Cryptography Reference
In-Depth Information
to get
∗
)
→ k
∗
→
(
C
)
∗
)
G
Prin
k
(
C
)
G
H
1
(
G,
H
1
(
G,
(
C
)
∗
)
k
→
→
k
→
k
1
(
∗
)
.
H
1
(
G,
Prin
H
2
(
G,
→
→
k
(
C
))
k
∗
)
(
C
)
∗
)
G
and
H
1
(
G,
Since
(
k
= k
(
C
)
(Theorem
7.
8.
3
)
k
=
0
(Hilbert
90)
we
have
∗
)
(
C
)
G
Prin
k
(
C
). Further,
H
2
(
G,
Prin
=
k
=
0 when
k
is finite (see Section X.7 of [
488
])
k
and
H
1
(
G,
(
C
)
∗
)
0 (see Silverman Exercise X.10). Hence,
H
1
(
G,
Prin
k
=
(
C
))
=
0.
k
Now, take Galois cohomology of the exact sequence
Div
0
k
Pic
0
k
→
→
→
→
1
Prin
(
C
)
(
C
)
(
C
)
0
k
to get
Div
0
k
(
C
)
G
Pic
0
k
(
C
)
G
H
1
(
G,
Prin
k
(
C
))
Prin
k
(
C
)
→
→
→
=
0
.
Now, Div
0
k
(
C
)
G
Div
0
k
=
(
C
) by definition and so the result follows.
We minimise the use of the word Jacobian in this topic; however, we make a few
remarks here. We have associated to a curve
C
over a field
k
the divisor class group
Pic
0
k
(
C
). This group can be considered as an algebraic group. To be precise there is a
variety
J
C
(called the
Jacobian variety
of
C
) that is an algebraic group (i.e., there is a
morphism
6
+
:
J
C
×
J
C
→
J
C
) and such that, for any extension
K
/
k
, there is a bijective
map between Pic
0
K
K
(
C
) and
J
C
(
) that is a group homomorphism.
One can think of Pic
0
k
as a functor that, given a curve
C
over
, associates with every
k
/
a group Pic
0
field extension
k
k
(
C
). The Jacobian variety of the curve is a variety
J
C
k
-rational points
J
C
(
k
) are in one-to-one correspondence with the elements
over
k
whose
of Pic
0
k
/
. For most applications it is sufficient to work in the language of
divisor class groups rather than Jacobians (despite our remarks about algebraic groups in
Chapter
4
).
k
(
C
) for all
k
7.9 Elliptic curves
The goal of this section is to show that the 'traditional'
chord-and-tangent rule
for elliptic
curves does give a group operation. Our approach is to show that this operation coincides
with addition in the divisor class group of an elliptic curve. Hence, elliptic curves are an
algebraic group.
First we state the chord-and-tangent rule without justifying any of the claims or assump-
tions made in the description. The results later in the section will justify these claims (see
Remark
7.9.4
). For more details about the chord-and-tangent rule see Washington [
560
],
Cassels [
114
], Reid [
447
] or Silverman and Tate [
508
].
Let
P
1
=
(
x
1
,y
1
) and
P
2
=
(
x
2
,y
2
) be points on the affine part of an elliptic curve
E
.
Draw the line
l
(
x,y
)
=
0 between
P
1
and
P
2
(if
P
1
=
P
2
then this is called a chord; if
6
To make this statement precise requires showing that
J
C
×
J
C
is a variety.
