A Grain-like structure is a cascade connection of a primitive LFSR into an NFSR. It is well known that such structures generate sequences with periods multiples of the period of primitive LFSR sequences. In this paper, we study weak Grain-like structures, i.e., Grain-like structures generating at least one sequence with minimum period. Assume the orders of LFSR and NFSR are $n$ and $m$ respectively. We prove that weak Grain-like structures always exist when $m>n$ . For $m=n$ , we give three classes of weak Grain-like structures. Then we extend the method in the second class to prove that weak Grain-like structures exist when $m\ge n-\lg n+4$ . Moreover, our experimental data shows that, the ratio of weak Grain-like structures approximates a value which is a little more than 63% for small $m=n$ , and weak Grain-like structures exist with $m=3$ or 4 when $n\le 18$ .

中文翻译：

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弱粒状结构

类粒结构是原始 LFSR 到 NFSR 的级联连接。众所周知，这种结构产生的序列的周期是原始LFSR序列周期的倍数。在本文中，我们研究弱粒状结构，即产生至少一个周期最小序列的粒状结构。假设 LFSR 和 NFSR 的顺序是 $n$ 和 百万美元 分别。我们证明弱粒状结构总是存在时 $m>n$ . 为了 $m=n$ ，我们给出了三类弱粒状结构。然后我们扩展第二类中的方法来证明弱Grain-like结构存在时 $m\ge n-\lg n+4$ . 此外，我们的实验数据表明，弱粒状结构的比例接近一个值，对于小 $m=n$ , 弱粒状结构存在于 $m=3$ 或 4 时 $n\le 18$ .