Weak Grain-Like Structures

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≥ 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≤ 18.