Cryptography Reference
In-Depth Information
a uniformiser at
a
. In terms of the projective equation, one has
t
a
=
(
x
−
az
)
/z
being a
1
one again works with the corresponding
uniformiser. For the point
∞=
(1 : 0)
∈
U
0
⊆ P
ϕ
−
1
0
point 0
∈
(
U
0
). The uniformiser is
t
a
=
z
which, projectively, is
t
a
=
z/x
. A common
ϕ
−
1
1
1
abuse of notation is to say that 1
/x
is a uniformiser at
∞
on
A
=
(
U
1
).
Example 7.3.4
We determine uniformisers for the points on an elliptic curve. First, consider
points (
x
P
,y
P
) on the affine equation
−
x
3
a
6
.
y
2
a
2
x
2
E
(
x,y
)
=
+
a
1
xy
+
a
3
y
+
+
a
4
x
+
Without loss of generality, we can translate the point to
P
0
=
(0
,
0), in which case write
a
1
,...,a
6
for the coefficients of the translated equation
E
(
x,y
)
0 (i.e.,
E
(
x,y
)
=
=
y
P
)). One can verify that
a
6
=
0,
a
3
=
(
∂E/∂y
)(
P
) and
a
4
=
E
(
x
+
x
P
,y
+
(
∂E/∂x
)(
P
).
(
x,y
) and, since the curve is not singular, at least one of
a
3
or
a
4
is non-zero.
Then m
P
0
=
If
a
3
=
0 then
3
x
(
x
2
a
2
x
a
4
−
a
1
y
)
y
2
.
+
+
=
Since (
x
2
a
2
x
a
4
−
a
1
y
)(
P
0
)
a
4
=
0wehave(
x
2
a
2
x
a
4
−
a
1
y
)
−
1
+
+
=
+
+
∈
O
P
0
and
so
y
2
(
a
4
+
a
2
x
x
2
a
1
y
)
−
1
.
x
=
+
−
(
y
2
)
⊆
m
P
0
In other words,
x
∈
and
y
is a uniformiser at
P
0
.
Similarly, if
a
4
=
0 then
y
(
a
3
+
a
1
x
x
2
(
x
a
2
) and so
y
(
x
2
)
m
P
0
+
y
)
=
+
∈
⊆
and
x
is
a uniformiser at
P
0
.If
a
3
,a
4
=
0 then either
x
or
y
can be used as a uniformiser. (Indeed, any
a
4
x
can be used as a uniformiser; geometrically,
any line through
P
, except the line which is tangent to the curve at
P
, is a uniformiser.)
Now consider the point at infinity
by
except
a
3
y
linear combination
ax
+
−
O
E
=
=
=
(
x
:
y
:
z
)
(0 : 1 : 0) on
E
. Taking
y
1
transforms the point to (0
,
0) on the affine curve
a
3
z
2
x
3
a
2
x
2
z
a
4
xz
2
a
6
z
3
.
z
+
a
1
xz
+
=
+
+
+
(7.6)
It follows that
a
2
x
2
a
6
z
2
)
x
3
z
(1
+
a
1
x
+
a
3
z
−
−
a
4
xz
−
=
m
P
and so
x
is a uniformiser. This is written as
x/y
in homogeneous
and so
z
∈
(
x
3
)
⊆
coordinates.
In practice, it is not necessary to move
P
to (0
,
0) and compute the
a
i
.Wehaveshown
that if
P
=
(
x
P
,y
P
) then
t
P
=
x
−
x
P
is a uniformiser unless
P
=
O
E
, in which case
ι
(
P
),
4
t
P
=
x/y
,or
P
=
in which case
t
P
=
y
−
y
P
.
Lemma
7.
3.5
Let C be a curve over
k
,letP
∈
C
(
k
)
and let t
P
be a uniformiser at P. Let
σ
∈
Gal(
k
/
k
)
. Then σ
(
t
P
)
is a uniformiser at σ
(
P
)
.
3
We will see later that
a
3
=
0 implies (0
,
0) has order 2 (since
−
(
x,y
)
=
(
x,
−
y
−
a
1
x
−
a
3
)).
4
i.e., has order 2.