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

TL;DR

This paper analyzes the linear complexity of generalized cyclotomic quaternary sequences with length 2pq over GF(r), showing they are suitable for cryptography and can achieve maximal complexity under certain conditions.

Contribution

It determines the linear complexity of these sequences and identifies conditions for maximal complexity, advancing their cryptographic applicability.

Findings

Minimal linear complexity is (5pq + p + q + 1)/4, exceeding half the period.

Sequences can reach maximal linear complexity equal to their length under specific field conditions.

Sequences are suitable for cryptography based on their linear complexity properties.

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 $r\notin \lbrace p,q\rbrace$. 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.