Overview
Research published on arXiv presents a method for constructing two new classes of bent functions. These functions map from $\mathbb{F}_p^{2n}$ to $\mathbb{F}_p$, where $p$ is an odd prime, $n$ is a positive integer, and $m$ is an integer that divides $n$. The construction utilizes $m$-dimensional partial spreads within the vector space $\mathbb{F}_p^{2n}$.
Research Context
The work directly builds upon and generalizes prior research in the field of bent functions. Specifically, it extends the classes of $p$-ary $\mathcal{PS}^{-}$ and $\mathcal{PS}^{+}$ bent functions. These original classes were introduced by P. Lison\v ek and H. Y. Lu, and published in Des. Codes Cryptogr. 73 (2014), 209–216.
Approach
The construction methodology involves the application of $m$-dimensional partial spreads. These partial spreads are defined within the vector space $\mathbb{F}_p^{2n}$. The parameters for this construction are specific: $p$ must be an odd prime, $n$ must be a positive integer, and $m$ must be an integer divisor of $n$. This framework provides the basis for deriving the new classes of bent functions.
Findings
The research successfully constructed two distinct classes of bent functions. These functions operate on the domain $\mathbb{F}_p^{2n}$ and yield outputs in $\mathbb{F}_p$. The construction is characterized by its use of $m$-dimensional partial spreads of $\mathbb{F}_p^{2n}$. A key attribute of these newly constructed functions is their generalization of the existing $p$-ary $\mathcal{PS}^{-}$ and $\mathcal{PS}^{+}$ bent function classes.