Cryptography Reference
In-Depth Information
Exercise 10.1.18
Let
C
be a hyperelliptic curve in ramified model. Show that
v
∞
(
x
)
=−
2.
Show that if the curve has equation
y
2
1 then
x
g
/y
is an
=
F
(
x
) where deg(
F
(
x
))
=
2
g
+
alternative uniformiser at infinity.
Now suppose
C
is given as a split or inert model. Show that
v
∞
+
(
x
)
=
v
∞
−
(
x
)
=−
1.
Exercise 10.1.19
Let
C
be a hyperelliptic cu
rv
e (ramified, split or inert). If
u
(
x
)
=
(
x
−
x
0
)
∞
+
)
is a function on
C
and
P
0
=
(
x
0
,y
0
)
∈
C
(
k
) then div(
u
(
x
))
=
(
P
0
)
+
(
ι
(
P
0
))
−
(
−
∞
−
).
(
Exercise 10.1.20
Let
C
be a hyperelliptic curve of genus
g
. Show that if
C
is in rami-
fied model then
v
∞
(
y
)
=−
(2
g
+
1) and if
C
is in split model then
v
∞
+
(
y
)
=
v
∞
−
(
y
)
=
−
(
g
+
1).
Exercise 10.1
.2
1
Let
C
be a hyperelliptic curve. Let
A
(
x
)
,B
(
x
)
∈ k
[
x
] and let
P
=
(
x
P
,y
P
)
∈
C
(
k
) be a point on the affine curve. Show that
v
P
(
A
(
x
)
−
yB
(
x
)) is equal to
e
x
P
)
e
(
A
(
x
)
2
F
(
x
)
B
(
x
)
2
).
where (
x
−
+
H
(
x
)
A
(
x
)
B
(
x
)
−
We now describe a polynomial that will be crucial for arithmetic on hyperelliptic curves
with a split model. Essentially,
G
+
(
x
) is a function that cancels the pole of
y
at
∞
+
.This
∞
+
for these models.
leads to another choice of uniformiser at
Exercise 10.1.22
Let
C
:
y
2
+
H
(
x
)
y
=
F
(
x
) be a hyperelliptic curve with split model
of genus
g
.Let
α
+
,α
−
∈ k
be the roots of
Y
2
over
k
+
H
d
Y
−
F
2
d
. Show that there exists
a polynomial
G
+
(
x
)
α
+
x
d
1 such that deg(
G
+
(
x
)
2
=
+···∈k
[
x
]ofdegree
d
=
g
+
+
H
(
x
)
G
+
(
x
)
−
≤
−
=
g
. Similarly, show that there is a polynomial
G
−
(
x
)
=
F
(
x
))
d
1
α
−
x
d
+···
such that deg(
G
−
(
x
)
2
+
H
(
x
)
G
−
(
x
)
−
≤
−
=
F
(
x
))
d
1
g
. Indeed, show that
G
−
(
x
)
G
+
(
x
)
=−
−
H
(
x
).
Exercise 10.1.23
Let
C
:
y
2
+
H
(
x
)
y
=
F
(
x
) be a hyperelliptic curve with split model
of genus
g
and let
G
+
(
x
)beasinExercise
10.1.22
. Show that
v
∞
+
(
y
G
+
(
x
))
over
k
−
≥
1.
10.1.3 The genus of a hyperelliptic curve
In Lemma
10.1.6
and Lemma
10.1.8
we showed that some hyperelliptic equations
y
2
F
(
x
) with
deg(
F
(
x
))
<
deg(
f
(
x
)) or deg(
H
(
x
))
<
deg(
h
(
x
)). Hence, it is natural to suppose that
the geometry of the curve
C
imposes a lower bound on the degrees of the polynomials
H
(
x
) and
F
(
x
) in its curve equation. The right measure of the complexity of the geometry
is the genus.
Indeed, the Riemann-Roch theorem implies that if
C
is a hyperelliptic curve over
+
h
(
x
)
y
=
f
(
x
) are birational to hyperelliptic equations
y
2
+
H
(
x
)
y
=
k
of genus
g
and there is a function
x
∈ k
(
C
)ofdegree2then
C
is birational over
k
to
an equation of the form
y
2
+
H
(
x
)
y
=
F
(
x
) with deg(
H
(
x
))
≤
g
+
1 and deg(
F
(
x
))
≤
2. Furthermore, the Hurwitz genus formula shows that if
y
2
2
g
+
+
H
(
x
)
y
=
F
(
x
)is
