Article

# The Structure of Linear Congruential Sequences

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## Abstract

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. ...
Article
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. ...
Article
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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. ...
Article
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 , . . . ...
Preprint
Full-text available
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. ...
Article
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 ), ...
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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]. ...
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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). ...
Thesis
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. ...
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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]. ...
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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. ...
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... 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. ...
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... .in this case. See [1], [3, Chapter 3], and [5]. For our purposes, the following characterization is convenient. ...
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... 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). ...
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... 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). ...
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Chapter
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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.
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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.
Chapter
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.
Chapter
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.
Chapter
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.
Article
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.
Article
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.
Article
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.