Linear Complexity over F_q of Generalized Cyclotomic Quaternary Sequences with Period 2p

Abstract

Let r be an odd prime, such that r≥5 and r≠p, m be the order of r modulo p. Then, there exists a 2p^th root of unity in the extension field F_r^m. Let G(x) be the generating polynomial of the considered quaternary sequences over F_q[x] with q=r^m. By explicitly computing the number of zeros of the generating polynomial G(x) over F_r^m, we can determine the degree of the minimal polynomial, of the quaternary sequences which in turn represents the linear complexity. In this paper, we show that the minimal value of the linear complexity is equal to $ \frac{1}{2}(3p-1) $ which is more than p, the half of the period 2p. According to Berlekamp-Massey algorithm, these sequences viewed as enough good for the use in cryptography.