Article

# Coding the Lehmer pseudo number generator

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## Abstract

An algorithm and coding technique is presented for quick evaluation of the Lehmer pseudo-random number generator modulo 2 ** 31 - 1, a prime Mersenne number which produces 2 ** 31 - 2 numbers, on a p-bit (greater than 31) computer. The computation method is extendible to limited problems in modular arithmetic. Prime factorization for 2 ** 61 - 2 and a primitive root for 2 ** 61 - 1, the next largest prime Mersenne number, are given for possible construction of a pseudo-random number generator of increased cycle length.

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... We will assume α = 32. Then, it uses a PRNG technique such as defined in [26] to build N coefficients c 1 , . . . , c N . ...
... The server will send through each RSU a random sequence of such blocks. Note that by using a simple PRNG such as a Lehmer [26], there is no need of any kind of synchronisation between the server and the receivers. ...
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This paper introduces a new approach for infrastructure based content distribution in a vehicular network. It is built on broadcasting and pseudo random network coding. Its main strength is that, being broadcast based, it does not need any feedback channel and thus uses less data rate. Data is transmitted exploiting network coding, multiple linear combinations of data are sent. A vehicle needs to receive a defined number of independent linear combinations to decode the data. The server will send a larger number of different linear combinations. The unreliability of broadcast is thus neutralized through a useful redundancy rather than through re-transmission. Finally, computation of the linear combination coefficients is done so that the overhead is the same as it would be without network coding. Depending on the infrastructure deployed, this technique can be a content distribution solution per se or the first step in a more general solution, the second step being based on collaborative download. The high diversity in transmissions will then be a key feature for the performance of such an application.
... We will assume α = 32. Then, it uses a PRNG technique such as defined in [26] to build N coefficients c 1 , . . . , c N . ...
... The server will send through each RSU a random sequence of such blocks. Note that by using a simple PRNG such as a Lehmer [26], there is no need of any kind of synchronisation between the server and the receivers. ...
Article
Full-text available
This paper introduces a new approach for infrastructure based content distribution in a vehicular network. It is built on broadcasting and pseudo random network coding. Its main strength is that, being broadcast based, it does not need any feedback channel and thus uses less data rate. Data is transmitted exploiting network coding, multiple linear combinations of data are sent. A vehicle needs to receive a defined number of independent linear combinations to decode the data. The server will send a larger number of different linear combinations. The unreliability of broadcast is thus neutralized through a useful redundancy rather than through re-transmission. Finally, computation of the linear combination coefficients is done so that the overhead is the same as it would be without network coding. Depending on the infrastructure deployed, this technique can be a content distribution solution per se or the first step in a more general solution, the second step being based on collaborative download. The high diversity in transmissions will then be a key feature for the performance of such an application.
... It would be better if this kind of generator provides a longer cycle length to meet the current needs of large-scale simulations. Payne et al. [1969] predicted that, due to increases in computer speed and the next Mersenne prime of 2 31 Ϫ 1 being 2 61 Ϫ 1, multiplicative congruential generators with modulus 2 61 Ϫ 1 would be needed. The time has now arrived. ...
... The division operation (mod m) in multiplicative congruential generators with (prime) modulus m ϭ 2 p Ϫ 1 can be performed by shifting and addition [Payne et al. 1969]. To further replace multiplication with shifting and addition, multiplier a must be in the form of simple expressions of 2 k . ...
Article
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The demand for random numbers in scientific applications is increasing. However, the most widely used multiplicative, congruential random-number generators with modulus 231 − 1 have a cycle length of about 2.1 × 109. Moreover, developing portable and efficient generators with a larger modulus such as 261 − 1 is more difficult than those with modulus 231 − 1. This article presents the development of multiplicative, congruential generators with modulus m = 2p − 1 and four forms of multipliers: 2k1 &minus 2k2, 2k1 + 2k2, m − 2k1 + 2k2, and m − 2k1 − 2k2, k1 > k2. The multipliers for modulus 231 − 1 and 261 − 1 are measured by spectral tests, and the best ones are presented. The generators with these multipliers are portable and vary fast. They have also passed several empirical tests, including the frequency test, the run test, and the maximum-of-t test.
... The generation of normally distributed random numbers is required for many numerical applications. Oppositely, most of pseudo-random number generators used for computations produce uniform distributed numbers via bit operations [107,108]. Gladly, there are methods to convert these uniformly distributed numbers into which-ever distribution we might be interested. Most modern programming languages offer these algorithms implicitly, but I thought it would be useful to describe one of these Box-Muller algorithm. ...
Thesis
Understanding chromatin organization and its role in gene regulation is of major importance, however its underlying dynamics has been overseen up to recent years. Here I present results regarding dynamical properties of chromatin in diverse stages of the cell cycle and a possible connection between gene activity and local diffusion properties. I develop a new computational framework based on Gaussian processes and fractional Brownian motion called GP-FBM. This method infers apparent diffusion and anomalous coefficients more accurately than other popular methods and corrects for confound background movement. I further introduce a new biopolymer model using a mean-field approach in which Hi-C maps are used to model chromatin long-range interactions. Further, ChIP-seq data is used to calibrate local properties of the nuclear environment. This model was able to recapitulate experimental results for specific loci of the HoxA domain in mouse cells.
... • Mother RNG, available in Marsaglia's website (MOT, Marsaglia, 1994); • Multiple with carry RNG (MWC, Marsaglia, 1994); • Combo RNG (COM, Marsaglia, 1994); • Lehmer RNG (LEH, Payne et al., 1969); • Fractional Brownian motion (fBm) and fractional Gaussian noise (fGn); refer to Bardet et al. (2003); • Coloured noise with power spectrum f Àk with k ≥ 0 (Larrondo, 2012); • Linear congruential generator (LCG, Knuth, 1997). ...
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... We assume that the assignment operator, all arithmetic operations, and all comparison operations take one time unit. Let the rnd function be calculated by the Lehmer pseudo-random number generator [33] using the following equation: ...
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... We assume that the assignment operator, all arithmetic operations, and all comparison operations take one time unit. Let the rnd function be calculated by the Lehmer pseudo-random number generator [33] using the following equation: ...
Preprint
This paper presents a two-dimensional mathematical model of compound eye vision. Such a model is useful for solving navigation issues for autonomous mobile robots on the ground plane. The model is inspired by the insect compound eye that consists of ommatidia, which are tiny independent photoreception units, each of which combines a cornea, lens, and rhabdom. The model describes the planar binocular compound eye vision, focusing on measuring distance and azimuth to a circular feature with an arbitrary size. The model provides a necessary and sufficient condition for the visibility of a circular feature by each ommatidium. On this basis, an algorithm is built for generating a training data set to create two deep neural networks (DNN): the first detects the distance, and the second detects the azimuth to a circular feature. The hyperparameter tuning and the configurations of both networks are described. Experimental results showed that the proposed method could effectively and accurately detect the distance and azimuth to objects.
... We will assume α = 32. It then uses a PRNC technique such as defined in [18] to build N coefficients c 1 , . . . , c N . ...
Conference Paper
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This paper introduces PRAVDA, a new approach for infrastructure based content distribution in a vehicular network. PRAVDA is built on broadcasting and pseudo random network coding. Its first strength is that, being broadcast based, it does not need any feedback channel and thus consumes very little throughput. Data is transmitted through network coding: multiple linear combinations of data are sent. A vehicle needs to receive a defined number of independent linear combinations to decode the data. The server will send a larger number of different linear combinations. The unreliability of broadcast is thus fought through a useful redundancy rather than through re-transmission. Finally, computation of the linear combination coefficients is done so that the overhead is the same as it would be without network coding.
... The above conditions can be satisfied if mod m is a full repetend prime or proper prime in base a [16]. A number m is said to be a full repetend prime, if the remainder of (2) repeats after a period of m -1. ...
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... Ag and Au atoms in the alloy nanoparticles were generated randomly using the build-in (pseudo-) random number generator of LAMMPS. 46,48 The physical quantities of alloy NPs calculated, including potential energies and number of atoms ejected, were also averaged using five replicates with different randomly-generated structures. ...
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... The random number generators (RNGs) are kept thread-private and are initiated with independent seeds, which are provided by a different type of RNG [e.g., 16 807 RNG (Ref. 30)] in our implementation. ...
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Purpose: The clinical commissioning of IMRT subject to a magnetic field is challenging. The purpose of this work is to develop a GPU-accelerated Monte Carlo dose calculation platform based on penelope and then use the platform to validate a vendor-provided MRIdian head model toward quality assurance of clinical IMRT treatment plans subject to a 0.35 T magnetic field.
... In this way, the smart card successfully checks the identity and password of user U i . The smart card generates a random number r using a pseudorandom number generator function [43]. For example, r jþ1 ¼ ðar j þ bÞ mod n where a, b are user defined parameters; 1 a n À 1 and 0 b n À 1. r 0 is seed value which is also defined by user. ...
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... We will assume α = 32. It then uses a PRNC technique such as defined in [18] to build N coefficients c 1 , . . . , c N . ...
Article
Full-text available
This paper introduces PRAVDA, a new approach for infrastructure based content distribution in a vehicular network. PRAVDA is built on broadcasting and pseudo random network coding. Its first strength is that, being broadcast based, it does not need any feedback channel and thus consumes very little throughput. Data is transmitted through network coding: multiple linear combinations of data are sent. A vehicle needs to receive a defined number of independent linear combinations to decode the data. The server will send a larger number of different linear combinations. The unreliability of broadcast is thus fought through a useful redundancy rather than through re-transmission. Finally, computation of the linear combination coefficients is done so that the overhead is the same as it would be without network coding.
... We suggest a uniform distribution algorithm such as multiplicative congruential algorithm [45,46], which is the basis for many of the random number generators in use today. Lehmer's generators [47] involve three integer parameters, r, s, and m, and an initial value, x0, called the seed. A sequence is generated by the following modified formula: ...
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Full-text available
Wireless sensor network (WSN) consists of many hosts called sensors. These sensors can sense a phenomenon (motion, temperature, humidity, average, max, min, etc.) and represent what they sense in a form of data. There are many applications for WSNs including object tracking and monitoring where in most of the cases these objects need protection. In these applications, data privacy itself might not be as important as the privacy of source location. In addition to the source location privacy, sink location privacy should also be provided. Providing an efficient end-to-end privacy solution would be a challenging task to achieve due to the open nature of the WSN. The key schemes needed for end-to-end location privacy are anonymity, observability, capture likelihood, and safety period. We extend this work to allow for countermeasures against multi-local and global adversaries. We present a network model protected against a sophisticated threat model: passive /active and local/multi-local/global attacks. This work provides a solution for end-to-end anonymity and location privacy as well. We will introduce a framework called fortified anonymous communication (FAC) protocol for WSN.
... The Mersenne twister [28], for example, developed in 1997, generates very high-quality pseudorandom numbers. Others exist such as the Lehmer random number generator [29], a variant of the linear congruential generator. ...
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... The random distribution we use is of uniform rate over the entire area between the intervals a, b. The main generators of pseudo-random numbers used today are called linear congruence generators introduced by Lehmer in 1951 [3][4][5]. A congruential method starts with an initial value (seed) x 0 , and successive values xn, n ≥ 1 are obtained recursively using the following formula: ...
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Considerable attention has recently been directed at developing simpler and faster algorithms for generating gamma random variates (with general, not necessarily integral, shape parameter α) on digital computers. This paper surveys the current state of the art, which includes fifteen gamma algorithms applicable for α≥1 and six that are applicable for α<1. These algorithms are compared according to the criteria of speed and simplicity. General random variate generation techniques are explained with reference to these gamma algorithms. Computer simulation experiments on DEC and CDC computers are reported. Guidelines for some specific applications are given.
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Exploring Monte Carlo Methods is a basic text that describes the numerical methods that have come to be known as "Monte Carlo. " The book treats the subject generically through the first eight chapters and, thus, should be of use to anyone who wants to learn to use Monte Carlo. The next two chapters focus on applications in nuclear engineering, which are illustrative of uses in other fields. Five appendices are included, which provide useful information on probability distributions, general-purpose Monte Carlo codes for radiation transport, and other matters. The famous "Buffon's needle problem " provides a unifying theme as it is repeatedly used to illustrate many features of Monte Carlo methods. This book provides the basic detail necessary to learn how to apply Monte Carlo methods and thus should be useful as a text book for undergraduate or graduate courses in numerical methods. It is written so that interested readers with only an understanding of calculus and differential equations can learn Monte Carlo on their own. Coverage of topics such as variance reduction, pseudo-random number generation, Markov chain Monte Carlo, inverse Monte Carlo, and linear operator equations will make the book useful even to experienced Monte Carlo practitioners. Provides a concise treatment of generic Monte Carlo methods Proofs for each chapter Appendixes include Certain mathematical functions; Bose Einstein functions, Fermi Dirac functions, Watson functions.
Article
One of the general methods for implementing a multiple recursive generator is recursive reduction method (RRM). This paper provides an analysis of its number of operations. We also propose a new method that modifies the RRM by considering the stop conditions and the order of multiplier and multiplicand. Some empirical comparisons reveal that the new algorithm is more efficient with a reduction of iterative numbers in the range of 13.1416%-20.5118% depending on the moduli and multipliers being used.
Article
Linear congruential random number generators must have large moduli to attain maximum periods, but this creates integer overflow during calculations. Several methods have been suggested to remedy this problem while obtaining portability. Approximate factoring is the most common method in portable implementations, but there is no systematic technique for finding appropriate multipliers and an exhaustive search is prohibitively expensive. We offer a very efficient method for finding all portable multipliers of any given modulus value. Letting M=AB+CM=AB+C, the multiplier AA gives a portable result if B−CB−C is positive. If it is negative, the portable multiplier can be defined as A=⌊M/B⌋A=⌊M/B⌋. We also suggest a method for discovering the most fertile search region for spectral top-quality multipliers in a two-dimensional space. The method is extremely promising for best generator searches in very large moduli: 64-bit sizes and above. As an application to an important and challenging problem, we examined the prime modulus 263−25263−25, suitable for 64-bit register size, and determined 12 high quality portable generators successfully passing stringent spectral and empirical tests.
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This paper considers a Gaussian first-order autoregressive process with unknown intercept where the initial value of the variable is a known constant. Monte Carlo simulations are used to investigate the sampling distribution of the t statistic for the autoregressive parameter when its value is in the neighborhood of unity. A small sigma asymptotic result is exploited in the construction of exact non-similar tests. The powers of non-similar tests of the random walk and other hypotheses are estimated for sample sizes typical in economic applications.
Article
The present method generates machine-Independent uniform random sequences of real numbers in the interval (0.,1.) excluding 1. It uses a generalization of mulltiplicative linear gongruential generators working with prime numbers as moduli whose values have been fixed according to the positive integer arithmetic storage available from the system, and one or their corresponding primitive elements as multipliers to complete independently each full cycle.The periodicity can be considered as infinite: O (1092) for a 16-bit machine and O (10174) for a 32-bit machine and their respective integer arithmetic; the periodicity can be adjusted if it is required by the user in the normal version or statistically reaching the maximum in the enhanced 'stagger' version.An implementation of the method is available in the form of structured Fortran 77 functions and gives bettr results in term of velocity and periodicity than the other transportable functions compared with good quality of randomness.
Article
Two simple and easily implemented algorithms are presented for obtaining random variables from the exponential power distribution with parameter α. Both algorithms are based on a generalization of Von Neumann's rejection technique. In the first algorithm, the first-stage sampling is from the double-exponential distribution, while the second algorithm uses the normal distribution. These two algorithms are applicable for all values of α, α ≥ 1 and α ≥ 2, respectively.
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For three symmetric distributions and six sample sizes, this article presents estimates of the small-sample variance of Pitman's location-invariant and location-scale-invariant estimators of location. It then compares these two estimators and investigates the closeness of the Cramér-Rao bound when estimators are required to be invariant. Expressions of the form c/(n − d) prove quite effective in fitting variances of Pitman estimators, and d can be interpreted as the amount of Fisher information “lost.” In terms of relative efficiency, maximum-likelihood estimators and linear combinations of order statistics offer computationally attractive alternatives to the Pitman estimators.
Article
The effects of different monetary rules on the rates of inflation and unemployment are studied by stochastic simulation of the Federal Reserve Board-MIT-Pennsylvania (FMP) Econometric Model and the St. Louis “Monetarist” Model. A number of heuristic and more formal statistical methods are used in evaluating the results. It is shown that simple feedback control rules—involving proportional and derivative controls—reduce the variability of the target variables relative to the rule in which the money supply is increased at a constant rate. The improvement is considerably greater in the St. Louis model than in the FMP model.
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Die Konvergenzordnung eines Differenzenverfahrens ist von den Differenzierbarkeitseigenschaften der Lösung der Differentialgleichung abhängig. Bei parabolischen Anfangsrandwertaufgaben werden diese durch die Anfangsfunktion bestimmt. Es wird das Verhalten der Konvergenzordnung in Abhängigkeit von der Anfangsfunktion untersucht.
Article
Pseudo-random number generators of the power residue (sometimes called congruential or multiplicative) type are discussed and results of statistical tests performed on specific examples of this type are presented. Tests were patterned after the methods of MacLaren and Marsaglia (M&M). The main result presented is the discovery of several power residue generators which performed well in these tests. This is important because, of all the generators using standard methods (including power residue) that were tested by M&M, none gave satisfactory results. The overall results here provide further evidence for their conclusion that the types of tests usually encountered in the literature do not provide an adequate index of the behavior of n-tuples of consecutively generated numbers. In any Monte Carlo or simulation problem where n supposedly independent random numbers are required at each step, this behavior is likely to be important. Finally, since the tests presented here differ in certain details from those of M&M, some of their generators were retested as a check. A cross-check shows that results are compatible; in particular, if a generator failed one of their tests badly, it also failed the present author's corresponding test badly.
Random number generation on the BRL highspeed computing machines, by M
• LEHMER D.H