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={ti | ti=fk(Tr(ωi)+A)}, where ω is a primitive element in the extension field $F{p}{m}$, Tr(⋅) is the trace function that maps $F{p}{m} ightarrow {p}$, A is a non-zero scalar in the prime field ${p}$, and fk(⋅) 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.

This paper proposes a new approach for generating pseudo random multi-valued (including binary-valued) sequences. The approach uses a primitive polynomial over an odd characteristic prime field ${p}$, where p is an odd prime number. Then, for the maximum length sequence of vectors generated by the primitive polynomial, the trace function is used for mapping these vectors to scalars as elements in the prime field. Power residue symbol (Legendre symbol in binary case) is applied to translate the scalars to k-value scalars, where k is a prime factor of p-1. Finally, a pseudo random k-value sequence is obtained. Some important properties of the resulting multi-valued sequences are shown, such as their period, autocorrelation, and linear complexity together with their proofs and small examples.