Cryptography Reference
In-Depth Information
Chapter 3
Torsion Points
The torsion points, namely those whose orders are finite, play an important
role in the study of elliptic curves. We'll see this in Chapter 4 for elliptic
curves over finite fields, where all points are torsion points, and in Chapter
8, where we use 2-torsion points in a procedure known as descent. In the
present chapter, we first consider the elementary cases of 2- and 3-torsion,
then determine the general situation. Finally, we discuss the important Weil
and Tate-Lichtenbaum pairings.
3.1 Torsion Points
Let
E
be an elliptic curve defined over a field
K
.Let
n
be a positive integer.
We are interested in
E
[
n
]=
{P ∈ E
(
K
)
| nP
=
∞}
(recall that
K
= algebraic
cl
osure of
K
). We emphasize that
E
[
n
]contains
points with coordinates in
K
, not just in
K
.
When the characteristic of
K
is not 2,
E
can be put in the form
y
2
= cubic,
and it is easy to determine
E
[2]. Let
y
2
=(
x − e
1
)(
x − e
2
)(
x − e
3
)
,
with
e
1
,e
2
,e
3
∈
if and only if the tangent line
at
P
is vertical. It is easy to see that this means that
y
=0,so
K
.Apoint
P
satisfies 2
P
=
∞
E
[2] =
{∞
,
(
e
1
,
0)
,
(
e
2
,
0)
,
(
e
3
,
0)
}
.
As an abstract group, this is isomorphic to
Z
2
⊕
Z
2
.
The situation in characteristic 2 is more subtle. In Section 2.8 we showed
that
E
can be assumed to have one of the following two forms:
y
2
+
xy
+
x
3
+
a
2
x
2
+
a
6
=0
y
2
+
a
3
y
+
x
3
+
a
4
x
+
a
6
=0
.
(
I
)
or
(
II
)
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