Xorshift

{{short description|Class of pseudorandom number generators}} thumb|right|Example random distribution of Xorshift128 '''Xorshift''' random number generators, also called '''shift-register generators''', are a class of pseudorandom number generators that were invented by George Marsaglia.<ref name="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.<ref name="brent" /> 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.<ref name="panne" />

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.{{r|panne|p=360}}

==Example implementation== 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: <!-- "unsigned long" and "unsigned long long" in the original paper changed to explicit sized types from <stdint.h> --> <syntaxhighlight lang="c"> #include <stdint.h>

typedef struct { uint32_t a; } XorShift32State;

// The state must be initialized to non-zero uint32_t xorshift32(XorShift32State* 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; }

typedef struct { uint64_t a; } XorShift64State;

// The state must be initialized to non-zero uint64_t xorshift64(XorShift64State* state) { uint64_t x = state->a; x ^= x << 13; x ^= x >> 7; x ^= x << 17; return state->a = x; }

// XorShift128State can alternatively be defined as a pair // of uint64_t or a uint128_t where supported typedef struct { uint32_t x[4]; } XorShift128State;

// The state must be initialized to non-zero uint32_t xorshift128(XorShift128State* 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); } </syntaxhighlight>

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.<ref name="brent" /><ref>{{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).</ref> <syntaxhighlight lang="c"> #include <stdint.h>

typedef struct { uint64_t a; } XorShift64State;

uint64_t xorshift64(XorShift64State* state) { uint64_t x = state->a; x ^= x << 7; x ^= x >> 9; return state->a = x; } </syntaxhighlight>

==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 {{mono|+}} and {{mono|*}} still leave weakness in the low bits,<ref name=Lemire19/> so they are intended for floating point use, where the lowest bits of floating-point numbers have a smaller impact on the interpreted value.<ref>{{cite web |title=ISO/IEC 60559:2020 |url=https://www.iso.org/standard/80985.html |website=ISO |language=en}}</ref> For general purpose, the scrambler {{mono|**}} (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: <syntaxhighlight lang="c"> #include <stdint.h>

typedef struct { uint32_t x[5]; uint32_t counter; } XorWowState;

// The state array must be initialized to not be all zero in the first four words uint32_t xorwow(XorWowState* 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; } </syntaxhighlight> This performs well, but fails a few tests in BigCrush.<ref>{{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 }}</ref> This generator is the default in Nvidia's CUDA toolkit.<ref>{{cite web |url=https://docs.nvidia.com/cuda/curand/testing.html |title=cuRAND Testing |publisher=Nvidia |access-date=2017-11-02 }}</ref>

===xorshift*=== A {{em|xorshift*}} generator applies an invertible multiplication (modulo the word size) as a non-linear transformation to the output of a {{em|xorshift}} generator, as suggested by Marsaglia.<ref name="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).<ref name="vigna"/>

The following 64-bit generator has a maximal period of 2⁶⁴−1.<ref name="vigna" /> <syntaxhighlight lang="c"> #include <stdint.h>

// 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; } </syntaxhighlight>

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''<ref name="NR"/> as <code>RanQ1</code> also fails the ''BirthdaySpacings'' test.

Vigna<ref name="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. <syntaxhighlight lang="c"> #include <stdint.h>

// The state must be seeded so that there is at least one non-zero element in array typedef struct { uint64_t x[16]; int index; } XorShift1024sState;

uint64_t xorshift1024s(XorShift1024sState* state) { int index = state->index; const uint64_t 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; } </syntaxhighlight>

===xorshift+=== A {{em|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.<ref name="shootout"/>

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.<ref name="xsadd"/> 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.<ref name="vigna2"/>

{{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,<ref name="vigna2"/> 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.<ref name="shootout"/>

The following {{em|xorshift128+}} generator uses 128 bits of state and has a maximal period of 2¹²⁸−1. <syntaxhighlight lang="c"> #include <stdint.h>

typedef struct { uint64_t x[2]; } XorShift128pState;

// The state must be seeded so that it is not all zero uint64_t xorshift128p(XorShift128pState* state) { uint64_t t = state->x[0]; const uint64_t 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; } </syntaxhighlight>

===xorshiftr+=== {{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.<ref name="Cabuk2017"/> 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.<ref name="shootout"/><ref name="Cabuk2017"/>

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. <syntaxhighlight lang="c"> #include <stdint.h>

typedef struct { uint64_t s[2]; // seeds } XorShiftR128PlusState;

// The state must be seeded so that it is not all zero uint64_t xorshiftr128plus(XorShiftR128PlusState* state) { uint64_t x = state->s[0]; const uint64_t 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; } </syntaxhighlight>

==xoshiro== 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 <code source="C">jump</code> 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.<ref name="xoshiro">{{cite web|url=https://prng.di.unimi.it |title=xoshiro / xoroshiro generators and the PRNG shootout|access-date=2023-09-07}}</ref>

=== xoshiro256++ === xoshiro256++ is the family's general-purpose random 64-bit number generator.<syntaxhighlight lang="c"> // Adapted from the code included on Sebastiano Vigna's website

#include <stdint.h>

uint64_t rol64(uint64_t x, int k) { return (x << k) | (x >> (64 - k)); }

typedef struct { uint64_t s[4]; } Xoshiro256ppState;

uint64_t xoshiro256pp(Xoshiro256ppState* state) { uint64_t* s = state->s; const uint64_t result = rol64(s[0] + s[3], 23) + s[0]; const uint64_t 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; } </syntaxhighlight>

===xoshiro256**=== 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.<ref>{{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}}</ref> It is used in GNU Fortran compiler, Lua (as of Lua 5.4), and the .NET framework (as of .NET 6.0).<ref name="xoshiro"/>

<syntaxhighlight lang="c"> // Adapted from the code included on Sebastiano Vigna's website

#include <stdint.h>

uint64_t rol64(uint64_t x, int k) { return (x << k) | (x >> (64 - k)); }

typedef struct { uint64_t s[4]; } Xoshiro256ssState;

uint64_t xoshiro256ss(Xoshiro256ssState* state) { uint64_t* s = state->s; const uint64_t result = rol64(s[1] * 5, 7) * 9; const uint64_t 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; }</syntaxhighlight>

===xoshiro256+=== 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.

<syntaxhighlight lang="c"> #include <stdint.h>

uint64_t rol64(uint64_t x, int k) { return (x << k) | (x >> (64 - k)); }

typedef struct { uint64_t s[4]; } Xoshiro256pState;

uint64_t xoshiro256p(Xoshiro256pState* state) { uint64_t* s = state->s; const uint64_t result = s[0] + s[3]; const uint64_t 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; } </syntaxhighlight>

===xoroshiro=== 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 <samp>xorshift</samp> as a basic operation, <samp>xoroshiro128+</samp> 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.<ref>{{cite arXiv |last1=Blackman |first1=David |last2=Vigna|first2=Sebastiano|eprint=1805.01407|title= Scrambled Linear Pseudorandom Generators|year=2018 |class=cs.DS }}</ref>

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

==== Statistical quality ====

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

<blockquote> 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).<ref name="c-source">{{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}}</ref></blockquote>

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

Acknowledging the authors go on to say: <blockquote> We suggest to use a sign test to extract a random Boolean value<ref name="c-source" /></blockquote> Thus, programmers should prefer the highest bits (e.g., making a heads/tails by writing <code>random_number < 0</code> rather than <code>random_number & 1</code>). 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<ref>{{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}}</ref> and fails multiple tests in BigCrush even when the bits are reversed.<ref>{{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}}</ref>

==Initialization== 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.<ref name="xoshiro-paper"/><ref name="initialization"/> The authors specifically recommend using the SplitMix64 generator, from a 64-bit seed, as follows:

<syntaxhighlight lang="c"> #include <stdint.h>

typedef struct { uint64_t s; } SplitMix64State;

uint64_t splitmix64(SplitMix64State* state) { uint64_t result = (state->s += 0x9E3779B97F4A7C15); result = (result ^ (result >> 30)) * 0xBF58476D1CE4E5B9; result = (result ^ (result >> 27)) * 0x94D049BB133111EB; return result ^ (result >> 31); }

typedef struct { uint32_t x[4]; } XorShift128State;

// one could do the same for any of the other generators void xorshift128_init(XorShift128State* state, uint64_t seed) { SplitMix64State 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); } </syntaxhighlight>

== See also == * List of random number generators * Linear feedback shift register

==Notes== {{notelist}}

==References== <references>

<ref name="marsaglia">{{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 }}</ref> <ref name="brent">{{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 }}</ref> <ref name="panne">{{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 }}</ref> <ref name="NR">{{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 }}</ref> <ref name="shootout">{{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}}</ref> <ref name="vigna"> {{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<!--Actually article number--> | 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.</ref> <ref name="vigna2">{{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.</ref> <ref name="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}}</ref> <ref name="xoshiro-web">{{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}}</ref> <ref name="xoshiro-paper">{{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 }}</ref> <ref name="initialization">{{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 }}</ref> <ref name="PCG">{{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}}</ref> <ref name="Cabuk2017">{{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 |doi-access=free}}</ref> <ref name="Lemire19">{{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.}}</ref>

</references>

==Further reading== *{{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).

==External links== *{{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| archive-date=2018-05-04| archive-url=https://web.archive.org/web/20180504155117/http://vigna.di.unimi.it/xorshift/| url-status=dead}}

Category:Articles with example C code Category:Pseudorandom number generators