On the Linear Complexity of Generalized Cyclotomic Quaternary Sequences with Length $2pq$
This work addresses the cryptographic security of quaternary sequences for applications in communication systems, but it is incremental as it extends known results to a specific parameter set.
The paper determines the linear complexity over GF(r) of generalized cyclotomic quaternary sequences with period 2pq, finding a minimal value of (5pq+p+q+1)/4, which exceeds half the period, and shows that under specific conditions, the linear complexity can reach the maximal value equal to the sequence length.
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 $r\notin \lbrace p,q\rbrace$. The minimal value of the linear complexity is equal to $\tfrac{5pq+p+q+1}{4}$ 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 $\bigl(\tfrac{r}{p}\bigr) = \bigl(\tfrac{r}{q}\bigr) = -1$, $r\nmid 3pq-1$, and $r\nmid 2pq-4$, then the linear complexity can reach the maximal value equal to the length of the sequences.