1. Permutations satisfying (▫$P_1$▫) and (▫$P_2$▫) properties and ▫$\ell$▫-optimal bent functionsSadmir Kudin, Enes Pašalić, Alexandr Polujan, Fengrong Zhang, Haixia Zhao, 2026, original scientific article 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).
Keywords: bent functions, Maiorana-McFarland class, permutations
Published in RUP: 30.12.2025; Views: 425; Downloads: 2
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2. Almost Maiorana-McFarland bent functionsSadmir Kudin, Enes Pašalić, Alexandr Polujan, Fengrong Zhang, Haixia Zhao, 2025, original scientific article 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#.
Keywords: bent functions, Maiorana-McFarland class, M-subspaces
Published in RUP: 29.12.2025; Views: 241; Downloads: 2
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3. Vectorial negabent concepts : similarities, differences, and generalizationsNurdagül Anbar, Sadmir Kudin, Wilfried Meidl, Enes Pašalić, Alexandr Polujan, 2025, original scientific article Abstract: In Pasalic et al. (IEEE Trans Inf Theory 69:2702–2712, 2023), and in Anbar and Meidl (Cryptogr Commun 10:235–249, 2018), two different vectorial negabent and vectorial bent- negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from Fn 2 to the cyclic group Z2k . It is shown how to obtain nega-Z2k -bent functions from Z2k -bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of Z8 -bent functions employing per- mutations with the (Am ) property, and more generally we show that the inverse permutation gives rise to Z2k -bent functions.
Keywords: bent function, Maiorana-McFarland class, negabent functions
Published in RUP: 12.09.2025; Views: 388; Downloads: 12
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4. The algebraic characterization of M-subspaces of bent concatenations and its applicationSadmir Kudin, Enes Pašalić, Alexandr Polujan, Fengrong Zhang, 2025, original scientific article 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.
Keywords: bent function, Maiorana-McFarland class, M-subspaces
Published in RUP: 04.08.2025; Views: 538; Downloads: 6
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8. Konstrukcije novih superrazredov ukrivljenih funkcij in nadaljnje konstrukcije kriptografsko pomembnih preslikav izven M# : doktorska disertacijaAmar Bapić, 2022, doctoral dissertation Keywords: vecotorial bent function, class inclusion, complete Maiorana-McFaralnd class, MNBC functions, secondary constructions, weakly/almost strongly/strongly outside M#, 4-decomposition, SC and CD class, direct and indirect sum
Published in RUP: 12.12.2022; Views: 3748; Downloads: 35
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10. On characterization of transparency order for (n, m)-functionsYu Zhou, Yongzhuang Wei, Hailong Zhang, LuYang Li, Enes Pašalić, Wenling Wu, 2021, published scientific conference contribution Keywords: (n, m)-functions, transparency order, nonlinearity, cross-correlation
Published in RUP: 18.11.2021; Views: 2636; Downloads: 27
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