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Designs, Codes and Cryptography

, Volume 58, Issue 2, pp 123–134 | Cite as

Primitive polynomials, singer cycles and word-oriented linear feedback shift registers

  • Sudhir R. Ghorpade
  • Sartaj Ul Hasan
  • Meena Kumari
Article

Abstract

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng et al. (Word-Oriented Feedback Shift Register: σ-LFSR, 2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This conjecture is about the number of primitive σ-LFSRs of a given order over a finite field, and it generalizes a known formula for the number of primitive LFSRs, which, in turn, is the number of primitive polynomials of a given degree over a finite field. Moreover, this conjecture is intimately related to an open question of Niederreiter (Finite Fields Appl 1:3–30, 1995) on the enumeration of splitting subspaces of a given dimension.

Keywords

Primitive polynomial Linear Feedback Shift Register (LFSR) Primitive recursive vector sequence Singer cycle Singer subgroup Splitting subspaces 

Mathematics Subject Classification (2000)

11T06 11T71 20G40 94A60 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Sudhir R. Ghorpade
    • 1
  • Sartaj Ul Hasan
    • 1
    • 2
  • Meena Kumari
    • 2
  1. 1.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia
  2. 2.Scientific Analysis Group, Defense Research and Development OrganisationDelhiIndia

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