Cryptography Reference
In-Depth Information
3.9.2.3 Divisor of a Function
Let
f
be a function on a curve; then the divisor of the function is given by
=
å
()
()
div
f
ord
f
.(
P
)
(3.44)
P
P
i
where ord
p
(
f
)
∈
Z
and
()
f
is a point on the curve
E
.
Exa mple 3.3.
Let us consider the same elliptic curve equation
E
as in Example 3.2. Let
f
(
x, y
) be a rational function such as
æ
ö
÷
1
ç
==
2
fxy
(, )
x
y
+
÷
(3.45)
ç
ç
è
÷
2
x
1
Where
u
P
=
y
=
0 at point
P
=
(0, 0) and
g
(
P
)
=
(1/
x
2
+
1)
¹
0,
¥
at point
P
. Hence,
in this example, ord
P
(
x
)
=
2.
Lemma 3.1.
Let
f
be a rational function on an elliptic curve
E
such that
f
¹
0. Then
•
f
has only a finite number of zeros and poles.
•
[ ]
f
=
.
•
f
is a constant if the function does not have zeros and poles.
deg div(
)
0
=++=
pass through these three points. Then this function
f
1
has three zeros at
P
1
,
P
2
,
P
3
. And
if
b
¹
0, then
f
1
has three poles at
¥
. Hence, the divisor of the line function is
Let there be three points (
P
1
,
P
2
,
P
3
) on
E
and let a line
fxy
1
(, )
ax by c
0
++= + + -¥
d v(
ax
by
c
) [][][] [ ]
P
P
P
(3.46)
1
2
3
Let -
P
3
be a reflection of
P
3
(
x
3
,
y
3
). Then the line passing through
P
3
and -
P
3
is given
by
=- =
.
The divisor of
f
2
is given by
fxy
(, )
x x
0
2
3
-=+--¥
div (
xx
)
[
P
]
[
P
]
2[
]
(3.47)
3
3
3
Therefore,
æ
ö
ax
++
÷
by
c
ç
÷
=++--
div
ç
div (
ax
by
c
)
div (
x
x
)
(3.48)
÷
ç
÷
ç
-
3
è
xx
ø
3
æ
ö
ax
++
÷
by
c
ç
÷
=+---¥
div
ç
[][ ][
PP
P
][ ]
(3.49)
÷
ç
÷
ç
-
1
2
3
è
xx
ø
3
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