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 [square root of p] + 2 and that of an M-ary Sidel'nikov sequence of period [p.sup.m] - 1 is upper bounded by [square root of [p.sup.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. Index Terms--Cross correlation, cyclotomic numbers, Jacobi sum, multiplicative characters, power residue sequences, Sidel'nikov sequences.