Xorshift

Xorshift random number generators are a class of pseudorandom number generators that were discovered by George Marsaglia. Specifically, they are a subset of linear-feedback shift registers (LFSRs) which allow a particularly efficient implementation without using excessively sparse polynomials. They generate the next number in their sequence by repeatedly taking the exclusive or of a number with a bit-shifted version of itself. This makes them extremely fast on modern computer architectures. Like all LFSRs, the parameters have to be chosen very carefully in order to achieve a long period.

Xorshift generators are among the fastest non-cryptographically-secure random number generators, requiring very small code and state. Although they do not pass every statistical test without further refinement, this weakness is well-known and easily amended (as pointed out by Marsaglia in the original paper) by combining them with a non-linear function, resulting e.g. in a xorshift+ or xorshift* generator. A naive C implementation of a xorshift+ generator that passes all tests from the BigCrush suite (with an order of magnitude fewer failures than Mersenne Twister or WELL) typically takes fewer than 10 clock cycles on x86 to generate a random number, thanks to instruction pipelining.

Because plain xorshift generators (without a non-linear step) fail a few statistical tests, they have been accused of being unreliable.

A C/C++ version of two xorshift algorithms are given here. One has one 32-bit word of state, and period 2³²−1. The other has four words of state, and period 2¹²⁸−1. Both use three shifts and three exclusive-or operations:

The 128-bit algorithm passes the diehard tests. However, it fails the MatrixRank and LinearComp tests of the BigCrush test suite from the TestU01 framework.

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