Linear complexity of Ding-Helleseth generalized cyclotomic sequences of order eight

During the last two decades, many kinds of periodic sequences with good pseudorandom properties have been constructed from classical and generalized cyclotomic classes, and used as keystreams for stream ciphers and secure communications. Among them are a family DH-GCSd of generalized cyclotomic sequences on the basis of Ding and Helleseth’s generalized cyclotomy, of length pq and order d=gcd(p−1,q−1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d={\gcd }(p-1,q-1)$\end{document} for distinct odd primes p and q. The linear complexity (or linear span), as a valuable measure of unpredictability, is precisely determined for DH-GCS8 in this paper. Our approach is based on Edemskiy and Antonova’s computation method with the help of explicit expressions of Gaussian classical cyclotomic numbers of order 8. Our result for d = 8 is compatible with Yan’s low bound (pq − 1)/2 on the linear complexity for any order d, which is high enough to resist attacks of the Berlekamp–Massey algorithm. Finally, we include SageMath codes to illustrate the validity of our result by examples.