arXiv subjects

Charles J. Colbourn

Publications and source records attributed to Charles J. Colbourn.

18 recordsLinked to original sources

arXiv2023

Guaranteeing Anonymity in Attribute-Based Authorization

Attribute-based methods, such as attribute-based access control and attribute-based encryption, make decisions based on attributes possessed by a subject rather than the subject's identity. While this allows for anonymous authorization -- determining that a subject is authorized without knowing the identity of the subject -- it does not guarantee anonymity. If a policy can be composed such that few subjects possess attributes satisfying the policy, then when the policy is used for access control, in addition to making a grant or deny decision, the system can also guess with high probability the identity of the subject making the request. Other approaches to achieving anonymity in attribute-based authorization do not address this attribute distribution problem. Suppose polices contain conjunctions of at most $t$ attributes and the system must not be able to guess with probability greater than $\frac{1}{r}$ the identity of a subject using a policy for authorization. We say the anonymity guarantee is $r$ for maximum credential size $t$. An anonymizing array is a combinatorial array proposed as an abstraction to address the underlying attribute distribution problem by ensuring that any assignment of values to $t$ attributes appearing in the array appears at least $r$ times. Anonymizing arrays are related to covering arrays with higher coverage, but have an additional desired property, homogeneity, due to their application domain. In this work, we discuss the application of anonymizing arrays to guarantee anonymous authorization in attribute-based methods. Additionally, we develop metrics, local and global homogeneity, to compare anonymizing arrays with the same parameters.

cs.CR↗

arXiv2022

Sets of mutually orthogoval projective and affine planes

A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays.

math.CO↗

arXiv2021

Distributing hash families with few rows

Column replacement techniques for creating covering arrays rely on the construction of perfect and distributing hash families with few rows, having as many columns as possible for a specified number of symbols. To construct distributing hash families in which the number of rows is less than the strength, we examine a method due to Blackburn and extend it in three ways. First, the method is generalized from homogeneous hash families (in which every row has the same number of symbols) to heterogeneous ones. Second, the extension treats distributing hash families, in which only separation into a prescribed number of parts is required, rather than perfect hash families, in which columns must be completely separated. Third, the requirements on one of the main ingredients are relaxed to permit the use of a large class of distributing hash families, which we call fractal. Constructions for fractal perfect and distributing hash families are given, and applications to the construction of perfect hash families of large strength are developed.

math.CO↗

arXiv2021

Disjoint spread systems and fault location

When $k$ factors each taking one of $v$ levels may affect the correctness or performance of a complex system, a test is selected by setting each factor to one of its levels and determining whether the system functions as expected (passes the test) or not (fails). In our setting, each test failure can be attributed to at least one faulty (factor, level) pair. A nonadaptive test suite is a selection of such tests to be executed in parallel. One goal is to minimize the number of tests in a test suite from which we can determine which (factor, level) pairs are faulty, if any. In this paper, we determine the number of tests needed to locate faults when exactly one (or at most one) pair is faulty. To do this, we address an equivalent problem, to determine how many set partitions of a set of size $N$ exist in which each partition contains $v$ classes and no two classes in the partitions are equal.

math.CO↗

arXiv2020

New bounds on the maximum size of Sperner partition systems

An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems and use it to asymptotically determine $\mathrm{SP}(n,k)$ in many cases as $\frac{n}{k}$ becomes large. We also give a slightly improved upper bound for $\mathrm{SP}(n,k)$ and exhibit an infinite family of parameter sets $(n,k)$ for which this bound is tight.

math.CO↗

arXiv2019

Access Balancing in Storage Systems by Labeling Partial Steiner Systems

Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can be designed using dense partial Steiner systems in order to support fast reads, writes, and recovery of failed storage units. In order to ensure good performance, popularities of the data items should be taken into account and the frequencies of accesses to the storage units made as uniform as possible. A proposed combinatorial model ranks items by popularity and assigns data items to elements in a dense partial Steiner system so that the sums of ranks of the elements in each block are as equal as possible. By developing necessary conditions in terms of independent sets, we demonstrate that certain Steiner systems must have a much larger difference between the largest and smallest block sums than is dictated by an elementary lower bound. In contrast, we also show that certain dense partial $S(t, t+1, v)$ designs can be labeled to realize the elementary lower bound. Furthermore, we prove that for every admissible order $v$, there is a Steiner triple system $(S(2, 3, v))$ whose largest difference in block sums is within an additive constant of the lower bound.

cs.IT↗

arXiv2018

Constructions of optimal orthogonal arrays with repeated rows

We construct orthogonal arrays OA$_λ (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / λ$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k \geq n+1$, albeit with large $λ$. We also study basic OAs; these are optimal OAs in which $\gcd(m,λ) = 1$. We construct a basic OA with $n=2$ and $k =4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs wth $n=2$, modulo the Hadamard matrix conjecture.

math.CO↗

arXiv2018

Counting Subwords and Regular Languages

Let $x$ and $y$ be words. We consider the languages whose words $z$ are those for which the numbers of occurrences of $x$ and $y$, as subwords of $z$, are the same (resp., the number of $x$'s is less than the number of $y$'s, resp., is less than or equal). We give a necessary and sufficient condition on $x$ and $y$ for these languages to be regular, and we show how to check this condition efficiently.

cs.FL↗

arXiv2016

Two-stage algorithms for covering array construction

Modern software systems often consist of many different components, each with a number of options. Although unit tests may reveal faulty options for individual components, functionally correct components may interact in unforeseen ways to cause a fault. Covering arrays are used to test for interactions among components systematically. A two-stage framework, providing a number of concrete algorithms, is developed for the efficient construction of covering arrays. %Our framework divides the construction in two stages. In the first stage, a time and memory efficient randomized algorithm covers most of the interactions. In the second stage, a more sophisticated search covers the remainder in relatively few tests. In this way, the storage limitations of the sophisticated search algorithms are avoided; hence the range of the number of components for which the algorithm can be applied is extended, without increasing the number of tests. Many of the framework instantiations can be tuned to optimize a memory-quality trade-off, so that fewer tests can be achieved using more memory. The algorithms developed outperform the currently best known methods when the number of components ranges from 20 to 60, the number of options for each ranges from 3 to 6, and $t$-way interactions are covered for $t\in \{5,6\}$. In some cases a reduction in the number of tests by more than $50\%$ is achieved.

cs.DS↗

arXiv2016

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 $(1-ε)\binom{k}{t}$ 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.

math.CO↗

arXiv2016

Upper bounds on the size of covering arrays

Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, CAN$(t,k,v)$, in a covering array for given values of the parameters $t,k$ and $v$. Asymptotic upper bounds for CAN$(t,k,v)$ have earlier been established using the Stein-Lovász-Johnson strategy and the Lovász local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein-Lovász-Johnson bound is derived. Then using alteration, the Stein-Lovász-Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovász local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on CAN$(t,k,v)$ are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lovász local lemma and the conditional Lovász local lemma distribution.

math.CO↗

arXiv2015

Steiner Loops Satisfying the Statement of Moufang's Theorem

Andrew Rajah posed at the Loops'11 Conference in Trest, Czech Republic, the following conjecture: Is every variety of loops that satisfies Moufang's theorem contained in the variety of Moufang loops? This paper is motivated by that problem. We give a partial answer to this question and present two types of Steiner loops, one that satisfies Moufang's theorem and another that does not, and neither is Moufang loop.

math.GR↗

arXiv2015

A combinatorial approach to X-tolerant compaction circuits

Test response compaction for integrated circuits (ICs) with scan-based design-for-testability (DFT) support in the presence of unknown logic values (Xs) is investigated from a combinatorial viewpoint. The theoretical foundations of X-codes, employed in an X-tolerant compaction technique called X-compact, are examined. Through the formulation of a combinatorial model of X-compact, novel design techniques are developed for X-codes to detect a specified maximum number of errors in the presence of a specified maximum number of unknown logic values, while requiring only small fan-out. The special class of X-codes that results leads to an avoidance problem for configurations in combinatorial designs. General design methods and nonconstructive existence theorems to estimate the compaction ratio of an optimal X-compactor are also derived.

cs.IT↗

arXiv2014

Hierarchical Recovery in Compressive Sensing

A combinatorial approach to compressive sensing based on a deterministic column replacement technique is proposed. Informally, it takes as input a pattern matrix and ingredient measurement matrices, and results in a larger measurement matrix by replacing elements of the pattern matrix with columns from the ingredient matrices. This hierarchical technique yields great flexibility in sparse signal recovery. Specifically, recovery for the resulting measurement matrix does not depend on any fixed algorithm but rather on the recovery scheme of each ingredient matrix. In this paper, we investigate certain trade-offs for signal recovery, considering the computational investment required. Coping with noise in signal recovery requires additional conditions, both on the pattern matrix and on the ingredient measurement matrices.

cs.IT↗

arXiv2013

ATLAS: Adaptive Topology- and Load-Aware Scheduling

The largest strength of contention-based MAC protocols is simultaneously the largest weakness of their scheduled counterparts: the ability to adapt to changes in network conditions. For scheduling to be competitive in mobile wireless networks, continuous adaptation must be addressed. We propose ATLAS, an Adaptive Topology- and Load-Aware Scheduling protocol to address this problem. In ATLAS, each node employs a random schedule achieving its persistence, the fraction of time a node is permitted to transmit, that is computed in a topology and load dependent manner. A distributed auction (REACT) piggybacks offers and claims onto existing network traffic to compute a lexicographic max-min channel allocation. A node's persistence p is related to its allocation. Its schedule achieving p is updated where and when needed, without waiting for a frame boundary.We study how ATLAS adapts to controlled changes in topology and load. Our results show that ATLAS adapts to most network changes in less than 0.1s, with about 20% relative error, scaling with network size. We further study ATLAS in more dynamic networks showing that it keeps up with changes in topology and load sufficient for TCP to sustain multi-hop flows, a struggle in IEEE 802.11 networks. The stable performance of ATLAS supports the design of higher-layer services that inform, and are informed by, the underlying communication network.

cs.NI↗

arXiv2009

Drop cost and wavelength optimal two-period grooming with ratio 4

We study grooming for two-period optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the two-period grooming problem, during the first period of time, there is all-to-all uniform traffic among $n$ nodes, each request using $1/C$ of the bandwidth; and during the second period, there is all-to-all uniform traffic only among a subset $V$ of $v$ nodes, each request now being allowed to use $1/C'$ of the bandwidth, where $C' < C$. We determine the minimum drop cost (minimum number of ADMs) for any $n,v$ and C=4 and $C' \in \{1,2,3\}$. To do this, we use tools of graph decompositions. Indeed the two-period grooming problem corresponds to minimizing the total number of vertices in a partition of the edges of the complete graph $K_n$ into subgraphs, where each subgraph has at most $C$ edges and where furthermore it contains at most $C'$ edges of the complete graph on $v$ specified vertices. Subject to the condition that the two-period grooming has the least drop cost, the minimum number of wavelengths required is also determined in each case.

math.CO↗

arXiv2007

Optimal Memoryless Encoding for Low Power Off-Chip Data Buses

Off-chip buses account for a significant portion of the total system power consumed in embedded systems. Bus encoding schemes have been proposed to minimize power dissipation, but none has been demonstrated to be optimal with respect to any measure. In this paper, we give the first provably optimal and explicit (polynomial-time constructible) families of memoryless codes for minimizing bit transitions in off-chip buses. Our results imply that having access to a clock does not make a memoryless encoding scheme that minimizes bit transitions more powerful.

cs.AR↗

arXiv2007

Constructions for Difference Triangle Sets

Difference triangle sets are useful in many practical problems of information transmission. This correspondence studies combinatorial and computational constructions for difference triangle sets having small scopes. Our algorithms have been used to produce difference triangle sets whose scopes are the best currently known.

cs.IT↗