Cryptography Reference
In-Depth Information
high, then
n
is much higher than
n
−
k
and the gain in terms of computation
complexity is high.
To do this, we re-write the general term of
D
ν
(
l
)
in the form:
(
ρ
l
)
t
ν
(
l
)
=exp
{
t
ν
(
l
)ln
|
ρ
l
|}
exp
{
jπq
l
t
ν
(
l
)
}
1)
q
l
where
q
l
is such that
ρ
l
=(
−
|
ρ
l
|
.
We then have:
D
ν
(
l
)=exp
n−
1
)+(
jπq
l
t
ν
(
l
))
(
t
ν
(
l
)ln
|
ρ
l
|
l
=0
=exp
n−
1
)
exp
n−
1
(
jπq
l
t
ν
(
l
))
(
t
ν
(
l
)ln
|
ρ
l
|
l
=0
l
=0
Put:
)exp
jπ
n−
1
n−k−
1
F
ρ
(
w
)=
ln (
|
ρ
l
|
w
m
h
ml
l
=0
m
=0
2
n−k
for any integer
0
≤
w
≤
−
1
. We have, therefore:
n−
1
F
ρ
(
w
)=
ln (
|
ρ
l
|
)exp
{
jπt
w
(
l
)
}
l
=0
n
l
=0
−
1
with, in particular,
F
ρ
(0) =
ln (
|
ρ
l
|
)
.
1
−
exp
{
jπt
}
On the other hand, if
t
=0
or 1, then
=
t
and
2
n
−
1
F
ρ
(0)
−
F
ρ
(
ν
)
=
(
t
ν
(
l
)ln
|
ρ
l
|
)
.
2
l
=0
q
l
exp
jπ
w
m
h
ml
,wehave:
l
=0
m
=0
n
−
1
n
−
k
−
1
Likewise, if we put
F
q
(
w
)=
n−
1
F
q
(0)
−
F
q
(
ν
)
=
(
q
l
ln
|
ρ
l
|
)
2
l
=0
and therefore:
D
ν
(
l
)=exp
1
F
ρ
(
ν
))
exp
1
F
q
(
ν
))
2
(
F
ρ
(0)
−
2
jπ
(
F
q
(0)
−
The two terms
F
ρ
(
ν
)
and
F
q
(
ν
)
have a common expression of the form:
f
l
exp
jπ
n−
1
n−k−
1
F
(
w
)=
w
m
h
ml
l
=0
m
=0
