| Title: | Permutations satisfying (▫$P_1$▫) and (▫$P_2$▫) properties and ▫$\ell$▫-optimal bent functions |
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| Authors: | ID Kudin, Sadmir (Author)
ID Pašalić, Enes (Author)
ID Polujan, Alexandr (Author)
ID Zhang, Fengrong (Author)
ID Zhao, Haixia (Author) |
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| Files: |  RAZ_Kudin_Sadmir_2026.pdf (391,43 KB)
MD5: D4F9236C533F61160ABA06FA7E738688
 https://link.springer.com/article/10.1007/s00145-025-09562-5
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FAMNIT - Faculty of Mathematics, Science and Information Technologies
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| Abstract: | An 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). |
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| Keywords: | bent functions, Maiorana-McFarland class, permutations |
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| Publication date: | 11.11.2025 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 1-22 |
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| Numbering: | Vol. 39, iss. 1, article no. 5 |
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| PID: | 20.500.12556/RUP-22356  |
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| UDC: | 51 |
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| ISSN on article: | 0933-2790 |
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| DOI: | 10.1007/s00145-025-09562-5 |
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| COBISS.SI-ID: | 263051779 |
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| Publication date in RUP: | 30.12.2025 |
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| Views: | 61 |
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| Downloads: | 2 |
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