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    Vectorial bent functions and partial difference sets
    (Springer, 2021) Cesmelioglu, Ayca; Meidl, Wilfried; Pirsic, Isabel
    The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between p-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions F : F-p(n) -> F-p(s) are linear equivalent to l-forms, i.e., to functions satisfying F(beta x) = beta(l) F(x) for all beta is an element of F-p(s), we investigate properties of partial difference sets obtained from l-forms. We show that they are unions of cosets of F* p(s), which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from Fp(n) to Fp(s), we show that the preimage set of the squares of Fps forms a partial difference set. This extends earlier results on p-ary bent functions.
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    Vectorial bent functions in odd characteristic and their components
    (Springer, 2020) Cesmelioglu, Ayca; Meidl, Wilfried; Pott, Alexander
    Bent functions in odd characteristic can be either (weakly) regular or non-weakly regular. Furthermore one can distinguish between dual-bent functions, which are bent functions for which the dual is bent as well, and non-dual bent functions. Whereas a weakly regular bent function always has a bent dual, a non-weakly regular bent function can be either dual-bent or non-dual-bent. The classical constructions (like quadratic bent functions, Maiorana-McFarland or partial spread) yield weakly regular bent functions, but meanwhile one knows constructions of infinite classes of non-weakly regular bent functions of both types, dual-bent and non-dual-bent. In this article we focus on vectorial bent functions in odd characteristic. We first show that mostp-ary bent monomials and binomials are actually vectorial constructions. In the second part we give a positive answer to the question if non-weakly regular bent functions can be components of a vectorial bent function. We present the first construction of vectorial bent functions of which the components are non-weakly regular but dual-bent, and the first construction of vectorial bent functions with non-dual-bent components.