Cryptography Reference
In-Depth Information
to obtain
ω
2
=2
I
(
√
e
3
−
e
1
,
√
e
3
−
e
2
)
π
M
(
√
e
3
− e
1
,
√
e
3
− e
2
)
.
=
The proof of the formula for
ω
1
uses similar reasoning to obtain
e
2
K
(
1
− k
2
)
2
i
√
e
3
−
e
1
+
√
e
3
−
ω
1
=
2
i
√
e
3
− e
1
+
√
e
3
− e
2
I
(1
,k
)
=2
iI
(
√
e
3
− e
1
+
√
e
3
− e
2
,
√
e
3
− e
1
−
√
e
3
− e
2
)
.
=
If we let
a
=
√
e
3
− e
1
+
√
e
3
− e
2
,
b
=
√
e
3
− e
1
−
√
e
3
− e
2
,
then (9.20) yields
a
1
=
√
e
3
− e
1
,
1
=
√
e
2
− e
1
.
Proposition 9.24 therefore implies that
ω
1
=2
iI
(
√
e
3
− e
1
,
√
e
2
− e
1
)
πi
M
(
√
e
3
−
e
1
,
√
e
2
−
=
e
1
)
.
Example 9.2
Consider the elliptic curve
E
given by
y
2
=4
x
3
−
4
x.
Then
e
1
=
−
1
,e
2
=0
,e
3
=1,so
πi
M
(
√
2
,
1)
=
i
2
.
62205755429211981046483959
...
ω
1
=
π
M
(
√
2
,
1)
=2
.
62205755429211981046483959
....
ω
2
=
Therefore, the fundamental parallelogram for the lattice is a square. This also
follows from the fact that
E
has complex multiplication by
Z
[
i
]. See Chapter
10. The number 2
.
622
...
can be shown (see Exercise 9.8) to equal
1
dx
√
1
− x
4
=
Γ(1
/
4)Γ(1
/
2)
2Γ(3
/
4)
,
−
1
where Γ is the gamma function (for its definition, see Section 14.2). This is
a special case of the Chowla-Selberg formula, which expresses the periods of
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