Cryptography Reference
In-Depth Information
Definition 10.8.1
Let
k
be a field such that char(
k
)
=
p>
0 and let
C
be a curve of genus
g
such that #Pic
0
k
p
r
.
g
over
k
.The
p
-rank
of
C
is the integer 0
≤
r
≤
(
C
)[
p
]
=
An Abelian
v
ariety of dimension
g
over
F
q
is defined to be
su
pe
rsingular
if it is
F
q
to
E
g
where
E
is a supersingular elliptic curve over
isogenous over
F
q
. A curve
C
over
F
q
is
supersingular
if
J
C
is a supersingular Abelian variety. It follows that the
p
-rank of
a supersingular Abelian variety over
F
p
n
is zero. The converse is not true (i.e.,
p
-rank zero
does not imply supersingular) when the dimension is 3 or more; see Example
10.8.8
). If
the
p
-rank of a dimension
g
Abelian variety
A
over
F
p
n
is
g
then
A
is said to be
ordinary
.
F
q
and write P
A
(
T
)
f
or
the characteristic polynomial of Frobenius on A. The roots α of P
A
(
T
)
are such that α/
√
q
is a root of unity.
Lemma 10.8.2
Suppose A is a supersingular Abelian variety over
Proof
Since the isogeny to
E
g
is d
efin
ed over some finite extension
F
q
n
it follows from
part 4 of Theorem
9.11.2
that
α
n
/
√
q
n
is a root of unity. Hence,
α/
√
q
is a root of unity.
The converse of Lemma
10.8.2
follows from the Tate isogeny theorem.
Let
C
:
y
2
+
=
x
5
F
2
. One can check that #
C
(
F
2
)
=
Example 10.8.3
y
over
3 and
F
2
2
)
=
=
#
C
(
5 and so the characteristic polynomial of the 2-power Frobenius is
P
(
T
)
T
4
2). It follows from Theorem
10.7.13
(Tate's isogeny
theorem) that
J
C
is isogenous to
E
1
×
E
2
where
E
1
and
E
2
are supersingular curves
over
+
4
=
(
T
2
+
2
T
+
2)(
T
2
−
2
T
+
F
2
. The characteristic polynomial of the 2
2
-power Frobenius can be shown to be
T
4
8
T
2
(
T
2
4)
2
and it follows that
J
C
is isogenous over
+
+
16
=
+
F
2
2
to the square of a
supersingular elliptic curve. Hence,
C
is a supersingular curve.
Note that the endomorphism ring of
J
C
is non-commutative since the map
φ
(
x,y
)
=
(
ζ
5
x,y
), where
ζ
5
∈ F
2
4
is a root of
z
4
z
3
z
2
+
+
+
z
+
1
=
0, does not commute with the
2-power Frobenius map.
F
q
of genus 2 then #Pic
0
|
Exercise 10.8.4
Show that if
C
is a supersingular curve over
F
q
(
C
)
(
q
k
−
≤
≤
1) for some 1
k
12.
The following result shows that computing the
p
-rank and determining supersingularity
are easy when
P
(
T
)isknown.
Theorem 10.8.5
Let A be an Abelian variety of dimension g over
F
p
n
with characteristic
T
2
g
a
1
T
2
g
−
1
a
g
T
g
p
ng
.
polynomial of Frobenius P
(
T
)
=
+
+···+
+···+
1. The p-rank of A is the smallest integer
0
≤
r
≤
g such that p
|
a
i
for all
1
≤
i
≤
g
−
r.
(In other words, the p-rank is zero if p
|
a
i
for all
1
≤
i
≤
g and the p-rank is g if
a
1
.)
2. A is supersingular if and only if
p
p
in/
2
|
a
i
for all
1
≤
i
≤
g.