PS bent functions constructed from finite pre-quasifield spreads

Bent functions are of great importance in both mathematics and information science. The PS class of bent functions was introduced by Dillon in 1974, but functions belonging to this class that can be explicitly represented are only the PSap functions, which were also constructed by Dillon after his introduction of the PS class. In this paper, a technique of using finite pre-quasifield spread from finite geometry to construct PS bent functions is proposed. The constructed functions are in similar styles with the PSap 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-Muller pre-quasifield, the Knuth pre-semifield and the Kantor pre-semifield, three new subclasses of the PS class are obtained. They are the only sub-classes that can be explicitly constructed more than 30 years after the PSap subclass was introduced.