Permutations satisfying (▫$P_1$▫) and (▫$P_2$▫) properties and ▫$\ell$▫-optimal bent functionsKudin, Sadmir (Avtor) Pašalić, Enes (Avtor) Polujan, Alexandr (Avtor) Zhang, Fengrong (Avtor) Zhao, Haixia (Avtor) bent functionsMaiorana-McFarland classpermutationsAn important classification of permutations over Fm 2 , suitable for construct- ing Maiorana-McFarland bent functions on Fm 2 × Fm 2 with the unique M-subspace of maximal dimension, was recently considered in Pasalic et al (IEEE Trans Inf Theory 70:4464–4477, 2024). More precisely, two properties called (P1) and (P2) were in- troduced and a generic method of constructing permutations having the property (P1) was presented, whereas no such results were provided related to the (P2) property. In this article, we provide a deeper insight on these properties, their mutual relationship, and specify some explicit classes of permutations having these properties. Such per- mutations are then employed to generate a large variety of bent functions outside the completed Maiorana-McFarland class M# . We also introduce -optimal bent functions as bent functions with the lowest possible linearity index; such functions can be consid- ered as opposite to Maiorana-McFarland bent functions. We give explicit constructions of -optimal bent functions within the D0 class, which in turn can be employed in cer- tain secondary constructions of bent functions (Zhang et al in Inf Comput 297:105149, 2024) for providing even more classes of bent functions that are provably outside M# . Moreover, we demonstrate that a certain subclass of D0 has an additional property of having only 5-valued spectra decompositions, similarly to the only result in this direction concerning monomial bent functions (Canteaut and Charpin in IEEE Trans Inf Theory 498:2004–2019, 2003). Finally, we generalize the so-called swapping variables method introduced in Pasalic et al. (IEEE Trans Inf Theory 70:4464–4477, 2024) which then allows us to specify much larger families of bent functions outside M# compared to Pasalic et al (IEEE Trans Inf Theory 70:4464–4477, 2024). In this way, we give a better explanation of the origin of bent functions in dimension eight, since the vast majority of them is outside M# , as indicated in Langevin and Leander (Designs Codes Cryptogr 59:193–205, 2011).20262025-12-30 10:29:07Članek v reviji22356UDK: 51ISSN pri članku: 0933-2790DOI: 10.1007/s00145-025-09562-5COBISS.SI-ID: 263051779sl