Generalized bent functions -sufficient conditions and related constructions

Hodžić, Samir; Pašalić, Enes

doi:10.3934/amc.2017043

Cited by 3 publications

(15 citation statements)

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“…Employing conventional equivalence, we show that gbent functions and affine spaces of bent (semi-bent) functions with these properties, are identical objects. These results essentially completely resolve the case of gbent functions from V n to Z 2 k , using the approach based on Hadamard matrices introduced in [5].…”

Section: Introductionmentioning

confidence: 63%

“…Currently there is a lot of research activity regarding the construction and analysis of gbent functions, see for instance [5,7,8,9,12,18,20,22]. The quaternary q = 4 and octal case q = 8 were investigated in [18,20] and [12,21], respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In these cases gbent functions were fully characterized by specifying both the related necessary and sufficient conditions. The first general characterization of gbent functions, in terms of the coordinate functions for any q even and n both even/odd, is given in [5,Theorem 4.1]. More precisely, for 2 k−1 < q ≤ 2 k , to any generalized function f : Z n 2 → Z q , one may associate a unique sequence of Boolean functions…”

Section: Introductionmentioning

confidence: 99%

“…2 , where a i are then called the coordinate functions of f . The gbent conditions derived in [5] were only sufficient and whether these are also necessary was left as an open problem. Necessary conditions for a function f ∈ GB 2 k n to be gbent are in [8] when n is even and also for odd n in [10].…”

Section: Introductionmentioning

confidence: 99%

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Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image

Hodžić¹,

Meidl²,

Pašalić³

2016

Preprint

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In difference to many recent articles that deal with generalized bent (gbent) functions f : Z n 2 → Z q for certain small valued q ∈ {4, 8, 16}, we give a complete description of these functions for both n even and odd and for any q = 2 k in terms of both the necessary and sufficient conditions their component functions need to satisfy. This enables us to completely characterize gbent functions as algebraic objects, namely as affine spaces of bent or semi-bent functions with interesting additional properties, which we in detail describe. We also specify the dual and the Gray image of gbent functions for q = 2 k . We discuss the subclass of gbent functions which corresponds to relative difference sets, which we call Z qbent functions, and point out that they correspond to a class of vectorial bent functions. The property of being Z q -bent is much stronger than the standard concept of a gbent function. We analyse two examples of this class of functions.

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Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image

Hodžić¹,

Meidl²,

Pašalić³

2016

Preprint

Self Cite

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“…Indeed, the dual of a sum of bent functions is in general not equal to the sum of duals of these functions, except in the cases when these functions are affinely related to each other (thus h i = h j ⊕ g, where g is an affine function) [1,Proposition 3]. A trivial method for satisfying these conditions, as indicated in [5,Example 3], is to select certain functions to be constant which then significantly limits the number of choices and consequently the cardinality of GB n q is quite small. The case n being odd appears to be even harder since apart from finding an affine space Λ of semi-bent functions, the condition (5) also implicitly involves the disjoint spectra property.…”

Section: Problem Descriptionmentioning

confidence: 99%

Construction methods for generalized bent functions

Hodžić

Pašalić

2018

Discrete Applied Mathematics

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Generalized bent (gbent) functions is a class of functions f : Z n 2 → Z q , where q ≥ 2 is a positive integer, that generalizes a concept of classical bent functions through their codomain extension. A lot of research has recently been devoted towards derivation of the necessary and sufficient conditions when f is represented as a collection of Boolean functions. Nevertheless, apart from the necessary conditions that these component functions are bent when n is even (respectively semi-bent when n is odd), no general construction method has been proposed yet for n odd case. In this article, based on the use of the well-known Maiorana-McFarland (MM) class of functions, we give an explicit construction method of gbent functions, for any even q > 2 when n is even and for any q of the form q = 2 r (for r > 1) when n is odd. Thus, a long-term open problem of providing a general construction method of gbent functions, for odd n, has been solved. The method for odd n employs a large class of disjoint spectra semi-bent functions with certain additional properties which may be useful in other cryptographic applications.

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