Cryptography Reference
In-Depth Information
This completes the proof of the second equality of Proposition 4.27.
We now show that the number of points on
v
2
=(
k/
4)
u
4
+1 can be ex-
pressed in terms of Jacobi sums. By separating out the terms with
u
=0and
the terms with
v
= 0, we obtain that the number of points is
#
{v | v
2
=1
}
+#
{u | u
4
=
−
4
/k}
+
a
+
b
=1
a,b
=0
v
2
=
a
u
4
=
#
{
v
|
}
#
{
u
|
−
4
b/k
}
1
3
1
3
χ
4
(
−
4
/k
)
+
a
+
b
=1
a,b
χ
2
(1)
j
+
χ
2
(
a
)
j
χ
4
(
−
4
b/k
)
=
j
=0
=0
j
=0
=0
=0
1
3
4
/k
)
+
b
3
χ
2
(1)
j
+
4
b/k
)
=
χ
4
(
−
χ
4
(
−
j
=0
=0
=0
,
1
=0
+
a
1
χ
2
(
a
)
j
−
(
p
−
2)
j
=0
=0
,
1
4
/k
)
2
J
(
χ
2
,χ
4
)+
χ
4
(
4
/k
)
3
J
(
χ
2
,χ
4
)
+
χ
4
(
−
−
4
/k
)
J
(
χ
2
,χ
4
)+
χ
4
(
−
(Separate out the terms with
j
=0and
= 0. These yield the sums over
and over
j
, respectively. The terms with
j
=
=0,whichsumto
p −
2, are
counted twice, so subtract
p −
2. The terms with
j,
= 0 contribute to the
Jacobi sums.)
1
3
χ
2
(
a
)
j
+
χ
4
(
−
4
b/k
)
=
−
(
p −
2)
j
=0
a
=0
=0
b
=0
−χ
2
(
−
4
/k
)+
χ
4
(
−
4
/k
)
J
(
χ
2
,χ
4
)+
χ
4
(
−
4
/k
)
3
J
(
χ
2
,χ
4
)
=(
p −
1) + (
p −
1)
−
(
p −
2)
−
4
/k
)
3
J
(
χ
2
,χ
4
)
χ
2
(
−
4
/k
)+
χ
4
(
−
4
/k
)
J
(
χ
2
,χ
4
)+
χ
4
(
−
(by Lemma 4.26)
=
p
+1
− δ
+
χ
4
(
−
4
/k
)
J
(
χ
2
,χ
4
)+
χ
4
(
−
4
/k
)
3
J
(
χ
2
,χ
4
)
.
For the last equality, we used the fact that
4
/k
)=1+
χ
2
(1
/k
)=
0if
k
is not a square
1+
χ
2
(
−
2if
k
is a square mod
p
,
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