Cryptography Reference
In-Depth Information
Taking into account the fact that
c
(
x
)
is a multiple of the generator poly-
nomial and that the addition and the subtraction can be merged in
F
2
,wecan
then write:
x
n−k
d
(
x
)=
q
(
x
)
g
(
x
)+
v
(
x
)
v
(
x
)
is therefore the remainder of the division of
x
n−k
d
(
x
)
by the generator
polynomial
g
(
x
)
. The codeword associated with dataword
d
(
x
)
is equal to
x
n−k
d
(
x
)
increased by the remainder of the division of
x
n−k
d
(
x
)
by the gener-
ator polynomial.
Figure 4.2 - Schematic diagram of a circuit divisor by
g
(
x
)
.
Example 4.8
To illustrate the computation of a codeword written in systematic form, let us
take the example of a
C
(7,4) code of generator polynomial
g
(
x
)=1+
x
+
x
3
and
let us determine the codeword
c
(
x
)
associated with message
d
(
x
)=1+
x
2
+
x
3
,
that is:
c
(
x
)=
x
3
d
(
x
)+
v
(
x
)
The remainder of the division of
x
3
d
(
x
)
by
g
(
x
)=1+
x
+
x
3
being equal to 1,
codeword
c
(
x
)
associated with dataword
d
(
x
)
is:
c
(
x
)=1+
x
3
+
x
5
+
x
6
Thus, with data block
d
, made up of 4 binary information symbols, is associated
codeword
c
with:
d
=
1011
→
c
=
1001011
To obtain the generator matrix, it suces to encode
d
(
x
)= 1
,x,x
2
,x
3
.
