Cryptography Reference
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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.
 
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