On the Linear Complexity of Generalized Cyclotomic Quaternary Sequences with Length $2pq$

Abstract

In this paper, the linear complexity over $\mathbf{GF}(r)$ of generalized cyclotomic quaternary sequences with period $2pq$ is determined, where $r$ is an odd prime such that $r\ge 5$ and $r\notin \{p,q\}$. The minimal value of the linear complexity is equal to $\frac{5pq+p+q+1}{4}$ which is greater than the half of the period $2pq$. According to the Berlekamp-Massey algorithm, these sequences are viewed as enough good for the use in cryptography. We show also that if the character of the extension field $\mathbf{GF}(r^m)$, $r$, is chosen so that $\left(\frac{r}{p}\right)=\left(\frac{r}{q}\right)=-1$, $r\nmid 3pq-1$, and $r\nmid 2pq-4$, then the linear complexity can reach the maximal value equal to the length of the sequences.

38pages