Cryptography Reference
In-Depth Information
For the permutation to really be a bijection, parameters
A
(
j
)
and
B
(
j
)
are
not just any parameters. To ensure the existence of the permutation, there is
one sucient condition: all the parameters have to be multiples of
C
.Th s
condition is not very restricting in relation to the eciency of the permutation.
(7.15)canthenberewrittenintheform:
Q
(
j
)=
C
(
α
(
j
)
P
+
β
(
j
))
(7.16)
where
α
(
j
)
and
β
(
j
)
are more often than not small integers, with values 0 to 8.
In addition, since the properties of a circular permutation are not modified by
a simple rotation, one of the
Q
(
j
)
values can systematically be 0.
Two typical sets of
Q
values, with cycle 4 and
α
=0
or
1
,aregivenbelow:
if
j
=0 mod4
,then
Q
=0
if
j
=1 mod4
,then
Q
=4
P
+4
β
1
if
j
=2 mod4
,then
Q
=4
β
2
if
j
=3 mod4
,then
Q
=4
P
+4
β
3
(7.17)
if
j
=0 mod4
,then
Q
=0
if
j
=1 mod4
,then
Q
=4
β
1
if
j
=2 mod4
,then
Q
=4
P
+4
β
2
if
j
=3 mod4
,then
Q
=4
P
+4
β
3
(7.18)
These models require the knowledge of only four parameters (
P
,
β
1
,
β
2
and
β
3
), which can be determined using the procedure described in [7.17]. The
utilization of
m
-binary codes (see Section 7.5), instead of binary codes, sim-
ply requires
k
to be replaced by
k/m
in Equation (7.14). In particular, the
permutations defined for double-binary turbo codes (
m
=2
) of the DVB-RCS,
DVB-RCT and WiMax standards are inspired by Equations (7.14) and (7.21).
Quadratic Permutation
Recently, Sun and Takeshita [7.41] proposed a new class of deterministic in-
terleavers based on permutation polynomials (PP) over integer rings. The use
of PP reduces the design of interleavers to simply a selection of polynomial
coecients. Furthermore, PP-based turbo codes have been shown to have a)
good distance properties [7.38] which are desirable for lowering the error floor
and b) a maximum contention-free property [7.43] which is desirable for parallel
processing to allow high-speed hardware implementation of iterative turbo
decoders.
