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Let
σ
-algebra
be generated by
F
. From lemma 2 we get that
=
{C ⊗
X
1
,C∈A
1
}
.
The next theorem shows how to use such a presentation of
.Let
σ
-algebra
, A
1
) - measurable function
F
⊆
be generated by an arbitrary (
,
.
P
0
P
1
Theorem 2.
Probability measures
and
,
which are generated by an arbitrary
(
, A
1
)-
measurable function
F
,
are perpendicular if and only if there are
A
0
and
A
1
of
,
that
A
0
∩ A
1
=
∅,
P
0
(
X \ A
0
)=0
,P
1
(
X \ A
1
)=0
.
Proof.
1)
Suciency.
If
A
0
and
A
1
of
, then there are
B
0
∈A
1
and
B
1
∈A
1
,
F
−
1
(
F
−
1
(
that
A
0
=
B
0
)
,A
1
=
B
1
). Then
P
0
(
X \ F
−
1
(
P
0
(
X
0
\ B
0
)=
F
−
1
(
X
0
\ B
0
)) =
P
0
(
B
0
)) =
P
0
(
X \ A
0
)=0
P
1
(
X \ A
1
)=0
,P
1
(
X
1
\ B
1
)=
=
.
F
−
1
(
F
−
1
(
It follows by definition
A
0
=
B
0
)
,A
1
=
B
1
)
,A
0
∩ A
1
=
∅.
Then
B
0
∩ B
1
=
The suciency is proved.
2)
Necessity.
If a statistical covert channel exists, then
∅.
B
0
,B
1
∈A
1
,B
0
∩B
1
=
∅,P
0
(
X
0
\ B
0
)=0
,P
1
(
X
0
\ B
1
)=0.Denote
F
−
1
(
A
0
,F
−
1
(
B
0
)=
B
1
)=
A
1
.
By definition
P
0
(
X \ F
−
1
(
F
−
1
(
X
0
\ B
0
)) =
P
0
(
B
0
)) = 0
,
P
1
(
X \ F
−
1
(
F
−
1
(
X
0
\ B
1
)) =
P
1
(
B
1
)) = 0
.
F
−
1
(
∩ F
−
1
(
As far as
B
0
∩ B
1
=
∅
,then
B
0
)
B
1
)=
∅
. Consequently, there
X
∞
\ A
0
)=0
X
∞
\ A
1
)=0.
are
A
0
,A
1
of
,that
A
0
∩ A
1
=
∅
and
P
0
(
,P
1
(
Theorem is proved.
Accordingtodefinition2the theorem 2 states that
U
can detect the statistical
KB
.
Let us consider another case of
KA
to
covert channel from
S
(
F
), when the possibility of
U
to control
the trac in
is limited due to shortage of the computational resources or/and
memory space available. As a result
S
U
has to draw his conclusions from some
subsequences of messages.
Let
be a binary sequence with an infinite number of 1 and an infinite
number of 0,
µ
γ
is the sequence of messages from
KA
to
KB
.
U
uses
µ
to make
sampling in
γ
. Every element in
γ
is taken away if it is in the position, where 0 is
X
∞
→ X
∞
.
Lemma 4.
For arbitrary sequence function
F
µ
is
(
A, A
)-
measurable
.
Proof.
in the sequence
µ
.Then
U
sees the sequence
δ
=
F
µ
(
γ
), where
F
µ
:
× X
∞
be an elementary cylindrical set of
Let
B
n
=(
δ
1
, ..., δ
n
)
A
.Then
F
−
1
µ
X
s
there is between
(
B
n
) is a cylindrical set of
A
,where
δ
i
and
δ
i
+1
.Here
s
denotes the number of binary zeros between
i
-th and (
i
+ 1)-th binary one
position in
µ
. Lemma is proved.
