Cryptography Reference
In-Depth Information
we have:
L
(
c
3
)=ln
1+exp(
L
(
c
2
)+
L
(
c
1
))
exp(
L
(
c
2
)) + exp(
L
(
c
1
))
L
(
c
1
)
⊕
L
(
c
2
)
(9.3)
Equation (9.3) enables us to define the switching operator
⊕
between the two
LLRs of the variables
c
1
and
c
2
.
Applying function
tanh (
x/
2) =
exp(
x
)
−
1
exp(
x
)+1
to Equation (9.3), the latter be-
comes:
tanh
L
(
c
3
)
2
=
exp(
L
(
c
0
))
1
exp(
L
(
c
0
))+1
×
−
exp(
L
(
c
1
))
1
exp(
L
(
c
1
))+1
−
j
=0
tanh
L
(
c
j
)
(9.4)
1
=
2
It is practical (and frequent) to separate the processing of the sign and the
magnitude in Equation (9.4) which can then be replaced by the following two
equations:
1
sgn
(
L
(
c
3
)) =
sgn
(
L
(
c
j
))
(9.5)
j
=0
tanh
|
=
tanh
|
1
L
(
c
3
)
|
L
(
c
j
)
|
(9.6)
2
2
j
=0
where the sign function sgn
(
x
)
is such that sgn
(
x
)=+1
if
x
≥
0
and sgn
(
x
)=
−
1
otherwise.
Processing the magnitude given by Equation (9.6) can be simplified by taking
the inverse of the logarithm of each of the terms of the equation, which gives:
=
f
−
1
⎛
⎞
⎝
j
=1
,
2
⎠
|
L
(
c
3
)
|
f
(
|
L
(
c
j
)
|
)
(9.7)
where function f, satisfying
f
−
1
(
x
)=
f
(
x
)
, is defined by:
f
(
x
)=ln(tanh(
x/
2))
(9.8)
These different aspects of the computation of function
⊕
will be developed in
the architecture part of this chapter.
Expression (9.6) in fact corresponds to the computation of (9.3) in the Fourier
domain. Finally, there is also a third writing of the LLR of variable
c
2
[9.65, 9.23]:
L
(
c
3
)=
sign
(
L
(
c
1
))
sign
(
L
(
c
2
)) min (
|
L
(
c
1
)
|
,
|
L
(
c
2
)
|
)
−
ln (1 + exp (
−|
L
(
c
1
)
−
L
(
c
2
)
|
))
(9.9)
+ln(1+exp(
−|
L
(
c
1
)+
L
(
c
2
)
|
))
