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Cubic bent functions outside the completed Maiorana-McFarland class. (English) Zbl 1453.94173

The bent functions are Boolean functions having the maximum Hamming distance from the set of all affine functions. They have been intensively studied in the last four decades, due to their broad applications to cryptography, coding theory and theory of difference sets. The cubic bent functions are studied quite exhaustively when the number of their variables is not too large. The classification and the enumeration of all cubic bent functions in six and eight variables is obtained, and it is proved that all these functions belong to the completed Maiorana-McFarland class \(M^{\sharp}\). It is an open question whether an \(n\)-variable cubic bent function can be outside the \(M^{\sharp}\) class whenever \(n \geq 10\).
In this paper, the known homogeneous cubic bent functions in ten and twelve variables from [C. Charnes et al., Des. Codes Cryptography 26, No. 1–3, 139–154 (2002; Zbl 1026.06015)] and [Q. Meng et al., Cryptology ePrint Archive, Report 2004/274 (2004)] are analysed and it is shown, that some of these functions do not belong to the \(M^{\sharp}\) class, and all of them are different from the primary construction of J. Seberry et al. [Australas. J. Comb. 22, 233–245 (2000; Zbl 0984.94043)]. It is worthwhile to notice that some of them have no affine derivatives.
Furthermore, these results are extended for infinite families, by showing, that proper direct sums of these functions inherit the properties of its summands, and so it is proved that for any \(n \geq 8\) there exist cubic Bent functions inside \(M^{\sharp}\), but different from the primary construction. Moreover, cubic Bent functions outside \(M^{\sharp}\), without affine derivatives, and homogeneous are considered, and it is shown that \(n\)-variable cubic bent functions with at least two of the three mentioned properties exist for all \(n \geq n_0\), where \(n_0\) depends on the selected combination of properties.
Finally, it is proved that cubic bent functions without affine derivatives exist outside \(M^{\sharp}\) class, which solves a recent open problem by [B. Mandal et al., “Cubic Maiorana-McFarland bent functions with no affine derivative”, Int. J. Comput. Math. Comput. Syst. Theory 2, No. 1, 14–27 (2017; doi:10.1080/23799927.2017.1304453)].

MSC:

94D10 Boolean functions
94A60 Cryptography
06E30 Boolean functions
14G50 Applications to coding theory and cryptography of arithmetic geometry
94C30 Applications of design theory to circuits and networks

Software:

Mathematica; Magma
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References:

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