Cryptography Reference
In-Depth Information
Figure 2.1(a) on page 10. The points don't appear on the graph
because
y
is imaginary. For the curve in Figure 2.1(b) on page 10,
x
moves to the left along the
x
-axis, from the point on the
x
-axis back
to the point at infinity, corresponding to the fact that
ω
1
=
2
ω
2
+
it
for appropriate
t
(see Exercise 9.5).
9.7 Define the
elliptic integral of the second kind
to be
E
(
k
)=
1
0
√
1
k
2
x
2
√
1
− x
2
−
dx,
−
1
<k<
1
.
(a) Show that
E
(
k
)=
π/
2
0
(1
− k
2
sin
2
θ
)
1
/
2
dθ.
(b) Show that the arc length of the ellipse
x
2
a
2
+
y
2
b
2
=1
with
b ≥ a>
0equals4
bE
(
1
−
(
a/b
)
2
).
This connection with ellipses is the origin of the name “elliptic inte-
gral.” The relation between elliptic integrals and elliptic curves, as in
Section 9.4, is the origin of the name “elliptic curve.”
For more on
elliptic integrals, see [78].
9.8 Let
E
be the elliptic curve
y
2
=4
x
3
−
4
x
. Show that
ω
2
=
∞
1
1
dx
x
(
x
2
−
1)
=
1
t
−
3
/
4
(1
t
)
−
1
/
2
dt
=
β
(1
/
4
,
1
/
2)
,
−
2
0
where
β
(
p, q
)=
0
t
p−
1
(1
t
)
q−
1
dt
is the
beta function
. A classical
−
result says that
β
(
p, q
)=
Γ(
p
)Γ(
q
)
Γ(
p
+
q
)
.
Therefore,
ω
2
=
1
Γ(1
/
4)Γ(1
/
2)
Γ(3
/
4)
.
2
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