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Title:
Applications of the indirect sum in the design of several special classes of bent functions outside the completed \mathcal{MM} class
Authors: Fengrong Zhang, Enes Pasalic, Amar Bapić, Baocang Wang
Abstract: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.
ePrint: https://eprint.iacr.org/2022/1587
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