Equivalence for negabent functions and their relative difference sets

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Official URL: http://dx.doi.org/10.1016/j.dam.2019.07.018

A bent function from Fn 2 to F2, n even, can be transformed into a negabent function, or slightly more general into a bent4, also called shifted bent function, by adding a certain quadratic term. If n is odd, then negabent functions similarly correspond to semibent functions with some additional property. Whereas bent functions induce relative difference sets in Fn 2 ×F2, negabent functions induce relative difference sets in Fn−1 2 ×Z4. We analyse equivalence of negabent functions respectively of their relative difference sets. We show that equivalent bent functions can correspond to inequivalent negabent functions, hence one can obtain inequivalent relative difference sets in Fn−1 2 ×Z4 with EA-equivalence. We also show that this is not the case when n is odd. Finally we analyse the class of semibent functions that corresponds to negabent functions and show that though partially bent semibent functions always can be shifted to negabent or bent4 functions, there are many semibent functions which do not correspond to negabent and bent4 functions.