Cryptography Reference
In-Depth Information
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,
¥
.
Search WWH ::
Custom Search