Trace Representation and Linear Complexity of Binary $e$th Power Residue Sequences of Period $p$

Let
$p=ef+1$
be an odd prime for some
$e$
and
$f$
, and let
$F_{p}$
be the finite field with
$p$
elements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period
$p$
) of all the cyclic difference sets
$D$
which are some union of cosets of
$e$
th powers
$H_{e}$
in
$F_{p}^{\ast }(\triangleq F_{p}\backslash \{0\})$
for
$e\leq 12$
. For this, we define
$e$
th power residue sequences of period
$p$
, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of
$H_{e}$
in
$F_{p}^{\ast }$
at some primitive
$p$
th root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known
$e$
th residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant
$e$
th power residue sequence for any
$e$
to be either
$p-1$
or
$p$
whenever
$(e,(p-1)/n)=1$
, where
$n$
is the order of 2 mod
$p$
.