Vectorial bent functions and partial difference sets

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(β x) = β ^lF(x) for all β∈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 𝔽_p^n to 𝔽_p^s , we show that the preimage set of the squares of 𝔽_p^s forms a partial difference set. This extends earlier results on p -ary bent functions.