<s>
Random	O
number	O
generators	O
are	O
important	O
in	O
many	O
kinds	O
of	O
technical	O
applications	O
,	O
including	O
physics	O
,	O
engineering	O
or	O
mathematical	O
computer	O
studies	O
(	O
e.g.	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
)	O
,	O
cryptography	O
and	O
gambling	O
(	O
on	O
game	B-Application
servers	I-Application
)	O
.	O
</s>
<s>
Whenever	O
using	O
a	O
pseudorandom	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
,	O
keep	O
in	O
mind	O
John	O
von	O
Neumann	O
's	O
dictum	O
"	O
Anyone	O
who	O
considers	O
arithmetical	O
methods	O
of	O
producing	O
random	O
digits	O
is	O
,	O
of	O
course	O
,	O
in	O
a	O
state	O
of	O
sin.	O
"	O
</s>
<s>
The	O
following	O
algorithms	O
are	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
.	O
</s>
<s>
Germond	O
,	O
eds.	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
Method	I-Algorithm
,	O
National	O
Bureau	O
of	O
Standards	O
Applied	O
Mathematics	O
Series	O
,	O
vol	O
.	O
</s>
<s>
H	O
.	O
LehmerOne	O
of	O
the	O
very	O
earliest	O
and	O
most	O
influential	O
designs.Linear	O
congruential	O
generator	O
(	O
LCG	B-Algorithm
)	O
1958W	O
.	O
</s>
<s>
E	O
.	O
Thomson	O
;	O
A	O
.	O
Rotenberg	O
A	O
generalisation	O
of	O
the	O
Lehmer	B-Algorithm
generator	I-Algorithm
and	O
historically	O
the	O
most	O
influential	O
and	O
studied	O
generator.Lagged	O
Fibonacci	O
generator	O
(	O
LFG	O
)	O
1958G	O
.	O
</s>
<s>
27.Linear-feedback	O
shift	O
register	O
(	O
LFSR	B-Architecture
)	O
1965R	O
.	O
</s>
<s>
A	O
.	O
Wichmann	O
and	O
D	O
.	O
I	O
.	O
HillA	O
combination	O
of	O
three	O
small	O
LCGs	B-Algorithm
,	O
suited	O
to	O
16-bit	B-Device
CPUs	I-Device
.	O
</s>
<s>
it	O
is	O
used	O
in	O
Excel	B-Application
2003	I-Application
and	O
later	O
versions	O
for	O
the	O
Excel	B-Application
function	O
RAND	O
and	O
it	O
was	O
the	O
default	O
generator	O
in	O
the	O
language	O
Python	B-Language
up	O
to	O
version	O
2.2.Rule	O
301983S	O
.	O
</s>
<s>
K	O
.	O
Park	O
and	O
K	O
.	O
W	O
.	O
MillerA	O
specific	O
implementation	O
of	O
a	O
Lehmer	B-Algorithm
generator	I-Algorithm
,	O
widely	O
used	O
because	O
it	O
is	O
included	O
in	O
C++	B-Language
as	O
the	O
function	O
minstd_rand0	O
from	O
C++11	B-Language
onwards.ACORN	O
generator(discovered 1984 )	O
1989R	O
.	O
</s>
<s>
S	O
.	O
WikramaratnaWikramaratna	O
,	O
R.S.	O
</s>
<s>
MIXMAX	B-Algorithm
generator1991G	O
.	O
</s>
<s>
K	O
.	O
Savvidy	O
and	O
N	O
.	O
G	O
.	O
Ter-Arutyunyan-SavvidyIt	O
is	O
a	O
member	O
of	O
the	O
class	O
of	O
matrix	O
linear	B-Algorithm
congruential	I-Algorithm
generator	I-Algorithm
,	O
a	O
generalisation	O
of	O
LCG	B-Algorithm
.	O
</s>
<s>
The	O
rationale	O
behind	O
the	O
MIXMAX	B-Algorithm
family	O
of	O
generators	O
relies	O
on	O
results	O
from	O
ergodic	O
theory	O
and	O
classical	O
mechanics.Add-with-carry	O
(	O
AWC	O
)	O
1991G	O
.	O
</s>
<s>
MarsagliaPrototypical	O
example	O
of	O
a	O
combination	O
generator.Multiply-with-carry	O
(	O
MWC	O
)	O
1994G	O
.	O
</s>
<s>
Marsaglia	O
;	O
C	O
.	O
KoçPost	O
by	O
George	O
Marsaglia	O
on	O
the	O
newsgroup	O
sci.stat.math	O
dated	O
1	O
August	O
2018	O
with	O
title	O
'	O
Yet	O
another	O
RNG	O
 '	O
.Complementary-multiply-with-carry	O
(	O
CMWC	O
)	O
1997R	O
.	O
</s>
<s>
Matsumoto	O
and	O
T	O
.	O
NishimuraClosely	O
related	O
with	O
LFSRs	B-Architecture
.	O
</s>
<s>
In	O
its	O
MT19937	B-Algorithm
implementation	O
is	O
probably	O
the	O
most	O
commonly	O
used	O
modern	O
PRNG	O
.	O
</s>
<s>
Default	O
generator	O
in	O
R	B-Language
and	O
the	O
Python	B-Language
language	I-Language
starting	O
from	O
version	O
2.3.Xorshift2003G	O
.	O
</s>
<s>
MarsagliaIt	O
is	O
a	O
very	O
fast	O
sub-type	O
of	O
LFSR	B-Architecture
generators	O
.	O
</s>
<s>
Marsaglia	O
also	O
suggested	O
as	O
an	O
improvement	O
the	O
xorwow	O
generator	O
,	O
in	O
which	O
the	O
output	O
of	O
a	O
xorshift	B-Algorithm
generator	O
is	O
added	O
with	O
a	O
Weyl	O
sequence	O
.	O
</s>
<s>
The	O
xorwow	O
generator	O
is	O
the	O
default	O
generator	O
in	O
the	O
CURAND	O
library	O
of	O
the	O
nVidia	B-Architecture
CUDA	I-Architecture
application	O
programming	O
interface	O
for	O
graphics	O
processing	O
units.Well	O
equidistributed	O
long-period	O
linear	O
(	O
WELL	O
)	O
2006F	O
.	O
</s>
<s>
Panneton	O
,	O
P	O
.	O
L'Ecuyer	O
and	O
M	O
.	O
MatsumotoA	O
LFSR	B-Architecture
closely	O
related	O
with	O
Mersenne	B-Algorithm
Twister	I-Algorithm
,	O
aiming	O
at	O
remedying	O
some	O
of	O
its	O
shortcomings.A	O
small	O
noncryptographic	O
PRNG	O
(	O
JSF	O
)	O
2007Bob	O
JenkinsAdvanced	O
Randomization	O
System	O
(	O
ARS	O
)	O
2011J	O
.	O
</s>
<s>
Salmon	O
,	O
M	O
.	O
Moraes	O
,	O
R	B-Language
.	O
Dror	O
and	O
D	O
.	O
ShawA	O
simplified	O
version	O
of	O
the	O
AES	B-Algorithm
block	I-Algorithm
cipher	I-Algorithm
,	O
leading	O
to	O
very	O
fast	O
performance	O
on	O
systems	O
supporting	O
the	O
AES-NI.Threefry2011J	O
.	O
</s>
<s>
Salmon	O
,	O
M	O
.	O
Moraes	O
,	O
R	B-Language
.	O
Dror	O
and	O
D	O
.	O
ShawA	O
simplified	O
version	O
of	O
the	O
Threefish	B-Algorithm
block	O
cipher	B-Application
,	O
suitable	O
for	O
GPU	O
implementations.Philox2011J	O
.	O
</s>
<s>
Salmon	O
,	O
M	O
.	O
Moraes	O
,	O
R	B-Language
.	O
Dror	O
and	O
D	O
.	O
ShawA	O
simplification	O
and	O
modification	O
of	O
the	O
block	O
cipher	B-Application
Threefish	B-Algorithm
with	O
the	O
addition	O
of	O
an	O
S-box.WELLDOC	O
2013L	O
.	O
</s>
<s>
Balkova	O
,	O
M	O
.	O
Bucci	O
,	O
A	O
.	O
de	O
Luca	O
,	O
J	O
.	O
Hladky	O
,	O
S	O
.	O
PuzyninaAperiodic	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
based	O
on	O
infinite	O
words	O
technique.SplitMix2014G	O
.	O
</s>
<s>
E	O
.	O
O'NeillA	O
modification	O
of	O
LCG.Random	O
Cycle	O
Bit	O
Generator	O
(	O
RCB	O
)	O
2016R	O
.	O
</s>
<s>
CookmanRCB	O
is	O
described	O
as	O
a	O
bit	O
pattern	O
generator	O
made	O
to	O
overcome	O
some	O
of	O
the	O
shortcomings	O
with	O
Mersenne	B-Algorithm
Twister	I-Algorithm
and	O
short	O
periods/bit	O
length	O
restriction	O
of	O
shift/modulo	O
generators.Middle-Square	O
Weyl	O
Sequence	O
RNG	O
(	O
see	O
also	O
middle-square	B-Algorithm
method	I-Algorithm
)	O
2017B	O
.	O
</s>
<s>
WidynskiA	O
variation	O
on	O
John	O
von	O
Neumann	O
's	O
original	O
middle-square	B-Algorithm
method	I-Algorithm
,	O
this	O
generator	O
may	O
be	O
the	O
fastest	O
RNG	O
that	O
passes	O
all	O
the	O
statistical	O
tests.Xoroshiro128	O
+2018D	O
.	O
</s>
<s>
Blackman	O
,	O
S	O
.	O
VignaA	O
modification	O
of	O
Marsaglia	O
's	O
Xorshift	B-Algorithm
generators	O
,	O
one	O
of	O
the	O
fastest	O
generators	O
on	O
modern	O
64-bit	B-Device
CPUs	I-Device
.	O
</s>
<s>
Related	O
generators	O
include	O
xoroshiro128**	O
,	O
xoshiro256+	B-Algorithm
and	O
xoshiro256**	B-Algorithm
.64-bit	O
MELG	O
(	O
MELG-64	O
)	O
2018S	O
.	O
</s>
<s>
Harase	O
,	O
T	O
.	O
KimotoAn	O
implementation	O
of	O
64-bit	B-Device
maximally	O
equidistributed	O
F2-linear	O
generators	O
with	O
Mersenne	O
prime	O
period.Squares	O
RNG2020B	O
.	O
</s>
<s>
WidynskiA	O
counter-based	B-Algorithm
version	O
of	O
Middle-Square	O
Weyl	O
Sequence	O
RNG	O
.	O
</s>
<s>
Similar	O
to	O
Philox	B-Algorithm
in	O
design	O
but	O
significantly	O
faster	O
.	O
</s>
<s>
Cipher	B-Application
algorithms	O
and	O
cryptographic	B-Algorithm
hashes	I-Algorithm
can	O
be	O
used	O
as	O
very	O
high-quality	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
.	O
</s>
<s>
Stream	B-Algorithm
ciphers	I-Algorithm
.	O
</s>
<s>
Popular	O
choices	O
are	O
Salsa20	B-Algorithm
or	O
ChaCha	O
(	O
often	O
with	O
the	O
number	O
of	O
rounds	O
reduced	O
to	O
8	O
for	O
speed	O
)	O
,	O
ISAAC	B-Algorithm
,	O
HC-128	B-Algorithm
and	O
RC4	O
.	O
</s>
<s>
Block	O
ciphers	B-Application
in	O
counter	O
mode	O
.	O
</s>
<s>
Common	O
choices	O
are	O
AES	B-Algorithm
(	O
which	O
is	O
very	O
fast	O
on	O
systems	B-Algorithm
supporting	I-Algorithm
it	I-Algorithm
in	I-Algorithm
hardware	I-Algorithm
)	O
,	O
TwoFish	B-Algorithm
,	O
Serpent	B-Algorithm
and	O
Camellia	B-Algorithm
.	O
</s>
<s>
A	O
few	O
cryptographically	O
secure	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
do	O
not	O
rely	O
on	O
cipher	B-Application
algorithms	O
but	O
try	O
to	O
link	O
mathematically	O
the	O
difficulty	O
of	O
distinguishing	O
their	O
output	O
from	O
a	O
`true	O
 '	O
random	O
stream	O
to	O
a	O
computationally	O
difficult	O
problem	O
.	O
</s>
<s>
These	O
approaches	O
combine	O
a	O
pseudo-random	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
(	O
often	O
in	O
the	O
form	O
of	O
a	O
block	O
or	O
stream	B-Algorithm
cipher	I-Algorithm
)	O
with	O
an	O
external	O
source	O
of	O
randomness	O
(	O
e.g.	O
,	O
mouse	O
movements	O
,	O
delay	O
between	O
keyboard	O
presses	O
etc	O
.	O
</s>
<s>
RDRAND	B-Language
instructions	O
(	O
called	O
Intel	B-Language
Secure	I-Language
Key	I-Language
by	O
Intel	O
)	O
,	O
available	O
in	O
Intel	O
x86	O
CPUs	O
since	O
2012	O
.	O
</s>
<s>
They	O
use	O
the	O
AES	B-Algorithm
generator	O
built	O
into	O
the	O
CPU	O
,	O
reseeding	O
it	O
periodically	O
.	O
</s>
