Cryptography Reference
In-Depth Information
C
B
A
A'
B'
C'
Figure 2.5
Pappus's Theorem
2.5 Other Equations for Elliptic Curves
In this topic, we are mainly using the Weierstrass equation for an elliptic
curve. However, elliptic curves arise in various other guises, and it is worth-
while to discuss these briefly.
2.5.1
Legendre Equation
This is a variant on the Weierstrass equation. Its advantage is that it
allows us to express all elliptic curves over an algebraically closed field (of
characteristic not 2) in terms of one parameter.
PROPOSITION 2.16
Let
K
be a fi eld of characteristicnot2and et
y
2
=
x
3
+
ax
2
+
bx
+
c
=(
x
−
e
1
)(
x
−
e
2
)(
x
−
e
3
)
be an ellipticcurve
E
over
K
with
e
1
,e
2
,e
3
∈ K
.Let
λ
=
e
3
− e
1
e
1
)
−
1
(
x
e
1
)
−
3
/
2
y,
x
1
=(
e
2
−
−
e
1
)
,
1
=(
e
2
−
e
2
− e
1
.
Then
λ
=0
,
1
and
y
1
=
x
1
(
x
1
−
1)(
x
1
−
λ
)
.
PROOF
This is a straightforward calculation.
Search WWH ::
Custom Search