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.
 
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