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.