Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application

Li, Shuxing; Meidl, Wilfried; Polujan, Alexandr; Pott, Alexander; Riera, Constanza; Stănică, Pantelimon

doi:10.1109/tit.2020.3002993

Cited by 11 publications

(4 citation statements)

References 22 publications

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“…In this paper we extend the recent work [19,33], which studied incidence structures, arising from the linear codes C F and C ⊥ F of (n, n)-functions F . In particular, Li et al in [19], motivated by the study of CCZ-inequivalence of (n, n)-functions F , introduced the partial quadruple system VF(F ), called the vanishing flats of the (n, n)-function F , which captures a detailed combinatorial information about the given function. Formally, it is defined in the following way: VF(F ) = (P, VF F ), where the point set is given by P = {x : x ∈ F n 2 } and the block set VF F is given as follows…”

Section: Motivationmentioning

confidence: 79%

“…As a highlight we completed the characterization of one of the most important classes of cryptographic functions, namely, differentially 2-valued and plateaued functions in terms of the incidence structures. For instance, Li et al in [19] and Tang, Ding and Xiong in [33] showed, that the valency of vanishing flats reflects differential uniformity, from what follows that differentially two-valued (n, n)-functions F can be characterized in terms of vanishing flats VF(F ) having the property to be 2-designs. In this paper we showed that regularity of nonvanishing flats reflects another important cryptographic property, namely, plateauedness, and consequently we derived new characterizations of (n, m)-plateaued and (n, m)-bent functions in terms of nonvanishing flats having the property to be 1-designs and 2-designs, respectively.…”

Section: Conclusion and Open Problemsmentioning

confidence: 96%

“…At the same time, linear codes and incidence structures, constructed from Boolean and vectorial functions, are often used to characterize certain classes of functions or to distinguish different ones. Recently there has been a lot of work on (n, m)-functions, their designs and codes [12,13,19,25,33]. In this paper we continue the study of the interaction between Boolean and vectorial functions, linear codes and t-designs.…”

Section: Introductionmentioning

confidence: 92%

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Linear codes and incidence structures of bent functions and their generalizations

Meidl¹,

Polujan²,

Pott³

2020

Preprint

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In this paper we consider further applications of (n, m)-functions for the construction of 2-designs. For instance, we provide a new application of the extended Assmus-Mattson theorem, by showing that linear codes of APN functions with the classical Walsh spectrum support 2-designs. On the other hand, we use linear codes and combinatorial designs in order to study important properties of (n, m)-functions. In particular, we give a new design-theoretic characterization of (n, m)-plateaued and (n, m)-bent functions and provide a coding-theoretic as well as a design-theoretic interpretation of the extendability problem for (n, m)-bent functions.

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Section: Motivationmentioning

confidence: 79%

Section: Conclusion and Open Problemsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 92%

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Linear codes and incidence structures of bent functions and their generalizations

Meidl¹,

Polujan²,

Pott³

2020

Preprint

Self Cite

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“…For instance, any new construction of bent functions may lead to a new construction of certain designs. On the other hand, combinatorial invariants of incidence structures constructed from functions over finite fields serve as good distinguishers between inequivalent functions and even classes of functions [10,15,28]. Before we briefly mention the main constructions of designs from bent functions and their most notable applications, we would like to point the reader's attention, that the notation we use below for translation and addition designs of bent functions will be introduced in details in the following sections.…”

Section: Introductionmentioning

confidence: 99%

On design-theoretic aspects of Boolean and vectorial bent functions

Polujan,

Pott

2020

Preprint

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There are two construction methods of designs from Boolean and vectorial bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean and vectorial bent functions is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial bent functions and isomorphism of addition designs are the same concepts. Further, we show that extended-affine inequivalent Boolean bent functions in n variables, whose translation designs are isomorphic exist for all n ≥ 6. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomena for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial bent functions in six variables.

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Vectorial negabent concepts: similarities, differences, and generalizations

Anbar,

Kudin,

Meidl

et al. 2024

Des. Codes Cryptogr.

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In Pasalic et al. (IEEE Trans Inf Theory 69:2702–2712, 2023), and in Anbar and Meidl (Cryptogr Commun 10:235–249, 2018), two different vectorial negabent and vectorial bent-negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from $${\mathbb {F}}_2^n$$ F 2 n to the cyclic group $${\mathbb {Z}}_{2^k}$$ Z 2 k . It is shown how to obtain nega-$${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions from $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of $${\mathbb {Z}}_8$$ Z 8 -bent functions employing permutations with the $$({\mathcal {A}}_m)$$ ( A m ) property, and more generally we show that the inverse permutation gives rise to $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions.

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Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application

Cited by 11 publications

References 22 publications

Linear codes and incidence structures of bent functions and their generalizations

Linear codes and incidence structures of bent functions and their generalizations

On design-theoretic aspects of Boolean and vectorial bent functions

Vectorial negabent concepts: similarities, differences, and generalizations

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