Stochastic simulation requires a reliable source of randomness. Inversive methods are an interesting and very promising new approach to produce uniform pseudorandom numbers. In this paper, we present evidence that these methods are an important contribution to our toolbox. We survey the outstanding performance of inversive pseudorandom number generators in theoretical and empirical tests, in comparison to linear generators. In addition, this paper contains tables of parameters to implement inversive congruential generators. More empirical results as well as an implementation of inversive generators in C are available in the Internet from our Web-site http:// random.mat.sbg.ac.at. 1 INTRODUCTION Pseudorandom number generators are essential elements in the toolbox of stochastic simulation. Their task is to simulate realizations of independent, identically U([0; 1[)-distributed random variables. Other distributions will be obtained by transformation methods, see Devroye (1986), and the...

Inversive congruential generators have many attractive theoretical properties (see the contribution of Entacher and Hellekalek [6] in this volume). However, it is still unknown if and in what respect the new generators are superior to the traditional ones in real-world stochastic simulations. In this paper, we study the behavior of linear congruential and inversive congruential random number generators in stringent statistical tests with varying parameters. We find that the performance of linear generators depends heavily on the choice of the testparameters, while that of inversive generators does not. We conclude that stochastic simulations should not be based on linear generators alone but also on inversive generators. Contents 1 Introduction 2 2 Basic digit test 3 2.1 Empirical results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 3 Extended digit test 5 3.1 Empirical results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 4 A run tes...