Cryptography Reference
In-Depth Information
Replacing the probability densities by their respective expression we obtain:
exp
s
jp
)
2
π
j
1
N
p
=1
(
r
p
−
N
1
N
0
s
j
(
t
)
if
−
√
πN
0
exp
s
np
)
2
>π
n
1
N
p
=1
(
r
p
−
N
1
N
0
−
√
πN
0
After simplification:
N
N
s
j
(
t
)
if
r
p
s
jp
+
C
j
>
r
p
s
np
+
C
n
∀
n
=
jn
=1
,
2
,
···
,M
(2.59)
p
=1
p
=1
p
=1
N
E
j
2
N
2
(
s
jp
)
2
.
where
C
j
=
ln(
π
j
)
−
with
E
j
=
Noting that:
T
T
N
N
r
(
t
)
s
j
(
t
)
dt
=
r
p
ν
p
(
t
)
s
jm
ν
m
(
t
)
dt
p
=1
m
=1
0
0
and recalling that the functions
ν
p
(
t
)
are normed and orthogonal, we obtain:
T
N
r
(
t
)
s
j
(
t
)
dt
=
r
p
s
jp
p
=1
0
In the same way:
T
T
N
N
s
j
(
t
)
dt
=
s
jp
ν
p
(
t
)
s
jm
ν
m
(
t
)
dt
p
=1
m
=1
0
0
and finally:
T
N
s
j
(
t
)
dt
=
s
jp
p
=1
0
Taking into account the above, the MAP criterion can again be written in the
form:
0
T
0
T
s
j
(
t
)
if
r
(
t
)
s
j
(
t
)
dt
+
C
j
>
r
(
t
)
s
n
(
t
)
dt
+
C
n
(2.60)
∀
n
=
jn
=1
,
2
,
···
,M
0
T
E
j
2
N
2
s
j
(
t
)
dt
.
where
C
j
=
ln(
π
j
)
−
with
E
j
=