Cryptography Reference
In-Depth Information
By (8.4),
4
j
(
h
(2
j
P
)
−
4
h
(2
j−
1
P
))
≤
c
1
4
j
,
so the infinite sum converges. Therefore,
h
(
P
) exists. Since
1
∞
c
1
4
j
=
c
1
3
,
j
=1
h
(
P
)
−
2
h
(
P
)
|≤c
1
/
6. It is clear from the definitions that
h
(
P
)
≥
0
1
we obtain
|
for all
P
.
Proofofpart(3):
If
h
(
P
)
≤ c
,then
h
(
P
)
≤
2
c
+
c
3
. There are only finitely
many
P
satisfying this inequality.
P roof of part (5):
We have
1
c
1
4
n
|h
(2
n
P
+2
n
Q
)+
h
(2
n
P −
2
n
Q
)
−
2
h
(2
n
P
)
−
2
h
(2
n
Q
)
|≤
4
n
.
Letting
n →∞
yields the result.
Proofofpart(4):
Since the height depends only on the
x
-coordinate,
h
(
P
)=
h
(
P
).
0. The cases
m
=0
,
1
are trivial. Letting
Q
=
P
in part (5) yields the case
m
= 2. Assume that we
know the result for
m −
1and
m
.Then
h
((
m
+1)
P
)=
−
Therefore, we may assume
m
≥
h
((
m
1)
P
)+2
h
(
mP
)+2
h
(
P
) (by part (5))
−
−
=
−
1)
2
+2
m
2
+2
h
(
P
)
=(
m
+1)
2
h
(
P
)
.
(
m
−
By induction, the result is true for all
m
.
Proofofpart(6):
If
mP
=
∞
,then
m
2
h
(
P
)=
h
(
mP
)=
h
(
∞
)=0,so
h
(
P
) = 0. Conversely, if
h
(
P
)=0,then
h
(
mP
)=
m
2
h
(
P
) = 0 for all
m
.
Since there are only finitely many points of height 0, the set of multiples
of
P
is finite. Therefore,
P
is a torsion point. This completes the proof of
Theorem 8.18.
P roof of P roposition 8.19
. It remains to prove Proposition 8.19. It can be
restated as saying that there exist constants
c
,c
such that
c
≤
2
h
(
P
)+2
h
(
Q
)
−
h
(
P
+
Q
)+
h
(
P
−
Q
)
(8.6)
2
h
(
P
)+2
h
(
Q
)+
c
h
(
P
+
Q
)+
h
(
P
−
Q
)
≤
(8.7)
for all
P, Q
. These two inequalities will be proved separately. We'll start with
the second one.
Let the elliptic curve
E
be given by
y
2
=
x
3
+
Ax
+
B
with
A, B ∈
Z
.Let
P
=(
a
1
Q
=(
a
2
b
1
,y
1
)
,
b
2
,y
2
)
,
P
+
Q
=(
a
3
P− Q
=(
a
4
b
3
,y
3
)
,
b
4
,y
4
)
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