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
(, )
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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