arXiv subjects

Arne Winterhof

Publications and source records attributed to Arne Winterhof.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the distribution of the Rudin-Shapiro function for finite fields

Let $q=p^r$ be the power of a prime $p$ and $(β_1,\ldots ,β_r)$ be an ordered basis of $\mathbb{F}_q$ over $\mathbb{F}_p$. For $$ ξ=\sum\limits_{j=1}^r x_jβ_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$ R(ξ)=\sum\limits_{i=1}^{r-1} x_ix_{i+1}, \quad ξ\in \mathbb{F}_q. $$ For a non-constant polynomial $f(X)\in \mathbb{F}_q[X]$ and $c\in \mathbb{F}_p$ we study the number of solutions $ξ\in \mathbb{F}_q$ of $R(f(ξ))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\rightarrow \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.

math.NT↗

arXiv2020

Additive double character sums over some structured sets and applications

We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace products and sum-product equations and improve results of the first author and of Gyarmati and Sárközy.

math.NT↗

arXiv2020

On the Index of Diffie-Hellman Mapping

Let $γ$ be a generator of a cyclic group $G$ of order $n$. The least index of a self-mapping $f$ of $G$ is the index of the largest subgroup $U$ of $G$ such that $f(x)x^{-r}$ is constant on each coset of $U$ for some positive integer~$r$. We determine the index of the univariate Diffie-Hellman mapping $d(γ^a)=γ^{a^2}$, $a=0,1,\ldots,n-1$, and show that any mapping of small index coincides with~$d$ only on a small subset of $G$. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping $D(γ^a,γ^b)=γ^{ab}$, $a,b=0,1,\ldots,n-1$. In the special case that $G$ is a subgroup of the multiplicative group of a finite field we present improvements.

math.CO↗

arXiv2020

A Note on the Cross-Correlation of Costas Permutations

We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime $p\ge 5$, the maximal cross-correlation of the family of the $φ(p-1)$ different Welch Costas permutations of $\{1,\ldots,p-1\}$ is $(p-1)/t$, where $t$ is the smallest prime divisor of $(p-1)/2$ if $p$ is not a safe prime and at most $1+p^{1/2}$ otherwise. Here $φ$ denotes Euler's totient function and a prime $p$ is a safe prime if $(p-1)/2$ is also prime. Second, for a prime power $q\ge 4$ the maximal cross-correlation of a subfamily of Golomb Costas permutations of $\{1,\ldots,q-2\}$ is $(q-1)/t-1$ if $t$ is the smallest prime divisor of $(q-1)/2$ if $q$ is odd and of $q-1$ if $q$ is even provided that $(q-1)/2$ and $q-1$ are not prime, and at most $1+q^{1/2}$ otherwise. Note that we consider a smaller family than Drakakis et al. Our family is of size $φ(q-1)$ whereas there are $φ(q-1)^2$ different Golomb Costas permutations. The maximal cross-correlation of the larger family given in the tables of Drakakis et al. is larger than our bound (for the smaller family) for some $q$.

math.NT↗

arXiv2020

Binary sequences derived from differences of consecutive quadratic residues

For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $t_n=0$ otherwise, $n=0,1,\ldots,(p-5)/2$. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $(p-3)/2$. Studying the linear complexity of $(d_n)$ was motivated by heuristics of Caragiu et al. However, $(d_n)$ is not balanced and we show that a period of $(d_n)$ contains about $1/3$ zeros and $2/3$ ones if $p$ is sufficiently large. In contrast, $(t_n)$ is not only essentially balanced but also all longer patterns of length $s$ appear essentially equally often in the vector sequence $(t_n,t_{n+1},\ldots,t_{n+s-1})$, $n=0,1,\ldots,(p-5)/2$, for any fixed $s$ and sufficiently large $p$.

math.NT↗

arXiv2020

The Spherical Kakeya Problem in Finite Fields

We study subsets of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a sphere for each first coordinate of the center. We call such sets radii spherical Kakeya sets and center spherical Kakeya sets, respectively. For $n\ge 4$ we prove a general lower bound on the size of any set containing $q-1$ different spheres which applies to both kinds of spherical Kakeya sets. We provide constructions which meet the main terms of this lower bound. We also give a construction showing that we cannot get a lower bound of order of magnitude~$q^n$ if we take lower dimensional objects such as circles in $\mathbb{F}_q^3$ instead of spheres, showing that there are significant differences to the line Kakeya problem. Finally, we study the case of dimension $n=1$ which is different and equivalent to the study of sum and difference sets that cover $\mathbb{F}_q$.

math.CO↗

arXiv2019

A note on Hall's sextic residue sequence: correlation measure of order $k$ and related measures of pseudorandomness

It is known that Hall's sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$. Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$, which is close to the expected value for a random sequence of length $p$. Moreover, we derive from this bound a lower bound on the $N$th maximum order complexity of order of magnitude $\log p$, which is the expected order of magnitude for a random sequence of length $p$.

math.NT↗

arXiv2019

On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence

Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the $N$th maximum order complexity is of order of magnitude $\log N$ whereas it is easy to find families of sequences with $N$th expansion complexity exponential in $\log N$. This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapiro sequence with very small expansion complexity but very large maximum order complexity. More precisely, we prove explicit formulas for their $N$th maximum order complexity which are both of largest possible order of magnitude $N$. We present the result on the Rudin-Shapiro sequence in a more general form as a formula for the maximum order complexity of certain pattern sequences.

math.CO↗

arXiv2019

On the maximum order complexity of subsequences of the Thue-Morse and Rudin-Shapiro sequence along squares

Automatic sequences such as the Thue-Morse sequence and the Rudin-Shapiro sequence are highly predictable and thus not suitable in cryptography. In particular, they have small expansion complexity. However, they still have a large maximum order complexity. Certain subsequences of automatic sequences are not automatic anymore and may be attractive candidates for applications in cryptography. In this paper we show that subsequences along the squares of certain pattern sequences including the Thue-Morse sequence and the Rudin-Shapiro sequence have also large maximum order complexity but do not suffer a small expansion complexity anymore.

math.CO↗

arXiv2019

Conical Kakeya and Nikodym Sets in Finite Fields

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.) Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

math.CO↗

arXiv2018

Codes correcting restricted errors

We study the largest possible length $B$ of $(B-1)$-dimensional linear codes over $\mathbb{F}_q$ which can correct up to $t$ errors taken from a restricted set $\mathcal{A}\subseteq \mathbb{F}_q^*$. Such codes can be applied to multilevel flash memories. Moreover, in the case that $q=p$ is a prime and the errors are limited by a constant we show that often the primitive $\ell$th roots of unity, where $\ell$ is a prime divisor of $p-1$, define good such codes.

cs.IT↗

arXiv2017

On the $N$th linear complexity of automatic sequences

The $N$th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large $N$th linear complexity. However, in this paper we show that for $q$-automatic sequences over $\mathbb{F}_q$ the converse is not true. We prove that any (not ultimately periodic) $q$-automatic sequence over $\mathbb{F}_q$ has $N$th linear complexity of order of magnitude $N$. For some famous sequences including the Thue--Morse and Rudin--Shapiro sequence we determine the exact values of their $N$th linear complexities. These are non-trivial examples of predictable sequences with $N$th linear complexity of largest possible order of magnitude.

math.NT↗

arXiv2017

On the pseudorandomness of automatic sequences

We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating functions or from their generating finite automatons, respectively.

math.NT↗

arXiv2017

Packing Sets

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge (q-1)/|A/A|$ and show that this bound is in general optimal. The case that $q=p$ is a prime and $A=\{1,2,\ldots,λ\}$ for some positive integer $λ$ is particularly interesting in view of the construction of limited-magnitude error correcting codes. Here we construct a packing set $B$ of size $|B|\gg p (λ\log p)^{-1}$ for any $λ\le c p^{1/2}$ for some explicitly calcuable constant $c$. This result is optimal up to the logarithmic factor.

math.CO↗

arXiv2017

Maximum-order Complexity and Correlation Measures

We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order $k$ cannot have very small maximum-order complexity.

math.NT↗

arXiv2017

Analogues of the Balog--Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions

Balog and Wooley have recently proved that any subset $A$ of either real numbers or of a prime finite field can be decomposed into two parts $U$ and $V$, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function $f$ both the additive energies of $U$ and $f(V)$ are small. Our method is based on bounds of character sums which leads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. The bound is optimal, up to logarithmic factors, when $\# A \geq q^{9/13}$. Using $f(X)=X^{-1}$ we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions $ab+ac+bc$ with variables $a,b,c$ running through three arbitrary subsets of a finite field.

math.NT↗

arXiv2016

Carlitz Rank and Index of Permutation Polynomials

Carlitz rank and index are two important measures for the complexity of a permutation polynomial $f(x)$ over the finite field $\F_q$. In particular, for cryptographic applications we need both, a high Carlitz rank and a high index. In this article we study the relationship between Carlitz rank $Crk(f)$ and index $Ind(f)$. More precisely, if the permutation polynomial is neither close to a polynomial of the form $ax$ nor a rational function of the form $ax^{-1}$, then we show that $Crk(f)>q- \max\{3 Ind(f),(3q)^{1/2}\}$. Moreover we show that the permutation polynomial which represents the discrete logarithm guarantees both a large index and a large Carlitz rank.

math.CO↗

arXiv2016

Expansion complexity and linear complexity of sequences over finite fields

The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo $p$. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field.

math.NT↗