Cryptography Reference
In-Depth Information
3.8 Torsion Points
Every p o int on an elliptic curve E has a finite order or an infinite order. Let
PE Î
where K is the algebraic closure of K , and let n be a small integer; then there exists a
finite subgroup such that
( ,
(3.35)
to En
[] {
P EKnP
( ) |
}
En
[]
Î
Hence, n is the order of P . On the contrary, if no such n exists for a
PEK
()
(we will
never get by adding P to itself ), then the order of the point is infinite.
2
3
Example 3.1. Let us consider the elliptic curve
EK y
/:
=+ where K = F 11
x
3,
x
(Figure 3.9).
The set of rational points on E is {[0,0],[1,9],[1,2],[2,5],[2,6],[3,5],[3,6],[6,5],[6,6],
[7,1],[7,10], } Therefore, # E ( K ) = 12. Referring to Eq. (3.20), the value of t = 0.
Hence the elliptic curve
2
3
=+ defined over F 11 is supersingular. Table
3.4 shows the order of the torsion points in
EK y
/:
x
3
x
2
3
EK y
/:
=+ whe re K = F 11 .
x
3
x
Let
Ey
=++ be an elliptic curve and let
x
³
Ax B
Î be the roots of
(, , )
rr r
123
the cubic equation E . Then
2
y
=- - -
(
x
r
)(
x
r
)(
x
r
)
(3.36)
1
2
3
If a point P reaches infinity at 2 P , then a line through the P is a tangent (Figure
3.5). Hence, we get the subgroup with second-order torsion points.
(3.37)
E
[2]
=
{( ,0),(
r
r
,0),(
r
,0),
}
tor
1
2
3
12
y
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
11
12
x
EKy x x where K = F 11
/:
2
3
3,
Figure 3.9.
=+
Search WWH ::




Custom Search