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

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.

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

Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasi-Monte 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.

We give the results of a computer search for maximally-equidistributed combined linear feedback shift register (or Tausworthe) random number generators, whose components are trinomials of degrees slightly less than 32 or 64. These generators are fast and have good statistical properties.

This paper discusses certain classes of uniform random number generators which have been studied and better understood in the recent few years. Most of the attention is devoted to combined generators. We also mention others and point out some pitfalls. Combination is a good way to obtain fast and reliable generators, but the structural properties of the combined generator should be carefully examined before it could be recommended. Nonlinear generators offer some promise, but still require deeper investigation before specific instances can be safely recommended. 1. INTRODUCTION Random numbers are the nuts and bolts of all stochastic simulations. Simple linear congruential generators (LCGs) (Bratley, Fox, and Schrage 1987; Knuth 1981) are still in widespread use for generating uniform random numbers, mainly because of their simplicity and ease of implementation. However, LCGs have several well-known defects and no longer satisfy the requirements of today's computerintensive simulations...

This paper studies the combination of generators of the latter class. It shows that such combination offers a good way of obtaining a fast and reliable generator, and improves significantly upon the combination of simple LCGs. We also provide a specific generator with its computer implementation