Introduction to Finite Fields and Bases
This introductory chapter contains some basic results on bases for finite fields that will be of interest or use throughout the book. The concentration is on the existence of certain types of bases, their duals and their enumeration. There has been considerable activity in this area in the past decade and while many of the questions are resolved, a few of the important ones remain open. The presentation here tries to complement that of Lidl and Niederreiter  although there is some unavoidable overlap. For a more extensive treatment of the topics covered in this chapter, we recommend the recent book by D. Jungnickel . For the remainder of this section some basic properties of the trace and norm functions are recalled.
KeywordsFinite Field Normal Basis Dual Basis Primitive Element Polynomial Basis
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- T. Beth, W. Geiselmann and D. Jungnickel, “A note on orthogonal circulant matrices over finite fields” , preprint, 1992.Google Scholar
- K. Byrd and T. Vaughan, “Counting and constructing orthogonal circulants” , J. of Combinatorial Theory, A 24 (1978), 34–49.Google Scholar
- D. Gollmann, Algorithmenentwurf in der Kryptographie, Habilitationsschrift, FB Informatik der Universität Karlsruhe, 1990.Google Scholar
- I. Imamura “On self-complementary bases of GF (qn) over GF (q)” , Trans. IECE Japan (Section E), 66 (1983), 717–721.Google Scholar
- K. Imamura and M. Morii, “Two classes of finite fields which have no self-complementary normal bases”, IEEE Int’l Symp. Inform. Theory, Brighton, EnglAnd, June, 1985.Google Scholar
- D. Jungnickel, “Trace-orthogonal normal bases”, Discrete Applied Math., to appear.Google Scholar
- R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1987.Google Scholar
- A. Menezes, Representations in Finite Fields, M.Math. thesis, Department of Combinatorics and Optimization, University of Waterloo, 1989.Google Scholar