On generalized spread bent partitions

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Official URL: https://dx.doi.org/10.1007/s12095-023-00670-2

Generalized semifield spreads are partitions Γ={U,A1,…,Apk} of Fpm×Fpm obtained from (pre)semifields with a certain additional property, which generalize semifield spreads. In particular, a generalized semifield spread is a bent partition, i.e., every function f:Fpm×Fpm→Fp , which is constant on every set of Γ , such that every c∈ Fp has the same number pk-1 of Ai in the preimage set, is a bent function. We show that from generalized semifield spreads one obtains not only p-ary and vectorial bent functions, but also bent functions f:Fpm×Fpm→B for any abelian group of order ps , s≤ k . We investigate the effect of (pre)semifield isotopisms on generalized semifield spreads. We observe that isotopisms can destroy the bent partition property, and derive conditions on an isotopism between two (pre)semifields such that the corresponding partitions are equivalent bent partitions. Most notably, we show that with some other class of isotopisms, one can obtain inequivalent bent partitions, hence different classes of bent functions. This is in contrast to the situation for classical semifield spreads. The spreads of two isotopic (pre)semifields are always equivalent. Employing the 2-rank of Boolean functions we confirm that generalizations of the Desarguesian spread bent functions, which we call generalized PS ap functions, are in general not in the Maiorana-McFarland class. The generalized PS ap class contains functions which are not Maiorana-McFarland nor partial spread bent functions for any partial spread. Explicitly we determine the 2-rank of some Maiorana-McFarland functions in the generalized PS ap class in terms of Fibonacci numbers.