Paper 2022/1587

Applications of the indirect sum in the design of several special classes of bent functions outside the completed $\mathcal{MM}$ class

Fengrong Zhang, Xidian University
Enes Pasalic, University of Primorska
Amar Bapić, University of Primorska
Baocang Wang

Two main secondary constructions of bent functions are the direct and indirect sum methods. We show that the direct sum, under more relaxed conditions compared to those in \cite{PolujanandPott2020}, can generate bent functions provably outside the completed Maiorana-McFarland class ($\mathcal{MM}^\#$). We also show that the indirect sum method, though imposing certain conditions on the initial bent functions, can be employed in the design of bent functions outside $\mathcal{MM}^\#$. Furthermore, applying this method to suitably chosen bent functions we construct several generic classes of homogenous cubic bent functions (considered as a difficult problem) that might posses additional properties (namely without affine derivatives and/or outside $\mathcal{MM}^\#$). Our results significantly improve upon the best known instances of this type of bent functions given by Polujan and Pott \cite{PolujanandPott2020}, and additionally we solve an open problem in \cite[Open Problem 5.1]{PolujanandPott2020}. More precisely, we show that one class of our homogenous cubic bent functions is non-decomposable (inseparable) so that $h$ under a non-singular transform $B$ cannot be represented as $h(xB)=f(y)\oplus g(z)$. Finally, we provide a generic class of vectorial bent functions strongly outside $\mathcal{MM}^\#$ of relatively large output dimensions, which is generally considered as a difficult task.

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Publication info
Bent functions Direct and indirect sum Completed classes Homogenous bent functions Strongly outside
Contact author(s)
amar bapic @ famnit upr si
2022-11-17: approved
2022-11-15: received
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      author = {Fengrong Zhang and Enes Pasalic and Amar Bapić and Baocang Wang},
      title = {Applications of the indirect sum in the design of several special classes of bent functions outside the completed $\mathcal{MM}$ class},
      howpublished = {Cryptology ePrint Archive, Paper 2022/1587},
      year = {2022},
      note = {\url{}},
      url = {}
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