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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Sequences, Bent Functions and Jacobsthal Sums
verfasst von : Tor Helleseth, Alexander Kholosha
Erschienen in: Sequences and Their Applications – SETA 2010
Verlag: Springer Berlin Heidelberg
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The
p
-ary function
f
(
x
) mapping GF(
p
4
k
) to GF(
p
) and given by
$f(x)={\rm Tr}_{4k}\big(ax^d+bx^2\big)$
with
a
,
b
∈ GF(
p
4
k
) and
d
=
p
3
k
+
p
2
k
−
p
k
+ 1 is studied with the respect to its exponential sum. In the case when either
$a^{p^k(p^k+1)}\neq b^{p^k+1}$
or
a
2
=
b
d
with
b
≠ 0, this sum is shown to be three-valued and the values are determined. For the remaining cases, the value of the exponential sum is expressed using Jacobsthal sums of order
p
k
+ 1. Finding the values and the distribution of those sums is a long-lasting open problem.