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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Akbik, “Normal generators of finite fields”, J. Number Theory, 41 (1992), 146–149.
A.A. Albert, Fundamental Concepts of Higher Algebra, Univ. of Chicago Press, Chicago, 1956.
E. Artin, Galois Theory, University of Notre Dame Press, South Bend, Ind., 1966.
D. Ash, I. Blake and S. Vanstone, “Low complexity normal bases”, Discrete Applied Math., 25 (1989), 191–210.
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).
I. Blake, S. Gao and R. Mullin, “On normal bases in finite fields”, preprint, 1992.
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.
P.M. Cohn, Algebra, vol. 3, Wiley, Toronto, 1982.
S. Gao, Normal Bases over Finite Fields, Ph.D. thesis, Department of Combinatorics and Optimization, University of Waterloo, in preparation.
J. von zur Gathen and M. Giesbrecht, “Constructing normal bases in finite fields”, J. Symbolic Computation, 10 (1990), 547–570.
K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J., 1971.
N. Jacobson, Basic Algebra I, 2nd ed., W.H. Freeman, New York, 1985.
D. Jungnickel, “Trace-orthogonal normal bases”, Discrete Applied Math., to appear.
S. Lang, Algebra, 2nd ed., Addison-Wesley, Menlo Park, California, 1984.
H.W. Lenstra, “Finding isomorphisms between finite fields”, Math. Comp., 56 (1991), 329–347.
R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1987.
R. Mullin, I. Onyszchuk, S. Vanstone and R. Wilson, “Optimal normal bases in Gf(qn)”, Discrete Applied Math., 22 (1988/1989), 149–161.
O. Ore, “Contributions to the theory of finite fields”, Trans. Amer. Math. Soc., 36 (1934), 243–274.
D. Pei, C. Wang and J. Omura, “Normal bases of finite field Gf(2rn)”, IEEE Trans. Info. Th., 32 (1986), 285–287.
S. Perlis, “Normal bases of cyclic fields of prime-power degree”, Duke Math. J., 9 (1942), 507–517.
A. Pincin, “Bases for finite fields and a canonical decomposition for a normal basis generator”, Communications in Algebra, 17 (1989), 1337–1352.
L. Rédei, Algebra, Pergamon Press, Oxford, New York, 1967.
S. Schwarz, “Construction of normal bases in cyclic extensions of a field”, Czechslovak Math. J., 38 (1988), 291–312.
S. Schwarz, “Irreducible polynomials over finite fields with linearly independent roots”, Math. Slovaca, 38 (1988), 147–158.
G.E. Séguin, “Low complexity normal bases for F 2 mn ”, Discrete Applied Math., 28 (1990), 309–312.
I.A. Semaev, “Construction of polynomials irreducible over a finite field with linearly independent roots”, Math. USSR Sbornik, 63 (1989), 507–519.
B. Van Der Waerden, Algebra, vol. 1, Springer-Verlag, Berlin, 1966.
M. Wang, I. Blake and V. Bhargava, “Normal bases and irreducible polynomials in the finite field Gf(22r)”, preprint, 1990.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Blake, I.F., Gao, X., Mullin, R.C., Vanstone, S.A., Yaghoobian, T. (1993). Normal Bases. In: Menezes, A.J. (eds) Applications of Finite Fields. The Springer International Series in Engineering and Computer Science, vol 199. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2226-0_4
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2226-0_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5130-4
Online ISBN: 978-1-4757-2226-0
eBook Packages: Springer Book Archive