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Random number generation
"... Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables dis ..."
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Cited by 136 (30 self)
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Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval
Recent Advances In Randomized QuasiMonte Carlo Methods
"... We survey some of the recent developments on quasiMonte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a highdimensional inte ..."
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Cited by 59 (12 self)
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We survey some of the recent developments on quasiMonte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a highdimensional integral. We review several QMC constructions and dierent randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of gures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration. 1 2 1.
Improved longperiod generators based on linear recurrences modulo 2
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 2006
"... Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed v ..."
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Cited by 38 (7 self)
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Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The hugeperiod generators proposed so far are not quite optimal in that respect. In this paper, we propose new generators of that form, with better equidistribution and “bitmixing ” properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.
High Quality Uniform Random Number Generation Through LUT Optimised Linear Recurrences
"... This paper describes a class of FPGAspecific uniform random number generators with a 2 k − 1 length period, which can provide k random bits percycle for the cost of k Lookup Tables (LUTs) and k flipflops. The generator is based on a binary linear recurrence, but with a recurrence matrix optimised ..."
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Cited by 22 (16 self)
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This paper describes a class of FPGAspecific uniform random number generators with a 2 k − 1 length period, which can provide k random bits percycle for the cost of k Lookup Tables (LUTs) and k flipflops. The generator is based on a binary linear recurrence, but with a recurrence matrix optimised for LUT based architectures. It avoids many of the problems and inefficiencies associated with LFSRs and Tausworthe generators, while retaining the ability to efficiently skip ahead in the sequence. In particular we show that this class of generators produce the highest sample rate for a given area compared to LFSR and Tausworthe generators. The statistical quality of this type of generators is very good, and can be used to create small and fast generators with long periods which pass all common empirical tests, such as Diehard, Crush, BigCrush and the NIST cryptographic tests. 1.
Random Number Generators Based on Linear Recurrences In ...
 AND QUASIMONTE CARLO METHODS 2002
, 2004
"... This paper explores new ways of constructing and implementing random number generators based on linear recurrences in a finite field with 2 elements, for some integer w. Two types of constructions are examined. Concrete parameter sets are provided for generators with good equidistribution propert ..."
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Cited by 18 (6 self)
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This paper explores new ways of constructing and implementing random number generators based on linear recurrences in a finite field with 2 elements, for some integer w. Two types of constructions are examined. Concrete parameter sets are provided for generators with good equidistribution properties and whose speed is comparable to that of the fastest generators currently available. The implementations use precomputed tables to speed up computations in F2 w
TestU01: A Software Library in ANSI C for Empirical Testing of Random Number Generators
, 2007
"... This document describes the software library TestU01, implemented in the ANSI C language, and offering a collection of utilities for the (empirical) statistical testing of uniform random number generators (RNG). The library implements several types of generators in generic form, as well as many spec ..."
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Cited by 18 (2 self)
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This document describes the software library TestU01, implemented in the ANSI C language, and offering a collection of utilities for the (empirical) statistical testing of uniform random number generators (RNG). The library implements several types of generators in generic form, as well as many specific generators proposed in the literature or found in widelyused software. It provides general implementations of the classical statistical tests for random number generators, as well as several others proposed in the literature, and some original ones. These tests can be applied to the generators predefined in the library and to userdefined generators. Specific tests suites for either sequences of uniform random numbers in [0, 1] or bit sequences are also available. Basic tools for plotting vectors of points produced by generators are provided as well. Additional software permits one 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.
Empirical Evidence concerning AES
 ACM TRANSACTIONS ON MODELING AND COMPUTER SIMULATION
, 2003
"... ..."
Random Number Generators: Selection Criteria and Testing
, 1998
"... this paper, we shall assume that the sequence is purely periodic, in the sense that the initial state s 0 is always revisited. In other words, the sequence has no transient part. The goal is to make it hard to distinguish between the output of the generator and a typical realization of an i.i.d. uni ..."
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Cited by 15 (7 self)
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this paper, we shall assume that the sequence is purely periodic, in the sense that the initial state s 0 is always revisited. In other words, the sequence has no transient part. The goal is to make it hard to distinguish between the output of the generator and a typical realization of an i.i.d. uniform sequence over U . In
Construction of Equidistributed Generators based on linear recurrences modulo 2
, 2000
"... Random number generators based on linear recurrences modulo 2 are widely used and appear in dierent forms, such as the simple and combined Tausworthe generators, the GFSR, and the twisted GFSR generators. Lowdiscrepancy point sets for quasiMonte Carlo integration can also be constructed based on t ..."
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Cited by 13 (6 self)
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Random number generators based on linear recurrences modulo 2 are widely used and appear in dierent forms, such as the simple and combined Tausworthe generators, the GFSR, and the twisted GFSR generators. Lowdiscrepancy point sets for quasiMonte Carlo integration can also be constructed based on these linear recurrences. The quality of these generators or point sets is usually measured by certain equidistribution criteria. Combining two or more recurrences and adding linear output transformations can be used to improve the equidistribution properties. In this
Randomized Polynomial Lattice Rules For Multivariate Integration And Simulation
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2001
"... Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasiMonte Carlo set of tools. A new class of lattice rules, defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, and a theoretical fra ..."
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Cited by 12 (3 self)
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Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasiMonte Carlo set of tools. A new class of lattice rules, defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, and a theoretical framework for these polynomial lattice rules is developed. A randomized version is studied, implementations and criteria for selecting the parameters are discussed, and examples of its use as a variance reduction tool in stochastic simulation are provided. Certain types of digital net constructions, as well as point sets constructed by taking all vectors of successive output values produced by a Tausworthe random number generator, turn out to be special cases of this method.