arXiv subjects

Clemens Heuberger

Publications and source records attributed to Clemens Heuberger.

At least 37 records · Page 2Linked to original sources

arXiv2016

On the monoid generated by a Lucas sequence

A Lucas sequence is a sequence of the general form $v_n = (ϕ^n - \barϕ^n)/(ϕ-\barϕ)$, where $ϕ$ and $\barϕ$ are real algebraic integers such that $ϕ+\barϕ$ and $ϕ\barϕ$ are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenne numbers. We study the monoid that is generated by such a sequence; as it turns out, it is almost freely generated. We provide an asymptotic formula for the number of positive integers $\leq x$ in this monoid, and also prove Erdős-Kac type theorems for the distribution of the number of factors, with and without multiplicity. While the limiting distribution is Gaussian if only distinct factors are counted, this is no longer the case when multiplicities are taken into account.

math.NT↗

arXiv2016

Counting Zeros in Random Walks on the Integers and Analysis of Optimal Dual-Pivot Quicksort

We present an average case analysis of two variants of dual-pivot quicksort, one with a non-algorithmic comparison-optimal partitioning strategy, the other with a closely related algorithmic strategy. For both we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear, logarithmic, and constant terms. An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proven.

math.CO↗

arXiv2016

The Register Function and Reductions of Binary Trees and Lattice Paths

The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures ("branches") that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can be derived. We analyze this quantity, as well as the (cumulative) size of an (iteratively) reduced lattice path asymptotically.

math.CO↗

arXiv2016

Elliptic Curves with Isomorphic Groups of Points over Finite Field Extensions

Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we characterise for which finite field extensions $\mathbb{F}_{q^k}$, $k\geq 1$ (if any) the corresponding groups of $\mathbb{F}_{q^k}$-rational points are isomorphic, i.e. $E(\mathbb{F}_{q^k}) \cong E'(\mathbb{F}_{q^k})$.

math.NT↗

arXiv2016

On the Higher Dimensional Quasi-Power Theorem and a Berry-Esseen Inequality

Hwang's quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry-Esseen inequality is proved, generalising a two-dimensional version of Sadikova.

math.PR↗

arXiv2016

Automata in SageMath---Combinatorics meet Theoretical Computer Science

The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit expansion, which was not known before.

cs.FL↗

arXiv2016

Analysis of Carries in Signed Digit Expansions

The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the $(q, d)$-system and the symmetric signed digit expansion. The results include expectation, variance, covariance between the positive and negative carries and a central limit theorem. Dependencies between the digits require determining suitable transition probabilities to obtain equidistribution on all expansions of given length. A general procedure is described to obtain such transition probabilities for arbitrary regular languages. The number of iterations in von Neumann's parallel addition method for the symmetric signed digit expansion is also analyzed, again including expectation, variance and convergence to a double exponential limiting distribution. This analysis is carried out in a general framework for sequences of generating functions.

math.CO↗

arXiv2015

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in their Maximum

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.

math.CO↗

arXiv2015

Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).

math.NT↗

arXiv2015

Application of Smirnov Words to Waiting Time Distributions of Runs

Consider infinite random words over a finite alphabet where the letters occur as an i.i.d. sequence according to some arbitrary distribution on the alphabet. The expectation and the variance of the waiting time for the first completed $h$-run of any letter (i.e., first occurrence of $h$ subsequential equal letters) is computed. The expected waiting time for the completion of $h$-runs of $j$ arbitrary distinct letters is also given.

math.CO↗

arXiv2015

The height of multiple edge plane trees

Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.

math.CO↗

arXiv2015

Output sum of transducers: Limiting distribution and periodic fluctuation

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in $[0, N)$. Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework. Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.

math.CO↗

arXiv2014

A statistical noise model for a class of Physically Unclonable Functions

The interest in "Physically Unclonable Function"-devices has increased rapidly over the last few years, as they have several interesting properties for system security related applications like, for example, the management of cryptographic keys. Unfortunately, the output provided by these devices is noisy and needs to be corrected for these applications. Related error correcting mechanisms are typically constructed on the basis of an equal error probability for each output bit. This assumption does not hold for Physically Unclonable Functions, where varying error probabilities can be observed. This results in a generalized binomial distribution for the number of errors in the output. The intention of this paper is to discuss a novel Bayesian statistical model for the noise of an especially wide-spread class of Physically Unclonable Functions, which properly handles the varying output stability and also reflects the different noise behaviors observed in a collection of such devices. Furthermore, we compare several different methods for estimating the model parameters and apply the proposed model to concrete measurements obtained within the CODES research project in order to evaluate typical correction and stabilization approaches.

stat.AP↗

arXiv2014

Variances and Covariances in the Central Limit Theorem for the Output of a Transducer

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent. Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width-$w$ non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence.

math.CO↗

arXiv2014

Analysis of the binary asymmetric joint sparse form

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e., the number of nonzero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are for instance used in elliptic curve cryptography. If the digit set is a set of contiguous integers containing the zero, a special syntactical condition is known to minimize the weight. We analyze the optimal weight of all non-negative integer vectors with maximum entry less than N. The expectation and the variance are given with a main term and a periodic fluctuation in the second order term. Finally, we prove asymptotic normality.

math.NT↗

arXiv2012

Analysis of Width-$w$ Non-Adjacent Forms to Imaginary Quadratic Bases

We consider digital expansions to the base of $τ$, where $τ$ is an algebraic integer. For a $w \geq 2$, the set of admissible digits consists of 0 and one representative of every residue class modulo $τ^w$ which is not divisible by $τ$. The resulting redundancy is avoided by imposing the width $w$-NAF condition, i.e., in an expansion every block of $w$ consecutive digits contains at most one non-zero digit. Such constructs can be efficiently used in elliptic curve cryptography in conjunction with Koblitz curves. The present work deals with analysing the number of occurrences of a fixed non-zero digit. In the general setting, we study all $w$-NAFs of given length of the expansion. We give an explicit expression for the expectation and the variance of the occurrence of such a digit in all expansions. Further a central limit theorem is proved. In the case of an imaginary quadratic $τ$ and the digit set of minimal norm representatives, the analysis is much more refined: We give an asymptotic formula for the number of occurrence of a digit in the $w$-NAFs of all elements of $\Z[τ]$ in some region (e.g. a disc). The main term coincides with the full block length analysis, but a periodic fluctuation in the second order term is also exhibited. The proof follows Delange's method. We also show that in the case of imaginary quadratic $τ$ and $w \geq 2$, the digit set of minimal norm representatives leads to $w$-NAFs for \emph{all} elements of $\Z[τ]$. Additionally some properties of the fundamental domain are stated.

math.NT↗

arXiv2012

Existence and Optimality of $w$-Non-adjacent Forms with an Algebraic Integer Base

We consider digital expansions in lattices with endomorphisms acting as base. We focus on the $w$-non-adjacent form ($w$-NAF), where each block of $w$ consecutive digits contains at most one non-zero digit. We prove that for sufficiently large $w$ and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a $w$-NAF. If the eigenvalues of the endomorphism are large enough and $w$ is sufficiently large, then the $w$-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.

math.NT↗

arXiv2011

Optimality of the Width-$w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width\nbd-$w$ non-adjacent form (\wNAF): each block of $w$ consecutive digits contains at most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the \wNAF{}-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of \wNAF{}s can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal \wNAF{}-expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $τ=\pm1\pm i$, optimality depends on the parity of $w$. Computational results for small trace are given.

math.NT↗