There are infinitely many bent functions for which the dual is not bent

Abstract: Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs since their duals are also bent functions. In general this does not apply to non-weaky regular bent functions. However, the first known construction of non-weakly regular bent functions by Ce\c{s}melio\u{g}lu et {\it al.}, 2012, yields bent functions for which the dual is also bent. In this paper the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions non-dual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Ce\c{s}melio\u{g}lu et {\it al.} one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known.