Elliptic Curve Cryptography - Security for Wireless Sensor Networks Using Identity-Based Cryptography - page 51

æ ö ÷

ç

åå

÷

ç

[]

=

sum aP ÷ aP (3.40)

ç

÷

ç

i i i i

÷

ç è

i i

is principal if and only if (

)

= å

å and =

A divisor A a [] P deg A a [ P ] 0

i i i i

i i

( )

å

= =¥

sum A a [ P ] . A group of divisors of E , denoted as Div( E ), is an abelian

i i

i

group formed by the points on E . In addition, divisors of degree 0 [Div 0 ( E )] form a
subgroup of Div( E ), which is of particular interest to us.

3.9.2 Rational Function

Î

A function on the curve E can be viewed as a rational function fxy (, ) Kxy (, ) if it

Î ; then we define f ( P ) = f ( x, y ). is defined for at least one point in E . Let Pxy (, )

Example 3.2. Let E be a curve y 2 = x 3

+

x and let f ( x, y ) = x/y . Note that the function
f ( x, y ) is not defined for P = (0, 0). However, on the curve E , the function f ( x, y ) is

x y

== +

fxy (, ) (3.41)

2

y x 1

and equates to 0 P . Furthermore, the f ( x, y ) can be transformed to

y y

== +

fxy (, ) (3.42)

2

x x 1

and becomes ¥ at P .

3.9.2.1 Zeros and Poles

A rational function on the curve is said to have a zero at point P if the function takes
the value 0 and point P , f ( P ) = 0. And a function on the curve is said to have a pole at
point P if the function takes the value ¥ at point P .

3.9.2.2 Order of Poles and Zeros

It can be shown that there is a function u p , called the Uniformizer at P , with u ( P ) = 0,
and such that every rational function f ( x, y ) can be written in the form

( r

=

f ug (3.43)

p

with r ∈ Z and g ( P ) ¹ 0, ¥ .