Cryptography Reference
In-Depth Information
Chapter 12
Isogenies
Isogenies, which are homomorphisms between elliptic curves, play a funda-
mental role in the theory of elliptic curves since they allow us to relate one
elliptic curve to another. In the first section, we describe the analytic theory
over the complex numbers. In subsequent sections, we obtain similar results
in the algebraic setting. Finally, we sketch how isogenies can be used to count
points on elliptic curves over finite fields.
12.1 The Complex Theory
Let
E
1
=
C
/L
1
and
E
2
=
C
/L
2
be elliptic curves over
C
.Let
α
∈
C
be
such that
αL
1
⊆
L
2
.Then
[
α
]:
E
1
−→
E
2
z
−→
αz
gives a homomorphism from
E
1
to
E
2
(we need
αL
1
⊆
L
2
to make the map
well-defined). A map of the form [
α
]with
α
= 0 is called an
isogeny
from
E
1
to
E
2
. If there exists an isogeny from
E
1
to
E
2
,wesaythat
E
1
and
E
2
are
isogenous
.
LEMMA 12.1
If
α
=0
,then
αL
1
is offiniteindex in
L
2
.
Let
{ω
(
k
)
,ω
(
k
)
PROOF
}
be a basis for
L
k
,for
k
=1
,
2. Write
1
2
αω
(1)
=
a
i
1
ω
(2)
+
a
i
2
ω
(2)
i
1
2
with
a
ij
∈
Z
.Ifdet(
a
ij
)=0then(
a
11
,a
12
) is a rational multiple of (
a
21
,a
22
),
which implies that
αω
(1)
is a rational multiple of
αω
(1)
. This is impossible
1
2
since
ω
(1)
and
ω
(1)
are linearly independent over
R
.
1
2
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