Design and analysis of bent functions using ℳ-subspaces
share
In this article, we provide the first systematic analysis of bent functions f on 𝔽_2^n in the Maiorana-McFarland class ℳℳ regarding the origin and cardinality of their ℳ-subspaces, i.e., vector subspaces on which the second-order derivatives of f vanish. By imposing restrictions on permutations π of 𝔽_2^n/2, we specify the conditions, such that Maiorana-McFarland bent functions f(x,y)=x·π(y) + h(y) admit a unique ℳ-subspace of dimension n/2. On the other hand, we show that permutations π 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 ℳ-subspaces is invariant under equivalence. Additionally, we give several generic methods of specifying permutations π so that f∈ℳℳ admits a unique ℳ-subspace. Most notably, using the knowledge about ℳ-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 𝔽_2^n outside the completed Maiorana-McFarland class ℳℳ^# for any even n≥ 8. Remarkably, with our construction methods it is possible to obtain inequivalent bent functions on 𝔽_2^8 not stemming from two primary classes, the partial spread class 𝒫𝒮 and ℳℳ. 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 𝒫𝒮 and ℳℳ, whereas the total number of bent functions on 𝔽_2^8 is approximately 2^106.
READ FULL TEXT
