Cryptography Reference
In-Depth Information
Definition 8.5.1
Let
C
be a curve over
k
.A
derivation
on
k
(
C
)isa
k
-linear map (treating
k
(
C
)asa
k
-vector space)
δ
:
k
(
C
)
→ k
(
C
) such that
δ
(
f
1
f
2
)
=
f
1
δ
(
f
2
)
+
f
2
δ
(
f
1
).
Lemma 8.5.2
Let δ
:
k
(
C
)
→ k
(
C
)
be a derivation. Then:
1. If c
∈ k
then δ
(
c
)
=
0
.
then δ
(
x
n
)
nx
n
−
1
δ
(
x
)
.
2. If x
∈ k
(
C
)
and n
∈ Z
=
(
C
)
then δ
(
x
p
)
3. If
char(
k
)
=
p and x
∈ k
=
0
.
(
C
)
then δ
(
f
)
4. If h
∈ k
=
hδ
(
f
)
is a derivation.
xδ
(
y
))
/y
2
.
5. If x,y
∈ k
(
C
)
then δ
(
x/y
)
=
(
yδ
(
x
)
−
∈ k
∈ k
=
+
6. If x,y
(
C
)
and F
(
u,v
)
[
u,v
]
is a polynomial then δ
(
F
(
x,y
))
(
∂F/∂x
)
δ
(
x
)
(
∂F/∂y
)
δ
(
y
)
.
Exercise 8.5.3
Prove Lemma
8.5.2
.
Definition 8.5.4
Let
C
be a curve over
k
. A function
x
∈ k
(
C
)isa
separating element
(or
separating variable
)if
k
(
C
) is a finite separable extension of
k
(
x
).
Note that if
x
∈ k
(
C
) is such that
x
∈ k
then
k
(
C
)
/
k
(
x
) is finite; hence, the non-trivial
condition is that
k
(
C
)
/
k
(
x
) is separable.
1
(
1
)
(
x
)) and
x
p
is not
Example 8.5.5
For
P
F
p
),
x
is a separating element (since
k
(
P
= k
1
)
/
(
x
p
)
(
x
p
) is not separable). The mapping
a separating element (since
k
(
P
k
= k
(
x
)
/
k
δ
(
f
)
=
∂f/∂x
is a derivation.
The following exercise shows that separating elements exist for elliptic and hyperelliptic
curves. For general curves we need Lemma
8.5.7
.
Exercise 8.5.6
Let
k
be any field and let
C
be a curve given by an equation of the
form
y
2
+
H
(
x
)
y
=
F
(
x
) with
H
(
x
)
,F
(
x
)
∈ k
[
x
]. Show that if either
H
(
x
)
=
0orif
char(
k
)
=
2 then
x
is a separating element of
k
(
C
).
Lemma 8.5.7
Let C be a curve over
k
, where
k
is a perfect field. Then there exists a
∈ k
separating element x
(
C
)
.
Proof
See the full version of the topic or Proposition III.9.2 of Stichtenoth [
529
].
Lemma 8.5.8
Let C be a curve over
k
,letP
∈
C
(
k
)
and let t
P
be a uniformiser at P. Then
t
p
is a separating element of
k
(
C
)
.
Proof
Proposition III.9.2 of Stichtenoth [
529
].
Suppose now that
C
is a curve over
k
and
x
is a separating element. We wish to extend
δ
(
f
)
=
∂f/∂x
from
k
(
x
) to the whole of
k
(
C
). The natural approach is to use property
6 of Lemma
8.5.2
:If
f
∈ k
(
C
) then
k
(
x,f
)
/
k
(
x
) is finite and separable; write
F
(
T
)for
the minimal polynomial of
f
over
k
(
x
)in
k
(
C
); since the extension is separable we have
