Cryptography Reference
In-Depth Information
Exercise 10.2.7
Let
p>
2 be a prime and
C
:
y
2
x
n
=
p
the equation is singular). Show that the subgroup of Aut(
C
) consisting of automorphisms
that fix infinity has order 2
n
.
=
+
1 over
F
p
with
n
=
p
(when
n
Exercise 10.2.8
Let
p
F
p
. Write
ζ
8
∈ F
p
for
a primitive 8th root of unity. Show that
ζ
8
∈ F
p
4
. Show that
ψ
(
x,y
)
≡
1(mod8)andlet
C
:
y
2
=
x
5
+
Ax
over
(
ζ
8
x,ζ
8
y
)isan
=
automorphism of
C
. Show that
ψ
4
=
ι
.
10.3 Effective affine divisors on hyperelliptic curves
This section is about how to represent effective divisors on affine hyperelliptic curves, and
algorithms to compute with them. A convenient way to represent divisors is using Mumford
representation, and this is only possible if the divisor is semi-reduced.
2
Definition 10.3.1
Let
C
be a hyperelliptic curve over
k
and denote by
C
∩ A
the affine
curve. An
effective affine divisor
on
C
is
D
=
n
P
(
P
)
P
∈
(
C
∩A
2
)(
k
)
where
n
P
≥
0 for only finitely many
P
).
A
divisor on
C
is
semi-
reduced
if it is an effective affine divisor and for all
P
0 (and, as always,
n
P
=
2
)(
∈
(
C
∩ A
k
)wehave:
1. If
P
=
ι
(
P
) then
n
P
=
1.
2. If
P
=
ι
(
P
) then
n
P
>
0 implies
n
ι
(
P
)
=
0.
2
.
We slightly adjust the notion of equivalence for divisors on
C
∩ A
Definition 10.3.2
Let
C
be a hyperelliptic curve over a field
k
and let
f
∈ k
(
C
). We define
2
div(
f
)
∩ A
=
v
P
(
f
)(
P
)
.
P
∈
(
C
∩A
2
)(
k
)
Two
d
ivisors
D,D
on
C
2
D
, if there is some function
∩ A
are
equivalent
, written
D
≡
D
+
2
.
f
∈ k
(
C
) such that
D
=
div(
f
)
∩ A
2
Lemma 10.3.3
Let C be a hyperelliptic curve. Every divisor on C
∩ A
is equivalent to a
semi-reduced divisor.
=
P
∈
C
∩A
Proof
Let
D
2
n
P
(
P
). By Exercise
10.1.19
the
f
unction
x
−
x
P
has divisor
2
.If
n
P
<
0forsome
P
2
)(
(
P
)
+
(
ι
(
P
)) on
C
∩ A
∈
(
C
∩ A
k
) then, by adding an appro-
priate multiple of div(
x
−
x
P
), one can arrange that
n
P
=
0 (this will increase
n
ι
(
P
)
).
Similarly, if
n
P
>
0 and
n
ι
(
P
)
>
0(orif
P
=
ι
(
P
) and
n
P
≥
2) then subtracting a mul-
tiple of div(
x
x
P
) lowers the values of
n
P
and
n
ι
(
P
)
. Repeating this process yields a
semi-reduced divisor.
−
