Cryptography Reference
In-Depth Information
Therefore,
D
=[
P
]
−
[
Q
]+div(
g
1
)
.
Also,
sum(
D
)=
P − Q
+ sum(div(
g
1
)) =
P − Q.
Suppose sum(
D
)=
∞
.Then
P − Q
=
∞
,so
P
=
Q
and
D
= div(
g
1
).
Conversely, suppose
D
=div(
f
) for some function
f
.Then
[
P
]
−
[
Q
]=div(
f/g
1
)
.
The following lemma implies that
P
=
Q
, and hence sum(
D
)=
∞
.This
completes the proof of Theorem 11.2.
LEMMA 11.
3
Let
P, Q ∈ E
(
K
)
and suppose there existsafunction
h
on
E
with
div(
h
)=[
P
]
−
[
Q
]
.
Then
P
=
Q
.
Since the proof is slightly long, we postpone it until the end of this section.
COROLLARY 11.4
Themap
sum : Div
0
(
E
)
(principal divisors)
−→ E
(
K
)
isanisom orphism of groups.
]) =
P
, the sum map from Div
0
(
E
)to
E
(
K
)is
surjective. The theorem says that the kernel is exactly the principal divisors.
PROOF
Since sum([
P
]
−
[
∞
Corollary 11.4 shows that the group law on
E
(
K
) corresponds to the very
natural group law on Div
0
(
E
) mod principal divisors.
Example 11.4
The proof of the theorem gives an algorithm for finding a function with a
given divisor (of degree 0 and sum equal to
∞
). Consider the elliptic curve
E
over
F
11
given by
y
2
=
x
3
+4
x.
Let
D
=[(0
,
0)] + [(2
,
4)] + [(4
,
5)] + [(6
,
3)]
−
4[
∞
]
.
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