Cryptography Reference
In-Depth Information
12.3 Velu's Formulas
We now consider the algebraic version of Proposition 12.4. Since it is often
convenient to translate a point in the kernel of an isogeny to the origin, for
example, we work with the general Weierstrass form. The explicit formulas
given in the theorem are due to Velu [123].
THEOREM 12.16
Let
E
be an elliptic curve given by the generalized W eierstra ss equ a tion
y
2
+
a
1
xy
+
a
3
y
=
x
3
+
a
2
x
2
+
a
4
x
+
a
6
,
withall
a
i
insomefie d
K
.Let
C
be a finitesubgroupof
E
(
K
)
.Thenthere
existsanellipticcurve
E
2
and a separableisogeny
α
fro m
E
to
E
2
su ch that
C
=Ker
α
.
For a point
Q
=(
x
Q
,y
Q
)
∈ C
with
Q
=
∞
,define
g
Q
=3
x
2
Q
+2
a
2
x
Q
+
a
4
− a
1
y
Q
g
Q
=
−
2
y
Q
− a
1
x
Q
− a
3
v
Q
=
g
Q
(if 2
Q
=
∞
)
2
g
Q
− a
1
g
Q
(if 2
Q
=
∞
)
u
Q
=(
g
Q
)
2
.
Let
C
2
be the pointsoforder2in
C
. C hoose
R ⊂ C
su ch that w e have a
disjointunion
C
=
{∞} ∪ C
2
∪ R ∪
(
−R
)
(inother w ords, for each pair o f n o n -2 -torsion points
P,−P ∈ C
,putexactly
one of them in
R
). L et
S
=
R ∪ C
2
.Set
v
=
Q
w
=
Q
v
Q
,
(
u
Q
+
x
Q
v
Q
)
.
∈
S
∈
S
Then
E
2
has the equation
Y
2
+
A
1
XY
+
A
3
Y
=
X
3
+
A
2
X
2
+
A
4
X
+
A
6
,
where
A
1
=
a
1
,
A
2
=
a
2
,
A
3
=
a
3
(
a
1
+4
a
2
)
v
A
4
=
a
4
−
5
v,
A
6
=
a
6
−
−
7
w.
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