Characterization of Basic 5-Value Spectrum Functions Through Walsh-Hadamard Transform

Hodžić, Samir; Horák, Peter; Pašalić, Enes

doi:10.1109/tit.2020.3044059

Cited by 8 publications

(3 citation statements)

References 30 publications

(60 reference statements)

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“…Furthermore, it was noted in [17] that different orderings of S f , both with respect to the choice of v so that S f = v ⊕ E as well as representing it differently so that S f = v ⊕ E (with v = v and E = E ), essentially give affine equivalent duals f * and f * , see Section 5 in [17] for further details. Nevertheless, all these results use the assumption that item (i) in Lemma 3.1 in [16] is satisfied.…”

Section: Plateaued Functions and Their Dualsmentioning

confidence: 99%

“…We first recall certain notations, introduced in [14] and also used in [17], useful in handling a 5-valued spectra Boolean function which has two different non-zero absolute values.…”

Section: Specifying 5-valued Spectra Functions Through Dualsmentioning

confidence: 99%

“…Hence, without loss of generality, we can take α = (α (1) , α (2) , α (3) , 0, 0) and β = (β (1) , β (2) , β (3) , 0, 0) for some distinct nonzero α 12 , β 12 . From (17), for w = 0 2 and α (4) = β (4) = (0, 0), we have that…”

Section: Bent Functions Outside MM # Using Semi-bent Functions With S...mentioning

confidence: 99%

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

Pašalić

Bapić

Zhang

et al. 2023

Des. Codes Cryptogr.

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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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Section: Plateaued Functions and Their Dualsmentioning

confidence: 99%

“…We first recall certain notations, introduced in [14] and also used in [17], useful in handling a 5-valued spectra Boolean function which has two different non-zero absolute values.…”

Section: Specifying 5-valued Spectra Functions Through Dualsmentioning

confidence: 99%

See 1 more Smart Citation

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

Pašalić

Bapić

Zhang

et al. 2023

Des. Codes Cryptogr.

Self Cite

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Constructing totally disjoint spectra plateaued functions and searching five-value spectrum functions in odd variables

Zhang³

et al. 2022

Discrete Applied Mathematics

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Quadratic almost bent functions—Their partial characterization and design in the spectral domain

Bapić

Hodžić

Pašalić

2022

Discrete Applied Mathematics

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Characterization of Basic 5-Value Spectrum Functions Through Walsh-Hadamard Transform

Abstract: The first and the third authors recently introduced a spectral construction of plateaued and of 5-value spectrum functions. In particular, the design of the latter class requires a specification of integers {W (u) : u ∈ F n 2 }, where W (u) ∈ {0, ±2 n+s 1 2

Cited by 8 publications

References 30 publications

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

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

Constructing totally disjoint spectra plateaued functions and searching five-value spectrum functions in odd variables

Quadratic almost bent functions—Their partial characterization and design in the spectral domain

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