On the Cross-Correlation Distributions of $M$-ary Multiplicative Character Sequences

It is well known that the magnitude of the cross correlation between any distinct constant multiple sequences of an
M
-ary power residue sequence of period
p
is upper bounded by
radicp
+2 and that of an
M
-ary Sidel'nikov sequence of period
p

^m
-1 is upper bounded by
radic{p

^m
} +3, where
p
is a prime and
m
is a positive integer. In this paper, we first show that their cross-correlation functions are closely related to Jacobi sums and cyclotomic numbers. We then derive the cross-correlation distribution of constant multiple sequences of an
M
-ary power residue sequence. In the case of constant multiple sequences of an
M
-ary Sidel'nikov sequence, we get the possible cross-correlation values whose occurrence numbers are expressed in terms of the cyclotomic numbers of order
M
and are possibly zero.