Cryptography Reference
In-Depth Information
with
A, B
∈
R
an d assu m e there exists
r
∈
R
su ch that
(4
A
3
+27
B
2
)
r
−
1
∈
I.
Then the m ap
red
I
:
E
(
R
)
−→
E
(
R/I
)
(
x
:
y
:
z
)
→
(
x
:
y
:
z
)mod
I
is a group hom om orphism .
PROOF
The proof is the same as for Corollary 2.33, with
R
in place of
Z
and mod
I
in place of mod
n
. The condition that (4
A
3
+27
B
2
)
r −
1
∈ I
for some
r
is the requirement that 4
A
3
+27
B
2
is a unit in
R/I
,whichwas
required in the definition of an elliptic curve over the ring
R/I
.
Exercises
2.1
(a) Show that the constant term of a monic cubic polynomial is the
negative of the product of the roots.
(b) Use (a) to derive the formula for the sum of two distinct points
P
1
,P
2
in the case that the
x
-coordinates
x
1
and
x
2
are nonzero, as
in Section 2.2. Note that when one of these coordinates is 0, you
need to divide by zero to obtain the usual formula.
2.2 The point (3
,
5) lies on the elliptic curve
E
:
y
2
=
x
3
−
2, defined over
Q
. Find a point (not
∞
) with rational, nonintegral coordinates in (
Q
).
2.3 The points
P
=(2
,
9),
Q
=(3
,
10), and
R
=(
−
4
,
−
3) lie on the elliptic
curve
E
:
y
2
=
x
3
+ 73.
(a) Compute
P
+
Q
and (
P
+
Q
)+
R
.
(b) Compute
Q
+
R
and
P
+(
Q
+
R
). Your answer for
P
+(
Q
+
R
)
should agree with the result of part (a). However, note that one
computation used the doubling formula while the other did not use
it.
2.4 Let
E
be the elliptic curve
y
2
=
x
3
−
34
x
+ 37 defined over
Q
.Let
P
=(1
,
2) and
Q
=(6
,
7).
(a) Compute
P
+
Q
.
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