Periods on The Cascade Connection of an LFSR and an NFSR

We study the periods of sequences produced by the cascade connection of two Feedback shift registers (FSRs). The period of the cascade connection is the period of the longest sequences it produces. An upper bound for the period of the cascade connection of a Nonlinear feedback shift register (NFSR) into a Linear feedback shift register (LFSR) is established. In addition, the cascade connection of an n-stage maximum-length LFSR into an n-stage NFSR is called an (n + n)-stage Grain-like NFSR, and we propose two families of (n + n)- stage Grain-like NFSRs such that the minimal period 2ⁿ-1 is achievable for a positive integer n.