Cryptography Reference
In-Depth Information
encoder at each transition and
i
varies between 0 and 2, since 4 coded symbols
are possible (00, 01, 10, 11), with weights between 0 and 2.
The transfer function of the code
T
(
O, I
)
is then defined by:
T
(
O, I
)=
a
s
a
e
(5.9)
To establish this function, we have to solve the system of equations coming from
the relations between the 9 states (
ae
,
b
,
c
...
h
and
as
):
b
=
c
+
Od
c
=
Oe
+
f
d
=
h
+
Dg
e
=
O
2
Ia
e
+
OIb
f
=
O
2
Ic
+
OId
g
=
OIe
+
O
2
If
h
=
O
2
Ih
+
OIg
a
s
=
Ob
(5.10)
Using a formal computation tool, it is easy to arrive at the following result:
I
4
O
12
+(3
I
4
+
I
3
)
O
10
+(
3
I
4
3
I
3
)
O
8
+(
I
4
+2
I
3
)
O
6
+
IO
4
T
(
O, I
)=
−
−
−
I
4
O
10
+(
3
I
4
I
3
)
O
8
+(3
I
4
+4
I
3
)
O
6
+(
I
4
3
I
3
)
O
4
3
IO
2
+1
−
−
−
−
−
T
(
O, I
)
can then be developed as a series:
T
(
O, I
)=
IO
4
+(
I
4
+2
I
3
+3
I
2
)
O
6
+(4
I
5
+6
I
4
+6
I
3
)
O
8
+(
I
8
+5
I
7
+21
I
6
+24
I
5
+17
I
4
+
I
3
)
O
10
+(7
I
9
+30
I
8
+77
I
7
+73
I
6
+42
I
5
+3
I
4
)
O
12
+
(5.11)
···
This enables us to observe that an RTZ sequence with weight 4 is produced
by an input sequence with weight 1, that the RTZ sequences with weight 6 are
produced by a sequence with weight 4, by two sequences with weight 3 and by
three sequences with weight 2, etc.
In the case of the classical code mentioned above, the transfer function is:
T
(
O, I
)= (
I
3
+
I
)
O
6
+(2
I
6
+5
I
4
+3
I
2
)
O
8
+(4
I
9
+16
I
7
+21
I
5
+8
I
3
)
O
10
+(8
I
12
+44
I
10
+90
I
8
+77
I
6
+22
I
4
)
O
12
+(16
I
15
+ 112
I
13
+ 312
I
11
+ 420
I
9
+ 265
I
7
+60
I
5
)
O
14
+
(5.12)
···
