Completed GPS Covers All Bent Functions

In a recent paper, the so-called “generalized partial spread” (GPS) class which unifies almost all the known classes of bent functions is introduced. A necessary condition for a bent function to belong to GPS is that it takes the same value as its dual at the zero vector. In this paper, it is shown that the necessary condition above is sufficient. This proves that the completed class by composition with translations covers all the binary bent functions. Moreover, the elements of GPS are characterized in term of solutions of a quadratic Diophantine equation which may lead to count all bent functions. These results are presented in the general framework of partial bent functions which unify bent functions and r-dimensional vector space indicators.