<s>
The	O
ACORN	B-Algorithm
or	O
″Additive	O
Congruential	O
Random	O
Number″	O
generators	O
are	O
a	O
robust	O
family	O
of	O
PRNGs	O
(	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
)	O
for	O
sequences	O
of	O
uniformly	O
distributed	O
pseudo-random	O
numbers	O
,	O
introduced	O
in	O
1989	O
and	O
still	O
valid	O
in	O
2019	O
,	O
thirty	O
years	O
later	O
.	O
</s>
<s>
Introduced	O
by	O
R.S.Wikramaratna	O
,	O
ACORN	B-Algorithm
was	O
originally	O
designed	O
for	O
use	O
in	O
geostatistical	O
and	O
geophysical	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
,	O
and	O
later	O
extended	O
for	O
use	O
on	O
parallel	B-Operating_System
computers	I-Operating_System
.	O
</s>
<s>
Over	O
the	O
ensuing	O
decades	O
,	O
theoretical	O
analysis	O
(	O
formal	O
proof	O
of	O
convergence	B-Algorithm
and	O
statistical	O
results	O
)	O
,	O
empirical	O
testing	O
(	O
using	O
standard	O
test	O
suites	O
)	O
,	O
and	O
practical	O
application	O
work	O
have	O
continued	O
,	O
despite	O
the	O
appearance	O
and	O
promotion	O
of	O
other	O
better-known	O
[	O
but	O
not	O
necessarily	O
better	O
performing ]	O
PRNGs	O
.	O
</s>
<s>
The	O
main	O
advantages	O
of	O
ACORN	B-Algorithm
are	O
simplicity	O
of	O
concept	O
and	O
coding	O
,	O
speed	O
of	O
execution	O
,	O
long	O
period	O
length	O
,	O
and	O
mathematically	O
proven	O
convergence	B-Algorithm
.	O
</s>
<s>
In	O
addition	O
,	O
recent	O
research	O
has	O
shown	O
that	O
the	O
ACORN	B-Algorithm
generators	O
pass	O
all	O
the	O
tests	O
in	O
the	O
TestU01	B-Algorithm
test	O
suite	O
,	O
current	O
version	O
1.2.3	O
,	O
with	O
an	O
appropriate	O
choice	O
of	O
parameters	O
and	O
with	O
a	O
few	O
very	O
straightforward	O
constraints	O
on	O
the	O
choice	O
of	O
initialisation	O
;	O
it	O
is	O
worth	O
noting	O
,	O
as	O
pointed	O
out	O
by	O
the	O
authors	O
of	O
TestU01	B-Algorithm
,	O
that	O
some	O
widely-used	O
pseudo-random	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
fail	O
badly	O
on	O
some	O
of	O
the	O
tests	O
.	O
</s>
<s>
ACORN	B-Algorithm
is	O
particularly	O
simple	O
to	O
implement	O
in	O
exact	O
integer	O
arithmetic	O
,	O
in	O
various	O
computer	O
languages	O
,	O
using	O
only	O
a	O
few	O
lines	O
of	O
code	O
.	O
</s>
<s>
The	O
ACORN	B-Algorithm
generator	O
has	O
not	O
seen	O
the	O
wide	O
adoption	O
of	O
some	O
other	O
PRNGs	O
,	O
although	O
it	O
is	O
included	O
in	O
the	O
Fortran	B-Application
and	O
C	B-Language
library	I-Language
routines	O
of	O
NAG	O
Numerical	O
Library	O
.	O
</s>
<s>
Nevertheless	O
,	O
theoretical	O
and	O
empirical	O
is	O
ongoing	O
to	O
further	O
justify	O
the	O
continuing	O
use	O
of	O
ACORN	B-Algorithm
as	O
a	O
robust	O
and	O
effective	O
PRNG	O
.	O
</s>
<s>
In	O
testing	O
,	O
ACORN	B-Algorithm
performs	O
extremely	O
well	O
,	O
for	O
appropriate	O
parameters	O
.	O
</s>
<s>
However	O
in	O
its	O
present	O
form	O
,	O
ACORN	B-Algorithm
has	O
not	O
been	O
shown	O
to	O
be	O
suitable	O
for	O
cryptography	O
.	O
</s>
<s>
There	O
have	O
been	O
few	O
critical	O
appraisals	O
regarding	O
ACORN	B-Algorithm
.	O
</s>
<s>
Another	O
brief	O
reference	O
to	O
ACORN	B-Algorithm
simply	O
states	O
that	O
the	O
"	O
...	O
ACORN	B-Algorithm
generator	O
proposed	O
recently	O
 [ … ] 	O
is	O
in	O
fact	O
equivalent	O
to	O
a	O
MLCG	O
with	O
matrix	O
A	O
such	O
that	O
a	O
~	O
=	O
1	O
for	O
i	O
2	O
j	O
,	O
aq	O
=	O
0	O
otherwise	O
"	O
but	O
the	O
analysis	O
is	O
not	O
taken	O
further	O
.	O
</s>
<s>
ACORN	B-Algorithm
is	O
not	O
the	O
same	O
as	O
ACG	O
(	O
Additive	O
Congruential	O
Generator	O
)	O
and	O
should	O
not	O
be	O
confused	O
with	O
it	O
-	O
ACG	O
appears	O
to	O
have	O
been	O
used	O
for	O
a	O
variant	O
of	O
the	O
LCG	B-Algorithm
(	O
Linear	B-Algorithm
Congruential	I-Algorithm
Generator	I-Algorithm
)	O
described	O
by	O
Knuth	O
(	O
1997	O
)	O
.	O
</s>
<s>
Initially	O
,	O
ACORN	B-Algorithm
was	O
implemented	O
in	O
real	O
arithmetic	O
in	O
FORTRAN77	O
,	O
and	O
shown	O
to	O
give	O
better	O
speed	O
of	O
execution	O
and	O
statistical	O
performance	O
than	O
Linear	B-Algorithm
Congruential	I-Algorithm
Generators	I-Algorithm
and	O
Chebyshev	O
Generators	O
.	O
</s>
<s>
In	O
1992	O
,	O
further	O
results	O
were	O
published	O
,	O
implementing	O
the	O
ACORN	B-Algorithm
Pseudo-Random	B-Algorithm
Number	I-Algorithm
Generator	I-Algorithm
in	O
exact	O
integer	O
arithmetic	O
which	O
ensures	O
reproducibility	O
across	O
different	O
platforms	O
and	O
languages	O
,	O
and	O
stating	O
that	O
for	O
arbitrary	O
real-precision	O
arithmetic	O
it	O
is	O
possible	O
to	O
prove	O
convergence	B-Algorithm
of	O
the	O
ACORN	B-Algorithm
sequence	O
to	O
k-distributed	O
as	O
the	O
precision	O
increases	O
.	O
</s>
<s>
In	O
2000	O
,	O
ACORN	B-Algorithm
was	O
stated	O
to	O
be	O
a	O
special	O
case	O
of	O
a	O
Multiple	O
Recursive	O
Generator	O
(	O
and	O
,	O
therefore	O
,	O
of	O
a	O
Matrix	O
Generator	O
)	O
,	O
and	O
this	O
was	O
formally	O
demonstrated	O
in	O
2008	O
in	O
a	O
paper	O
which	O
also	O
published	O
results	O
of	O
empirical	O
Diehard	O
tests	O
and	O
comparisons	O
with	O
the	O
NAG	O
LCG	B-Algorithm
(	O
Linear	B-Algorithm
Congruential	I-Algorithm
Generator	I-Algorithm
)	O
.	O
</s>
<s>
In	O
2009	O
,	O
formal	O
proof	O
was	O
given	O
of	O
theoretical	O
convergence	B-Algorithm
of	O
ACORN	B-Algorithm
to	O
k-distributed	O
for	O
modulus	O
M	O
=	O
2m	O
as	O
m	O
tends	O
to	O
infinity	O
(	O
as	O
previously	O
alluded	O
to	O
in	O
1992	O
)	O
,	O
together	O
with	O
empirical	O
results	O
supporting	O
this	O
,	O
which	O
showed	O
that	O
ACORN	B-Algorithm
generators	O
are	O
able	O
to	O
pass	O
all	O
the	O
tests	O
in	O
the	O
standard	O
TESTU01	B-Algorithm
suite	O
for	O
testing	O
of	O
PRNGs	O
(	O
when	O
appropriate	O
order	O
and	O
modulus	O
parameters	O
are	O
selected	O
)	O
.	O
</s>
<s>
Since	O
2009	O
ACORN	B-Algorithm
has	O
been	O
included	O
in	O
the	O
NAG	O
(	O
Numerical	O
Algorithms	O
Group	O
)	O
FORTRAN	B-Application
and	O
C	B-Language
library	I-Language
routines	O
,	O
together	O
with	O
other	O
well-known	O
PRNG	O
routines	O
.	O
</s>
<s>
This	O
implementation	O
of	O
ACORN	B-Algorithm
works	O
for	O
arbitrarily	O
large	O
modulus	O
and	O
order	O
,	O
and	O
is	O
available	O
for	O
researchers	O
to	O
download	O
.	O
</s>
<s>
ACORN	B-Algorithm
was	O
also	O
implemented	O
in	O
GSLIB	O
GeoStatistical	O
modelling	O
and	O
simulation	O
library	O
.	O
</s>
<s>
More	O
recently	O
,	O
ACORN	B-Algorithm
was	O
presented	O
in	O
April	O
2019	O
at	O
a	O
poster	O
session	O
at	O
a	O
conference	O
on	O
Numerical	O
algorithms	O
for	O
high-performance	O
computational	O
science	O
at	O
the	O
Royal	O
Society	O
in	O
London	O
,	O
and	O
in	O
June	O
2019	O
at	O
a	O
Seminar	O
of	O
the	O
Numerical	O
Analysis	O
Group	O
at	O
the	O
Mathematical	O
Institute	O
of	O
the	O
University	O
of	O
Oxford	O
.	O
</s>
<s>
where	O
it	O
was	O
stated	O
that	O
statistical	O
performance	O
is	O
better	O
than	O
some	O
very	O
widely	O
used	O
generators	O
(	O
including	O
the	O
Mersenne	B-Algorithm
Twister	I-Algorithm
MT19937	I-Algorithm
)	O
and	O
comparable	O
to	O
the	O
best	O
currently	O
available	O
methods	O
"	O
and	O
that	O
ACORN	B-Algorithm
generators	O
have	O
been	O
shown	O
to	O
reliably	O
pass	O
all	O
the	O
tests	O
in	O
the	O
TestU01	B-Algorithm
,	O
while	O
some	O
other	O
generators	O
including	O
Mersenne	B-Algorithm
Twister	I-Algorithm
do	O
not	O
pass	O
all	O
these	O
tests	O
.	O
</s>
