Cryptography Reference
In-Depth Information
background. Gaudry [
224
] and Gaudry and Lubicz [
227
] have given fast algorithms for
computing with this algebraic group quotient.
10.1 Non-singular models for hyperelliptic curves
=
y
2
+
−
=
Consider the singular points on the affine curve
C
(
x,y
)
H
(
x
)
y
F
(
x
)
0. The
=
+
=
H
(
x
)
y
−
F
(
x
), so
partial derivatives are
∂C
(
x,y
)
/∂y
2
y
H
(
x
) and
∂C
(
x,y
)
/∂x
a singular point in particular satisfies 2
F
(
x
)
H
(
x
)
H
(
x
)
+
=
0. If
H
(
x
)
=
0 and if the
charact
er
istic of
k
is not 2 then
C
is non-singular over
k
if and only if
F
(
x
) has no repeated
root in
k
.
Exercise 10.1.1
Show that the curve
y
2
+
H
(
x
)
y
=
F
(
x
) over
k
has no affine singular
points if and only if one of the following conditions hold.
1. char(
k
)
=
2 and
H
(
x
) is a non-zero constant.
2,
H
(
x
) is a non-zero polynomial and gcd(
H
(
x
)
,F
(
x
)
2
F
(
x
)
H
(
x
)
2
)
2. char(
k
)
=
−
=
1.
0 and gcd(
F
(
x
)
,F
(
x
))
3. char(
k
)
=
2,
H
(
x
)
=
=
1.
gcd(
H
(
x
)
2
4
F
(
x
)
,
2
F
(
x
)
H
(
x
)
H
(
x
))
4. char(
k
)
=
2,
H
(
x
)
=
0
and
+
+
=
1
(this
=
0or
H
(
x
)
=
applies even when
H
(
x
)
0).
We will now give a simple condition for when a hyperelliptic equation is geometrically
irreducible and of dimension 1. The proof also applies in many other cases. For the remaining
cases, one has to test irreducibility directly.
y
2
Lemma 10.1.2
Let C
(
x,y
)
be a hyperelliptic equati
on
. Sup-
pose that
deg(
F
(
x
))
is odd. Suppose also that there is no point P
=
+
H
(
x
)
y
−
F
(
x
)
over
k
=
(
x
P
,y
P
)
∈
C
(
k
)
such
that
(
∂C
(
x,y
)
/∂x
)(
P
)
0
. Then the affine algebraic set V
(
C
(
x,y
))
is geometrically irreducible. The dimension of V
(
C
(
x,y
))
is
1
.
=
(
∂C
(
x,y
)
/∂y
)(
P
)
=
P
roof
From Theorem
5.3.8
,
C
(
x,y
)
=
0is
k
-re
du
cible if and only if
C
(
x,y
) factors over
k
[
x,y
]. By considering
C
(
x,y
) as an element of
k
(
x
)[
y
] it follows that
su
ch a factorisation
must be of the form
C
(
x,y
)
=
(
y
−
a
(
x
))(
y
−
b
(
x
)) with
a
(
x
)
,b
(
x
)
∈ k
[
x
]. Since deg(
F
)
is odd it follows that deg(
a
(
x
))
=
deg(
b
(
x
)) and that at least one of
a
(
x
)
an
d
b
(
x
)is
non-constant. Hence,
a
(
x
)
−
b
(
x
) is a non-constant polynomial, so
l
et
x
P
∈ k
be a root
of
a
(
x
)
). It is then easy to
check that bot
h
partial derivatives vanish at
P
. Hence, under the conditions of the Lemma,
V
(
C
(
x,y
)) is
−
b
(
x
) and set
y
P
=
a
(
x
P
)
=
b
(
x
P
) so that (
x
P
,y
P
)
∈
C
(
k
-irreducible and so is an affine variety.
Now that
V
(
C
(
x,y
)) is known to be a variety we can consider the dimension. The
function field of the affine variety is
k
k
(
x
)(
y
), which is a quadratic algebraic extension of
k
(
x
) and so has transcendence degree 1. Hence, the dimension of is 1.
