Four-tap shift-register-sequence random-number generators

TL;DR

This paper demonstrates that increasing feedback taps from two to four in shift-register random-number generators significantly reduces correlations, resulting in higher-quality generators suitable for various applications.

Contribution

It introduces methods to design four-tap GFSR generators with maximal cycles and analyzes their correlation properties, showing improved randomness over traditional two-tap generators.

Findings

Four-tap generators with large offsets pass rigorous randomness tests

Two-tap generators generally fail the tests

Four-tap generators are suitable for high-quality applications

Abstract

It is shown how correlations in the generalized feedback shift-register (GFSR) random-number generator are greatly diminished when the number of feedback taps is increased from two to four (or more) and the tap offsets are lengthened. Simple formulas for producing maximal-cycle four-tap rules from available primitive trinomials are given, and explicit three- and four-point correlations are found for some of those rules. A number of generators are also tested using a simple but sensitive random-walk simulation that relates to a problem in percolation theory. While virtually all two-tap generators fail this test, four-tap generators with offset greater than about 500 pass it, have passed tests carried out by others, and appear to be good multi-purpose high-quality random-number generators.