Cryptography Reference
In-Depth Information
P
2
P
1
+
P
2
P
1
Figure 6.8.
Elliptic curve example over the real numbers.
Given P
=
(
x
,
y
)
, we define
−
P
=
(
x
,
−
y
)
and
−
O
=
O
. Given P
1
=
(
x
1
,
y
1
)
and
P
2
=
(
x
2
,
y
2
)
,ifP
2
=−
P
1
, we define P
1
+
P
2
=
O
. Otherwise, we let
y
2
−
y
1
x
2
−
x
1
if
x
1
=
x
2
λ
=
3
x
1
+
a
2
y
1
if
x
1
=
x
2
2
x
3
=
λ
−
x
1
−
x
2
y
3
=
(
x
1
−
x
3
)
λ
−
y
1
P
3
=
(
x
3
,
y
3
)
and P
1
+
P
2
=
P
3
. In addition, P
+
O
=
O
+
P
=
P and
O
+
O
=
O
.
16(4
a
3
27
b
2
)
and the j -
invariant
j
We further define the
discriminant
=−
+
=
1728(4
a
)
3
−
/
, which can also be expressed as
4
j
=
1728
4
+
27
b
2
/
a
3
when a
=
0
.
(Note that the
x
1
=
λ
formula implies that
y
1
=
y
2
, thus
P
1
=
x
2
case in the
P
2
, and
y
1
=
0. Otherwise we would have had
P
2
=−
P
1
.) The definition of point addition
may look quite odd. It can be geometrically illustrated as depicted in Fig. 6.8. Actually,
y
x
1
) is the equation of a straight line which contains
P
1
. It is the chord
which contains
P
2
when
P
1
=
−
y
1
=
λ
(
x
−
P
2
.
Since
E
a
,
b
is an algebraic curve of degree 3, it usually intersects straight lines on three
points. The intersection between this line and the curve is defined by
P
2
. It is the tangent to the elliptic curve when
P
1
=
x
1
))
2
x
3
(
y
1
+
λ
(
x
−
=
+
ax
+
b
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