Bent Functions in $$\mathcal C$$ and $$\mathcal D$$ Outside the Completed Maiorana-McFarland Class

Zhang, F.; Pašalić, Enes; Cepak, Nastja; Wei, Yongzhuang

doi:10.1007/978-3-319-55589-8_20

Cited by 10 publications

(4 citation statements)

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“…This also indicates that the choice of a non-affine Walsh support is not decisive for the class inclusion since the support S f in Theorem 3.5 is affine when t 1 = t 2 = 0. Since g ∈ B n is supposed to be a bent function outside MM # (with additional restriction that r -ind(g) < n/2 − 2), we can employ the class D 0 of Carlet [4] or certain families of bent functions in C and D that are provably outside MM # [18,26,28]. Alternatively g can be taken from the recent classes SC and CD [1,2], which are specified in Corollary 3 below.…”

Section: Remark 33mentioning

confidence: 99%

“…Alternatively g can be taken from the recent classes SC and CD [1,2], which are specified in Corollary 3 below. Notice that the subspaces L, E 1 , E 2 used to define g in Corollary 3 below, satisfy certain conditions with respect to the permutation π, see [4,26,28]. However, there exist efficient design methods for specifying bent functions in the above classes that are provably outside MM # [1,2,18,26,28].…”

Section: Remark 33mentioning

confidence: 99%

“…Notice that the subspaces L, E 1 , E 2 used to define g in Corollary 3 below, satisfy certain conditions with respect to the permutation π, see [4,26,28]. However, there exist efficient design methods for specifying bent functions in the above classes that are provably outside MM # [1,2,18,26,28]. On the other hand, for t 1 , t 2 ∈ B n we use certain indicators that preserve the bentness of g(x, y) ⊕ v 1 t 1 (x, y) ⊕ v 2 t 2 (x, y).…”

Section: Remark 33mentioning

confidence: 99%

“…A non-exhaustive list of various secondary constructions can be found in the following works [4,6,9,16,20,24,30]. However, the question regarding the class inclusion of bent functions stemming from these secondary construction methods is commonly left open, apart from a few works [1,4,18,20,[26][27][28] where some explicit families of bent functions provably outside the completed MM class are given. The main purpose of this article is to address the class inclusion more properly and thus also to contribute to a classification of bent functions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Explicit infinite families of bent functions outside the completed Maiorana–McFarland class

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et al. 2023

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During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland ($${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # ) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # and eventually we obtain many infinite families of bent functions that are provably outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . The fact that a bent function f is in/outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # if and only if its dual is in/outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # is employed in the so-called 4-decomposition of a bent function on $${\mathbb {F}}_2^n$$ F 2 n , which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an $$(n-2)$$ ( n - 2 ) -dimensional subspace V. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . For instance, for the elementary case of defining a bent function $$h(\textbf{x},y_1,y_2)=f(\textbf{x}) \oplus y_1y_2$$ h ( x , y 1 , y 2 ) = f ( x ) ⊕ y 1 y 2 on $${\mathbb {F}}_2^{n+2}$$ F 2 n + 2 using a bent function f on $${\mathbb {F}}_2^n$$ F 2 n , we show that h is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # if and only if f is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $$f_1||f_1||f_2||(1\oplus f_2)$$ f 1 | | f 1 | | f 2 | | ( 1 ⊕ f 2 ) also gives bent functions outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # if $$f_1$$ f 1 or $$f_2$$ f 2 is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # are provided.

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