Information Technology Reference
In-Depth Information
Consider two computer systems
KA
and
KB
connected by link
S
.Let
X
,
X<∞
, be a set of all possible messages, which can be sent from
KA
to
KB
through
S
. We describe the informational stream from
KA
to
KB
X
with a random infinite sequence of messages taken from
. Elements of any
N
sequence are numbered with natural numbers
. Denote a sequence space:
X
∞
.Let
{α
x
i
1
,x
i
2
, ..., x
i
n
, ...
,x
i
n
∈ X}
I
n
(
x
i
1
,x
i
2
, ..., x
i
n
, ...
=(
)
=
)bean
X
∞
and
elementary cylindrical set in
A
be a minimal
σ
-algebra, which is gen-
erated by all cylindrical sets. Let
{P
0
,t
1
,...t
n
(
x
t
1
,x
t
2
, ..., x
t
n
)
}−
(1)
be a consistent family of probability distributions on cylindrical sets. Then there
is the only probability measure
X
∞
, A
), generated by
(1) which describes the normal behavior of the system. Imagine that adversary
hardware/software agent
P
0
on measurable space (
KA
functions independently in computer system
KA
KB
in
and tries to send illegally a message to his partner
KB
through the link
S
. It is not allowed and they need a covert channel [7] on the base of the legal
transmission from
KA
to
KB
. We characterize the existence of transmission
KA
in legal trac with a consistent family of probability distributions on
cylindrical sets of
from
X
∞
{P
1
,t
1
,...t
n
(
x
t
1
,x
t
2
, ..., x
t
n
)
},
(2)
which generates the only probability measure
P
1
on measurable space (
X
∞
, A
).
When seeing the trac from
KA
agent
KB
KA
to
KB
to detect the signal from
should test a hypothesis
H
0
:
{P
0
,t
1
,...t
n
}
versus an alternative
H
1
:
{P
1
,t
1
,...t
n
}
.
KA
to
KB
in
Definition 1.
There is a statistical covert channel from
S
if and only if measures
P
0
and
P
1
are mutually perpendicular [6]. That is
∃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
(
.
If
P
0
and
P
1
are mutually perpendicular then there exists a consistent testing
of
H
0
versus alternative
H
1
.
Considering Warden existence define a set of alternatives
H
11
instead of
the single alternative
H
1
. It consists of all probability measures
{P
1
θ
,θ ∈ Θ}
where each
P
1
θ
is perpendicular to
P
0
(
Θ
is an arbitrary parameterization of
alternatives in
H
11
). Let examine an existence of a consistent testing of
H
0
versus alternatives
H
11
.
To prove an existence of the consistent tests we should consider more compli-
cated model. It is convenient to see the finite set
as a topological space where
every point is an open set and a closed set simultaneously. Then every subset of
X
X
X
∞
is a Tychonoff product
[6], if its basis of open sets consists of cylindrical sets of
is an open set and a closed set. Topological space
X
∞
. The topological
X
∞
is compactum
space
X
is a separable space and compactum. Then the space
X
∞
equals to
and Baire
σ
-algebra [6] on
σ
-algebra
A
[6]. Let us define a Borel
