Abstract
Suppose that n is even. Let \({\mathbb{F}_2}\) denote the two-element field and \({\mathbb{Z}}\) the set of integers. Bent functions can be defined as ± 1-valued functions on \({\mathbb{F}_2^n}\) with ± 1-valued Fourier transform. More generally we call a mapping f on \({\mathbb{F}_2^n}\) a \({\mathbb{Z}}\) -bent function if both f and its Fourier transform \({\widehat{f}}\) are integer-valued. \({\mathbb{Z}}\) -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and \({\widehat{f}}\) . It is shown how \({\mathbb{Z}}\) -bent functions of lower level can be built up recursively by gluing together \({\mathbb{Z}}\) -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of \({\mathbb{Z}}\) -bent functions and give some guidelines for further research.
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Canteaut, A., Charpin, P., Dobbertin, H.: Weight divisibility of cyclic codes, highly nonlinear functions on GF(2n) and crosscorrelation of maximum-length sequences. Siam J. Discrete Math. 13, 105–138 (2000)
Canteaut, A., Charpin, P., Dobbertin, H.: Three-valued crosscorrelation for binary m-sequences: a proof of Welch’s conjecture. IEEE Trans. Inform. Theory 46, 4–8 (2000)
Carlet, C., Dobbertin, H., Leander, G.: Normal extensions of bent functions. IEEE Trans. Inform. Theory 50, 2880–2885 (2004)
Carlet, C., Guillot, P.: A characterization of binary bent functions. J. Comb. Theory Ser. A 76, 328–335 (1996)
Carlet, C., Guillot, P.: An alternate characterization of the bentness of binary functions, with uniqueness. Des. Codes Cryptogr. 14, 133–140 (1998)
Cusick, T.W., Dobbertin, H.: Some new three-valued crosscorrelation functions for binary m-sequences. IEEE Trans. Inform. Theory 42, 1238–1240 (1996)
Dillon J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1972).
Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields Appl. 10, 342–389 (2004)
Dobbertin, H., Felke, P., Helleseth, T., Rosendahl, P.: Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums. IEEE Trans. Inform. Theory 52, 613–627 (2006)
Dobbertin H., Leander G.: A survey of some recent results on bent functions. In: Proceedings of the Workshop on Sequences and Their Applications 2004 (SETA ’04), Seoul, Korea, October 2004. Lecture Notes on Computer Science, LNCS 3486, pp. 1–29. Springer Verlag (2005).
Helleseth, T.: Some results about the cross-correlation function between two maximal linear sequences. Discrete Math. 16, 209–232 (1976)
Leander G.: Normality of bent functions, monomial and binomial bent functions. Ph.D. thesis, Ruhr-University of Bochum (2004).
Niho Y.: Multivalued cross-correlation functions between two maximal linear recursive sequences. Ph.D. thesis, University of Southern California (1972).
Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A 20, 300–305 (1976)
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This is a posthumous publication, Hans Dobbertin finished this paper in early 2005 and was not able to publish it anymore. He died on February 2, 2006.
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Dobbertin, H., Leander, G. Bent functions embedded into the recursive framework of \({\mathbb{Z}}\) -bent functions. Des. Codes Cryptogr. 49, 3–22 (2008). https://doi.org/10.1007/s10623-008-9189-3
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DOI: https://doi.org/10.1007/s10623-008-9189-3