Cryptography Reference
In-Depth Information
are done modulo 2 in the Galois field). We then continue the development
of
m
α
(
x
)
and finally we have:
m
α
(
x
)=
x
4
+
x
+1
For the computation of
m
α
3
(
x
)
, the roots to take into account are
α
3
,α
6
,α
12
,α
24
=
α
9
(
α
15
=1
), and the other powers of
α
3
(
α
48
,α
96
,
)
give the previous roots again. The minimal polynomial
m
α
3
(
x
)
is therefore
equal to:
···
m
α
3
(
x
)=(
x
+
α
3
)(
x
+
α
6
)(
x
+
α
12
)(
x
+
α
9
)
which after development and simplification gives:
m
α
3
(
x
)=
x
4
+
x
3
+
x
2
+
x
+1
The S.C.M. of polynomials
m
α
(
x
)
and
m
α
3
(
x
)
is obviously equal to the
product of these two polynomials since they are irreducible and thus, the
polynomial generator is equal to:
g
(
x
)=(
x
4
+
x
+1)(
x
4
+
x
3
+
x
2
+
x
+1)
Developing this, we obtain:
g
(
x
)=
x
8
+
x
7
+
x
6
+
x
4
+1
Finally the parameters of this BCH code are:
m
=4;
n
= 15;
n
−
k
=8;
k
=7;
t
=2
The numerical values of parameters
(
n, k, t
)
of the main BCH codes and
the associated generator polynomials have been put table form and can be
found in [4.2]. As an example, we give in Table 4.2 the parameters and
the generator polynomials, expressed in octals, of some BCH codes with
error correction capability
t
=1
(Hamming codes).
Note :
g
(
x
)=13
in octals gives 1011 in binary, that is,
g
(
x
)=
x
3
+
x
+1
•
Primitive BCH code with
l
=0
The generator polynomial of a primitive BCH code correcting at least
t
errors (constructed distance
d
=2
t
+2
)has(
2
t
+1
) roots of the form:
α
0
,α
1
,
,α
2
t
; that is, one root more (
α
0
) than when
l
=1
.
Taking into account the fact that the minimal polynomials
m
α
j
(
x
)
and
m
α
2
j
(
x
)
have the same roots, generator polynomial
g
(
x
)
is equal to:
,α
j
,
···
···
g
(
x
)=
S.C.M.
(
m
α
0
(
x
)
,m
α
1
(
x
)
,m
α
3
(
x
)
,
···
,m
α
2
t−
1
(
x
))
