| Title: | The algebraic characterization of M-subspaces of bent concatenations and its application |
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| Authors: | ID Kudin, Sadmir (Author)
ID Pašalić, Enes (Author)
ID Polujan, Alexandr (Author)
ID Zhang, Fengrong (Author) |
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| Files: |  RAZ_Kudin_Sadmir_2025.pdf (349,74 KB)
MD5: 44FA811FD1EC3875F81453B8F9995D8F
 https://ieeexplore.ieee.org/document/10909699
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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: | Every Boolean bent function f can be written either as a concatenation f = f1|| f2 of two complementary semi-bent functions f1, f2; or as a concatenation f = f1|| f2|| f3|| f4 of four Boolean functions f1, f2, f3, f4, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to specify conditions for these bent concatenations so that f does (not) belong to the completed Maiorana-McFarland class M#. In this article, we resolve this question completely by providing the algebraic characterization of M-subspaces for the concatenation of the form f = f1|| f2 and f = f1|| f2|| f3|| f4, which allows us to estimate ind( f ), the linearity index of f, and consequently to establish the necessary and sufficient conditions so that f is outside M#. Based on these conditions, we propose several explicit and generic design methods of specifying bent functions outside M# in the special case when f = g||h||g||(h+1), where g and h are bent functions. Moreover, we show that it is possible to even decrease the linearity index of f = g||h||g||(h+1), compared to ind(g) and ind(h), if the largest dimension of a common M-subspace of g and h is small enough (less than min{ind(g), ind(h)} − 1). This also induces iterative methods of constructing bent functions outside M# with (controllable) low linearity index. Finally, we derive a lower bound on the 2-rank of f and show that this concatenation method can generate bent functions that are provably outside M# ∪ PS# ap. In difference to the approach of Weng et al. (2007) that uses the direct sum and a bent function g outside M#, our method employs g, h ∈ M# for the same purpose. |
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| Keywords: | bent function, Maiorana-McFarland class, M-subspaces |
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| Publication date: | 04.03.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 3999-4011 |
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| Numbering: | Vol. 71, no. 5 |
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| PID: | 20.500.12556/RUP-21513  |
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| UDC: | 51 |
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| ISSN on article: | 0018-9448 |
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| DOI: | 10.1109/TIT.2025.3547533 |
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| COBISS.SI-ID: | 244763651 |
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| Publication date in RUP: | 04.08.2025 |
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| Views: | 511 |
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| Downloads: | 6 |
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