Cryptography Reference
In-Depth Information
Exercise 5.7.5
Let
p
≡
3 (mod 4). Write down the Weil restriction of scalars of
X
=
V
(
x
2
1
−
2
i
)
⊂ A
with respect to
F
p
2
/
F
p
.
3 (mod 4). Write down the Weil restriction of scalars of
V
(
x
1
+
Exercise 5.7.6
Let
p
≡
x
2
−
2
(1
+
2
i
))
⊂ A
with respect to
F
p
2
/
F
p
.
n
be an affine algebraic set over
mn
be the Weil
Theorem 5.7.7
Let X
⊆ A
F
q
m
. Let Y
⊆ A
restriction of X. Let k
∈ N
be coprime to m. Then there is a bijection between X
(
F
q
mk
)
and
Y
(
F
q
k
)
.
Proof
When gcd(
k,m
)
1 it is easily checked that the map
φ
of Lemma
5.7.1
gives a a
one-to-one correspondence between
=
nm
(
n
(
A
F
q
k
) and
A
F
q
mk
).
Now, let
P
=
(
x
1
,...,x
n
)
∈
X
and write
Q
=
(
y
1
,
1
,...,y
n,m
) for the corresponding
mn
. For any
f
point in
A
∈
S
we have
f
(
P
)
=
0. Writing
f
1
,...,f
m
for the polynomials
in equation (
5.4
)wehave
f
1
(
Q
)
θ
1
+
f
2
(
Q
)
θ
2
+···+
f
m
(
Q
)
θ
m
=
0
.
Since
{
θ
1
,...,θ
m
}
is also a vector space basis for
F
q
mk
over
F
q
k
we have
f
1
(
Q
)
=
f
2
(
Q
)
=···=
f
m
(
Q
)
=
0
.
S
and so
Q
Hence,
f
(
Q
)
=
0 for all
f
∈
∈
Y
. Similarly, if
Q
∈
Y
then
f
j
(
Q
)
=
0 for all
=
∈
such
f
j
and so
f
(
P
)
0 for all
f
S
.
Note that, as the following example indicates, when
k
is not coprime to
m
then
X
(
F
q
mk
)
is not usually in one-to-one correspondence with
Y
(
F
q
k
).
Exercise 5.7.8
Consider the algebraic set
X
from Exercise
5.7.5
. Show that
X
(
F
p
4
)
=
{
1
+
i,
−
1
−
i
}
.Let
Y
be the Weil restriction of
X
with respect to
F
p
2
/
F
p
. Show that
Y
(
F
p
2
)
={
(1
,
1)
,
(
−
1
,
−
1)
,
(
i,
−
i
)
,
(
−
i,i
)
}
.
n
with respect to
Note that the Weil restriction of
P
F
q
m
/
F
q
is not the projective closure
mn
. For example, considering the case
n
1
1
,
of
A
=
1,
P
has one point not contained in
A
m
has an (
m
whereas the projective closure of
A
−
1)-dimensional algebraic set of points at
infinity.
1
Exercise 5.7.9
Recall from Exercise
5.5.14
that there is a morphism from
P
to
Y
=
V
(
x
2
y
2
2
. Determine the Weil restriction of scalars of
Y
with respect to
+
−
1)
⊆ A
F
p
2
/
F
p
.
1
It makes sense to call this algebraic set the Weil restriction of
P
with respect to
F
p
2
/
F
p
.
