Cryptography Reference
In-Depth Information
3.5.5 Group Law
2
3
=++
defined over the field
K
, then there exists a
chord-and-tangent rule for performing an addition operation over the elements of the
field
K
. Since the set of points
E
(
K
)
forms an abelian group where
serves as its iden-
tity, addition of two points in
E
(
K
)
will result in a third point in
E
(
K
). Figure 3.6 shows
a geometric representation of the addition rule. Let
1
If
E
is an elliptic curve
y
x
x
B
Px
=
(, )
y
and
Px
=
(, )
y
be
1
1
2
2
2
two distinct points on an elliptic curve
E
. Then the sum
Px
=
can be obtained
by drawing a line between
P
1
and
P
2
, which will intersect the curve on
(, )
3
3
3
P
¢
. Reflection
of
P
¢
about the
x
axis results in
P
3
.
In addition, Figure 3.7 shows the geometric representation of doubling a
P
1
. In this
case, we draw a tangent line to the elliptic curve on
P
. This line intersects the curve at
point -
R
. The result of
R
is the reflection of -
R
along the
x
axis. As a special case, if
the line is a tangent, it is assumed that it intersect at the point to infinity (Figure 3.8).
In such cases,
P
=
Cryptographic applications use two families of elliptic curves:
prime curves
and
binary curves
.
2
.
3.5.5.1 Group Law for
EK y
/:
² ³
=++
,
char(
x ax b
K
¹
)
2,3
()
1. Point addition. Let
Px
=Î
(, )
y
EK
and
Px
=
(, )
y
Î
EK
()
, where
P
¹
1
1
1
2
2
2
P
Then
+=
1
2
.
PP xy
, where
(, )
2
3
3
10
y
P'
3
8
6
4
P
2
2
-10
-8
-6
-4
-2
2
4
6
8
10
x
P
1
-2
-4
-6
-8
P
3
-10
Figure 3.6.
Geometric Representation of Addition Rule over an Elliptic Curve
E
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