arXiv subjects

Wilfried Meidl

Publications and source records attributed to Wilfried Meidl.

18 recordsLinked to original sources

arXiv2024

Vectorial Negabent Concepts: Similarities, Differences, and Generalizations

In Pasalic et al., IEEE Trans. Inform. Theory 69 (2023), 2702--2712, and in Anbar, Meidl, Cryptogr. Commun. 10 (2018), 235--249, 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 $\mathbb{F}_2^n$ to the cyclic group $\mathbb{Z}_{2^k}$. It is shown how to obtain nega-$\mathbb{Z}_{2^k}$-bent functions from $\mathbb{Z}_{2^k}$-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 $\mathbb{Z}_8$-bent functions employing permutations with the $(\mathcal{A}_m)$ property, and more generally we show that the inverse permutation gives rise to $\mathbb{Z}_{2^k}$-bent functions.

math.CO↗

arXiv2023

Linear codes and incidence structures of bent functions and their generalizations

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.

cs.IT↗

arXiv2022

P$\wp$N functions, complete mappings and quasigroup difference sets

We investigate pairs of permutations $F,G$ of $\mathbb{F}_{p^n}$ such that $F(x+a)-G(x)$ is a permutation for every $a\in\mathbb{F}_{p^n}$. We show that necessarily $G(x) = \wp(F(x))$ for some complete mapping $-\wp$ of $\mathbb{F}_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear (P$\wp$N) function. If $\wp(x) = cx$, then $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on $\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain a quasigroup, and show that the graph of a P$\wp$N function $F$ is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P$\wp$N functions, respectively, the difference sets in the corresponding quasigroup.

cs.IT↗

arXiv2020

On functions with the maximal number of bent components

A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where $\tilde{F}(x)=(f_1(x),\ldots,f_m(x))$ is a vectorial bent function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, and $a_i$, $1\le i\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $F(x) = x^{2^r}{\rm Tr^n_m}(Λ(x))$, where $Λ$ is a linearized permutation of $\mathbb{F}_{2^m}$. In the first part of this article it is shown that plateaued functions with $2^n-2^m$ bent components can have nonlinearity at most $2^{n-1}-2^{\lfloor\frac{n+m}{2}\rfloor}$, a bound which is attained by the example $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$ (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form $x^{2^r}{\rm Tr^n_m}(Λ(x))$. We show that for odd $m$, only $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$, has maximal nonlinearity, whereas there are more of them for even $m$, of which we present one more infinite class explicitly. In detail, we investigate Walsh spectrum, differential spectrum and their relations for the functions $x^{2^r}{\rm Tr^n_m}(Λ(x))$. Our results indicate that this class contains many nontrivial EA-equivalence classes of functions with the maximal number of bent components, if $m$ is even, several with maximal possible nonlinearity.

math.NT↗

arXiv2020

Bent and $\mathbb Z_{2^k}$-bent functions from spread-like partitions

Bent functions from a vector space $V_n$ over $\mathbb F_2$ of even dimension $n=2m$ into the cyclic group $\mathbb Z_{2^k}$, or equivalently, relative difference sets in $V_n\times\mathbb Z_{2^k}$ with forbidden subgroup $\mathbb Z_{2^k}$, can be obtained from spreads of $V_n$ for any $k\le n/2$. In this article, existence and construction of bent functions from $V_n$ to $\mathbb Z_{2^k}$, which do not come from the spread construction is investigated. A construction of bent functions from $V_n$ into $\mathbb Z_{2^k}$, $k\le n/6$, (and more generally, into any abelian group of order $2^k$) is obtained from partitions of $\mathbb F_{2^m}\times\mathbb F_{2^m}$, which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitions is always the support of a Boolean bent function.

math.NT↗

arXiv2020

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

For a function $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$, the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an "intermediate" concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it convey detailed structural information about $f$. We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.

cs.IT↗

arXiv2019

Determining the Walsh spectra of Taniguchi's and related APN-functions

We introduce a method based on Bezout's theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on $\mathbb{F}_{2^{2m}}$. Among those are the functions of Taniguchi 2019, Carlet 2011, and Zhou and Pott 2013, all of which are APN under certain conditions. This approach helps to understand why the majority of the functions in those classes have solely bent and semibent components, which in the case of APN functions is called the classical spectrum. More precisely, we show that all Taniguchi functions have the classical spectrum independent from being APN. We determine the nonlinearity of all functions belonging to Carlet's class and to the class of Zhou and Pott, which also confirms with comparatively simple proofs earlier results on the Walsh spectrum of APN-functions in these classes. Using the Hasse-Weil bound, we show that some simple sufficient conditions for the APN-ness of the Zhou-Pott functions, which are given in the original paper, are also necessary.

math.NT↗

arXiv2017

On the normality of $p$-ary bent functions

Depending on the parity of $n$ and the regularity of a bent function $f$ from $\mathbb F_p^n$ to $\mathbb F_p$, $f$ can be affine on a subspace of dimension at most $n/2$, $(n-1)/2$ or $n/2- 1$. We point out that many $p$-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) $n/2$-normal, i.e. affine on a $n/2$-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not $n/2$- normal. We develop an algorithm for testing normality for functions from $\mathbb F_p^n$ to $\mathbb F_p$. Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.

math.NT↗

arXiv2016

Modified planar functions and their components

Zhou 2013 introduced modified planar functions to describe $(2^n,2^n,2^n,1)$ relative difference sets $R$ as a graph of a function on the finite field $\F_{2^n}$, and pointed out that projections of $R$ are difference sets that can be described by negabent or bent$_4$ functions, which are Boolean functions given in multivariate form. Objective of this paper is to contribute to the understanding of these component functions of modified planar functions. We first completely describe a multivariate version of modified planar functions in terms of their bent$_4$ components. In the second part we characterize the component functions of (univariate) modified planar functions in terms of appropriate generalizations of the Walsh-Hadamard transform, with respect to which they have a flat spectrum. We hereby obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function.

math.NT↗

arXiv2016

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

In difference to many recent articles that deal with generalized bent (gbent) functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued $q\in \{4,8,16 \}$, 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 $\mathbb{Z}_q$-bent functions, and point out that they correspond to a class of vectorial bent functions. The property of being $\mathbb{Z}_q$-bent is much stronger than the standard concept of a gbent function. We analyse two examples of this class of functions.

cs.IT↗

arXiv2016

Decomposing generalized bent and hyperbent functions

In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. For odd $n$, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$, where the associated Boolean functions are bent.

cs.IT↗

arXiv2016

Idempotent and p-potent quadratic functions: Distribution of nonlinearity and co-dimension

The Walsh transform $\widehat{Q}$ of a quadratic function $Q:F_{p^n}\rightarrow F_p$ satisfies $|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$ for all $b\in F_{p^n}$, where $0\le s\le n-1$ is an integer depending on $Q$. In this article, we study the following three classes of quadratic functions of wide interest. The class $\mathcal{C}_1$ is defined for arbitrary $n$ as $\mathcal{C}_1 = \{Q(x) = Tr(\sum_{i=1}^{\lfloor (n-1)/2\rfloor}a_ix^{2^i+1})\;:\; a_i \in F_2\}$, and the larger class $\mathcal{C}_2$ is defined for even $n$ as $\mathcal{C}_2 = \{Q(x) = Tr(\sum_{i=1}^{(n/2)-1}a_ix^{2^i+1}) + {\rm Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in F_2\}$. For an odd prime $p$, the subclass $\mathcal{D}$ of all $p$-ary quadratic functions is defined as $\mathcal{D} = \{Q(x) = Tr(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1})\;:\; a_i \in F_p\}$. We determine the distribution of the parameter $s$ for $\mathcal{C}_1, \mathcal{C}_2$ and $\mathcal{D}$. As a consequence we obtain the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $p > 2$, our results yield the distribution of the co-dimensions for the rotation symmetric quadratic $p$-ary functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the subcodes of the second order Reed-Muller codes corresponding to $\mathcal{C}_1$ and $\mathcal{C}_2$.

math.NT↗

arXiv2015

There are infinitely many bent functions for which the dual is not bent

Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs since their duals are also bent functions. In general this does not apply to non-weaky regular bent functions. However, the first known construction of non-weakly regular bent functions by Ceşmelioğlu et {\it al.}, 2012, yields bent functions for which the dual is also bent. In this paper the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions non-dual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Ceşmelioğlu et {\it al.} one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known.

cs.IT↗

arXiv2015

Partial Spread and Vectorial Generalized Bent Functions

In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$. Explicitly, we describe gbent functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$, which can be seen as a gbent version of Dillon's $PS_{ap}$ class. For the first time, we also introduce the concept of a vectorial gbent function from $\F_2^n$ to $\Z_q^m$, and determine the maximal value which $m$ can attain for the case $q=2^t$. Finally we point to a relation between vectorial gbent functions and relative difference sets.

cs.IT↗

arXiv2015

Generalized bent functions and their Gray images

In this paper we prove that generalized bent (gbent) functions defined on $\mathbb{Z}_2^n$ with values in $\mathbb{Z}_{2^k}$ 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 $\mathbb{Z}_{16}$, which extends earlier results on gbent functions with values in $\mathbb{Z}_4$ and $\mathbb{Z}_8$. We also show that the Gray images of gbent functions with values in $\mathbb{Z}_{2^k}$ are semibent/plateaued when $k=3,4$.

cs.IT↗

arXiv2014

Multisequences with high joint nonlinear complexity

We introduce the new concept of joint nonlinear complexity for multisequences over finite fields and we analyze the joint nonlinear complexity of two families of explicit inversive multisequences. We also establish a probabilistic result on the behavior of the joint nonlinear complexity of random multisequences over a fixed finite field.

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arXiv2013

(Not) weakly regular univariate bent functions

In this article a procedure to construct bent functions from $\F_{p^n}$ to $\F_p$ by merging plateaued functions which are bent on ($n-2$)-dimensional subspaces of $\F_{p^n}$ is presented. Taking advantage of such classes of plateaued functions with a simple representation as monomials and binomials, we obtain infinite classes of bent functions with a fairly simple representation. In particular we present the first direct construction of univariate not weakly regular bent functions, and give one class explicitly in a simple representation with binomials.

math.NT↗

arXiv2010

A Construction of Weakly and Non-Weakly Regular Bent Functions

In this article a technique for constructing $p$-ary bent functions from near-bent functions is presented. Two classes of quadratic $p$-ary functions are shown to be near-bent. Applying the construction of bent functions to these classes of near-bent functions yields classes of non-quadratic bent functions. We show that one construction in even dimension yields weakly regular bent functions. For other constructions, we obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions.

math.CO↗