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<p>Nonlinear feedback shift registers (NFSRs) are widely used in the design of stream ciphers and the cycle structure of an NFSR is a fundamental problem still open. In this paper, a new configuration of Galois NFSRs, called F-Ring NFSRs, is proposed. It is shown that an <i>n</i>-bit F-Ring NFSR generates <i>n</i> sequences with the same period simultaneously, that is, sequences from all bit registers have the same period. Recall that the ring-like cascade connection proposed by Zhao et al. (Des Codes Cryptogr 86:2775–2790, 2018) also has such period property. But it is abnormal that if every component shift register is nonsingular, then the ring-like cascade connection is <i>singular</i>. F-Ring NFSRs proposed in this paper could fix this weakness. Moreover, it is proved that when an <i>n</i>-stage <i>m</i>-sequence is input to the internal state of an F-Ring NFSR by xor, the periods of its internal state are multiples of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1616_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(2^n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. At last, two toy examples are given to illustrate the new configuration.</p>

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On the cycle structure of a class of Galois NFSRs: component sequences possessing identical periods

  • 期刊论文

  • 发表日期：

    2025年3月29日

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  • Xiao-juan Wang,
  • Tian Tian,
  • Wen-feng Qi

摘要

Nonlinear feedback shift registers (NFSRs) are widely used in the design of stream ciphers and the cycle structure of an NFSR is a fundamental problem still open. In this paper, a new configuration of Galois NFSRs, called F-Ring NFSRs, is proposed. It is shown that an n-bit F-Ring NFSR generates n sequences with the same period simultaneously, that is, sequences from all bit registers have the same period. Recall that the ring-like cascade connection proposed by Zhao et al. (Des Codes Cryptogr 86:2775–2790, 2018) also has such period property. But it is abnormal that if every component shift register is nonsingular, then the ring-like cascade connection is singular. F-Ring NFSRs proposed in this paper could fix this weakness. Moreover, it is proved that when an n-stage m-sequence is input to the internal state of an F-Ring NFSR by xor, the periods of its internal state are multiples of \(2^n-1\) 2 n - 1 . At last, two toy examples are given to illustrate the new configuration.