Cryptography Reference
In-Depth Information
If the encoder is initialized to state 0 (
s
0
=0
), the final state
s
k
obtained at the
endofaframeoflength
k
is:
k
s
k
=
A
j−
1
Bd
k−j
(5.16)
j
=1
When it is initialized in any state
s
c
, the final state
s
k
is expressed as follows:
k
s
k
=
A
k
s
c
+
A
j−
1
Bd
k−j
(5.17)
j
=1
For this state
s
k
to be equal to the departure state
s
c
and for the latter therefore
to become the circulation state, it is necessary and sucient that:
k
I
A
k
s
c
=
A
j−
1
Bd
k−j
−
(5.18)
j
=1
where
I
is the identity matrix of dimension
ν
×
ν
.
Thus, by introducing state
s
k
of the encoder after initialization to 0
s
c
=
I
A
k
−
1
s
k
−
(5.19)
A
k
)
is
invertible
:thisis
This is only possible on condition that the matrix (
I
−
the condition for the existence of the circulation state.
In conclusion, if the circulation state exists, it can be obtained in two steps.
The first involves coding the frame of
k
bits at the input after initializing the
encoder to state zero and keeping the termination state. The second is to simply
deduce the circulation state from the previous termination state and from a table
(obtained by inverting
I
A
k
)
.
Take for example the recursive systematic code used above. Since it is a
binary code, addition and subtraction are equivalent: the circulation state exists
if
I
+
A
k
−
is invertible. Matrix
A
is such that
A
7
is equal to
I
.Thus,if
k
is a
multiple of
7(= 2
ν
1)
,
I
+
A
k
is null and therefore non-invertible: this case
should be avoided. Another consequence is that
A
k
=
A
k
mod 7
: it suces to
calculate once and for the 6 state transformation tables associated with the 6
possible values of (
I
−
A
k
)
−
1
, to store them and to read the right table, after
calculating
k
mod 7
. The table of the CRSC code with 8 states of generator
polynomials
[1
,
(1 +
D
2
+
D
3
)
/
(1 +
D
+
D
3
)]
is given in Table 5.2.
A simple method for encoding according to a circular trellis can be summa-
rized in five steps, after checking the existence of the circulation state:
−
1. Initialize the encoder to state 0
2. Code the frame to obtain the final state
s
k
;
