Perfect hash families, identifiable parent property codes and covering arrays

In letzter Zeit haben einige kombinatorische Strukturen und Codes eine Vielzahl verschiedener Anwendungen in der Kommunikationstechnik, Kryptographie, Netzwerktechnik und der Informatik gefunden. Der Zweck dieser Dissertation ist, offene Probleme im Zusammenhang mit verschiedenen kombinatorischen Objekten zu lösen, welche durch praktische Anwendungen im Bereich der Informatik und Kryptographie motiviert sind. Genauer gesagt, untersuchen wir perfect hash families, identifiable parent property codes und covering arrays. Perfect hash families sind kombinatorische Strukturen, die verschiedene praktische Anwendungen haben, so wie Compilerbau, Probleme der Komplexität von Schaltkreisen, Datenbank-Verwaltung, Betriebssysteme, derandomization probabilistischer Algorithmen und broadcast encryption. Wir konzentrieren uns auf explizite Konstruktionsverfahren für perfect hash families. Erstens liefern wir eine explizite rekursive Konstruktion einer unendlichen Klasse von perfect hash families mit dem besten bekannten asymptotischen Verhalten unter allen ähnlichen, bekannten Klassen. Zum zweiten stellen wir ein neues rekursives Konstruktionsverfahren vor, mit dessen Hilfe man gute perfect hash families für kleine Parameter erzeugen kann. Durch diese Methode erhalten wir eine unendliche Klasse von perfect hash families, die eine sehr große Menge von Parameter-Werten abdeckt. Weiterhin leiten wir eine neue untere Schranke für die minimale Anzahl von Hash-Funktionen her. Ein Vergleich der existierenden Schranken zeigt, dass unsere Schranke für einige Parameter-Bereiche schärfer ist als andere bekannte Schranken. Identifiable parent property codes (IPP) wurden entwickelt für die Anwendung in Verfahren, die urheberrechtlich geschützte digitale Daten gegen unerlaubte Kopien schützen, die gemeinsam von mehreren berechtigten Nutzern hergestellt werden. TA codes sind eine gut erforschte Teilmenge der IPP-Codes. Wir stellen zwei neue Konstruktionen für IPP-Codes vor. Unsere erste Konstruktion bietet eine unendlichen Klasse von IPP-Codes mit dem besten bekannten asymptotischen Verhalten unter allen ähnlichen Klassen in der Literatur. Weiterhin beweisen wir, dass diese Codes ein Verfahren zum Finden von Verrätern mit im Allgemeinen Laufzeit O(M) erlauben, wobei M die Code-Größe ist. Man beachte, dass vorher außer den TA-Codes keine IPP-Codes mit dieser Eigenschaft bekannt waren. Für einige unendliche Unterklassen dieser Codes kann man sogar noch schnellere Verfahren zum Aufspüren von Verrätern finden, mit Laufzeit poly(logM). Außerdem wird eine neue unendliche Klasse von IPP-Codes konstruiert, die gute IPP-Codes für nicht zu große Werte von n liefert, wobei n die Code-Länge bezeichnet. Diese Klasse von IPP-Codes deckt einen großen Bereich von Parameter-Werten ab. Weiterhin konstruieren wir eine große Klasse von w-TA-Codes, die eine positive Antwort auf ein offenes Existenzproblem geben. Covering arrays sind von vielen Wissenschaftlern intensiv untersucht worden, aufgrund ihrer zahlreichen Anwendungen in der Informatik, so wie Software- oder Schaltkreis-Testen, switching networks, Datenkompressions-Probleme, und etliche mathematische Anwendungen, so wie Differenz-Matrizen, Such-Theorie und Wahrheits-Funktionen. Wir untersuchen explizite Konstruktions-Methoden für t-covering arrays. Zuerst benutzen wir den Zusammenhang zwischen perfect hash families und covering arrays, um unendliche Familien von t-covering arrays zu finden, für die wir beweisen, dass sie besser sind als die augenblicklich bekannten probabilistischen Schranken für covering arrays. Diese Familien haben ein sehr gutes asymptotisches Verhalten. Zum zweiten liefern wir, angeregt durch ein Ergebnis von Roux und auch von einem kürzlich erzielten Ergebnis von Chateauneuf und Kreher für 3-covering arrays, verschiedene neue Konstruktionen für t-covering arrays, t >_ 4, die als eine Verallgemeinerung dieser Ergebnisse gesehen werden können

Partial Covering Arrays: Algorithms and Asymptotics

A covering array $\mathsf{CA}(N;t,k,v)$ is an $N\times k$ array with entries in $\{1, 2, \ldots , v\}$, for which every $N\times t$ subarray contains each $t$-tuple of $\{1, 2, \ldots , v\}^t$ among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound $\mathsf{CAN}(t,k,v)$, the minimum number $N$ of rows of a $\mathsf{CA}(N;t,k,v)$. The well known bound $\mathsf{CAN}(t,k,v)=O((t-1)v^t\log k)$ is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set $\{1, 2, \ldots , v\}^t$ need only be contained among the rows of at least $(1-\epsilon)\binom{k}{t}$ of the $N\times t$ subarrays and (2) the rows of every $N\times t$ subarray need only contain a (large) subset of $\{1, 2, \ldots , v\}^t$. In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time

Perfect Hash Families: The Generalization to Higher Indices

Perfect hash families are often represented as combinatorial arrays encoding partitions of kitems into v classes, so that every t or fewer of the items are completely separated by at least a specified number of chosen partitions. This specified number is the index of the hash family. The case when each t-set must be separated at least once has been extensively researched; they arise in diverse applications, both directly and as fundamental ingredients in a column replacement strategy for a variety of combinatorial arrays. In this paper, construction techniques and algorithmic methods for constructing perfect hash families are surveyed, in order to explore extensions to the situation when each t-set must be separated by more than one partition.https://digitalcommons.usmalibrary.org/books/1029/thumbnail.jp

An Asymptotically Optimal Bound for Covering Arrays of Higher Index

A \emph{covering array} is an $N \times k$ array ($N$ rows, $k$ columns) with each entry from a $v$-ary alphabet, and for every $N\times t$ subarray, all $v^t$ tuples of size $t$ appear at least $\lambda$ times. The \emph{covering array number} is the smallest number $N$ for which such an array exists. For $\lambda = 1$, the covering array number is asymptotically logarithmic in $k$, when $v, t$ are fixed. Godbole, Skipper, and Sunley proved a bound of the form $\log k + \lambda \log \log k$ for the covering array number for arbitrary $\lambda$ and $v,t$ constant. The author proved a similar bound via a different technique, and conjectured that the $\log \log k$ term can be removed. In this short note we answer the conjecture in the affirmative with an asymptotically tight upper bound. In particular, we employ the probabilistic method in conjunction with the Lambert $W$ function

Algorithmic Methods for Covering Arrays of Higher Index

Covering arrays are combinatorial objects used in testing large-scale systems to increase confidence in their correctness. To do so, each interaction of at most a specified number t of factors is represented in at least one test; that is, the covering array has strength t and index 1. For certain systems, the outcome of running a test may be altered by variability of the interaction effect or by measurement error of the test result. To improve the efficacy of testing, one can ensure that each interaction of t or fewer factors is represented in at least λ tests. When λ \u3e 1, this leads to covering arrays of higher index. We explore two algorithmic methods for constructing covering arrays of higher index. One is based on the in-parameter-order algorithm, and the other employs a conditional expectation paradigm. We compare these two by performing experiments on real-world benchmarks and on uniform parameter sets

Linear time Constructions of some dd-Restriction Problems

We give new linear time globally explicit constructions for perfect hash families, cover-free families and separating hash functions

The Design and Analysis of Hash Families For Use in Broadcast Encryption

abstract: Broadcast Encryption is the task of cryptographically securing communication in a broadcast environment so that only a dynamically specified subset of subscribers, called the privileged subset, may decrypt the communication. In practical applications, it is desirable for a Broadcast Encryption Scheme (BES) to demonstrate resilience against attacks by colluding, unprivileged subscribers. Minimal Perfect Hash Families (PHFs) have been shown to provide a basis for the construction of memory-efficient t-resilient Key Pre-distribution Schemes (KPSs) from multiple instances of 1-resilient KPSs. Using this technique, the task of constructing a large t-resilient BES is reduced to finding a near-minimal PHF of appropriate parameters. While combinatorial and probabilistic constructions exist for minimal PHFs with certain parameters, the complexity of constructing them in general is currently unknown. This thesis introduces a new type of hash family, called a Scattering Hash Family (ScHF), which is designed to allow for the scalable and ingredient-independent design of memory-efficient BESs for large parameters, specifically resilience and total number of subscribers. A general BES construction using ScHFs is shown, which constructs t-resilient KPSs from other KPSs of any resilience ≤w≤t. In addition to demonstrating how ScHFs can be used to produce BESs , this thesis explores several ScHF construction techniques. The initial technique demonstrates a probabilistic, non-constructive proof of existence for ScHFs . This construction is then derandomized into a direct, polynomial time construction of near-minimal ScHFs using the method of conditional expectations. As an alternative approach to direct construction, representing ScHFs as a k-restriction problem allows for the indirect construction of ScHFs via randomized post-optimization. Using the methods defined, ScHFs are constructed and the parameters' effects on solution size are analyzed. For large strengths, constructive techniques lose significant performance, and as such, asymptotic analysis is performed using the non-constructive existential results. This work concludes with an analysis of the benefits and disadvantages of BESs based on the constructed ScHFs. Due to the novel nature of ScHFs, the results of this analysis are used as the foundation for an empirical comparison between ScHF-based and PHF-based BESs . The primary bases of comparison are construction efficiency, key material requirements, and message transmission overhead.Dissertation/ThesisM.S. Computer Science 201

Interaction Testing, Fault Location, and Anonymous Attribute-Based Authorization

abstract: This dissertation studies three classes of combinatorial arrays with practical applications in testing, measurement, and security. Covering arrays are widely studied in software and hardware testing to indicate the presence of faulty interactions. Locating arrays extend covering arrays to achieve identification of the interactions causing a fault by requiring additional conditions on how interactions are covered in rows. This dissertation introduces a new class, the anonymizing arrays, to guarantee a degree of anonymity by bounding the probability a particular row is identified by the interaction presented. Similarities among these arrays lead to common algorithmic techniques for their construction which this dissertation explores. Differences arising from their application domains lead to the unique features of each class, requiring tailoring the techniques to the specifics of each problem. One contribution of this work is a conditional expectation algorithm to build covering arrays via an intermediate combinatorial object. Conditional expectation efficiently finds intermediate-sized arrays that are particularly useful as ingredients for additional recursive algorithms. A cut-and-paste method creates large arrays from small ingredients. Performing transformations on the copies makes further improvements by reducing redundancy in the composed arrays and leads to fewer rows. This work contains the first algorithm for constructing locating arrays for general values of $d$ and $t$. A randomized computational search algorithmic framework verifies if a candidate array is $(\bar{d},t)$-locating by partitioning the search space and performs random resampling if a candidate fails. Algorithmic parameters determine which columns to resample and when to add additional rows to the candidate array. Additionally, analysis is conducted on the performance of the algorithmic parameters to provide guidance on how to tune parameters to prioritize speed, accuracy, or a combination of both. This work proposes anonymizing arrays as a class related to covering arrays with a higher coverage requirement and constraints. The algorithms for covering and locating arrays are tailored to anonymizing array construction. An additional property, homogeneity, is introduced to meet the needs of attribute-based authorization. Two metrics, local and global homogeneity, are designed to compare anonymizing arrays with the same parameters. Finally, a post-optimization approach reduces the homogeneity of an anonymizing array.Dissertation/ThesisDoctoral Dissertation Computer Science 201