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Bent polynomials over finite fields

Published online by Cambridge University Press:  17 April 2009

Robert S. Coulter  and
Rex W. Matthews
Robert S. Coulter
Affiliation:
Department of Computer ScienceThe University of QueenslandQueensland 4072Australia e-mail: shrub@cs.uq.edu.au
Rex W. Matthews
Affiliation:
Department of MathematicsUniversity of TasmaniaGPO Box 252CHobart Tas 7000Australia e-mail: galois@hilbert.maths.utas.edu.au
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Abstract

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The definition of bent is redefined for any finite field. Our main result is a complete description of the relationship between bent polynomials and perfect non-linear functions over finite fields: we show they are equivalent. This result shows that bent polynomials can also be viewed as the generalisation to several variables of the class of polynomials known as planar polynomials. An explicit method for obtaining large sets of not necessarily distinct maximal orthogonal systems using bent polynomials is given and we end with a short discussion on the existence of bent polynomials over finite fields.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 429 - 437
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Ambrosimov, A.S., ‘Properties of bent functions of q-valued logic over finite fields’, Discrete Math. Appl. 4 (1994), 341–350.Google Scholar
[2]Coulter, R.S., Bent functions and their applications, Honour's thesis (Department of Computer Science, University of Queensland, Australia, 1993).Google Scholar
[3]Coulter, R.S. and Matthews, R.W., ‘Planar functions and planes of Lenz-Barlotti class II’, Des. Codes Cryptogr. 10 (1997), 167–184.CrossRefGoogle Scholar
[4]Dembowski, P. and Ostrom, T.G., ‘Planes of order n with collineation groups of order n 2’, Math. Z. 103 (1968), 239–258.CrossRefGoogle Scholar
[5]Kumar, P.V., Scholtz, R.A., and Welch, L.R., ‘Generalized bent functions and their properties’, J. Combin. Theory Ser. A 40 (1985), 90–107.CrossRefGoogle Scholar
[6]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia Math. Appl. 20 (Addison-Wesley, Reading, 1983).Google Scholar
[7]Niederreiter, H., ‘Orthogonal systems of polynomials in finite fields’, Proc. Amer. Math. Soc. 28 (1971), 415–422.CrossRefGoogle Scholar
[8]Nyberg, K., ‘Constructions of bent functions and difference sets’, in Eurocrypt '90, Lecture Notes in Comput. Sci. 473 (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 151–160.Google Scholar
[9]Rónyai, L. and Szőonyi, T., ‘Planar functions over finite fields’, Combinatorica 9 (1989), 315–320.CrossRefGoogle Scholar
[10]Rothaus, O.S., ‘On “Bent” functions’, J. Combin. Theory Ser. A 20 (1976), 300–305.CrossRefGoogle Scholar