Cryptography Reference
In-Depth Information
Since gcd(
a, b
)=1,wehave
d
|
4Δ, so
d
≤|
4Δ
|
.Since
H
(2
R
)=Max
|
,
h
1
|
d
|
h
2
|
d
,
we have
H
(
R
)
4
=
4
|
4Δ
|
|
4Δ
||
b
|
≤
C
1
Max(
|
h
1
|
,
|
h
2
|
)
Max(
|
h
1
|
d
|
h
2
|
d
≤
C
1
|
4Δ
|
,
)
≤
C
1
|
4Δ
|
H
(2
R
)
.
Dividing by
|
4Δ
|
and taking logs yields
4
h
(
R
)
≤ h
(2
R
)+
C
2
for some constant
C
2
, independent of
R
.
Thecasewhere
|
a
|≥|
b
|
is similar. This completes the proof of Lemma 8.22.
Changing
P
to
P
+
Q
and
Q
to
P
−
Q
in (8.9) yields
h
(2
P
)+
h
(2
Q
)
≤
2
h
(
P
+
Q
)+2
h
(
P − Q
)+
c
.
By Lemma 8.22,
4
h
(
P
)+4
h
(
Q
)
−
2
C
2
≤ h
(2
P
)+
h
(2
Q
)
.
Therefore,
c
≤
2
h
(
P
)+2
h
(
Q
)
−
h
(
P
+
Q
)+
h
(
P
−
Q
)
for some constant
c
. This completes the proof of Proposition 8.19.
8.4 Examples
The Mordell-Weil theorem says that if
E
is an elliptic curve defined over
Q
,then
E
(
Q
) is a finitely generated abelian group. The structure theorem
for such groups (see Appendix B) says that
E
(
Q
)
T ⊕
Z
r
,
where
T
is a finite group (the
torsion subgroup
)and
r ≥
0 is an integer,
called the
rank
of
E
(
Q
).
In Section 8.1, we showed how to compute
T
.
Search WWH ::
Custom Search