Let NFSR(f ) denote the nonlinear feedback shift register (NFSR) with characteristic function f = x₀ ⊕ g(x₁,x₂,…,x_n− 1) ⊕ x_n. In this paper, the cycle structure of NFSR(f^d) 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(f^d) can be completely determined. Moreover, with these results, three applications of the cycle structure of NFSR(f^d) are presented: Firstly, the cycle structure of NFSR(f^d) 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(f^d) when f is a characteristic function of de Bruijn sequences and d = 2^k. 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 Çalik 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).

中文翻译：

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NFSR（fd）的循环结构及其应用

让NFSR（˚F）与特性函数表示所述非线性反馈移位寄存器（NFSR）˚F = X ₀ ⊕克（X ₁，X ₂，...，X _{Ñ - 1}）⊕ X _Ñ。本文讨论了NFSR（f ^d）的循环结构，其中\（f ^ {d} = x_ {0} \ oplus g（x_ {d}，x_ {2d}，{\ ldots}，x_ { （n-1）d}）\ oplus x_ {nd} \）也是由f和给定整数d确定的特征函数。如果NFSR的循环结构（˚F已知），则表明可以完全确定NFSR（f ^d）的循环结构。此外，利用这些结果，提出了NFSR（f ^d）的循环结构的三个应用：首先，讨论了当f属于一类对称特征函数时NFSR（f ^d）的循环结构。与以前的工作相比，我们的结果可以涵盖更多的案例，而证明则更为直接。其次，我们展示了当f是de Bruijn序列的特征函数且d = 2 ^k时NFSR（f ^d）的循环结构。最后，一个新的必要条件f是de Bruijn序列的特征函数，它可以部分支持Çalik等人提出的观察。（IEICE Trans。Fund。Electron。Commun。Comput。Sci。E93 -A，（6），1226–1231 2012）和Chan等。（LECT。注释COMPUT。科学。809，166-173 1993）。