Cryptography Reference
In-Depth Information
(4) By (1) and (3),
1=
e
n
(
S
+
T,S
+
T
)=
e
n
(
S, S
)
e
n
(
S, T
)
e
n
(
T,S
)
e
n
(
T,T
)
=
e
n
(
S, T
)
e
n
(
T,S
)
.
Therefore
e
n
(
T,S
)=
e
n
(
S, T
)
−
1
.
(5) Let
σ
be an automorphism of
K
such that
σ
is the identity on the
coe
cients of
E
. Apply
σ
to everything in the construction of
e
n
.Then
div(
f
σ
)=
n
[
σT
]
−
n
[
∞
]
and similarly for
g
σ
,where
f
σ
and
g
σ
denote the functions obtained by ap-
plying
σ
to the coecients of the rational functions defining
f
and
g
(cf.
Section 8.9). Therefore,
σ
(
e
n
(
S, T
)) =
σ
g
(
P
+
S
)
g
(
P
)
=
g
σ
(
σP
+
σS
)
g
σ
(
σP
)
=
e
n
(
σS, σT
)
.
(6) Let
{
Q
1
,...,Q
k
}
=Ker(
α
).
Since
α
is a separable morphism,
k
=
deg(
α
) by Proposition 2.21. Let
div(
f
T
)=
n
[
T
]
−
n
[
∞
]
,
div(
f
α
(
T
)
)=
n
[
α
(
T
)]
−
n
[
∞
]
and
g
T
=
f
T
◦ n, g
α
(
T
)
=
f
α
(
T
)
◦ n.
As in (3), let
τ
Q
denote adding
Q
.Wehave
div(
f
T
◦ τ
−Q
i
)=
n
[
T
+
Q
i
]
− n
[
Q
i
]
.
Therefore,
[
T
]
− n
α
(
Q
)=
div(
f
α
(
T
)
◦ α
)=
n
[
Q
]
α
(
T
)=
α
(
T
)
∞
=
n
i
([
T
+
Q
i
]
−
[
Q
i
])
=div(
i
(
f
T
◦ τ
−Q
i
))
.
For each
i
, choose
Q
i
with
nQ
i
=
Q
i
.Then
g
T
(
P − Q
i
)
n
=
f
T
(
nP − Q
i
)
.
Consequently,
div
i
(
g
T
◦ τ
−Q
i
)
n
=div(
i
f
T
◦ τ
−Q
i
◦ n
)
=div(
f
α
(
T
)
◦
α
◦
n
)
=div(
f
α
(
T
)
◦
α
)
=div(
g
α
(
T
)
◦ α
)
n
.
n
◦
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