Equivalence for negabent functions and their relative difference sets
Anbar Meidl, Nurdagül and Meidl, Wilfried and Pott, Alexander (2019) Equivalence for negabent functions and their relative difference sets. Discrete Applied Mathematics, 270 . pp. 1-12. ISSN 0166-218X (Print) 1872-6771 (Online)
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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.
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