Cryptography Reference
In-Depth Information
9.7 The invariant differential
Let
E
over
k
be an elliptic curve. Recall the differential
dx
ω
E
=
(9.10)
2
y
+
a
1
x
+
a
3
on the Weierstrass equation for
E
, which was studied in Example
8.5.32
. We showed that
the divisor of
ω
E
is 0. Let
Q
∈
k
) and
τ
Q
be the translation map. Then
τ
Q
(
ω
E
)
∈
E
(
k
(
E
)
and so, by Theorem
8.5.21
,
τ
Q
(
ω
E
)
=
∈ k
fω
E
for some
f
(
E
). Lemma
8.5.36
implies
τ
Q
(div(
ω
E
))
0. It follows that
τ
Q
(
ω
E
)
∈ k
∗
.The
=
0 and so div(
f
)
=
=
cω
E
for some
c
following result shows that
c
1 and so
ω
E
is fixed by translation maps. This explains
why
ω
E
is called the
invariant differential
.
=
Theorem 9.7.1
Let E be an elliptic curve in Weierstra
ss
form and let ω
E
be the differential
in equation (
9.10
). Then τ
Q
(
ω
E
)
=
ω
E
for all Q
∈
E
(
k
)
.
Proof
See Proposition III.5.1 of [
505
].
An important fact is that the action of isogenies on differentials is linear.
Theorem 9.7.2
Let E,E be elliptic curves over
and ω
E
the invariant differential on E.
k
→
E are isogenies. Then
Suppose φ,ψ
:
E
ψ
)
∗
(
ω
E
)
φ
∗
(
ω
E
)
ψ
∗
(
ω
E
)
.
(
φ
+
=
+
Proof
See Theorem III.5.2 of [
505
].
A crucial application is to determine when certain isogenies are separable. In particular,
if
E
is an elliptic curve over
F
p
n
then [
p
] is inseparable on
E
, while
π
p
n
−
1 is separable
(where
π
p
n
is the
p
n
-power Frobenius).
Corollary 9.7.3
Let E be an elliptic curve over
k
. Let m
∈ Z
. Then
[
m
]
is separable if
and only if m is coprime to the characteristic of
k
. Let
k = F
q
and π
q
be the q-power
Frobenius. Let m,n
∈ Z
. Then m
+
nπ
q
is separable if and only if m is coprime to q.
Proof
(Sketch) Theorem
9.7.2
implies [
m
]
∗
(
ω
E
)
mω
E
.So[
m
]
∗
maps
k
(
E
)to
=
{
0
}
if
and only if the characteristic of
divides
m
. The first part then follows from Lemma
8.5.35
.
The second part follows by the same argument, using the fact that
π
q
is inseparable and so
π
q
(
ω
E
)
k
=
0. For full details see Corollaries III.5.3 to III.5.5 of [
505
].
This result has the following important consequence.
Theorem 9.7.4
Let E and E be elliptic curves over a finite field
→
E is an
F
q
.Ifφ
:
E
#
E
(
isogeny over
F
q
then
#
E
(
F
q
)
=
F
q
)
.
Proof
Let
π
q
be the
q
-power Frobenius map on
E
.For
P
∈
E
(
F
q
)wehave
π
q
(
P
)
=
P
if
and only if
P
∈
E
(
F
q
). Hence,
E
(
F
q
)
=
ker(
π
q
−
1). Since
π
q
−
1 is separable it follows
that #
E
(
F
q
)
=
deg(
π
q
−
1).
