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.

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Correspondence to Qunxiong Zheng.

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This work is supported by the National Natural Science Foundation of China (Grant Nos. 61872383 and 61502524). The work of Qunxiong Zheng was also supported by Young Elite Scientists Sponsorship Program by CAST (Grant No. 2016QNRC001) and by National Postdoctoral Program for Innovative Talents (Grant No. BX201600188) and by China Postdoctoral Science Foundation funded project (Grant No. 2017M611035)

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Wang, Z., Zheng, Q., Zhao, X. et al. Grain-like structures with minimal and maximal period sequences. Des. Codes Cryptogr. (2021). https://doi.org/10.1007/s10623-020-00839-3

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Keywords

  • Stream ciphers
  • Nonlinear feedback shift registers
  • Cascade connection of NFSRs
  • Grain-like structures

Mathematics Subject Classification

  • 11B50
  • 94A55
  • 94A60