*01:17*[Pub][ePrint] More PS and H-like bent functions, by C. Carlet

Two constructions/classes of bent functions are derived from the notion of spread. The first class, ${\\cal PS}$, gives a useful framework for designing bent functions which are constant (except maybe at 0) on the elements of a (partial) spread. Dillon has deduced the explicit class ${\\cal PS}_{ap}$ of bent functions obtained from the spread of all multiplicative cosets of ${\\Bbb F}_{2^m}^*$ (added with 0) in ${\\Bbb F}_{2^{2m}}^*$ (that we shall call the Dillon spread). The second class, $H$, later slightly modified into a class called ${\\cal H}$ so as to relate it to the so-called Niho bent functions, is up to addition of affine functions the set of bent functions whose restrictions to the spaces of the Dillon spread are linear. It has been observed that the functions in ${\\cal H}$ are related to o-polynomials, and this has led to several explicit classes of bent functions. In this paper we first apply the ${\\cal PS}$ construction to a larger class of spreads, well-known in the finite geometry domain and that we shall call Andr\\\'e\'s spreads, and we describe explicitly the ${\\cal PS}$ corresponding bent functions and their duals. We also characterize those bent functions whose restrictions to the spaces of an Andr\\\'e spread are linear. This leads to a notion extending that of o-polynomial. Finally, we obtain similar characterizations for the ${\\cal H}$-like functions derived from the spreads used by Wu to deduce ${\\cal PS}$ bent functions from the Demp\\-wolff-M\\\"uller pre-quasifield, the Knuth pre-semifield and the Kantor pre-semifield. In each case, this also leads to a new notion on polynomials.