The cycle structure of NFSR(fd) and its applications

Abstract

Let NFSR(f ) denote the nonlinear feedback shift register (NFSR) with characteristic function f = x0 ⊕ g(x1,x2,…,xn− 1) ⊕ xn. In this paper, the cycle structure of NFSR(fd) is discussed, where $f^{d}=x_{0}\oplus g(x_{d},x_{2d},{\ldots } ,x_{(n-1)d})\oplus x_{nd}$ is also a characteristic function determined by f and a given integer d. If the cycle structure of NFSR(f ) is known, then it is shown that the cycle structure of NFSR(fd) can be completely determined. Moreover, with these results, three applications of the cycle structure of NFSR(fd) are presented: Firstly, the cycle structure of NFSR(fd) is discussed when f belongs to a class of symmetric characteristic functions. Compared with the previous work, our result can cover more cases while the proof is more straightforward. Secondly, we show the cycle structure of NFSR(fd) when f is a characteristic function of de Bruijn sequences and d = 2k. At last, a new necessary condition for f to be a characteristic function of de Bruijn sequences is presented, which can partially support the observation proposed in Calik et al. (IEICE Trans. Fund. Electron. Commun. Comput. Sci. E93-A,(6), 1226–1231 2012) and Chan et al. (Lect. Notes Comput. Sci. 809, 166–173 1993).