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Lemma 1 ([1]).
For any decimation
d
with
gcd(
d,
2
k
1) = 1
the exponential
sum
S
(
a
)
defined in (1) satisfies the following moment identities
−
S
(
a
)=2
k
a∈
GF(2
k
)
∗
S
(
a
)
2
=2
2
k
(2
k
−
1)
a∈
GF(2
k
)
∗
S
(
a
)
3
=
2
4
k
+(
λ
+3)2
m
+
k
−
,
a∈
GF(2
k
)
∗
GF(2
m
)
∗
of the equation system
where
λ
is the number of solutions for
x
1
,x
2
∈
1+
x
1
+
x
2
=0
1+
x
d
(2
k
+1)
1
+
x
d
(2
k
+1)
2
=0
.
For the values of
d
that we consider it is easy to show that
λ
=2
gcd(
l,k
)
−
2.
2
The Ane Polynomial
A
a
(
x
)
In this section, we consider zeros in GF(2
k
) of the ane polynomial
A
a
(
x
)=
a
2
l
x
2
2
l
+
x
2
l
+
ax
+
c,
(2)
GF(2
e
), where
e
=gcd(
l, k
). Some additional condi-
tions on the parameters will be imposed later. The distribution of zeros in GF(2
k
)
of (2) will determine to a large extent the distribution of our cross-correlation
function. It is clear that
A
a
(
x
) does not have multiple roots if
a
GF(2
k
)and
c
where
a
∈
∈
= 0. Zeros of
the linearized homogeneous part of
A
a
(
x
) were previously studied in [6,7]. Some
results needed here will be cited from [7].
Let
k
=
ne
for some
n
and introduce a particular sequence of polynomials over
GF(2
k
) that will play a crucial role when finding zeros of (2). For any
u
∈
GF(2
k
)
denote
u
i
=
u
2
il
for
i
=0
,...,n
−
1so
A
a
(
x
)=
a
1
x
2
+
x
1
+
a
0
x
0
+
c
.Let
B
1
(
x
)=1
,
B
2
(
x
)=1
,
B
i
+2
(
x
)=
B
i
+1
(
x
)+
x
i
B
i
(
x
) or1
≤
i
≤
n
−
1
.
(3)
Observe the following recursive identity that can be seen as an equivalent defi-
nition of
B
i
(
x
) and which was proved in [7]
B
i
+2
(
x
)=
B
2
l
i
+1
(
x
)+
x
1
B
2
2
l
(
x
) or1
≤
i
≤
n
−
1
.
(4)
i
We also define polynomials
Z
n
(
x
)overGF(2
k
)as
Z
1
(
x
)=1and
Z
n
(
x
)=
B
n
+1
(
x
)+
xB
2
l
1
(
x
)
(5)
n−
for
n>
1. The following lemma describes zeros of
B
n
(
x
)inGF(2
k
).
