The Algebraic Characterization of ℳ-Subspaces of Bent Concatenations and Its Application

Every Boolean bent function f can be written either as a concatenation $f=f_{1}|| f_{2}$ of two complementary semi-bent functions $f_{1},f_{2}$ ; or as a concatenation $f=f_{1}|| f_{2}|| f_{3}|| f_{4}$ of four Boolean functions $f_{1},f_{2},f_{3},f_{4}$ , all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to specify conditions for these bent concatenations so that f does (not) belong to the completed Maiorana-McFarland class ${\mathcal {M}}^{\#}$ . In this article, we resolve this question completely by providing the algebraic characterization of $\mathcal {M}$ -subspaces for the concatenation of the form $f=f_{1}|| f_{2}$ and $f=f_{1}|| f_{2}|| f_{3}|| f_{4}$ , which allows us to estimate ${\rm {ind}}(f)$ , the linearity index of f, and consequently to establish the necessary and sufficient conditions so that f is outside ${\mathcal {M}}^{\#}$ . Based on these conditions, we propose several explicit and generic design methods of specifying bent functions outside ${\mathcal {M}}^{\#}$ in the special case when $f=g||h||g||(h+1)$ , where g and h are bent functions. Moreover, we show that it is possible to even decrease the linearity index of $f = g||h||g||(h+1)$ , compared to ${\rm {ind}}(g)$ and ${\rm {ind}}(h)$ , if the largest dimension of a common $\mathcal {M}$ -subspace of g and h is small enough (less than $\min \{{\rm {ind}}(g), {\rm {ind}}(h)\} - 1$ ). This also induces iterative methods of constructing bent functions outside ${\mathcal {M}}^{\#}$ with (controllable) low linearity index. Finally, we derive a lower bound on the 2-rank of f and show that this concatenation method can generate bent functions that are provably outside ${\mathcal {M}}^{\#} \cup {\mathcal {PS}}_{ap}^{\#}$ . In difference to the approach of Weng et al. (2007) that uses the direct sum and a bent function g outside ${\mathcal {M}}^{\#}$ , our method employs $g, h \in {\mathcal {M}}^{\#}$ for the same purpose.