Cryptography Reference
In-Depth Information
Figure 4.7 - Schematic diagram of the circuit implementing the Chien algorithm.
performs the computation of the left-hand part of expression (4.42) for
j
=2
.If
the result of this computation is equal to 1,
α
n−
2
is a root of the error locator
polynomial and the error that concerned symbol
r
n−
2
is then corrected. The
algorithm continues in the same way for the following clock pulses.
4.4.3
Iterative method
Decoding RS codes or binary BCH codes with the iterative method uses two
polynomials, error locator polynomial
Λ(
x
)
and error evaluator polynomial
Γ(
x
)
.
These two polynomials are defined respectively by:
t
Λ(
x
)=
(1 +
Z
j
x
)
(4.43)
j
=1
t
Λ(
x
)
1+
Z
i
x
Γ(
x
)=
e
i
Z
i
x
(4.44)
i
=1
The error locator polynomial whose roots are
Z
−
j
enables the position of the
errors to be determined and the error evaluator polynomial enables the value
of the error
e
j
to be determined.
Indeed, taking into account the fact that
Λ(
Z
−
1
j
)=0
, the polynomial
Γ(
x
)
taken in
Z
−
1
j
is equal to:
)=
e
j
p
=
j
Γ(
Z
−
1
j
(1 +
Z
p
Z
−
1
j
)
e
j
Z
−
1
j
Λ
(
Z
−
1
j
=
)
where
Λ
(
x
)=
d
dx
(
x
)
.
