| Title: | Almost Maiorana-McFarland 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_2025.pdf (370,37 KB)
MD5: C4221A8F538AC0351713DF98C9E8A482
 https://ieeexplore.ieee.org/document/11180145
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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: | In this article, we study bent functions on F2m 2 of the form f (x, y) = x·φ(y)+h(y), where x ∈ Fm−1 2 and y ∈ Fm+1 2 , which form the generalized Maiorana-McFarland class (denoted by GMMm+1) and are referred to as almost Maiorana-McFarland bent functions. We provide a complete characterization of the bent property for such functions and determine their duals. Specifically, we show that f is bent if and only if the mapping φ partitions Fm+1 2 into 2-dimensional affine subspaces, on each of which the function h has odd weight. While the partition of Fm+1 2 into 2-dimensional affine subspaces is crucial for the bentness, we demonstrate that the algebraic structure of these subspaces plays an even greater role in ensuring that the constructed bent func- tions f are excluded from the completed Maiorana-McFarland class M# (the set of bent functions that are extended-affine equivalent to bent functions from the Maiorana-McFarland class M). Consequently, we investigate which properties of mappings φ : Fm+1 2 → Fm−1 2 lead to bent functions of the form f (x, y) = x · φ(y) + h(y) both inside and outside M# and provide construction methods for suitable Boolean functions h on Fm+1 2 . As part of this framework, we present a simple algorithm for constructing partitions of the vector space Fm+1 2 together with appropriate Boolean functions h that generate bent functions outside M#. When 2m = 8, we explicitly identify many such partitions that produce at least 278 distinct bent functions on F8 2 that do not belong to M#, thereby generating more bent functions outside M# than the total number of 8-variable bent functions in M# (whose cardinality is approximately 277). Additionally, we demonstrate that concatenating four almost Maiorana-McFarland bent functions outside M#, i.e., defining f = f1|| f2|| f3|| f4 where fi < M#, can result in a bent function f ∈ M#. This finding essentially answers an open problem posed recently in Kudin et al. (IEEE Trans. Inf. Theory 71(5): 3999- 4011, 2025). Conversely, using a similar approach to concatenate our functions f1|| f2|| f3|| f4, where each fi ∈ M#, we generate bent functions that are provably outside M#. |
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| Keywords: | bent functions, Maiorana-McFarland class, M-subspaces |
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| Publication version: | Version of Record |
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| Publication date: | 25.09.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 9698-9713 |
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| Numbering: | Vol. 71, no. 12 |
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| PID: | 20.500.12556/RUP-22352  |
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| UDC: | 51 |
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| ISSN on article: | 0018-9448 |
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| DOI: | 10.1109/TIT.2025.3614379 |
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| COBISS.SI-ID: | 263044867 |
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| Publication date in RUP: | 29.12.2025 |
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| Views: | 54 |
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| Downloads: | 2 |
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