More permutations and involutions for constructing bent functions

Abstract

Bent functions are extremal combinatorial objects with several applications, such as coding theory, maximum length sequences, cryptography, the theory of difference sets, etc. Based on C. Carlet’s secondary construction, S. Mesnager proposed in 2014 an effective method to construct bent functions in their bivariate representation by employing three permutations of the finite field ${\mathbb {F}}_{2^{m}}$ satisfying an algebraic property $(\mathcal {A}_{m})$. This paper is devoted to constructing permutations that satisfy the property $(\mathcal {A}_{m})$ and then obtaining some explicit bent functions. Firstly, we construct one class of involutions from vectorial functions and further obtain some explicit bent functions by choosing some triples of these involutions satisfying the property $(\mathcal {A}_{m})$. We then investigate some bent functions by involutions from trace functions and linearized polynomials. Furthermore, based on several triples of permutations (not all involutions) that satisfy the property $(\mathcal {A}_{m})$ constructed by D. Bartoli et al., we give some more general results and extend most of their work. Then we also find several general triples of permutations that can also satisfy the property $(\mathcal {A}_{m})$.