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Duals of non-weakly regular bent functions are not weakly regular and generalization to plateaued functions

Özbudak, Ferruh
It is known that the dual of a weakly regular bent function is again weakly regular. On the other hand, the dual of a nonweakly regular bent function may not even be a bent function. In 2013, cesmelioglu, Meidl and Pott pointed out that the existence of a non-weakly regular bent function having weakly regular bent dual is an open problem. In this paper, we prove that for an odd prime p and n is an element of Z+, if f : F-p(n)-> F-p is a non-weakly regular bent function such that its dual f* is bent, then f**(-x) = f(x), and f* is non-weakly regular, which solves the open problem. We also generalize our results to plateaued functions.