Cubic bent functions outside the completed Maiorana-McFarland class
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Cubic bent functions outside the completed Maiorana-McFarland class Alexandr A. Polujan1
· Alexander Pott1
Received: 3 September 2019 / Revised: 21 December 2019 / Accepted: 24 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we prove that in opposite to the cases of 6 and 8 variables, the MaioranaMcFarland construction does not describe the whole class of cubic bent functions in n variables for all n ≥ 10. Moreover, we show that for almost all values of n, these functions can simultaneously be homogeneous and have no affine derivatives. Keywords Cubic bent functions · Homogeneous functions · Affine derivatives · Equivalence of Boolean functions · Completed Maiorana-McFarland class Mathematics Subject Classification 05B10 · 06E30 · 14G50 · 94C30
1 Introduction Bent functions, introduced by Rothaus in [35], are Boolean functions having the maximum Hamming distance from the set of all affine functions. Being extremal combinatorial objects, they have been intensively studied in the last four decades, due to their broad applications to cryptography, coding theory and theory of difference sets. Cubic bent functions, i.e. bent functions of algebraic degree three, attracted a lot of attention from researchers, partly because small algebraic degree of these functions allows to investigate them exhaustively, when the number of variables is not too large. For instance, all cubic bent functions in six and eight variables are well-understood: the classification is given in [3,35], the enumeration was obtained in [23,33], and all these functions belong to
The first version of this work [30] was presented in the “Eleventh International Workshop on Coding and Cryptography (WCC 2019)”. This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography 2019”.
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Alexandr A. Polujan [email protected]; [email protected] Alexander Pott [email protected]
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Otto von Guericke University, Universitätsplatz 2, 39106 Magdeburg, Germany
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A. A. Polujan, A. Pott
the completed Maiorana-McFarland class M# [3,10]. A couple of infinite families of cubic bent functions were constructed recently, however, some of them [5,24] are proved to be the members of M# , while some of them are not analyzed yet [14,28]. Therefore, it is not clear, whether an n-variable cubic bent function can be outside the M# class whenever n ≥ 10. At the same time, cubic bent functions, which are homogeneous or have no affine derivatives, are of a special interest. A cubic function has no affine derivatives, if all its non-trivial first-order derivatives are quadratic, what makes cryptographic systems with such components more resistant to certain differential attacks. It is well-known that cubic bent functions without affine derivatives exist for all even n ≥ 6, n = 8, as it was shown in in [4,20]. Recently Mandal, Gangopadhyay and St˘anic˘a in [26] constructed two classes of cubic bent functions without affine deriv
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