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


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.


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bourbaki N.: Algèbre. Chapitres 4 à 7. Masson, Paris (1981).zbMATHGoogle Scholar
  2. 2.
    Cossidente A., de Resmini M.J.: Remarks on Singer cyclic groups and their normalizers. Des. Codes Cryptogr. 32, 97–102 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Crabb M.C.: Counting nilpotent endomorphisms. Finite Fields Appl. 12, 151–154 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Darafsheh M.R.: Order of elements in the groups related to the general linear group. Finite Fields Appl. 11, 738–747 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fine N.J., Herstein I.N.: The probability that a matrix be nilpotent. Illinois J. Math. 2, 499–504 (1958)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Gerstenhaber M.: On the number of nilpotent matrices with coefficients in a finite field. Illinois J. Math. 5, 330–333 (1961)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Ghorpade S.R., Ram S.: Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields (in preparation).Google Scholar
  8. 8.
    Golomb S.W.: Shift Register Sequences. Holden-Day, San Francisco (1967)zbMATHGoogle Scholar
  9. 9.
    Huppert B.: Endliche Gruppen I. Springer, Berlin (1967)zbMATHGoogle Scholar
  10. 10.
    Jacobson N.: Basic Algebra I, 2nd edn. W. H. Freeman, New York (1985)zbMATHGoogle Scholar
  11. 11.
    Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1983)zbMATHGoogle Scholar
  12. 12.
    Niederreiter H.: Factorization of polynomials and some linear-algebra problems over finite fields. Linear Algebra Appl. 192, 301–328 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Niederreiter H.: The multiple-recursive matrix method for psedorandom number generation. Finite Fields Appl. 1, 3–30 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Niederreiter H.: Psedorandom vector generation by the multiple-recursive matrix method. Math. Comp. 64, 279–294 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Niederreiter H.: Improved bound in the multiple-recursive matrix method for psedorandom number and vector generation. Finite Fields Appl. 2, 225–240 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Preneel B.: Introduction to the Proceedings of the Second Workshop on Fast Software Encryption. (Leuven, Belgium, Dec 1994). Lecture Notes in Comput. Sci., vol. 1008, pp. 1–5. Springer, Berlin (1995).Google Scholar
  17. 17.
    Reiner I.: On the number of matrices with given characteristic polynomial. Illinois J. Math. 5, 324–329 (1961)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Singer J.: A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43, 377–385 (1938)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Zeng G., Han W., He K.: Word-Oriented Feedback Shift Register: σ-LFSR. (Cryptology ePrint Archive: Report 2007/114).
  20. 20.
    Zeng G., Han W., He K., Fan S.: High Efficiency Feedback Shift Register: σ-LFSR. preprint (2008).Google Scholar

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

Personalised recommendations