Cryptography Reference
In-Depth Information
Example 2.1
The calculations of Chapter 1 can now be interpreted as adding points on
elliptic curves. On the curve
y
2
=
x
(
x
+ 1)(2
x
+1)
6
,
we have
(0
,
0) + (1
,
1) = (
1
1
2
)
,
(
1
1
2
)+(1
,
1) = (24
,
2
,
−
2
,
−
−
70)
.
On the curve
y
2
=
x
3
−
25
x,
we have
2(
−
4
,
6) = (
−
4
,
6) + (
−
4
,
6) =
1681
.
62279
1728
144
, −
We also have
(0
,
0) + (
−
5
,
0) = (5
,
0)
,
2(0
,
0) = 2(
−
5
,
0) = 2(5
,
0) =
∞
.
The fact that the points on an elliptic curve form an abelian group is be-
hind most of the interesting properties and applications. The question arises:
what can we say about the groups of points that we obtain? Here are some
examples.
1. An elliptic curve over a finite field has only finitely many points with
coordinates in that finite field. Therefore, we obtain a finite abelian
group in this case. Properties of such groups, and applications to cryp-
tography, will be discussed in later chapters.
2. If
E
is an elliptic curve defined over
Q
,then
E
(
Q
) is a finitely generated
abelian group. This is the Mordell-Weil theorem, which we prove in
Chapter 8.
Such a group is isomorphic to
Z
r
⊕ F
for some
r ≥
0
and some finite group
F
. The integer
r
is called the
rank
of
E
(
Q
).
Determining
r
is fairly di
cult in general. It is not known whether
r
can be arbitrarily large. At present, there are elliptic curves known with
rank at least 28. The finite group
F
is easy to compute using the Lutz-
Nagell theorem of Chapter 8. Moreover, a deep theorem of Mazur says
that there are only finitely many possibilities for
F
,as
E
ranges over all
elliptic curves defined over
Q
.
3. An elliptic curve over the complex numbers
C
is isomorphic to a torus.
This will be proved in Chapter 9. The usual way to obtain a torus is as
C
/L
,where
L
is a lattice in
C
. The usual addition of complex numbers
induces a group law on
C
/L
that corresponds to the group law on the
elliptic curve under the isomorphism between the torus and the elliptic
curve.
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