Digital Nonlinear Random Numbers

Random number generators ("RNGs") are the basic tools of stochastic modeling. Bad random number generators may ruin a simulation. There are several pitfalls to be avoided. The facts are as follows: FACT ONE: With random number generators, there are no guarantees, only predictions. This is not because the word "randomness" is involved but because the finitely many random numbers we produce and their transformed variates cannot fit every imaginable distribution well enough. Every generator has its regularities which, occasionally, may become deficiencies. Hence, in a given application, even reliable generators may fail. FACT TWO: Although there are no guarantees, there are mathematical safety-measures against wrong simulation results due to inappropriate random number generators. The safety-measures consist of theoretical analysis, empirical tests, and of the assessment of the practical aspects of a RNG. In this project, we aim at constructing nonlinear counterparts to digital RNGs like the GFSR (i.e. generalized feedback shift register method), and the tGFSR (i.e. twisted GFSR) of Matsumoto et. al. We will assess our algorithms by a comprehensive theoretical analysis that deals with the questions of period length, the correlation properties of the random numbers we produce, and structural analysis. Further, we will use a series of empirical tests with great variations in the parameters to compare the performance of the new algorithms to known RNGs in a setting of practical relevance. In several projects funded by the Austrian Science Foundation ("FWF"), we have established the theoretical background as well as the software and computing environment necessary to carry out this project, see our server http://random.mat.sbg.ac.at.