Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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110
and
caus
ed
by
the
self-collapse
indicated
by
E'2
the
state
01
= K
E'l
(00) will
be
transformed
to
provided
tr(Ol
.
TE'2)
>
O.
This
procedure
can
be
repeated
arbitrarily
often
- given
an
initial
state
00
and
a sequence (E'k)k=l one
obtains
this
way a sequence
(Ok)k=O
of
states
on
1i
and
a sequence (tr(Ok_l .
TE'k
))~l
of
probabilities
of
the
events (E'k)k=l
provided
these
expressions
exist
(i.
e.
if
tr(Ok_l
.
TE'k)
> 0 for k
~
1). A
necessary
condition
for
this
to
hold
is
that
the
sequence
(E'k)~l
has
to
be
in
some
sense increasing. More precisely, we
require
(rE{l,
...
,n})
(8)
for all
kEN.
The
relation
(8) defines a semi-ordering
of
the
elements
in
n.
We
writ
e
E'k
~
E'
k+l
to
indicate
relation
(8).
This
prop
e
rty
is in
accordance
with
the
requirements
our
model
should
fulfill.
If
once
partial
recog
nition
in
the
area
r
occurs
there
will
be
no
change
for
this
region
in
the
future,
only
further
zeros
may
be
transformed
into
1.
The
full recognition
of
the
signal would
be
obtained
if for some
kEN
we have
E'k
= (1,
..
.
,1).
The
sequence
(Ok)~O
is in
accordance
with
this
relation. Especially, we have
and
if
E'k
- l
~
E'k
does
not
hold,
'f
-k-l
<8
-k
1 E _ E .
(9)
(10)
The
above
identities
are
based
on the
following
properti
es
of
the
introduced
projection
the
easy
proof
of
which we will
omit.
Proposition
4.2.
T~T~
=
T~T~
=
T~,
T~T[
=
T[T~
= T[,
T~T[
=
T~T[
= 0
LT
E'
=
][H
implies
TE'l
T
E'2
=
TE'2
TE'l
= T
max{E'l
,E'2}
TE'l
TE'2
=
TE'2
TE'l
=
TE'2
(E'l
::;
E'2)
E'l
::;
E'2
(E'l,
E'2
E
n)
(E'l
::;
E'2)
111
Summarizing we
state
that
recognition process is characterised by
an
m-
creasing se
qu
ence (Ek)
k'=l
en
with
corres
ponding
sequence of
states
Unfortunat
ely,
this
sequence
(Ek)
k'=
l is unknown a
nd
seems
to
be
non-
computabl
e.
Only
in
the
case
the
sequences
(Ek)
~
l
would
be
known one
can
identify
the
process of recognition
with
the
sequence
(Ok)~O
of
states.
The
way
out
of
this
dilemma
is
to
consider
the
vectors
Ek
as
random
and
to
consider a classical Markov chain describing
this
process
of
recognition.
5. A
Markov
Chain
with
Discrete
Time
Since
the
sequence (Ek)k'=l is
not
known we replace
this
sequence by a
sequence of
random
ve
ctors
(Xk)~l'
Hereby, X
k
has values
in
0
and
is
interpr
e
ted
as
the
random
outcom
e in
the
k-th
step
of
the
recogntion
process.
The
sequence
(Ok)k'=O)
of
stat
es
we
replace by a new sequence
(f3k)~O
of
states
on
H given by
f3
k(-)
;=
L
P(Xk
=
E)
.
tr(K
e(Oo)(-))
(k
2:
1).
(11)
eE
fl
Hereby,
P(Xo
=
(0,
...
,0)) = 1
and
f3
0
= 00 is
the
initial
state
on
H.
Following
the
above
argum
e
nts
the
sequence
(Xk)~O
should form a classical
Markov chain
with
initial
distribution
P
Xo
=
15(0
, .. . ,0)
and
with
transition
probabilities
(k
2:
0)
(12)
where
EO
=
(0,
...
,0).
These
probabilities represent conditional probabili-
ties
of
passing in
the
k-th
step
to
Ek
starting
from
Ek
- l. We conclude
and
summarize
the
above considerations
with
the
following
theorem
the
proof
of
which
we
will
omit
.
Theorem
5.1.
Let 0 be a state on H (0 is the initial state).
(a) The
re
exists exactly one probability measure P
e
on
ON
(equipped
with the canonical
O"-algebra)
with finite-dimensional distributions
112
P;:
given
by
m
-0
-m
-0
rrm
tr({!TC'k-l
TC'k)
()
Pe(E,
...
,E
)=b(O,
...
,O)({E})k=l tr({!TC'k-l)
13
if
tr({!TC'l)
....
· tr({!TC'=-l) > 0
P
m (-0
-m)
0
e
E,
...
,E
=
otherwise.
(14)
(b) The sequence
(Xk)k=O
of
random vectors is increasing,
z.
e.
(k
;::::
0).
(c) The sequence
(Xk)k=O
is a homogenous Markov chain with initial
distribution
15(0,
... ,0) and with transition probabilities
(
-1
-2)
tr({!TC'2)
pEE
=
,
tr
({!TC'l
)
(15)
(d) The
m-steps
transition probabilities
are
given
by
(16)
(e)
(m;::::
1, E
EO).
(17)
Since 0 is a finite
set
and
the
sequence
(Xk)k=O
is increasing
it
definitely
reaches
its
maximum
Xmax
indicating
the
final
result
of
the
recognition
s
process.
This
vector
Xmax
<::::
(1,
...
,1)
may
be
non-random.
Further,
one
may
introduce
the
random
number
7]
of
steps
one needs
to
achieve X
max
.
One
has
7]
= 1 +
max
{k:
X k < X
max}.
In a
more
refined
model
one
has
to
pass
over
in
a sense of a scaling limit from
this
discrete
Markov
chain
to
a
homogenous
Markov
chain
with
continuous
time.
This
will
be
useful since
based
on
the
measuring
device one
cannot
measure
what
is going
on
in
the
single (very
short)
steps
of
the
recognition process. We will
deal
with
this
problem
in
a
forthcoming
paper.
113
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This page intentionally left blankThis page intentionally left blank
Quantum
Bio-Informatics
IV
eds. L.
Accardi,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
117-128)
STATISTICAL
ANALYSIS
OF
RANDOM
NUMBER
GENERATORS
LUIGI
ACCARDI
Universita
di
Roma
Tor Vergata, Centro Interdipartimentale Vito Volterra, Via
Columbia
2, 00133 Roma, Italy, E-Mail: accardi@volterra.uniroma2.it
MARKUS
GABLER
Brandenburg Technical University Cottbus, Department
of
Mathematics,
PO
box
101344, 03013 Cottbus, Germany, E-Mail: gaebler@math.tu-cottbus.de
In
many
a
pplications,
for
example
cryptogr
a
phy
and
Mont
e
Carlo
simulation,
there
is need for
random
numb
ers.
Any
procedur
e,
algorithm
or
device
which
is
int
en
ded
to
produce
such
is
called
a
random
number
generator
(RNG).
What
mak
es a
good
RNG?
This
paper
gives
an
overview
on
empirical
testing
of
th
e
statistical
prop
e
rties
of
the
sequences
produced
by
RNGs
and
special
software
packages
designed
for
that
purpos
e.
We
also
present
the
results
of
applying
a
particular
test
suite-T
estU
01-
to
a family
of
RNGs
currently
being
dev
eloped
at
the
Centro
Interdipartimentale
Vito
Vol
ter
ra
(CIVV)
,
Roma,
Italy.
1.
Introduction
Assume
Xl,
...
, X N is a
random
binary
sequence,
i.
e. for all 1
::;
k
::;
N
X
k
is a Bernoulli
random
variable
with
P(Xk
=
1)
=
Pk
E
(0,1)
. Such
a sequence is called
purely
random,
if
Xl"'"
X
N
are
independent
and
identically
distributed
(i.i.d.)
with
Pk
= P =
1/2.
In
many
applications, for
example
cryptography
and
Monte
Carlo
sim-
ulation,
there
is need for
random
numbers. Using
binary
expansion
and
transformation
methods
any
such
random
numbers
can
be
constructed
from
purely
random
binary
sequences
to
an
arbitrary
precision, ignoring
numerical difficulties for
the
moment.
So
many
ways have
been
invented
to
produce
,
or
at
least simulate, realizations
Xl,
. . . , X N
of
such sequences.
Repeated
flipping
of
a "fair" coin
and
recording "0" for "heads"
and
"I
"
for "tails" would
be
one
way,
but
surely
too
timeconsuming,
if
you needed,
say, 10
18
exponential
random
numbers
for some extensive
stochastic
simu-
lation.
Using
randomness
in physical
quantities
like noise
in
an
electrical
117
118
circuit
or
the
timing
of strokes
at
a user
keyboard
would
be
another.
Or
even
manipulation
of
a definitely
deterministic
process (like a small com-
puter
programm)
in
a way
that
the
outcomes
appear
to
be
purely
random
might
serve
just
as well.
Any
such procedure,
algorithm
or
device which
is
intended
to
produce
realizations
of
random
sequences, we call a
random
number
generator
(RNG).
Deterministic
RNGs
are
also sometimes called
pseudo-random
number
generators
(PRNG)
or
algorithmic
RNGs.
But
no
matter
what
the
nature
of
a
particular
RNG
is,
the
question arises
whether
the
sequences
it
produces
can
be
distinguished from
the
ones coming from
the
purely
random
theoretical
ideal.
This
paper
only deals
with
the
statistical
properties
of
binary
se-
quences (viewed as realizations of
random
binary
sequences).
But
while
good
statistical
properties
are
necessary,
there
are,
of
course,
other
impor-
tant
quality
criteria, including:
Efficiency
with
respect
to
time
and
memory.
Sufficiently
large
period: The
internal
state
of a
deterministic
RNG
runs
over a finite set
and
is therefore periodic, so
running
through
a signif-
icant
portion
of
the
whole cycle
should
be
beyond
reach
in
practice.
Repeatability:
the
ability
to
reproduce
the
same
sequence as
many
times
as
needed.
Non-deterministic
RNGs
are
not
repeatable.
Portability:
independence from software
and
hardware
environment.
Unpredictability:
The
next
output
bit
of
an
RNG
is
not
to
be
predicted
from knowing preceding ones
any
better
than
by
"tossing a fair coin". For
a
deterministic
RNG
this
means
that
it
should
not
be
possible
to
find
the
internal
state
and/or
the
transition
law from knowing
the
output
sequence,
at
least
not
in
reasonable
time
employing a reasonable
amount
of resources.
The
importance
of
these
criteria
mostly
depends
on
the
application
the
RNG
is
intended
for. So while for some uses
in
cryptography
nothing
less
than
an
unpredictable
RNG
will do, for
simulational
purposes
the
main
emphasis
might
be
on
high efficiency
and
repeatability.
Not
all
RNGs
are
designed
to
produce
binary
(bit) sequences. Besides
bit
generators, i.i.d. Uniform{O,
...
, 2
m
-
I}
and
i.i.d. Uniform[O,
1)
gen-
erators
are
also very common.
The
former
produce
m-bit-integers, where
31
and
32
bits
are
used
quite
frequently,
the
latter
approximate
reals, also
only
up
to
a
certain
precision (float, double, etc.),
of
course.
These
different
outcomes
can
be
transformed
one
into
the
other
quite
easily,
though.
Di-
119
viding a
bit
sequence
into
m-bit-blocks yields
the
binary
representation
of a
sequence
of
m-bit-integers.
An
m-bit-integer
divided
by
2
m
is a [0,1) real.
The
converse works
just
as well.
In
addition,
a
random
binary
sequence
is
purely
random
if
and
only if
the
corresponding
m-bit-integer
sequence
is i.i.d. Uniform{O,
...
, 2
m
-
I}.
Thus,
even
though
some
tests
need
bit
sequences
and
others,
say, 32-bit-integers as
input,
they
can
be
applied
to
any
RNG,
as long as
the
output
of
the
RNG
is
transformed
to
the
right
format.
Section 2 deals
with
statistical
tests
and
special software packages de-
signed for
testing
RNGs.
Then,
in
section
3,
the
results
of
applying
a
particular
test
suite
- TestUOl -
to
a family of
RNGs
currently
being
developed
at
the
Centro
Interdipartimentale
Vito
Volterra
(CIVV)
at
the
Universita
degli
studi
di
Roma
Tor Vergata, Italy,
are
presented.
2.
Statistical
Testing
of
RNGs
Bad
RNGs
are
those
that
fail simple
tests,
whereas
good
RNGs
fail
only complicated
tests
that
are
hard
to
find
and
run.
P.
L'Ecuyer
2.1.
Historical
Development
Over
the
years
many
statistical
tests
for
testing
random
number
genera-
tors
have
been
proposed.
One
of
the
first collections was found in earlier
editions of
Knuth
[1].
These
tests,
plus a few
others
designed for
test-
ing parallel generators, were
implemented
in
SPRNG:
a scalable library for
pseudorandom number generation
by
Mascagni
and
Srinivasan
[2].
New
and
more
stringent
a
tests,
compared
to
the
ones from
the
just
mentioned
Knuth
[1],
were
introduced
by
Marsaglia
in
1985
[3].
Most
of
these
tests
were
later
implemented
in
DIEHARD:
A
Battery
of
Tests
of
Randomness
by
Marsaglia
in
1995
[4],
probably
the
best-known
software
package for
RNG
testing. Because
of
its
very unflexible
setup
its
usefulness
has
become
rather
limited,
though,
by
now.
First
of
all,
the
sample
sizes
(as well
as
other
parameters)
are
fixed
in
the
package
and
not
very large
by
modern
standards,
which makes
the
test
results unreliable for
many
of
todays
applications. Second
the
sequence
must
be
provided
to
the
package
a A
test
Tl
is
considered
more
stringent
than
another
test
T2,
if
a
generator
passing
Tl
is also likely
to
pass
T2.