Cryptography Reference
In-Depth Information
9.6 Isogenies
We now return to more general maps between elliptic curves. Recall from Theorem
9.2.1
that a morphism
φ
:
E
1
→
=
O
E
2
is a group homo-
morphism. Hence, isogenies are group homomorphisms. Chapter
25
discusses isogenies
in further detail. In particular, Chapter
25
describes algorithms to compute isogenies
efficiently.
E
2
of elliptic curves such that
φ
(
O
E
1
)
Definition 9.6.1
Let
E
1
and
E
2
be elliptic curves over
k
.An
isogeny
over
k
is a morphism
φ
:
E
1
→
E
2
over
k
such that
φ
(
O
E
1
)
=
O
E
2
.Th
e
zero isogeny
is the constant map
φ
:
E
1
→
E
2
given by
φ
(
P
)
=
O
E
2
for all
P
∈
E
1
(
k
). If
φ
(
x,y
)
=
(
φ
1
(
x,y
)
,φ
2
(
x,y
)) is
an isogeny then define
(
X,Y
) denotes,
as
al
ways, the inverse for the group law. The
kernel
of an isogeny is ker(
φ
)
−
φ
by (
−
φ
)(
x,y
)
=−
(
φ
1
(
x,y
)
,φ
2
(
x,y
)). where
−
={
P
∈
E
1
(
.The
degree
of a non-zero isogeny is the degree of the morphism.
The degree of the zero isogeny is zero. If there is an isogeny (respectively, isogeny of
degree
d
) between two elliptic curves
E
1
and
E
2
then we say that
E
1
and
E
2
are
isogenous
(respectively,
d
-isogenous
). A non-zero isogeny is
separable
if it is separable as a morphism
(see Definition
8.1.6
). Denote by Hom
k
(
E
1
,E
2
) the set of isogenies from
E
1
to
E
2
defined
over
k
):
φ
(
P
)
=
O
E
2
}
k
. Denote by End
k
(
E
1
) the set of isogenies from
E
1
to
E
1
defined over
k
; this is called
the
endomorphism ring
of the elliptic curve.
Exercise 9.6.2
Show that if
φ
:
E
1
→
E
2
is an isogeny then so is
−
φ
.
Theorem 9.
6.3
Let E
1
an
d
E
2
be e
ll
iptic curves over
k
.Ifφ
:
E
1
→
E
2
is a non-zero
isogeny over
k
then φ
:
E
1
(
k
)
→
E
2
(
k
)
is surjective.
Proof
This follows from Theorem
8.2.4
.
We now relate the degree to the number of poi
n
ts in the kernel. First, we remark the
standard group theoretical fact that, for all
Q
), #
φ
−
1
(
Q
)
∈
E
2
(
k
=
#ker(
φ
) (this is just the
fact that all cosets have the same size).
Lemma 9.6.4
A non-zero separable isogeny φ
:
E
1
→
E
2
over
k
of degree d has
=
#ker(
φ
)
d.
Proof
It follows from Coro
lla
ry
8.2.10
that a separable degree
d
map
φ
has #
φ
−
1
(
Q
)
=
d
). Hence, by the above remark, #
φ
−
1
(
Q
)
for a generic point
Q
∈
E
2
(
k
=
d
for all points
Q
and # ker(
φ
)
=
d
. (Also see Proposition 2.21 of [
560
] for an elementary proof.)
A morphism of curves
φ
:
C
1
→
C
2
is called
unramified
if
e
φ
(
P
)
=
1 for all
P
∈
C
1
(
k
).
Let
φ
:
E
1
→
E
2
be a separable isogeny over
k
and let
P
∈
E
1
(
k
). Since
φ
(
P
)
=
φ
(
P
+
R
)
∈
for all
R
ker(
φ
) it follows that a separable morphism of elliptic curves is unramified (this
also follows from the Hurwitz genus formula).
Exercise 9.6.5
Let
E
1
and
E
2
be elliptic curves over
k
and suppose
φ
:
E
1
→
E
2
is an
isogeny over
k
. Show that ker(
φ
) is defined over
k
(in the sense that
P
∈
ker(
φ
) implies
σ
(
P
)
∈
ker(
φ
) for all
σ
∈
Gal(
k
/
k
)).
