xorshift

A C version{{efn|In C and most other C-based languages, {{code|^}} represents bitwise XOR, and {{code|<<}} and {{code|>>}} represent bitwise shifts.}} of three xorshift algorithms{{r|marsaglia|p=4,5}} 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:

1. include

struct xorshift32_state {

uint32_t a;

};

/* The state must be initialized to non-zero */

uint32_t xorshift32(struct xorshift32_state *state)

{

/* Algorithm "xor" from p. 4 of Marsaglia, "Xorshift RNGs" */

uint32_t x = state->a;

x ^= x << 13;

x ^= x >> 17;

x ^= x << 5;

return state->a = x;

}

struct xorshift64_state {

uint64_t a;

};

/* The state must be initialized to non-zero */

uint64_t xorshift64(struct xorshift64_state *state)

{

uint64_t x = state->a;

x ^= x << 13;

x ^= x >> 7;

x ^= x << 17;

return state->a = x;

}

/* struct xorshift128_state can alternatively be defined as a pair

of uint64_t or a uint128_t where supported */

struct xorshift128_state {

uint32_t x[4];

};

/* The state must be initialized to non-zero */

uint32_t xorshift128(struct xorshift128_state *state)

{

/* Algorithm "xor128" from p. 5 of Marsaglia, "Xorshift RNGs" */

uint32_t t = state->x[3];

uint32_t s = state->x[0]; /* Perform a contrived 32-bit shift. */

state->x[3] = state->x[2];

state->x[2] = state->x[1];

state->x[1] = s;

t ^= t << 11;

t ^= t >> 8;

return state->x[0] = t ^ s ^ (s >> 19);

}

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.{{Cite web|author=和田維作 |url=http://isaku-wada.my.coocan.jp/rand/rand.html |title=良い乱数・悪い乱数 |accessdate=2023-08-28}} The parameters are only (7,9) and (9.7).

1. include

struct xorshift64_state {

uint64_t a;

};

uint64_t xorshift64(struct xorshift64_state *state)

{

uint64_t x = state->a;

x ^= x << 7;

x ^= x >> 9;

return state->a = x;

}

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 {{mono|+}} and {{mono|*}} 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.{{cite web |title=ISO/IEC 60559:2020 |url=https://www.iso.org/standard/80985.html |website=ISO |language=en}} For general purpose, the scrambler {{mono|**}} (pronounced starstar) makes the LFSR generators pass in all bits.

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:

1. include

struct xorwow_state {

uint32_t x[5];

uint32_t counter;

};

/* The state array must be initialized to not be all zero in the first four words */

uint32_t xorwow(struct xorwow_state *state)

{

/* Algorithm "xorwow" from p. 5 of Marsaglia, "Xorshift RNGs" */

uint32_t t = state->x[4];

uint32_t s = state->x[0]; /* Perform a contrived 32-bit rotate. */

state->x[4] = state->x[3];

state->x[3] = state->x[2];

state->x[2] = state->x[1];

state->x[1] = s;

t ^= t >> 2;

t ^= t << 1;

t ^= s ^ (s << 4);

state->x[0] = t;

state->counter += 362437;

return t + state->counter;

}

This performs well, but fails a few tests in BigCrush.{{cite web

|title=XORWOW L'ecuyer TestU01 Results

|url=https://chasethedevil.github.io/post/xorwow-lecuyer-testu01-results/

|date=12 January 2011

|website=Chase The Devil (blog)

|first=Fabien |last=Le Floc'h

|access-date=2017-11-02

}} This generator is the default in Nvidia's CUDA toolkit.{{cite web

|url=https://docs.nvidia.com/cuda/curand/testing.html

|title=cuRAND Testing

|publisher=Nvidia

|access-date=2017-11-02

}}

An {{em|xorshift*}} generator applies an invertible multiplication (modulo the word size) as a non-linear transformation to the output of an {{em|xorshift}} generator, as suggested by Marsaglia. All {{em|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.

1. include

/* xorshift64s, variant A_1(12,25,27) with multiplier M_32 from line 3 of table 5 */

uint64_t xorshift64star(void) {

/* initial seed must be nonzero, don't use a static variable for the state if multithreaded */

static uint64_t x = 1;

x ^= x >> 12;

x ^= x << 25;

x ^= x >> 27;

return x * 0x2545F4914F6CDD1DULL;

}

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.{{r|PCG|p=7}} In fact, a reduced version with only 40 bits of internal state passes the suite, suggesting a large safety margin.{{r|PCG|p=19}} A similar generator suggested in Numerical Recipes as RanQ1 also fails the BirthdaySpacings test.

Vigna suggests the following {{em|xorshift1024*}} generator with 1024 bits of state and a maximal period of 2¹⁰²⁴−1; however, it does not always pass BigCrush.{{r|Lemire19}} xoshiro256** is therefore a much better option.

1. include

/* The state must be seeded so that there is at least one non-zero element in array */

struct xorshift1024s_state {

uint64_t x[16];

int index;

};

uint64_t xorshift1024s(struct xorshift1024s_state *state)

{

int index = state->index;

uint64_t const s = state->x[index++];

uint64_t t = state->x[index &= 15];

t ^= t << 31; // a

t ^= t >> 11; // b -- Again, the shifts and the multipliers are tunable

t ^= s ^ (s >> 30); // c

state->x[index] = t;

state->index = index;

return t * 1181783497276652981ULL;

}

An {{em|xorshift+}} generator can achieve an order of magnitude fewer failures than Mersenne Twister or WELL. A native C implementation of an 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 {{em|XSadd}} generator, which adds two consecutive outputs of an underlying {{em|xorshift}} generator based on 32-bit shifts. However, one disadvantage of adding consecutive outputs is that, while the underlying {{em|xorshift128}} generator is 2-dimensionally equidistributed, the {{em|xorshift128+}} generator is only 1-dimensionally equidistributed.

{{em|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 {{em|xorshift+}} family, based on 64-bit shifts. {{em|xorshift+}} generators, even as large as {{em|xorshift1024+}}, exhibit some detectable linearity in the low-order bits of their output;{{r|Lemire19}} it passes BigCrush, but doesn't when the 32 lowest-order bits are used in reverse order from each 64-bit word.{{r|Lemire19}} This generator is one of the fastest generators passing BigCrush.

The following {{em|xorshift128+}} generator uses 128 bits of state and has a maximal period of 2¹²⁸−1.

1. include

struct xorshift128p_state {

uint64_t x[2];

};

/* The state must be seeded so that it is not all zero */

uint64_t xorshift128p(struct xorshift128p_state *state)

{

uint64_t t = state->x[0];

uint64_t const s = state->x[1];

state->x[0] = s;

t ^= t << 23; // a

t ^= t >> 18; // b -- Again, the shifts and the multipliers are tunable

t ^= s ^ (s >> 5); // c

state->x[1] = t;

return t + s;

}

{{em|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 an 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 {{em|xorshift+}}, {{em|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 {{em|xorshiftr128+}} generator uses 128 bits of state (with two variables) and has a maximal period of 2¹²⁸−1.

1. include

struct xorshiftr128plus_state {

uint64_t s[2]; // seeds

};

/* The state must be seeded so that it is not all zero */

uint64_t xorshiftr128plus(struct xorshiftr128plus_state *state)

{

uint64_t x = state->s[0];

uint64_t const y = state->s[1];

state->s[0] = y;

x ^= x << 23; // shift & xor

x ^= x >> 17; // shift & xor

x ^= y; // xor

state->s[1] = x + y;

return x;

}

xoshiro (short for "xor, shift, rotate") and xoroshiro (short for "xor, rotate, shift, rotate") use rotations in addition to shifts. According to Vigna, they are faster and produce better quality output than xorshift.{{r|xoshiro-web|xoshiro-paper}}

This class of generator has variants for 32-bit and 64-bit integer and floating point output; for floating point, one takes the upper 53 bits (for binary64) or the upper 23 bits (for binary32), since the upper bits are of better quality than the lower bits in the floating point generators. The algorithms also include a jump function, which sets the state forward by some number of steps – usually a power of two that allows many threads of execution to start at distinct initial states.

For 32-bit output, xoshiro128** and xoshiro128+ are exactly equivalent to xoshiro256** and xoshiro256+, with {{mono|uint32_t}} in place of {{mono|uint64_t}}, and with different shift/rotate constants.

More recently, the {{mono|xoshiro++}} generators have been made as an alternative to the {{mono|xoshiro**}} generators. They are used in some implementations of Fortran compilers such as GNU Fortran, and in Java, and Julia.{{cite web|url=https://prng.di.unimi.it |title=xoshiro / xoroshiro generators and the PRNG shootout|access-date=2023-09-07}}

xoshiro256++ is the family's general-purpose random 64-bit number generator.

/* Adapted from the code included on Sebastiano Vigna's website */

1. include

uint64_t rol64(uint64_t x, int k) {

return (x << k) | (x >> (64 - k));

}

struct xoshiro256pp_state {

uint64_t s[4];

};

uint64_t xoshiro256pp(struct xoshiro256pp_state *state) {

uint64_t *s = state->s;

uint64_t const result = rol64(s[0] + s[3], 23) + s[0];

uint64_t const t = s[1] << 17;

s[2] ^= s[0];

s[3] ^= s[1];

s[1] ^= s[2];

s[0] ^= s[3];

s[2] ^= t;

s[3] = rol64(s[3], 45);

return result;

}

xoshiro256** uses multiplication rather than addition in its output function. It is worth noting, however, that the output function is invertible, allowing the underlying state to be trivially uncovered.{{Cite web |last=O'Neill |first=M. E. |date=2018-05-05 |title=A Quick Look at Xoshiro256** |url=https://www.pcg-random.org/posts/a-quick-look-at-xoshiro256.html |access-date=2024-10-04 |website=PCG, A Better Random Number Generator |language=en}} It is used in GNU Fortran compiler, Lua (as of Lua 5.4), and the .NET framework (as of .NET 6.0).

/* Adapted from the code included on Sebastiano Vigna's website */

1. include

uint64_t rol64(uint64_t x, int k) {

return (x << k) | (x >> (64 - k));

}

struct xoshiro256ss_state {

uint64_t s[4];

};

uint64_t xoshiro256ss(struct xoshiro256ss_state *state) {

uint64_t *s = state->s;

uint64_t const result = rol64(s[1] * 5, 7) * 9;

uint64_t const t = s[1] << 17;

s[2] ^= s[0];

s[3] ^= s[1];

s[1] ^= s[2];

s[0] ^= s[3];

s[2] ^= t;

s[3] = rol64(s[3], 45);

return result;

}

xoshiro256+ is approximately 15% faster than xoshiro256**, but the lowest three bits have low linear complexity; therefore, it should be used only for floating point results by extracting the upper 53 bits.

1. include

uint64_t rol64(uint64_t x, int k) {

return (x << k) | (x >> (64 - k));

}

struct xoshiro256p_state {

uint64_t s[4];

};

uint64_t xoshiro256p(struct xoshiro256p_state *state) {

uint64_t* s = state->s;

uint64_t const result = s[0] + s[3];

uint64_t const t = s[1] << 17;

s[2] ^= s[0];

s[3] ^= s[1];

s[1] ^= s[2];

s[0] ^= s[3];

s[2] ^= t;

s[3] = rol64(s[3], 45);

return result;

}

If space is at a premium, xoroshiro128** and xoroshiro128+ are equivalent to xoshiro256** and xoshiro256+. These have smaller state spaces, and thus are less useful for massively parallel programs. xoroshiro128+ also exhibits a mild dependency in the population count, generating a failure after {{val|5|ul=TB}} of output. The authors do not believe that this can be detected in real world programs. Instead of perpetuating Marsaglia's tradition of xorshift as a basic operation, xoroshiro128+ uses a shift/rotate-based linear transformation designed by Sebastiano Vigna in collaboration with David Blackman. The result is a significant improvement in speed and statistical quality.{{cite arXiv |last1=Blackman |first1=David |last2=Vigna|first2=Sebastiano|eprint=1805.01407|title= Scrambled Linear Pseudorandom Generators|year=2018 |class=cs.DS }}

xoroshiro64** and xoroshiro64* are equivalent to xoroshiro128** and xoroshiro128+. Unlike the xoshiro generators, they are not straightforward ports of their higher-precision counterparts.

The lowest bits of the output generated by xoroshiro128+ have low quality. The authors of xoroshiro128+ acknowledge that it does not pass all statistical tests, stating

This is xoroshiro128+ 1.0, our best and fastest small-state generator

for floating-point numbers. We suggest to use its upper bits for

floating-point generation, as it is slightly faster than

xoroshiro128**. It passes all tests we are aware of except for the four

lower bits, which might fail linearity tests (and just those), so if

low linear complexity is not considered an issue (as it is usually the

case) it can be used to generate 64-bit outputs, too; moreover, this

generator has a very mild Hamming-weight dependency making our test

(http://prng.di.unimi.it/hwd.php) fail after 5 TB of output; we believe

this slight bias cannot affect any application. If you are concerned,

use xoroshiro128** or xoshiro256+.

We suggest to use a sign test to extract a random Boolean value, and

right shifts to extract subsets of bits.

The state must be seeded so that it is not everywhere zero. If you have

a 64-bit seed, we suggest to seed a splitmix64 generator and use its

output to fill s.

NOTE: the parameters (a=24, b=16, c=37) of this version give slightly

better results in our test than the 2016 version (a=55, b=14, c=36).{{Cite web|url=http://prng.di.unimi.it/xoroshiro128plus.c|title=Original C source code implementation of xoroshiro128+|last1=Blackman|first1=David|last2=Vigna|first2=Sebastiano|author2-link=Sebastiano Vigna|date=2018|access-date=May 4, 2018}}

These claims about not passing tests can be confirmed by running PractRand on the input, resulting in output like that shown below:

RNG_test using PractRand version 0.93
RNG = RNG_stdin64, seed = 0xfac83126
test set = normal, folding = standard (64 bit)
rng=RNG_stdin64, seed=0xfac83126
length= 128 megabytes (2^27 bytes), time= 2.1 seconds
Test Name                         Raw       Processed     Evaluation
[Low1/64]BRank(12):256(2)         R= +3748  p~=  3e-1129    FAIL !!!!!!!!
[Low1/64]BRank(12):384(1)         R= +5405  p~=  3e-1628    FAIL !!!!!!!!
...and 146 test result(s) without anomalies

Acknowledging the authors go on to say:

We suggest to use a sign test to extract a random Boolean value

Thus, programmers should prefer the highest bits (e.g., making a heads/tails by writing random_number < 0 rather than random_number & 1). It must be noted, though, that the same test is failed by some instances of the Mersenne Twister and WELL.

The statistical problems extend far beyond the bottom few bits, because it fails the PractRand test even when truncated{{Cite web|url=https://www.pcg-random.org/posts/xoroshiro-fails-truncated.html|title=xoroshiro fails PractRand when truncated|first=M.E.|last=O'Neill|date=2018-03-25|access-date=Dec 30, 2020}} and fails multiple tests in BigCrush even when the bits are reversed.{{Cite web|url=https://lemire.me/blog/2017/09/08/the-xorshift128-random-number-generator-fails-bigcrush/ |title=The Xorshift128+ random number generator fails BigCrush |first=Daniel |last=Lemire |date=2017-09-08 |access-date=Dec 30, 2020}}

In the xoshiro paper, it is recommended to initialize the state of the generators using a generator which is radically different from the initialized generators, as well as one which will never give the "all-zero" state; for shift-register generators, this state is impossible to escape from. The authors specifically recommend using the SplitMix64 generator, from a 64-bit seed, as follows:

1. include

struct splitmix64_state {

uint64_t s;

};

uint64_t splitmix64(struct splitmix64_state *state) {

uint64_t result = (state->s += 0x9E3779B97f4A7C15);

result = (result ^ (result >> 30)) * 0xBF58476D1CE4E5B9;

result = (result ^ (result >> 27)) * 0x94D049BB133111EB;

return result ^ (result >> 31);

}

struct xorshift128_state {

uint32_t x[4];

};

// one could do the same for any of the other generators

void xorshift128_init(struct xorshift128_state *state, uint64_t seed) {

struct splitmix64_state smstate = {seed};

uint64_t tmp = splitmix64(&smstate);

state->x[0] = (uint32_t)tmp;

state->x[1] = (uint32_t)(tmp >> 32);

tmp = splitmix64(&smstate);

state->x[2] = (uint32_t)tmp;

state->x[3] = (uint32_t)(tmp >> 32);

}

• List of random number generators
• Linear feedback shift register

{{notelist}}

{{reflist|refs =

{{cite journal

| first=George | last=Marsaglia | author-link=George Marsaglia

| title=Xorshift RNGs

| journal=Journal of Statistical Software

| volume=8 | issue=14

| date=July 2003

| doi=10.18637/jss.v008.i14| doi-access=free

}}

{{cite journal

| first=Richard P. | last=Brent | author-link=Richard P. Brent

| title=Note on Marsaglia's Xorshift Random Number Generators

| journal=Journal of Statistical Software

| volume=11 | issue=5

| date=August 2004

| doi=10.18637/jss.v011.i05| doi-access=free

| hdl=1885/34049

| hdl-access=free

}}

{{cite journal

| first1=François | last1=Panneton

| first2=Pierre | last2=L'Ecuyer

| title=On the xorshift random number generators

| journal=ACM Transactions on Modeling and Computer Simulation

| volume=15 | issue=4

| pages=346–361

| date=October 2005

| url=https://www.iro.umontreal.ca/~lecuyer/myftp/papers/xorshift.pdf

| doi=10.1145/1113316.1113319| s2cid=11136098

}}

{{Cite book

| last1=Press | first1=WH | author-link1=William H. Press

| last2=Teukolsky | first2=SA |author-link2=Saul Teukolsky

| last3=Vetterling | first3=WT

| last4=Flannery | first4=BP

| year=2007

| title=Numerical Recipes: The Art of Scientific Computing

| edition=3rd

| publisher=Cambridge University Press

| location=New York

| isbn=978-0-521-88068-8

| chapter=Section 7.1.2.A. 64-bit Xorshift Method

| chapter-url=http://apps.nrbook.com/empanel/index.html#pg=345

}}

{{cite web

| last=Vigna | first=Sebastiano

| title=xorshift*/xorshift+ generators and the PRNG shootout

| url=http://prng.di.unimi.it

| access-date=2014-10-25}}

{{cite journal

| last=Vigna | first=Sebastiano

| title=An experimental exploration of Marsaglia's xorshift generators, scrambled

| journal=ACM Transactions on Mathematical Software

| volume=42 | issue=4 | date=July 2016 | page=30

| doi=10.1145/2845077

| arxiv=1402.6246

| s2cid=13936073

| url=http://vigna.di.unimi.it/ftp/papers/xorshift.pdf

}} Proposes xorshift* generators, adding a final multiplication by a constant.

{{cite journal

| last=Vigna | first=Sebastiano

| title=Further scramblings of Marsaglia's xorshift generators

| journal=Journal of Computational and Applied Mathematics

| volume=315 | issue=C | date=May 2017 | pages=175–181

| doi=10.1016/j.cam.2016.11.006

| arxiv=1404.0390

| s2cid=6876444

| url=http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf

}} Describes xorshift+ generators, a generalization of XSadd.

{{cite web

| title=XORSHIFT-ADD (XSadd): A variant of XORSHIFT

| first1=Mutsuo | last1=Saito

| first2=Makoto | last2=Matsumoto

| year=2014

| url=http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/XSADD/

| access-date=2014-10-25}}

{{cite web

| title=xoshiro/xoroshiro generators and the PRNG shootout

| last=Vigna | first=Sebastiano

| url=http://xoroshiro.di.unimi.it

| access-date=2019-07-07}}

{{cite journal

| title=Scrambled Linear Pseudorandom Number Generators

| last1=Blackman | first1=David

| last2=Vigna | first2=Sebastiano

| year=2018

| journal=Data Structures and Algorithms

| arxiv=1805.01407 }}

{{cite journal

| first1=Makoto | last1=Matsumoto

| first2=Isaku | last2=Wada

| first3=Ai | last3=Kuramoto

| first4=Hyo | last4=Ashihara

| title=Common defects in initialization of pseudorandom number generators

| journal=ACM Transactions on Modeling and Computer Simulation

| volume=17 | issue=4

| pages=15–es

| date=September 2007

| doi=10.1145/1276927.1276928| s2cid=1721554

}}

{{cite tech report

|title=PCG: A Family of Simple Fast Space-Efficient Statistically Good Algorithms for Random Number Generation

|first=Melissa E. |last=O'Neill

|publisher=Harvey Mudd College

|id=HMC-CS-2014-0905

|date=5 September 2014

|pages=6–8

|url=http://www.pcg-random.org/pdf/hmc-cs-2014-0905.pdf}}

{{Cite journal

|last=Çabuk

|first=Umut Can

|last2=Aydin

|first2=Ömer

|last3=Dalkiliç

|first3=Gökhan

|date=2017

|title=A random number generator for lightweight authentication protocols: xorshiftR+ |url=https://journals.tubitak.gov.tr/elektrik/vol25/iss6/31

|journal=Turkish Journal of Electrical Engineering and Computer Sciences |volume=25 |pages=4818–4828 |doi=10.3906/elk-1703-361}}

{{cite journal

| first1=Daniel | last1=Lemire

| first2=Melissa E. |last2=O’Neill

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

| journal=Computational and Applied Mathematics

| volume=350 |pages=139–142 |date=April 2019

| doi=10.1016/j.cam.2018.10.019 |arxiv=1810.05313

| s2cid=52983294

| quote=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.}}

}}

• {{Cite journal

| first=Richard P. | last=Brent | author-link=Richard P. Brent

| title=Some long-period random number generators using shifts and xors

| url=https://maths-people.anu.edu.au/~brent/pub/pub224.html

| date=July 2006

| journal=ANZIAM Journal

| volume=48 |ref=none

| pages=C188–C202}} Lists generators of various sizes with four shifts (two per feedback word).

• {{Cite web | url=http://vigna.di.unimi.it/xorshift/|title=xoshiro / xoroshiro generators and the PRNG shootout | year =2018 | first = Sebastiano | last = Vigna | access-date= 2018-05-04 }}

Category:Articles with example C code

Category:Pseudorandom number generators