Xorshift random number generators, also called shift-register generators, are a class of pseudorandom number generators that were invented by George Marsaglia. They are a subset of linear-feedback shift registers (LFSRs) which allow a particularly efficient implementation in software without the excessive use of 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 execution extremely efficient on modern computer architectures, but it does not benefit efficiency in a hardware implementation. Like all LFSRs, the parameters have to be chosen very carefully in order to achieve a long period. For execution in software, xorshift generators are among the fastest PRNGs, requiring very small code and state. However, they do not pass every statistical test without further refinement. This weakness is amended by combining them with a non-linear function, as described in the original paper. Because plain xorshift generators (without a non-linear step) fail some statistical tests, they have been accused of being unreliable.
Example implementation A C version of three xorshift algorithms is given here. The first has one 32-bit word of state, and period 232−1. The second has one 64-bit word of state and period 264−1. The last one has four 32-bit words of state, and period 2128−1. 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. All use three shifts and three or four exclusive-or operations:
In case of one 64-bit word of state, there exist parameters which hold period 264−1 with two pair of exclusive-or and shift.
Non-linear variations All xorshift generators fail some tests in the BigCrush test suite. This is true for all generators based on linear recurrences, such as the Mersenne Twister or WELL. However, it is easy to scramble the output of such generators to improve their quality. The scramblers known as + and * still leave weakness in the low bits, so they are intended for floating point use, where the lowest bits of floating-point numbers have a smaller impact on the interpreted value. For general purpose, the scrambler ** (pronounced starstar) makes the LFSR generators pass in all bits.
xorwow Marsaglia suggested scrambling the output by combining it with a simple additive counter modulo 232 (which he calls a "Weyl sequence" after Weyl's equidistribution theorem). This also increases the period by a factor of 232, to 2192−232:
This performs well, but fails a few tests in BigCrush. This generator is the default in Nvidia's CUDA toolkit.
xorshift* A xorshift* generator applies an invertible multiplication (modulo the word size) as a non-linear transformation to the output of a xorshift generator, as suggested by Marsaglia. All xorshift* generators emit a sequence of values that is equidistributed in the maximum possible dimension (except that they will never output zero for 16 calls, i.e. 128 bytes, in a row). The following 64-bit generator has a maximal period of 264−1.
The generator fails only the MatrixRank test of BigCrush, however if the generator is modified to return only the high 32 bits, then it passes BigCrush with zero failures. In fact, a reduced version with only 40 bits of internal state passes the suite, suggesting a large safety margin. A similar generator suggested in Numerical Recipes as RanQ1 also fails the BirthdaySpacings test. Vigna suggests the following xorshift1024* generator with 1024 bits of state and a maximal period of 21024−1; however, it does not always pass BigCrush. xoshiro256** is therefore a much better option.
xorshift+ A xorshift+ generator can achieve an order of magnitude fewer failures than Mersenne Twister or WELL. A native C implementation of a xorshift+ generator that passes all tests from the BigCrush suite can typically generate a random number in fewer than 10 clock cycles on x86, thanks to instruction pipelining. Rather than using multiplication, it is possible to use addition as a faster non-linear transformation. The idea was first proposed by Saito and Matsumoto (also responsible for the Mersenne Twister) in the XSadd generator, which adds two consecutive outputs of an underlying xorshift generator based on 32-bit shifts. However, one disadvantage of adding consecutive outputs is that, while the underlying xorshift128 generator is 2-dimensionally equidistributed, the xorshift128+ generator is only 1-dimensionally equidistributed. XSadd has some weakness in the low-order bits of its output; it fails several BigCrush tests when the output words are bit-reversed. To correct this problem, Vigna introduced the xorshift+ family, based on 64-bit shifts. xorshift+ generators, even as large as xorshift1024+, exhibit some detectable linearity in the low-order bits of their output; it passes BigCrush, but doesn't when the 32 lowest-order bits are used in reverse order from each 64-bit word. This generator is one of the fastest generators passing BigCrush. The following xorshift128+ generator uses 128 bits of state and has a maximal period of 2128−1.
xorshiftr+ xorshiftr+ (r stands for reduced; reads "xorshifter plus") generator was mainly based on xorshift+ yet incorporates modifications making it significantly faster (especially on lightweight devices) and more successful in randomness tests (including TestU01 BigCrush suite) compared to its predecessors. It is one of the fastest generators passing all tests in TestU01's BigCrush suite. Like xorshift+, a native C implementation of a xorshiftr+ generator that passes all tests from the BigCrush suite can typically generate a random number in fewer than 10 clock cycles on x86, thanks to instruction pipelining. Unlike xorshift+, xorshiftr+ does not return the sum of two variables derived from the state using xorshift-style steps, rather it returns a single variable with the very last operation in its cycle; however, it features an addition just before returning a value, namely in the phase of adjusting the seed for the next cycle; hence the "+" in the name of the algorithm. The variable sizes, including the state, can be increased with no compromise to the randomness scores, but performance drops may be observed on lightweight devices. The following xorshiftr128+ generator uses 128 bits of state (with two variables) and has a maximal period of 2128−1.
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