Further results on constructions of generalized bent Boolean functions

Zhang, Fengrong; Xia, Shixiong; Stănică, Pantelimon; Zhou, Yu

doi:10.1007/s11432-016-5543-7

Cited by 5 publications

(9 citation statements)

References 9 publications

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“…In recent years, gbent functions have become an active research topic. Many works focus on characterizing gbent functions in terms of (semi-) bent functions, such as for q = 4 [19,20,22] , for q = 8 [21][22][23] , and for general q [21,[24][25][26][27][28][29][30] . In detail, for any f ∈ GB 2 k n , f can be represented as…”

Section: A Function F ∈ Gb Qmentioning

confidence: 99%

“…In recent years, gbent functions have become an active research topic. Many works focus on characterizing gbent functions in terms of (semi‐) bent functions, such as for

q = 4

[ 19 , 20 , 22 ] , for

q = 8

[ 21 – 23 ] , and for general

q

[ 21 , 24 – 30 ] . In detail, for any

f \in {MJX-tex-caligraphicscriptGB}_{n}^{2^{k}}

f

can be represented as

\begin{matrix} true \\ right f (x) = a_{0} (x) + 2 a_{1} (x) + \dots + 2^{k ‐ 1} a_{k ‐ 1} (x) \end{matrix}

where

k \in {double-struckZ}^{+}, a_{j} \in {MJX-tex-caligraphicscriptB}_{n}, j = 0, 1, \dots, k ‐ 1

.…”

Section: Introductionmentioning

confidence: 99%

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New Secondary Constructions of Generalized Bent Functions

Zhiyao

Chen

2021

Chinese J of Electronics

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Three new secondary constructions of generalized bent functions are presented. We provide a secondary construction of generalized bent functions from indirect sum methods proposed by Carlet et al. A new secondary construction of generalized bent functions from four initial functions is also investigated. We demonstrate that many known constructions can be derived from our proposed construction as special cases by choosing proper initial functions and parameters. By modifying the new construction, a novel secondary construction of generalized bent functions from two initial generalized bent functions is obtained. For the binary case, the dual functions of the bent functions by our method are presented, which share the same formula as the indirect sum.

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Section: A Function F ∈ Gb Qmentioning

confidence: 99%

“…In recent years, gbent functions have become an active research topic. Many works focus on characterizing gbent functions in terms of (semi‐) bent functions, such as for

q = 4

[ 19 , 20 , 22 ] , for

q = 8

[ 21 – 23 ] , and for general

q

[ 21 , 24 – 30 ] . In detail, for any

f \in {MJX-tex-caligraphicscriptGB}_{n}^{2^{k}}

f

can be represented as

\begin{matrix} true \\ right f (x) = a_{0} (x) + 2 a_{1} (x) + \dots + 2^{k ‐ 1} a_{k ‐ 1} (x) \end{matrix}

where

k \in {double-struckZ}^{+}, a_{j} \in {MJX-tex-caligraphicscriptB}_{n}, j = 0, 1, \dots, k ‐ 1

.…”

Section: Introductionmentioning

confidence: 99%

New Secondary Constructions of Generalized Bent Functions

Zhiyao

Chen

2021

Chinese J of Electronics

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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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“…We know that the bent notion exists from any group A to another group B [22], defined via character theory. There are many generalizations of the bent concept and we mention here [9,10,11,14,15,16,17,19,20,24,25,28,29]. For example, a p-ary function F : F p n → F p is bent if the complex absolute value of the Walsh transforms is constant, namely,…”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

$C$-differential bent functions and perfect nonlinearity

Stănică¹,

Gangopadhyay²,

Geary³

et al. 2020

Preprint

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Drawing inspiration from Nyberg's paper [21] on perfect nonlinearity and the c-differential notion we defined in [8], in this paper we introduce the concept of c-differential bent functions in two different ways (thus extending Kumar et al. [11] classical definition). We further extend the notion of perfect c-nonlinear introduced in [8], also in two different ways, and show that, in both cases, the concepts of c-differential bent and perfect c-nonlinear are equivalent (under some natural restriction of the parameters). Some constructions of functions with these properties are also provided; one such construction provides a large class of PcN functions with respect to all c in some subfield of the field under consideration. We also show that both our classes of 0differential bents are supersets of permutation polynomials, and that Maiorana-McFarland bent functions are not differential bent (of the first kind).

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Further results on constructions of generalized bent Boolean functions

Cited by 5 publications

References 9 publications

New Secondary Constructions of Generalized Bent Functions

New Secondary Constructions of Generalized Bent Functions

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

$C$-differential bent functions and perfect nonlinearity

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