$\mathcal{P}\mathcal{S}$ bent functions constructed from finite pre-quasifield spreads

Bent functions are of great importance in both mathematics and information science. The $\mathcal{P}\mathcal{S}$ class of bent functions was introduced by Dillon in 1974, but functions belonging to this class that can be explicitly represented are only the $\mathcal{P}\mathcal{S}_{\text{ap}}$ functions, which were also constructed by Dillon after his introduction of the $\mathcal{P}\mathcal{S}$ class. In this paper, a technique of using finite pre-quasifield spread from finite geometry to construct $\mathcal{P}\mathcal{S}$ bent functions is proposed. The constructed functions are in similar styles with the $\mathcal{P}\mathcal{S}_{\text{ap}}$ functions. To explicitly represent them in bivariate forms, the main task is to compute compositional inverses of certain parametric permutation polynomials over finite fields of characteristic 2. Concentrated on the Dempwolff-M\"uller pre-quasifield, the Knuth pre-semifield and the Kantor pre-semifield, three new subclasses of the $\mathcal{P}\mathcal{S}$ class are obtained. They are the only sub-classes that can be explicitly constructed more than 30 years after the $\mathcal{P}\mathcal{S}_{\text{ap}}$ subclass was introduced.