Alev Topuzoğlu

Sabancı Üniversitesi

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Wilfried Meidl

University of Klagenfurt

Nurdagül Anbar

Sabancı Üniversitesi

Leyla Işık

Johns Hopkins University

Anna Somoza

Centre National de la Recherche Scientifique

Vandita Patel

University of Manchester

Arne Winterhof

Austrian Academy of Sciences

Luciane Quoos

Universidade Federal do Rio de Janeiro

AÇ

Ayça Çeşmelioğlu

Istanbul Bilgi University

Almasa Odžak

University of Sarajevo

Henning Stichtenoth

Sabancı Üniversitesi

Cooperative Institutions

Sabancı Üniversitesi

Austrian Academy of Sciences

Johann Radon Institute for Computational and Applied Mathematics

Darmstadt University of Applied Sciences

University of Salzburg

Otto-von-Guericke University Magdeburg

University of Sarajevo

Technical University of Darmstadt

Johannes Kepler University of Linz

Technical University of Denmark

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On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes

HAL (Le Centre pour la Communication Scientifique Directe) (2015)

Nurdagül Anbar Wilfried Meidl Alev Topuzoğlu

Weight distribution

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On components of vectorial permutations of Fqn

Finite Fields and Their Applications (2019)

Nurdagül Anbar Canan Kaşıkçı Alev Topuzoğlu

Cardinality (data modeling)

Value (mathematics)

10.1016/j.ffa.2019.03.006

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Citations (3)

A Note on Value Sets of Polynomials over Finite Fields

arXiv (Cornell University) (2017)

Leyla Işık Alev Topuzoğlu

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known. We consider a class $\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.

Coset

Cardinality (data modeling)

Degree (music)

Multiplicative group

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Irreducible factors of a class of permutation polynomials

Finite Fields and Their Applications (2020)

Tekgül Kalaycı Henning Stichtenoth Alev Topuzoğlu

10.1016/j.ffa.2020.101647

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Citations (1)

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

Designs Codes and Cryptography (2016)

Nurdagül Anbar Wilfried Meidl Alev Topuzoğlu

Quadratic form (statistics)

10.1007/s10623-016-0213-8

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The Carlitz rank of permutations of finite fields: A survey

Journal of Symbolic Computation (2013)

Alev Topuzoğlu

L. Carlitz proved that any permutation polynomial f of a finite field Fq is a composition of linear polynomials and the monomials xq−2. This result motivated the study of Carlitz rank of f, which is defined in 2009 to be the minimum number of inversions xq−2, needed to obtain f, by E. Aksoy et al. We give a survey of results obtained so far on natural questions related to this concept and indicate a variety of applications, which emerged recently.

Rank (graph theory)

10.1016/j.jsc.2013.07.004

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Citations (17)

On the period length of congruential pseudorandom number sequences generated by inversions

Journal of Computational and Applied Mathematics (1990)

Jürgen Eichenauer‐Herrmann Alev Topuzoğlu

Linear congruential generator

Period length

Sequence (biology)

Pseudorandom generator theorem

Periodic sequence

Lattice (music)

10.1016/0377-0427(90)90339-2

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Citations (30)

On u.d. mod 1 of sequences (anx)

Indagationes Mathematicae (Proceedings) (1981)

Alev Topuzoğlu

Mod

Irrational number

10.1016/1385-7258(81)90152-9

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Citations (3)

Permutation polynomials and factorization

Cryptography and Communications (2020)

Tekgül Kalaycı Henning Stichtenoth Alev Topuzoğlu

10.1007/s12095-020-00446-y

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Citations (1)

On functions of low differential uniformity in characteristic 2: A close look (I)

arXiv (Cornell University) (2024)

Nurdagül Anbar Tekgül Kalaycı Alev Topuzoğlu

We introduce a new concept, the APN-defect, which can be thought of as measuring the distance of a given function $G:\mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}$ to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity) $G$ using the so-called difference squares. We describe the relations between the APN-defect and other recent concepts of similar nature. Upper and lower bounds for the values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. Its exact values in some cases are also calculated. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on $n$ is evaluated and the implications are discussed. In the forthcoming second part of this work we further examine modifications of the inverse function. We also study modifications of classes of functions of low uniformity over infinitely many extensions of $\mathbb{F}_{2^n}$. We present quantitative results on their differential behaviour, especially in connection with their APN-defects.

10.48550/arxiv.2406.07468

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