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Arne Winterhof

Publications and source records attributed to Arne Winterhof.

47 records · Page 3Linked to original sources
arXiv2016

Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any $d\ge 2$ and any prime $p>(d^2-3d+4)^2$ there is no complete mapping polynomial in $\mathbb{F}_{p}[x]$ of degree $d$. For arbitrary finite fields $\mathbb{F}_{q}$, we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if $n<\lfloor q/2\rfloor$, then there is no complete mapping in $\mathbb{F}_{q}[x]$ of Carlitz rank $n$ of small linearity. We also determine how far permutation polynomials $f$ of Carlitz rank $n<\lfloor q/2\rfloor$ are from being complete, by studying value sets of $f+x.$ We provide examples of complete mappings if $n=\lfloor q/2\rfloor$, which shows that the above bound cannot be improved in general.

math.CO↗
arXiv2015

Digital inversive vectors can achieve strong polynomial tractability for the weighted star discrepancy and for multivariate integration

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not explode exponentially with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of Hölder continuous, absolute convergent Fourier- and cosine series, where the proposed method is successful.

math.NA↗
arXiv2015

On the linear complexity profile of some sequences derived from elliptic curves

For a given elliptic curve $\mathbf{E}$ over a finite field of odd characteristic and a rational function $f$ on $\mathbf{E}$ we first study the linear complexity profiles of the sequences $f(nG)$, $n=1,2,\dots$ which complements earlier results of Hess and Shparlinski. We use Edwards coordinates to be able to deal with many $f$ where Hess and Shparlinski's result does not apply. Moreover, we study the linear complexities of the (generalized) elliptic curve power generators $f(e^nG)$, $n=1,2,\dots$. We present large families of functions $f$ such that the linear complexity profiles of these sequences are large.

math.NT↗
arXiv2014

Family complexity and cross-correlation measure for families of binary sequences

We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al.\ in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family $(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N$, $i=1,\ldots,F$, of binary sequences of length $N$ in terms of the cross-correlation measure of its dual family $(e_{1,n},\ldots,e_{F,n})\in \{-1,+1\}^F$, $n=1,\ldots,N$. We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo $p$ with middle coefficient $0$, that is, $e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2}$ for $i=1,\ldots,(p-1)/2$, where $b$ is a quadratic nonresidue modulo $p$, showing that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order.

math.NT↗
arXiv2014

Non-Existence of Some Nearly Perfect Sequences, Near Butson-Hadamard Matrices, and Near Conference Matrices

In this paper we study the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matrices and (near) conference matrices. Firstly, we apply a result on vanishing sums of roots of unity and a result of Brock on the unsolvability of certain equations over a cyclotomic number field to derive non-existence results for near BH matrices and near conference matrices. Secondly, we refine the idea of Brock in the case of cyclotomic number fields whose ring of integers is not a principal ideal domains and get many new non-existence results.

math.NT↗
arXiv2014

Some highlights of Harald Niederreiter's work

In this paper we give a short biography of Harald Niederreiter and we spotlight some cornerstones from his wide-ranging work. We focus on his results on uniform distribution, algebraic curves, polynomials and quasi-Monte Carlo methods. In the flavor of Harald's work we also mention some applications including numerical integration, coding theory and cryptography.

math.HO↗
arXiv2014

Polynomial quotients: Interpolation, value sets and Waring's problem

For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of $\mathbb{F}_p$ as sum of values of polynomial quotients, we prove some lower bounds on the size of their value sets, and then we apply these lower bounds to prove some bounds on the Waring number using results from bounds on additive character sums and additive number theory.

math.NT↗
arXiv2013

Covering sets for limited-magnitude errors

For a set $\cM=\{-μ,-μ+1,\ldots, λ\}\setminus\{0\}$ with non-negative integers $λ,μ<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(λ,μ;q)$-\emph{covering set} if $$ \cM \cS=\{ms \bmod q : m\in \cM,\ s\in \cS\}=\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(λ,μ;q)$-covering set $\cS$ which is of the size $q^{1 + o(1)}\max\{λ,μ\}^{-1/2}$ for almost all integers $q\ge 1$ and of optimal size $p\max\{λ,μ\}^{-1}$ if $q=p$ is prime. Furthermore, using a bound on the fourth moment of character sums of Cochrane and Shi we prove the bound $$ω_{λ,μ}(q)\le q^{1+o(1)}\max\{λ,μ\}^{-1/2},$$ for any integer $q\ge 1$, however the proof of this bound is not constructive.

cs.IT↗
arXiv2008

Exact solutions to Waring's problem for finite fields

The Waring function $g(k,q)$ measures the difficulty of Waring's problem for $k$th powers in the field of $q$ elements. Its calculation seems to be difficult, and many partial results have been published, notably upper bounds for certain regions of the $k$-$q$-plane. In this paper, we compute the exact value of $g(k,q)$ for two infinite families of exponent-field pairs. In these, $k$ is large compared to $q$. We use a new method of proof that is mainly combinatorial in nature.

math.NT↗
arXiv2005

On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in $\C^n$ which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.

quant-ph↗