Generalized Bent Functions - Some General Construction Methods and Related Necessary and Sufficient Conditions

Hodžić, Samir; Pašalić, Enes

doi:10.1007/s12095-015-0126-9

Cited by 23 publications

(28 citation statements)

References 14 publications

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“…Generalizations of Boolean bent functions, like negabent functions and the more general class of gbent functions have lately attracted increasing attention, see e.g. [2,3,4,6,8,9,10,11,12,13,14] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Bent Functions and Their Gray Images

Martinsen

Meidl

Stănică

2016

Arithmetic of Finite Fields

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In this paper we prove that generalized bent (gbent) functions defined on Z n 2 with values in Z 2 k are regular, and find connections between the (generalized) Walsh spectrum of these functions and their components. We comprehensively characterize generalized bent and semibent functions with values in Z 16 , which extends earlier results on gbent functions with values in Z 4 and Z 8 . We also show that the Gray images of gbent functions with values in Z 2 k are semibent/plateaued when k = 3, 4.

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Section: Introductionmentioning

confidence: 99%

“…Remark 22. In [4], conditions are derived for the gbentness of some functions f ∈ GB q n of the form f (x) = q 2 a(x) + rb(x), r ∈ [q/4, 3q/4], a, b in GB q n or B n .…”

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confidence: 99%

Generalized Bent Functions and Their Gray Images

Martinsen

Meidl

Stănică

2016

Arithmetic of Finite Fields

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“…In this paper, we consider such a generalization for which the domain set remains the same as for classical Boolean functions but the range is the set of integers modulo a positive integer q ≥ 2. These generalized Boolean functions have evolved to an active area of research [11,12,14,16,17,18,20,23,24,25,26,27,29] due to several possible applications in communications and cryptography.…”

Section: Introductionmentioning

confidence: 99%

Gowers U_2 norm as a measure of nonlinearity for Boolean functions and their generalizations

Gangopadhyay¹,

Riera²,

Stănică³

2021

Advances in Mathematics of Communications

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In this paper, we investigate the Gowers U 2 norm for generalized Boolean functions, and Z-bent functions. The Gowers U 2 norm of a function is a measure of its resistance to affine approximation. Although nonlinearity serves the same purpose for the classical Boolean functions, it does not extend easily to generalized Boolean functions. We first provide a framework for employing the Gowers U 2 norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers U 2 norms, and an evaluation of the Gowers U 2 norm of functions that are affine over spreads. We also give an introduction to Z-bent functions, as proposed by Dobbertin and Leander [8], to provide a recursive framework to study bent functions. In the second part of the paper, we concentrate on Zbent functions and their U 2 norms. As a consequence of one of our results, we give an alternate proof to a known theorem of Dobbertin and Leander, and also find necessary and sufficient conditions for a function obtained by gluing Z-bent functions to be bent, in terms of the Gowers U 2 norms of its components.

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“…In particular, it was shown in [10] that some standard classes of bent functions such as Mariaona-McFarland class and Dillon's class naturally induce gbent functions. A particular class of the functions represented as f (x) = c 1 a(x) + c 2 b(x) were thoroughly investigated in terms of the imposed conditions on the coefficients c i ∈ Z q and the choice of the Boolean functions a and b, so that f is gbent [12]. A more interesting research challenge in this context is to propose some direct construction methods of functions from Z n 2 to Z q , which for suitable q may give a nontrivial decomposition into standard bent functions that possibly do not belong to the known classes of bent functions.…”

Section: Introductionmentioning

confidence: 99%

Generalized bent functions -sufficient conditions and related constructions

Hodžić¹,

Pašalić²

2017

Advances in Mathematics of Communications

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The necessary and sufficient conditions for a class of functions f : Z n 2 → Z q , where q ≥ 2 is an even positive integer, have been recently identified for q = 4 and q = 8. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of f , which then allows us to derive sufficient conditions that f is generalized bent for any even q. The case when q is not a power of two, which has not been addressed previously, is treated separately and a suitable representation in terms of the component functions is employed. Consequently, the derived results lead to generic construction methods of this class of functions. The main remaining task, which is not answered in this article, is whether the sufficient conditions are also necessary. There are some indications that this might be true which is also formally confirmed for generalized bent functions that belong to the class of generalized Maiorana-McFarland functions (GMMF), but still we were unable to completely specify (in terms of necessity) gbent conditions.

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Generalized Bent Functions - Some General Construction Methods and Related Necessary and Sufficient Conditions

Cited by 23 publications

References 14 publications

Generalized Bent Functions and Their Gray Images

Generalized Bent Functions and Their Gray Images

Gowers U_2 norm as a measure of nonlinearity for Boolean functions and their generalizations

Generalized bent functions -sufficient conditions and related constructions

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