Cryptography Reference
In-Depth Information
x
Q
)
2
. One now con-
Hence,
x
P
+
Q
−
x
Q
=
F
x
(
Q
)
/
(
x
P
−
x
Q
)
+
(
y
P
−
y
Q
)
F
y
(
Q
)
/
(
x
P
−
siders two cases: when [2]
Q
=
O
E
then
F
y
(
Q
)
=
0. When [2]
Q
=
O
E
then it is convenient
to consider
x
P
+
Q
−
x
Q
+
x
P
−
Q
−
x
−
Q
.
Now,
x
−
Q
=
x
Q
,
y
−
Q
=
y
Q
+
F
y
(
Q
),
F
x
(
−
Q
)
=
F
x
(
Q
)
−
a
1
F
y
(
Q
)
and
F
y
(
−
Q
)
=
−
F
y
(
Q
). The formula for
φ
1
(
x
) follows.
Now we sketch how to obtain the formula for the
Y
-coordinate of the isogeny in the
case char(
+
Q
[
t
(
Q
)
/
(
x
x
Q
)
2
] and so
k
)
=
2. Note that
φ
1
(
x
)
=
x
−
x
Q
)
+
u
(
Q
)
/
(
x
−
−
Q
[
t
(
Q
)
/
(
x
φ
1
(
x
)
=
x
Q
)
2
x
Q
)
3
]. Using
φ
3
(
x
)
1
−
+
2
u
Q
/
(
x
−
=
(
−
A
1
φ
1
(
x
)
−
A
3
+
a
3
)
φ
1
(
x
)
)
/
2 one computes
(
a
1
x
+
yφ
1
(
x
)
+
yφ
2
(
x
)
+
φ
3
(
x
)
=
φ
3
(
x
)
x
Q
)
3
1
x
Q
)
2
−
=
y
−
t
(
Q
)
/
(
x
−
+
2
u
(
Q
)
/
(
x
−
(
a
1
x
+
a
3
)
/
2
Q
a
1
Q
x
Q
)
2
x
Q
)
3
]
−
[
t
(
Q
)
/
(
x
−
+
2
u
(
Q
)
/
(
x
−
+
(
a
1
x
+
a
3
)
/
2
a
3
)
Q
x
Q
)
2
x
Q
)
3
]
+
(
a
1
x
+
[
−
t
(
Q
)
/
(
x
−
−
2
u
(
Q
)
/
(
x
−
t
(
Q
)
y
+
a
1
(
x
−
x
Q
)
−
y
Q
u
(
Q
)
2
y
+
a
1
x
+
a
3
=
y
−
+
(
x
−
x
Q
)
2
(
x
−
x
Q
)
3
Q
.
t
(
Q
)((
a
1
x
Q
+
a
3
)
/
2
+
y
Q
)
+
a
1
u
(
Q
)
/
2
+
(
x
−
x
Q
)
2
It
suffices
to
show
that
the
numerator
of
the
final
term
in
the
sum
is
equal
to
a
1
u
(
Q
)
−
F
x
(
Q
)
F
y
(
Q
). However, this follows easily by noting that (
a
1
x
Q
+
a
3
)
/
2
+
y
Q
=
F
y
(
Q
)
2
−
F
y
(
Q
)
/
2,
u
(
Q
)
=
and using the facts that
F
y
(
Q
)
=
0 when [2]
Q
=
O
E
and
=
−
t
(
Q
)
2
F
x
(
Q
)
a
1
F
y
(
Q
) otherwise.
Co
r
ollary 25.1.7
Let E be an elliptic curve defined over
k
and G a finite subgroup of
. Then there is an elliptic curve E
E
(
k
)
that is defined over
k
=
E/G defined over
k
and
→
E defined over
an isogeny φ
:
E
k
with
ker(
φ
)
=
G.
Proof
It suffices to show that the values
t
(
G
),
w
(
G
) and the rational functions
X
and
Y
in
Theorem
25.1.6
are fixed by any
σ
∈
Gal(
k
/
k
).
→
E be a separable isogeny of odd degree between
Corollary 25.1.8
Let φ
:
E
elliptic curves over
k
. Write φ
(
x,y
)
=
(
φ
1
(
x
)
,φ
2
(
x,y
))
, where φ
1
(
x
)
and φ
2
(
x,y
)
are
u
(
x
)
/v
(
x
)
2
, where
deg(
u
(
x
))
rational functions. Then φ
1
(
x,y
)
=
=
and
deg(
v
(
x
))
=
w
2
(
x
))
/v
(
x
)
3
, where
deg(
w
1
(
x
))
(
−
1)
/
2
.Also,φ
2
(
x,y
)
=
(
yw
1
(
x
)
+
≤
3(
−
1)
/
2
and
≤
−
deg(
w
2
(
x
))
(3
1)
/
2
.
Exercise 25.1.9
Prove Corollary
25.1.8
.
