properties of these generators. Shortly there-after Marsaglia (1968) showed that all such generators share a common theoretical flaw and Coveyou and MacPherson (1967), Beyer, Roof and Williamson (1971), Marsaglia (1972) and Smith (1971) proposed alternative procedures for rating the seriousness of this flaw for individual multipliers. Later Niederreiter (1976, 1977, 1978a,b) proposed a rating system based on the concept of discrepancy, a measure of error used in numerical integration. With regard to empirical evaluation, Fishman and Moore (1982) described a comprehensive battery of statistical tests and illustrated how they could be used to detect local departures from randomness in samples of moderate size taken from these generators.

In periodic sequences of pseudorandom numbers generated by multiplicative congruential schemes, terms at certain critical distances are strongly correlated. For powers of two moduli a fast arithmetic method for computing these distances is given and applied to several generators. For all of them the length of the sequence that can be safely used turns out to be much shorter than the period. These correlations should be taken into account in parallel computations when a single pseudorandom sequence is partitioned among concurrent processors.

The paper is devoted to asymptotical analysis of Linear Congruential Generators (LCGs) with increasing moduli and multiplicators. The study is performed within the scope of a stochastic model of LCGs, the technique of weak convergence of distributions is used. It is proved, f.e., that the classical spectral test can be described in terms of a special probability metric inducing a topology of weak convergence. A number of theoretical examples of LCGs with good and bad asymptotical equidistribution properties is presented.

This paper presents the results of an exhaustive analysis of all of the prime modulus multiplicative congruential random number (RN) generators with moduli smaller than 2. In an amount of around 20 million multipliers which are able to produce a full period of RNs, 239 multipliers have a good lattice structure. Among which 52 multipliers further pass a comprehensive battery of empirical tests. These 52 multipliers thus possess good local and global statistical properties. It is worthwhile to note that some empirically tested multipliers recommended in some previous studies are not on this list. The conclusion is that both theoretical and empirical tests are mandatory to sieve out good multipliers. To generate RNs of very long period, many existing techniques can be applied without further effort.

Stochastic simulation has become very popular since it is one of the easiest things one can do with a stochastic model and the computers needed for simulation methods are available everywhere. This paper is intended for all those who use these methods in their work and need some knowledge about pseudorandom numbers which are fundamental in stochastic simulation. Therefore, the most frequently used pseudorandom number generators are discussed and we describe how good generators can be selected and implemented correctly. The paper contains also several remarks on recent trends in pseudorandom number generation.

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.

There are many methods for the transformation of uniform random numbers into nonuniform random numbers. These methods are employed for pseudo-random numbers generated by computer programs. It is shown that the sensitivity to the pseudo-random numbers used can vary a lot between the transformation methods. A classification of the sensitivity of several transformation methods is given. Numerical examples are presented for various transformation methods.

Monte Carlo methods are among the most used and useful computational tools available today, providing efficient and practical algorithims to solve a wide range of scientific and engineering problems. Applications covered in this book include optimization, finance, statistical mechanics, birth and death processes, and gambling systems. Explorations in Monte Carlo Methods provides a hands-on approach to learning this subject. Each new idea is carefully motivated by a realistic problem, thus leading from questions to theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. Each chapter ends with a large collection of problems illustrating and directing the material. This book is suitable as a textbook for students of engineering and the sciences, as well as mathematics. The problem-oriented approach makes it ideal for an applied course in basic probability and for a more specialized course in Monte Carlo methods. Topics include probability distributions, counting combinatorial objects, simulated annealing, genetic algorithms, option pricing, gamblers ruin, statistical mechanics, sampling, and random number generation.

Performance evaluation through computer simulation consists of three separate phases once one has decided what questions must be answered. First of all, a model of the system, that is a simplification of its structure and its operation in time, has to be described in terms of a computer program. The basic principles of this description will be presented in Chapter II while the implementation languages will be reviewed in Chapter III. The second phase consists in collecting measurements during the execution of the program. This operation is very similar to the measurements of real systems and can be performed with a number of statistical techniques discussed in Chapter IV. Since “the purpose of a simulation experiment is to predict some aspect of reality” (Naylor), it is important to test this function to gain confidence in the results obtained. This problem and its various implications such as the modelling assumptions and random number generation are considered in the last chapter.

In 1951, D. H. Lehmer [4] first proposed the linear congruential technique, Xn+1 ≡ aXn + c (Mod M) as a source of random numbers. This technique has since become known as the multiplicative random number generator for c = 0 and as the mixed random number generator for c ≠ 0.

In this chapter, we review several of the approaches for generating pseudorandom numbers (PRNs) on the unit interval. These are numbers that exhibit many of the properties of actual random numbers but are generated using deterministic algorithms. Also discussed are many of the desired features of PRN generators, such as uniformity, portability, large periods, and efficiency. In particular, we consider linear and nonlinear congruential generators, linear feedback shift register generators, and generators based on cellular automata. Some specific PRN generators such as Park and Miller's “minimal standard congruential generator”, the Wichmann–Hill generator, the L'Ecuyer generator, the Tausworthe bit-level generator, and the Mersenne Twister generator are presented. In addition, some of the many tests that prove useful in evaluating PRN generators are considered. A brief summary of PRN generator development from 1991 to 2020 is presented in order to stress the intense interest in and diversity among PRN generators. This chapter is provided for completeness and in order to stress the necessity to use “good” generators but it is not essential for users to understand all of the contents in order to perform useful MC calculations.

The topics of the workshop were recent progress in the theory of uniform distribution theory (also known as discrepancy theory) and new developments in its applications in analysis, approximation theory, computer science, numerics, pseudo-randomness and stochastic simulation.

Monte Carlo simulations have become a common practice to evaluate a proposed statistical procedure, particularly when it is analytically intractable. Validity of any simulation study relies heavily on the goodness of random variate generators for some specified distributions, which in turn is based on the successful generation of independent variates from the uniform distribution. However, a typical computer-generated pseudo-random number generator (PRNG) is a deterministic algorithm and we know that no PRNG is capable of generating a truly random uniform sequence. Since the foundation of a simulation study is built on the PRNG used, it is extremely important to design a good PRNG. We review some recent developments on PRNGs with nice properties such as high-dimensional equi-distribution, efficiency, long period length, portability, and efficient parallel implementations. WIREs Comput Stat 2017, 9:e1404. doi: 10.1002/wics.1404. For further resources related to this article, please visit the WIREs website.

This paper focuses on coarse-grained (or mesoscopic) simulations of bidisperse cis-1,4-polyisoprene (cis-PI) melts with the EMSIPON (Engine for Mescoscopic Simulations for Polymer Networks) code that has been developed in the Computational Materials Science and Engineering group at the National Technical University of Athens. The code implements a hybrid particle-field Brownian dynamics/kinetic Monte Carlo method advanced in the course of the last four years. Structural, thermodynamic and rheological properties have been computed and are compared with the corresponding properties of monodisperse cis-PI melts at the same temperature. Moreover, the algorithm employed for creating initial configurations of bidisperse as well as polydisperse polymer melts is explained.

Random numbers play a key role in diverse fields of cryptography, stochastic simulations, gaming, etc. Random numbers used in cryptography must satisfy additional properties of forward secrecy. Chaotic systems have been a potential source of random number generators. Both lower (One-dimension) and higher (two, three-dimension) chaotic systems are popularized in the generation of random bit sequences. Higher-order chaotic systems have a higher resistance to attacks owing to multiple dimensional outputs. Logistic map initially designed in one-dimension has been extended to two- dimensions to improve security. This paper proposes to use the concept of biological Gene Dominance to further improvise the randomness of 2D Logistic map. The sequences X and Y are considered to be parent genes that determine the value of parameter ‘r’ for the next iteration. The scatter plot of the proposed 2D Logistic Map with Gene Dominance (2DLMGD) shows almost uniform distribution of points in the region. The generated sequences are statistically tested using NIST SP 800-22 test suite and the results show that all sequences pass the tests. The random sequences are analysed for key sensitivity, information entropy, linear complexity, correlation to verify their conformity for use in cryptographic applications.

Uma questão importante em análises com simulação é a qualidade dos números supostamente aleatórios utilizados. O artigo avalia estatisticamente por meio do teste BDS um gerador de números aleatórios e vinte e quatro de pseudo-aleatórios. O teste BDS é utilizado para selecionar e ordenar geradores de números aleatórios e pseudo-aleatórios (geradores Monte Carlo). O teste BDS rejeitou estatisticamente os geradores Monte Carlo do MS Visual Basic 6.3, SPSS 13 (Mersenne twister) e RATS 5.1. Uma importante vantagem do gerador físico de números aleatórios é que o seu comprimento do ciclo é infinito. Dessa forma, propõe-se o teste dos geradores físicos de números supostamente verdadeiramente aleatórios disponíveis no mercado.

Siegel determined a fundamental domain using the Minkowski reduction of quadratic forms. He gave all the details concerning this domain for genus 1. It is the determination of the Minkowski fundamental domain presented as the second condition and the maximal height condition, presented as the third condition, which prevents the exact determination of this domain for the general case. The latest results were obtained by Gottschling for the genus 2 in 1959. It has since remained unexplored and poorly understood, in particular the different regions of Minkowski reduction. In order to identify Siegel's fundamental domain for genus 3, we present some results concerning the third condition of this domain. Every abelian function can be written in terms of rational functions of theta functions and their derivatives. This allows the expression of solutions of integrable systems in terms of theta functions. Such solutions are relevant in the description of surface water waves, non linear optics. Because of these applications, Deconinck and Van Hoeij have developed and implemented al-gorithms for computing the Riemann matrix and Deconinck et al. have developed the computation of the corresponding theta functions. Deconinck et al. have used Siegel's algorithm to approximately reach the Siegel fundamental domain and have adopted the LLL reduction algorithm to nd the shortest lattice vector. However, we opt here to use a Minkowski algorithmup to dimension 5 and an exact determination of the shortest lattice vector for greater dimensions.

Opaque methods for the calculation of random numbers do not generally lead to good generators. Linear congruential generators turned out to be very good. New results about the lattice structure of these generators and an algorithm for the calculation of reduced lattice bases of arbitrary dimension give further facilities for the assessment of linear congruential generators.

Universal codes efficiently compress sequences generated by stationary and ergodic sources with unknown statistics, and they were originally designed for lossless data compression. In the meantime, it was realized that they can be used for solving important problems of prediction and statistical analysis of time series, and this book describes recent results in this area. The first chapter introduces and describes the application of universal codes to prediction and the statistical analysis of time series; the second chapter describes applications of selected statistical methods to cryptography, including attacks on block ciphers; and the third chapter describes a homogeneity test used to determine authorship of literary texts. The book will be useful for researchers and advanced students in information theory, mathematical statistics, time-series analysis, and cryptography. It is assumed that the reader has some grounding in statistics and in information theory.

In this part we consider the problem of detecting deviations of binary sequence from randomness in details, because this problem is very important for cryptography for the following reasons.
Random number (RNG) and pseudorandom number generators (PRNG) are widely used in cryptography and more generally in data security systems. That is why statistical tests for detecting deviations are of great interest for cryptography [26, 27, 33]. (Thus, the National Institute of Standards and Technology (NIST, USA) has elaborated “A statistical test suite for random and pseudorandom number generators for cryptographic applications” [33].)

The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method, both homogeneous and inhomogeneous, is estimated for parts of the period that are somewhat larger than the square root of the modulus. The analogous problem for an arbitrary linear congruential generator modulo a prime is also considered, the result being particularly interesting for maximal period sequences. It is shown that the discrepancy estimates in this paper are best possible apart from logarithmic factors.

This paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 232and 248. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which A--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 132 multipliers out of the more than 536 million candidate multipliers that exist for modulus 232and to only 42 multipliers in a sample of about 67.1 million tested among the more than 351 x 1011candidate multipliers for modulus 232.

In most applications of stochastic simulation the source of randomness is a sequence of standard pseudorandom numbers u
0, u
1, u
2,…, i.e. a sequence of numbers generated by a computer which “behave” as a realization of a sequence U
0, U
1, U
2,…. of independent identically distributed random variables having a uniform distribution on the unit interval [0,1]. Many users of simulation do not think about how such numbers are produced by their computers and apply the standard software at hand. This attitude is dangerous since the source of randomness is fundamental for all stochastic simulations and many pseudorandom number generators in use have serious defects. It is also evident that the results of simulation studies depend on the methods of generation of the pseudorandom numbers. The present paper gives a survey of methods for generating deterministic sequences of numbers which can be used as standard pseudorandom number sequences with emphasis on the generation methods studied by the author’s research group in Darmstadt.

This paper describes an empirical search for correlation in sample sequences produced by 16 multiplicative congruential random number generators with modulus 231 - 1. Each generator has a distinct multiplier. One multiplier is in common use in the LLRANDOM and IMSL random generation packages as well as in APL and SIMPL/1. A second is used in SIMSCRIPT II. Six multipliers were taken from a recent study that showed them to have the best spectral and lattice test properties among 50 multipliers considered. The last eight multipliers had the poorest spectral and lattice test properties for 2-tupes among the 50. A well known poor generator, RANDU, with modulus 231, was also tested to provide a benchmark for evaluating the empirical testing procedure.
A comprehensive analysis based on test statistics derived from cumulative periodograms computed for each multiplier for each of 512 independent replications of 16384 observations each showed evidence of excess high frequency variation in two multipliers and excess midrange frequency variation in three others, including RANDU. Also evidence exists for a bimodal spectral density function for yet another multiplier. An examination of the test results showed that the empirical evidence of a departure from independence did not significantly favor the eight poorest multipliers. This observation is in agreement with a similar observation made by the authors in an earlier study of these multipliers that principally concentrated on their distributional properties in one, two and three dimensions. This consistency raises some doubt as to how one should interpret the results of the spectral and lattice tests for a multiplier. Also, the three multipliers considered superior in the earlier study maintain that position in the current study.

Every Monte Carlo experiment relies on the availability of a procedure that supplies sequences of numbers from which arbitrarily selected nonoverlapping subsequences appear to behave like statistically independent sequences and where the variation in an arbitrarily chosen subsequence of length k (≥1) resembles that of a sample drawn from the uniform distribution on the k-dimensional unit hyper-cube \({\mathcal{I}^k}\). The words “appear to behave” and “resemble” alert the reader to yet another potential source of error that arises in Monte Carlo sampling. In practice, many procedures exist for generating these sequences. In addition to this error of approximation, the relative desirability of each depends on its computing time, on its ease of use, and on its portability By portability, we mean the ease of implementing a procedure or algorithm on a variety of computers, each with its own hardware peculiarities.

This paper suggests, as did an earlier one, [1] that points inn-space produced by congruential random number generators are too regular for general Monte Carlo use. Regularity was established in [1] 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
n
must contain at least two points with integral coordinates. The present paper takes a different approach to establishing the coarse lattice structure of congruential generators. It gives a simple, self-contained proof that points inn-space produced by the general congruential generatorr
i+1 ar
i
+b modm must fall on a lattice with unit-cell volume at leastm
n–1 There is no restriction ona orb; this means thatall 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 ann-lattice with unit-cell volume 1.

In this paper some generating methods of the multiplicative and mixed congruential types for obtaining pseudo-random numbers are reviewed. Procedures to determine autocorrelations are demonstrated and a few numerical results are also given.

Random sampling methods are valuable not only for providing solutions to problems involving probability but also for solving many problems that are deterministic in nature. Haphazard generation of numbers has several serious disadvantages, since the numbers used in the computation cannot be reproduced and thus rational ``debugging'' procedures cannot be developed. Over the past 20 years there has been a strong emphasis on arithmetic generators, which are based on recurrence relations involving integers.