Construction of de Bruijn Sequences From LFSRs With Reducible Characteristic Polynomials
Chaoyun Li, Xiangyong Zeng, Chunlei Li, Tor Helleseth, Ming Li · IEEE Transactions on Information Theory · 2015
In this paper, a family of new de Bruijn sequences is proposed through the construction of maximum-length nonlinear feedback shift registers (NFSRs). Let$k$be a positive integer and$p_{0}(x), p_{1}(x), \ldots , p_{k}(x)$be the primitive polynomials in$\mathbb {F}_{2}[x]$with their degrees strictly increasing and pairwise coprime. We determine the cycle structure and adjacency graphs of linear feedback shift registers (LFSRs) with characteristic polynomial$q(x)=\prod olimits _{i=0}^{k}p_{i}(x)$. In the case that$p_{0}(x)=1+x$, an algorithm is proposed to produce maximum-length NFSRs from these LFSRs, and it is shown that the algorithm can generate$O(2^{(2^{k}-1)n})~n$-stage maximum-length NFSRs with memory complexity$O(2^{k}kn)$and time complexity$O(2^{n-d_{k}}kn)$, where$n$and$d_{k}$are the degrees of$q(x)$and$p_{k}(x)$, respectively. Finally, we illustrate the proposed algorithm in the case of$k=2$. In this case, we prove that for any integer$n\geq 8$, the algorithm can produce$n$-stage maximum-length NFSRs with time complexity as low as$O(n^{{\rm {log}{log}}(n)}$).