Some linear and nonlinear methods for pseudorandom number generation

Two principal classes of methods for the generation of uniform pseudorandom numbers can nowadays be distinguished, namely linear and nonlinear methods, and contributions to both types of methods are presented. A very general linear method, the multiple-recursive matrix method, was recently introduced and analyzed by the author. This method includes as special cases several classical methods, and also the twisted GFSR method. New theoretical results on the multiple-recursive matrix method are discussed. Among nonlinear methods, the digital inversive method recently introduced by Eichenauer-Herrmann and the author is highlighted. This method combines real and finite-field arithmetic and, in contrast to other inversive methods, allows a very fast implementation, while still retaining the advantages of inversive methods.