Grain-like structures with minimal and maximal period sequences

Abstract

Nonlinear feedback shift registers (NFSRs) are important building blocks for stream ciphers. The cascade connection of an n-stage full-length linear feedback shift register (LFSR) into an m-stage NFSR is called a Grain-like structure. In this paper, we focus on Grain-like structures which can generate minimal and maximal possible period sequences. The existence of Grain-like structures which can generate minimal possible period sequences is proved for the cases $$m=n$$ and $$m>n$$ . The number of such Grain-like structures is estimated in both cases. Two necessary conditions are presented for Grain-like structures to generate maximal possible period sequences. Moreover, some interesting properties of such Grain-like structures are discussed.