Xorshift1024*, Xorshift1024+, Xorshift128+ and Xoroshiro128+ Fail Statistical Tests for Linearity

TL;DR

This paper evaluates the statistical robustness of scrambled Xorshift-based random number generators, revealing they fail certain linearity tests when analyzing specific bits in reverse order, highlighting potential weaknesses.

Contribution

It uncovers systematic failures of scrambled Xorshift generators in linearity tests, challenging assumptions about their statistical quality.

Findings

Fail Big Crush linearity tests on lower 32 bits in reverse order

Scrambled generators are not statistically robust in certain configurations

Highlights need for improved testing of RNGs' linearity properties

Abstract

L'Ecuyer & Simard's Big Crush statistical test suite has revealed statistical flaws in many popular random number generators including Marsaglia's Xorshift generators. Vigna recently proposed some 64-bit variations on the Xorshift scheme that are further scrambled (i.e., Xorshift1024*, Xorshift1024+, Xorshift128+, Xoroshiro128+). Unlike their unscrambled counterparts, they pass Big Crush when interleaving blocks of 32 bits for each 64-bit word (most significant, least significant, most significant, least significant, etc.). We report that these scrambled generators systematically fail Big Crush---specifically the linear-complexity and matrix-rank tests that detect linearity---when taking the 32 lowest-order bits in reverse order from each 64-bit word.