Design and analysis of bent functions using ā³-subspaces
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In this article, we provide the first systematic analysis of bent functions f on š½_2^n in the Maiorana-McFarland class ā³ā³ regarding the origin and cardinality of their ā³-subspaces, i.e., vector subspaces on which the second-order derivatives of f vanish. By imposing restrictions on permutations Ļ of š½_2^n/2, we specify the conditions, such that Maiorana-McFarland bent functions f(x,y)=xĀ·Ļ(y) + h(y) admit a unique ā³-subspace of dimension n/2. On the other hand, we show that permutations Ļ with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of ā³-subspaces is invariant under equivalence. Additionally, we give several generic methods of specifying permutations Ļ so that fāā³ā³ admits a unique ā³-subspace. Most notably, using the knowledge about ā³-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions, one can in a generic manner generate bent functions on š½_2^n outside the completed Maiorana-McFarland class ā³ā³^# for any even nā„ 8. Remarkably, with our construction methods it is possible to obtain inequivalent bent functions on š½_2^8 not stemming from two primary classes, the partial spread class š«š® and ā³ā³. In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction, of which size is about 2^76, stems from š«š® and ā³ā³, whereas the total number of bent functions on š½_2^8 is approximately 2^106.
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