Four-tap shift-register-sequence random-number generators
Robert M. Ziff (University of Michigan, Department of Chemical, Engineering, Ann Arbor)

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.
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