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P 0 and
P 1
Then for every
µ
the function
F µ generates probability measures
X , A
on (
).
Theorem 3.
P 0
P 1
Measures
and
are perpendicular then
∃A 0 , ∃A 1 ,A 0 ,A 1
A,A 0 ∩ A 1 =
,
X \ A 0 )=0
X \ A 1 )=0
P 0 (
,P 1 (
,
and
F µ (
A 0 )
∩ F µ (
A 1 )=
.
Proof.
The proof follows from the proof of the theorem 2.
P 0
P 1
Example.
If
and
are not perpendicular then the warden
U
cannot de-
tect a covert transmission reliably. Let
µ
be an arbitrary binary sequence and
ω )=1
ω . Assume
ω ). Despite
P 0 (
ω
)=1
,P 1 (
=
F µ (
ω
)=
F µ (
P 0
and
P 1
are perpendicular there is no consistent test for
U
to identify a hidden mes-
KA to
KB . That means
sage from
U
can discover the hidden message if
the agents do not use such
P 1 ,that
∀A 0 , ∀A 1 ,A 0 ,A 1 ∈A,A 0 ∩ A 1
=
,if
X \ A 0 )=0
X \ A 1 )=0,then
P 0 (
,P 1 (
F µ (
A 0 )
∩ F µ (
A 1 )
=
∅.
4
Conclusions
The obtained results show that the manipulation of the probability distribution
of the messages in communicational link in order to send hidden messages can
be revealed. The warden should know well enough the normal properties of the
communication link and its probabilistic characteristics, e.g. in lemma 1'. Then it
is possible to construct a consistent test for detection of hidden message. That is
why it is a problem to make a statistical covert channel invisible for the warden.
Even if the warden's resources are limited the detection of a hidden message
most often can be done reliable enough.
We plan to research the necessary conditions to detect hidden transmission
by the warden.
We didn't touch the problem whether the construction of a consistent test is
a hard task. Most probably it is.
References
1. Anderson, R.J., Petitcolas, F.A.P.: On the Limits of Steganography. IEEE Journal
of Selected Areas in Communications, 16(4) (May 1998) 474-481
2. Axelson S.: The Base-Rate Fallacy and its Implications for the Diculty Of Intrusion
Detection. Proc. of the 6th Conference on Computer and Communications Security
(November 1999)
3. Grusho A.: Consistent revelation conditions for rare events search a sample from
the uniform distribution. In: Probabilistic problems of discrete mathematic. Moscow
Institute of Electronic mechanical engineering (1987) (in Russian)
4. Denning, D.: An Intrusion Detection Model. Proceedings of the IEEE Symposium
on Security and Privacy (May 1986) 119-131
5. Lee W., Xiang D.: Information-Theoretic Meaasures for Anomaly Detection. IEEE
Symposium on Security and Privacy (2001) 130-143
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