A multi-value sequence generated by power residue symbol and trace function over odd characteristic field

This paper proposes a new multi-value sequence generated by utilizing primitive element, trace, and power residue symbol over odd characteristic finite field. In detail, let p and k be an odd prime number as the characteristic and a prime factor of p - 1, respectively. Our proposal generates k-value sequence T = t_i j t_i = f_k(Tr(ωⁱ) + A), where ω is a primitive element in the extension field F_p^m, Tr(·) is the trace function that maps F_p^m → F_p, A is a non-zero scalar in the prime field F_p, and f_k(·) is a certain mapping function based on k-th power residue symbol. Thus, the proposed sequence has four parameters as p, m, k, and A.Then, this paper theoretically shows its period, autocorrelation, and cross-correlation. In addition, this paper discusses its linear complexity based on experimental results. Then, these features of the proposed sequence are observed with some examples.