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Normal Bases

  • Ian F. Blake
  • XuHong Gao
  • Ronald C. Mullin
  • Scott A. Vanstone
  • Tomik Yaghoobian
Chapter
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 199)

Abstract

Interest in normal bases over finite fields stems from both purely mathematical curiosity and practical applications. The practical aspects of normal bases will be treated in Chapter 5. In the present chapter, we discuss the theoretical aspects of normal bases over finite fields.

Keywords

Finite Field Characteristic Polynomial Null Space Normal Basis Minimal Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S. Akbik, “Normal generators of finite fields”, J. Number Theory, 41 (1992), 146–149.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A.A. Albert, Fundamental Concepts of Higher Algebra, Univ. of Chicago Press, Chicago, 1956.zbMATHGoogle Scholar
  3. [3]
    E. Artin, Galois Theory, University of Notre Dame Press, South Bend, Ind., 1966.Google Scholar
  4. [4]
    D. Ash, I. Blake and S. Vanstone, “Low complexity normal bases”, Discrete Applied Math., 25 (1989), 191–210.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    E. Bach, J. Driscoll and J. Shallit, “Factor refinement”, Proceedings of the First Annual Acm-SIAM Symposium on Discrete Algorithms (1990), 202–211 (full version to appear in J. of Algorithms). Google Scholar
  6. [6]
    I. Blake, S. Gao and R. Mullin, “On normal bases in finite fields”, preprint, 1992.Google Scholar
  7. [7]
    I. Blake, S. Gao and R. Mullin, “Factorization of cx q+1 + dax q -ax- b and normal bases over GF(q)”, Research Report Corr 91–26, Faculty of Mathematics, University of Waterloo, 1991.Google Scholar
  8. [8]
    P.M. Cohn, Algebra, vol. 3, Wiley, Toronto, 1982.zbMATHGoogle Scholar
  9. [9]
    S. Gao, Normal Bases over Finite Fields, Ph.D. thesis, Department of Combinatorics and Optimization, University of Waterloo, in preparation.Google Scholar
  10. [10]
    J. von zur Gathen and M. Giesbrecht, “Constructing normal bases in finite fields”, J. Symbolic Computation, 10 (1990), 547–570.zbMATHCrossRefGoogle Scholar
  11. [11]
    K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J., 1971.zbMATHGoogle Scholar
  12. [12]
    N. Jacobson, Basic Algebra I, 2nd ed., W.H. Freeman, New York, 1985.zbMATHGoogle Scholar
  13. [13]
    D. Jungnickel, “Trace-orthogonal normal bases”, Discrete Applied Math., to appear.Google Scholar
  14. [14]
    S. Lang, Algebra, 2nd ed., Addison-Wesley, Menlo Park, California, 1984.zbMATHGoogle Scholar
  15. [15]
    H.W. Lenstra, “Finding isomorphisms between finite fields”, Math. Comp., 56 (1991), 329–347.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1987.Google Scholar
  17. [17]
    R. Mullin, I. Onyszchuk, S. Vanstone and R. Wilson, “Optimal normal bases in Gf(qn)”, Discrete Applied Math., 22 (1988/1989), 149–161.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    O. Ore, “Contributions to the theory of finite fields”, Trans. Amer. Math. Soc., 36 (1934), 243–274.MathSciNetCrossRefGoogle Scholar
  19. [19]
    D. Pei, C. Wang and J. Omura, “Normal bases of finite field Gf(2rn)”, IEEE Trans. Info. Th., 32 (1986), 285–287.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    S. Perlis, “Normal bases of cyclic fields of prime-power degree”, Duke Math. J., 9 (1942), 507–517.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    A. Pincin, “Bases for finite fields and a canonical decomposition for a normal basis generator”, Communications in Algebra, 17 (1989), 1337–1352.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    L. Rédei, Algebra, Pergamon Press, Oxford, New York, 1967.zbMATHGoogle Scholar
  23. [23]
    S. Schwarz, “Construction of normal bases in cyclic extensions of a field”, Czechslovak Math. J., 38 (1988), 291–312.Google Scholar
  24. [24]
    S. Schwarz, “Irreducible polynomials over finite fields with linearly independent roots”, Math. Slovaca, 38 (1988), 147–158.MathSciNetzbMATHGoogle Scholar
  25. [25]
    G.E. Séguin, “Low complexity normal bases for F 2 mn ”, Discrete Applied Math., 28 (1990), 309–312.zbMATHCrossRefGoogle Scholar
  26. [26]
    I.A. Semaev, “Construction of polynomials irreducible over a finite field with linearly independent roots”, Math. USSR Sbornik, 63 (1989), 507–519.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    B. Van Der Waerden, Algebra, vol. 1, Springer-Verlag, Berlin, 1966.zbMATHGoogle Scholar
  28. [28]
    M. Wang, I. Blake and V. Bhargava, “Normal bases and irreducible polynomials in the finite field Gf(22r)”, preprint, 1990.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Ian F. Blake
    • 1
  • XuHong Gao
    • 1
  • Ronald C. Mullin
    • 1
  • Scott A. Vanstone
    • 1
  • Tomik Yaghoobian
    • 1
  1. 1.University of WaterlooCanada

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