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then
Tr
β
1
α
d
1
i
+
β
2
α
d
2
i
where
d
1
,d
2
satisfy
d
1
d
−
1
=
d
, does not have perfect
2
autocorrelation.
Proof.
The second item follows immediately by Theorem 5 and Proposition 2.
If
d
is AB with
Θ
d
(1)
= 0 then the result follows from Proposition 2. Now
assume
d
is AB with
Θ
d
(1) = 0. Then
|
Im
(
ψ
d
)
|
|
Im
(
φ
d
)
|
=2
m−
1
−
=
1, since
=2
m−
1
d
is AB, and in turn APN. But
|
Ker
(
Θ
d
)
\{
0
,
1
}|
−
2, and therefore
=2
m−
1
if
d
is AB (see for instance [2]).
Im
(
ψ
d
)
Question 2.
If
m
is odd, are there sequences
Tr
β
1
α
d
1
i
+
β
2
α
d
2
i
with perfect
autocorrelation where
d
1
,d
2
satisfy
d
1
d
−
1
⊆
Ker
(
Θ
d
), since
|
Ker
(
Θ
d
)
|
=
d
,and
gcd
(
d, q
−
1) = 1?
2
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