Cryptography Reference
In-Depth Information
describe the line
ax
+
by
+
cz
=0in
P
2
K
. (It is easy to see that the
points (
x
:
y
:
z
) lie on the line. The main point is that each point on
the line corresponds to a pair (
u, v
).)
(a) Put the Legendre equation
y
2
=
x
(
x −
1)(
x − λ
) into Weierstrass
form and use this to show that the
j
-invariant is
2.13
j
=2
8
(
λ
2
− λ
+1)
3
λ
2
(
λ −
1)
2
.
(b) Show that if
j
=0
,
1728 then there are six distinct values of
λ
giving this
j
,andthatif
λ
is one such value then the full set is
{λ,
1
1
λ
1
,
λ
−
1
λ
,
1
− λ,
λ
,
}.
1
−
λ
−
λ
(c) Show that if
j
= 1728 then
λ
=
−
1
,
2
,
1
/
2, and if
j
=0then
λ
2
− λ
+1=0.
2.14 Consider the equation
u
2
− v
2
= 1, and the point (
u
0
,v
0
)=(1
,
0).
(a) Use the method of Section 2.5.4 to obtain the parameterization
u
=
m
2
+1
m
2
2
m
m
2
−
1
,
v
=
−
1
.
(b) Show that the projective curve
u
2
− v
2
=
w
2
has two points at
infinity, (1 : 1 : 0) and (1 :
−
1:0).
(c) The parameterization obtained in (a) can be written in projective
coordinates as (
u
:
v
:
w
)=(
m
2
+1:2
m
:
m
2
−
1) (or (
m
2
+
n
2
:
2
mn
:
m
2
− n
2
) in a homogeneous form). Show that the values
m
=
±
1 correspond to the two points at infinity. Explain why this
is to be expected from the graph (using real numbers) of
u
2
−v
2
=1.
(
Hint:
Where does an asymptote intersect a hyperbola?)
2.15 Suppose (
u
0
,v
0
,w
0
)=(
u
0
,
0
,
0) lies in the intersection
au
2
+
bv
2
=
e,
cu
2
+
dw
2
=
f.
(a) Show that the procedure of Section 2.5.4 leads to an equation of
the form “square = degree 2 polynomial in
m
.”
(b) Let
F
=
au
2
+
bv
2
=
e
and
G
=
cu
2
+
dw
2
=
f
. Show that the
Jacobian matrix
F
u
F
v
F
w
at (
u
0
,
0
,
0) has rank 1. Since the
G
u
G
v
G
w
rank is less than 2, this means that the point is a singular point.
2.16 Show that the cubic equation
x
3
+
y
3
=
d
can be transformed to the
elliptic curve
y
1
=
x
1
−
432
d
2
.
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