Cryptography Reference
In-Depth Information
25
y
20
15
10
5
0
0
2
4
6
8
10
12
14
16
18
20
22
24
x
( )
3
Figure 3.5.
y
²mod
p
=+
x
2
x
mod ,
p where p = 23
( )
=+
3
²mod
y
p
x
2
x
mod
p
(3.24)
( )
=+
3
Points of elliptic curve
y
²mod
p
x
2
x
mod
p
, where p = 23, are
{(0,0),(1,16),(1, 7),(2,9),(2,14),(4,16),(4, 7),(7,9),(7,14),(10,13),(10,10),
12, 2 , 12, 21 , 14,9 , 14,14 , 15,1 , 15, 22 , 17,18 , 17,5 , 18,16 , 18, 7 ,
( )( )( )( )( )( )( )( )( )( )
}
where refers to point to infinity .
Basic operations include addition, subtraction, multiplication, division, and
inversion.
3.5.4 Binary Curves
Binary curves are defined over the finite field F (2 m ), where the field consists of 2 m ele-
ments. In this case we use a cubic equation of the form
+=+ +
Eaby yx x b
( ,): ²
³
²
(3.25)
2
where the coefficients and variables take on the values in GF (2 m ).
Search WWH ::




Custom Search