Cryptography Reference
In-Depth Information
#Aut(
E
)
=
12
if j
(
E
)
=
0
and
char(
k
)
=
3
,
#Aut(
E
)
=
24
if j
(
E
)
=
0
and
char(
k
)
=
2
.
(Note that when
char(
k
)
=
2
or
3
then
0
=
1728
in
k
.)
Proof
See Theorem III.10.1 and Proposition A.1.2 of [
505
].
Exercise 9.4.5
Con
si
der
E
:
y
2
x
3
over
∈ F
2
satisfy
u
3
+
y
=
F
2
.Let
u
=
1,
s
∈ F
2
satisfy
s
4
∈ F
2
satisfy
t
2
s
6
. Show that
u,s
+
s
=
0 and
t
+
t
=
∈ F
2
2
,
t
∈ F
2
4
and that
(
u
2
x
s
2
,y
u
2
sx
φ
(
x,y
)
=
+
+
+
t
)
is an automorphism of
E
. Show that every automorphism arises this way and so #Aut(
E
)
=
Aut(
E
) then either
φ
2
1or
φ
3
24. Show that if
φ
∈
=±
=±
1. Show that Aut(
E
) is non-
Abelian.
9.5 Twists
Twists of elliptic curves have several important applications such as point compression
in pairing-based cryptography, and efficient endomorphisms on elliptic curves (see Exer-
cise
11.3.24
).
.A
t
wi
st
of
E
is an elliptic curve
E
over
Definition 9.5.1
Let
E
be an elliptic curve over
k
→
E
over
k
such that there is an isomorphism
φ
:
E
k
of pointed curves (i.e., such that
=
O
E
). Two twists
E
1
and
E
2
of
E
are
equivalent
if there is an isomorphism from
E
1
to
E
2
defined over
φ
(
O
E
)
. A twist
E
of
E
is called a
trivial twist
if
E
is equivalent to
E
.
Denote by Twist(
E
) the set of equivalence classes of twists of
E
.
k
2. Let
E
:
y
2
x
3
Example 9.5.2
Let
k
be a field such that char(
k
)
=
=
+
a
4
x
+
a
6
over
k
∈ k
∗
. Define the elliptic curve
E
(
d
)
:
y
2
=
x
3
+
d
2
a
4
x
+
d
3
a
6
.Themap
φ
(
x,y
)
=
and let
d
(
dx,d
3
/
2
y
) is an iso
m
orphism from
E
to
E
(
d
)
. Hence,
E
(
d
)
is a twist of
E
. Note that
E
(
d
)
is a trivial twist if
√
d
∈ k
∗
.
then there are infinitely many non-equivalent twists
E
(
d
)
, since one can let
d
run over the square-free elements in
If
k = Q
N
.
Exercise 9.5.3
Let
q
be an odd prime power and let
E
:
y
2
x
3
=
+
a
4
x
+
a
6
over
F
q
.Let
∈ F
q
. Show that the twist
E
(
d
)
d
of
E
by
d
is not isomorphic over
F
q
to
E
if and only if
d
is a non-square (i.e., the equation
u
2
=
d
has no solution in
F
q
). Show also that if
d
1
and
d
2
F
q
then
E
(
d
1
)
and
E
(
d
2
)
are isomorphic over
are non-squares in
F
q
. Hence, there is a unique
F
q
-isomorphism class of elliptic curves arising in this way. Any curve in this isomorphism
class is called a
quadratic twist
of
E
.
Exercise 9.5.4
Show that if
E
:
y
2
x
3
=
+
a
4
x
+
a
6
over
F
q
has
q
+
1
−
t
points then a
quadratic twist of
E
has
q
+
1
+
t
points over
F
q
.
