Cryptography Reference
In-Depth Information
Using (9.35), (9.37), (9.38), and (9.39), we find that (9.34) equals
⎛
⎞
8
∞
⎝
⎠
q
N
.
d
−
d
−
d
(9.40)
N
=1
d
|
N, N/d
odd
d
|
N, d
odd
d
|
N, N/d
even
We claim that for all
N
≥
1,
d
−
d
−
d
=0
.
d
|
N, N/d
odd
d
|
N, d
odd
d
|
N, N/d
even
Write
N
=2
a
u
with
a ≥
0and
u
odd. Then
d
=
d
1
2
a
d
1
d
|
N, N/d
odd
|
u
d
=
d
d
d
|
N, d
odd
|
u
d
=
d
2
|
d
2
.
d
|
N, N/d
even
2
a−
1
u
If
a
= 0, the last sum is interpreted to be 0. In this case, the claim is easily
seen to be true. If
a
1, then the divisors of 2
a−
1
u
are of the form 2
j
d
3
with
≥
0
≤
j
≤
a
−
1and
d
3
|
u
. Therefore,
d
2
=
d
3
−
1)
d
3
(1+2+2
2
+
···
+2
a−
1
)
d
3
=(2
a
d
3
.
2
a−
1
u
d
2
|
|
u
|
u
The claim follows easily. This completes the proof of the lemma.
Since
C
= 0, the proof of the proposition is complete.
Example 9.3
Consider the curve
E
:
y
2
=
x
3
−
58347
x
+ 3954150
.
We have 4
A
2
+27
B
2
372386507784192, which factors as 2
18
3
17
11, al-
though we do not need this factorization. Since 11 divides this number, we
skip 11 and start with
p
1
= 13. The number of points in
E
(
F
13
) is 10. The
number of points in
E
(
F
17
) is also 10. Either of these facts implies that the
number of torsion points in
E
(
Q
) divides 10. Using the AGM, we calculate
=
−
ω
1
=
i
0
.
156713
...,
ω
2
=0
.
198602
···
.
This yields
τ
=
i
0
.
78908
···
and
q
=0
.
0070274
···
.Wecalculate
℘
(
ω
2
/
10) = 2539
.
82553
...,
Search WWH ::
Custom Search