Cryptography Reference
In-Depth Information
that is defined for at least one point in
E
(
K
) (so, for example, the rational
fu
nction 1
/
(
y
2
x
3
−
−
Ax
−
B
) is not allowed). The function takes values in
K
.
There is a technicality that is probably best described by an example. Sup-
pose
y
2
=
x
3
∪{∞}
− x
is the equation of the elliptic curve. The function
f
(
x, y
)=
x
y
is not defined at (0
,
0). However, on
E
,
x
y
y
=
−
1
,
x
2
which is defined and takes on the value 0 at (0
,
0). Similarly, the function
y/x
can be changed to (
x
2
−
1)
/y
, which takes on the value
∞
at (0
,
0). It can
be shown that a function can always be transformed in this manner so as to
obtain a
n
expression that is not 0
/
0 and hence gives a uniquely determined
value in
K
.
A function is said to have a
zero
at a point
P
if it takes the value 0 at
P
,
and it has a
pole
at
P
if it takes the value
∪{∞}
at
P
. However, we need more
refined information, namely the order of the zero or pole. Let
P
be a point.
It can be shown that there is a function
u
P
, called a
uniformizer
at
P
,with
u
(
P
) = 0 and such that every function
f
(
x, y
)canbewrittenintheform
∞
f
=
u
P
g,
with
r
∈
Z
and
g
(
P
)
=0
,
∞
.
Define the
order
of
f
at
P
by
ord
P
(
f
)=
r.
Example 11.1
On
y
2
=
x
3
−
x
, it can be shown that the function
y
is a uniformizer at (0
,
0).
We have
1
x
=
y
2
−
1
,
x
2
and 1
/
(
x
2
−
1) is nonzero and finite at (0
,
0). Therefore,
ord
(0
,
0)
(
x
)=2
,
and
ord
(0
,
0)
(
x/y
)=1
.
This latter fact agrees with the above computation that showed that
x/y
vanishes at (0
,
0).
Example 11.2
At any finite point
P
=(
x
0
,y
0
) on an elliptic curve, the uniformizer
u
P
can
be taken from the equation of a line that passes through
P
but is not tangent
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