Further constructions of bent functions and their duals

Abstract

In 2012, Carlet et al. developed two secondary constructions of bent functions (Advances in Mathematics of Communications, 6: 305-314) and proposed some applications for their constructions. However, the duals of bent functions in their constructions were not presented. In order to find more general applications to these constructions and obtain new classes of bent functions, an open problem was proposed by Carlet in 2014. Hence, in this study, a class of PS vectorial bent functions for answering that open problem, which also addresses another open problem on vectorial bent functions proposed by Mesnager in 2014, is constructed. In addition, a new secondary construction of bent functions that generalises one of Carlet et al.'s constructions in 2012 is presented. Based on that, two new classes of bent functions were obtained and their duals were presented explicitly. In particular, some self-dual bent functions are constructed. Moreover, it can be proved that our bent functions can be EA-inequivalent to those constructed by Carlet et al. in 2012.