Cryptography Reference
In-Depth Information
The integer
r
is harder to compute. In this section, we show how to use the
methods of the previous sections to compute
r
in some cases. In Section 8.8,
we'll give an example that shows why the computation of
r
is sometimes
di
cult.
Example 8.7
Let
E
be the curve
y
2
=
x
3
−
4
x.
In Section 8.2, we showed that
E
(
Q
)
/
2
E
(
Q
)=
{∞,
(0
,
0)
,
(2
,
0)
,
(
−
2
,
0)
}
(more precisely, the points on the right are representatives for the cosets on
the left). Moreover, an easy calculation using the Lutz-Nagell theorem shows
that the torsion subgroup of
E
(
Q
)is
T
=
E
[2]
.
From Theorem 8.15, we have
E
(
Q
)
T ⊕
Z
r
,so
E
(
Q
)
/
2
E
(
Q
)
(
T/
2
T
)
⊕
Z
2
=
T ⊕
Z
2
.
Since
E
(
Q
)
/
2
E
(
Q
) has order 4, we must have
r
= 0. Therefore,
E
(
Q
)=
E
[2] =
{∞,
(0
,
0)
,
(2
,
0)
,
(
−
2
,
0)
}.
Example 8.8
Let
E
be the curve
y
2
=
x
3
−
25
x.
This curve
E
appeared in Chapter 1, where we found the points
(0
,
0)
,
(5
,
0)
,
(
−
5
,
0)
,
(
−
4
,
6)
.
We also calculated the point
2(
−
4
,
6) = (
41
2
12
2
,
−
62279
1728
)
.
Since 2(
−
4
,
6) does not have integer coordinates, (
−
4
,
6) cannot be a torsion
point, by Theorem 8.7. In fact, a calculation using the Lutz-Nagell theorem
shows that the torsion subgroup is
T
=
{∞,
(0
,
0)
,
(5
,
0)
,
(
−
5
,
0)
}
Z
2
⊕
Z
2
.
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