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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