Cryptography Reference
In-Depth Information
2.17
y
) is a group homomorphism from
E
to
itself, for any elliptic curve in Weierstrass form.
(b) Show that (
x, y
)
(a) Show that (
x, y
)
→
(
x,
−
y
), where
ζ
is a nontrivial cube root of
1, is an automorphism of the elliptic curve
y
2
=
x
3
+
B
.
(c) Show that (
x, y
)
→
(
−x, iy
), where
i
2
=
−
1, is an automorphism
of the elliptic curve
y
2
=
x
3
+
Ax
.
→
(
ζx,
−
2.18 Let
K
have characteristic 3 and let
E
be defined by
y
2
=
x
3
+
a
2
x
2
+
a
4
x
+
a
6
.The
j
-invariant in this case is defined to be
a
2
a
2
a
4
− a
2
a
6
− a
4
j
=
(this formula is false if the characteristic is not 3).
(a) Show that either
a
2
=0or
a
4
= 0 (otherwise, the cubic has a triple
root, which is not allowed).
(b) Show that if
a
2
= 0, then the change of variables
x
1
=
x −
(
a
4
/a
2
)
yields an equation of the form
y
1
=
x
1
+
a
2
x
1
+
a
6
. This means
that we may always assume that exactly one of
a
2
and
a
4
is 0.
(c) Show that if two elliptic curves
y
2
=
x
3
+
a
2
x
2
+
a
6
and
y
2
=
x
3
+
a
2
x
2
+
a
6
have the same
j
-invariant, then there exists
μ ∈ K
×
such that
a
2
=
μ
2
a
2
and
a
6
=
μ
6
a
6
.
(d) Show that if
y
2
=
x
3
+
a
4
x
+
a
6
and
y
2
=
x
3
+
a
4
x
2
+
a
6
are
two elliptic curves (in characteristic 3), the
n t
here is a c
han
ge of
variables
y
K
×
and
c
→
ay
,
x
→
bx
+
c
, with
a, b
∈
∈
K
,that
changes one equation into the other.
(e) Observe that if
a
2
=0then
j
=0andif
a
4
=0then
j
=
a
2
/a
6
.
Show that every element of
K
appears as the
j
-invariant of a curve
defined over
K
.
(f) Show that if two curves
ha
ve the same
j
-invariant then there is a
change of variables over
K
that changes one into the other.
−
2.19 Let
α
(
x, y
)=(
p
(
x
)
/q
(
x
)
,y
s
(
x
)
/t
(
x
)) be an endomorphism of the ellip-
tic curve
E
given by
y
2
=
x
3
+
Ax
+
B
,where
p, q, s, t
are polynomials
such that
p
and
q
have no common root and
s
and
t
have no common
root.
·
(a) Using the fact that (
x, y
)and
α
(
x, y
) lie on
E
, show that
(
x
3
+
Ax
+
B
)
s
(
x
)
2
t
(
x
)
2
u
(
x
)
q
(
x
)
3
=
for some polynomial
u
(
x
) such that
q
and
u
have no common root.
(
Hint:
Show that a common root of
u
and
q
must also be a root of
p
.)
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