Cryptography Reference
In-Depth Information
functions
R
1
,R
2
such that if
α
(
x
1
,y
1
)=(
x
2
,y
2
), then
x
2
=
R
1
(
x
1
,y
1
)
,
2
=
R
2
(
x
1
,y
1
)
for all but finitely many (
x
1
,y
1
)
E
1
(
K
). The technicalities for the points
where
R
1
and
R
2
are not defined are dealt with in the same way as for
endomorphisms, as in Section 2.9. In fact, when
E
1
=
E
2
, an isogeny is a
nonzero endomorphism.
As in Section 2.9, we may write
α
in the form
∈
(
x
2
,y
2
)=
α
(
x
1
,y
1
)=(
r
1
(
x
1
)
,y
1
r
2
(
x
1
))
,
where
r
1
,r
2
are rational functions. If the coe
cients of
r
1
,r
2
lie in
K
,wesay
that
α
is
defined over
K
.Write
r
1
(
x
)=
p
(
x
)
/q
(
x
)
with polynomials
p
(
x
)and
q
(
x
) that do not have a common factor. Define
the
degree
of
α
to be
deg(
α
)=Max
{
deg
p
(
x
)
,
deg
q
(
x
)
}
.
If the derivative
r
1
(
x
) is not identically 0, we say that
α
is
separable
.
PROPOSITION 12.8
Let
α
:
E
1
→ E
2
be an isogeny. If
α
is separable, then
deg
α
=#Ker(
α
)
.
If
α
is not separable, then
deg
α>
#Ker(
α
)
.
In pa rticular, the kernel of an isogeny is a finitesubgroupof
E
1
(
K
)
.
PROOF
The proof is identical to the proof of Proposition 2.21.
PROPOSITION 12.9
Let
α
:
E
1
→ E
2
be an isogeny. T hen
α
:
E
1
(
K
)
→ E
2
(
K
)
issurjective.
PROOF
The proof is identical to the proof of Theorem 2.22.
Example 12.2
Let
p
be an odd prime, let
A
1
,B
1
be in a field of characteristic
p
,andlet
E
1
:
y
1
=
x
1
+
A
1
x
1
+
B
1
and
E
2
:
y
2
=
x
2
+
A
1
x
2
+
B
1
. Define
φ
by
(
x
2
,y
2
)=
φ
(
x
1
,y
1
)=(
x
1
,y
1
)
.
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