Cryptography Reference
In-Depth Information
2
(
For example, consider the hyperbola
xy
=
1in
A
R
). Projective geometry allows an
interpretation of the behaviour of the curve at
x
=
0or
y
=
0; see Example
5.2.7
.
k
Definition 5.2.1 Projective space
over
of dimension
n
is
n
(
n
+
1
(
P
k
)
={
lines through (0
,...,
0) in
A
k
)
}
.
n
(
A convenient way to represent points of
P
k
)isusing
homogeneous coordinates
.Let
a
0
,a
1
,...,a
n
∈ k
with not all
a
j
=
0 and define (
a
0
:
a
1
:
···
:
a
n
) to be the equivalence
class of (
n
+
1)-tuples under the equivalence relation
(
a
0
,a
1
,
···
,a
n
)
≡
(
λa
0
,λa
1
,
···
,λa
n
)
∈ k
∗
.
n
(
for
any
λ
Thus
P
k
)
={
(
a
0
:
···
:
a
n
):
a
i
∈ k
for 0
≤
i
≤
n
and
a
i
=
n
n
(
0forsome0
≤
i
≤
n
}
. Write
P
= P
k
).
···
:
a
n
) is the set of points on the line between
(0
,...,
0) and (
a
0
,...,a
n
) with the point (0
,...,
0) removed.
There is a map
ϕ
:
In other words, the equivalence class (
a
0
:
A
n
→ P
n
given by
ϕ
(
x
1
,...,x
n
)
=
(
x
1
:
···
:
x
n
: 1). Hence,
A
n
is
n
.
identified with a subset of
P
1
(
1
(
Example 5.2.2
The
projective line
P
k
) is in one-to-one correspondence with
A
k
)
∪
{∞}
P
1
(
k
={
(
a
0
:1):
a
0
∈ k}∪{
}
P
2
(
k
since
)
(1 : 0)
.The
projective plane
) is in one-to-
A
2
(
k
∪ P
1
(
k
one correspondence with
)
).
n
(
Defi
n
ition 5.2.3
A point
P
=
(
P
0
:
P
1
:
···
:
P
n
)
∈ P
k
)is
defined ov
er
k
if there is some
∗
such that
λP
j
∈ k
n
and
σ
λ
∈ k
for all 0
≤
j
≤
n
.If
P
∈ P
∈
Gal(
k
/
k
) then
σ
(
P
)
=
(
σ
(
P
0
):
···
:
σ
(
P
n
)).
Exercise 5.2.4
Show that
P
is defined over
k
if and only if there is some 0
≤
i
≤
n
such
n
(
that
P
i
=
0 and
P
j
/P
i
∈ k
for all 0
≤
j
≤
n
. Show that
P
k
) is equal to the set of points
n
(
P
∈ P
k
) that are defined over
k
. Show that
σ
(
P
) in Definition
5.2.3
is well-defined (i.e.,
P
=
(
P
0
,...,P
n
) then
σ
(
P
)
σ
(
P
)).
if
P
=
(
P
0
,...,P
n
)
≡
≡
n
(
Lem
m
a 5.2.5
A point P
∈ P
k
)
is defined over
k
if and only if σ
(
P
)
=
P for all σ
∈
Gal(
k
/
k
)
.
n
(
Proof
Let
P
=
(
P
0
:
···
:
P
n
)
∈ P
k
) and suppose
σ
(
P
)
≡
P
for all
σ
∈
Gal(
k
/
k
). Then
∗
such that
σ
(
P
i
)
there is some
ξ
:Gal(
k
/
k
)
→ k
=
ξ
(
σ
)
P
i
for all 0
≤
i
≤
n
. One can
∗
. It follows by Theorem
A.7.2
(Hilbert 90) that
ξ
(
σ
)
verify
2
that
ξ
is a 1-c
oc
ycle in
k
=
∗
. Hence,
σ
(
P
i
/γ
)
σ
(
γ
)
/γ
for some
γ
∈ k
=
P
i
/γ
for all 0
≤
i
≤
n
and all
σ
∈
Gal(
k
/
k
).
Hence,
P
i
/γ
∈ k
for all 0
≤
i
≤
n
and the proof is complete.
Recall that if
f
is a homogeneous polynomial of degree
d
then
f
(
λx
0
,...,λx
n
)
=
λ
d
f
(
x
0
,...,x
n
) for all
λ
n
+
1
(
∈ k
and all (
x
0
,...,x
n
)
∈ A
k
).
2
At least, one can verify the formula
ξ
(
στ
)
=
σ
(
ξ
(
τ
))
ξ
(
σ
). The topological condition also holds, but we do not discuss this.
