57 research outputs found
Combinatorial batch codes
In this paper, we study batch codes, which were introduced by Ishai, Kushilevitz, Ostrovsky and Sahai in [4]. A batch code specifies a method to distribute a database of [n] items among [m] devices (servers) in such a way that any [k] items can be retrieved by reading at most [t] items from each of the servers. It is of interest to devise batch codes that minimize the total storage, denoted by [N] , over all [m] servers.
We restrict out attention to batch codes in which every server stores a subset of the items. This is purely a combinatorial problem, so we call this kind of batch code a ''combinatorial batch code''. We only study the special case [t=1] , where, for various parameter situations, we are able to present batch codes that are optimal with respect to the storage requirement, [N] . We also study uniform codes, where every item is stored in precisely [c] of the [m] servers (such a code is said to have rate [1/c] ). Interesting new results are presented in the cases [c = 2, k-2] and [k-1] . In addition, we obtain improved existence results for arbitrary fixed [c] using the probabilistic method
Distinct difference configurations: multihop paths and key predistribution in sensor networks
A distinct difference configuration is a set of points in Z2 with the property that the vectors (difference vectors) connecting any two of the points are all distinct. Many specific examples of these configurations have been previously studied: the class of distinct difference configurations includes both Costas arrays and sonar sequences, for example. Motivated by an application of these structures in key predistribution for wireless sensor networks, we define the k-hop coverage of a distinct difference configuration to be the number of distinct vectors that can be expressed as the sum of k or fewer difference vectors. This is an important parameter when distinct difference configurations are used in the wireless sensor application, as this parameter describes the density of nodes that can be reached by a short secure path in the network. We provide upper and lower bounds for the k-hop coverage of a distinct difference configuration with m points, and exploit a connection with Bh sequences to construct configurations with maximal k-hop coverage. We also construct distinct difference configurations that enable all small vectors to be expressed as the sum of two of the difference vectors of the configuration, an important task for local secure connectivity in the application
Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes
A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid
Optimal constructions for ID-based one-way-function key predistribution schemes realizing specified communication graphs
We study a method for key predistribution in a network of n users where pairwise keys are
computed by hashing usersâ IDs along with secret information that has been (pre)distributed to
the network users by a trusted entity. A communication graph G can be specified to indicate
which pairs of users should be able to compute keys. We determine necessary and sufficient
conditions for schemes of this type to be secure. We also consider the problem of minimizing
the storage requirements of such a scheme; we are interested in the total storage as well as
the maximum storage required by any user. Minimizing the total storage is NP-hard, whereas
minimizing the maximum storage required by a user can be computed in polynomial time
Disjoint difference families and their applications
Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and showed that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families
On the equivalence of authentication codes and robust (2,2)-threshold schemes
In this paper, we show a "direct" equivalence between certain authentication
codes and robust secret sharing schemes. It was previously known that
authentication codes and robust secret sharing schemes are closely related to
similar types of designs, but direct equivalences had not been considered in
the literature. Our new equivalences motivate the consideration of a certain
"key-substitution attack." We study this attack and analyze it in the setting
of "dual authentication codes." We also show how this viewpoint provides a nice
way to prove properties and generalizations of some known constructions
Circular external difference families, graceful labellings and cyclotomy
(Strong) circular external difference families (which we denote as CEDFs and
SCEDFs) can be used to construct nonmalleable threshold schemes. They are a
variation of (strong) external difference families, which have been extensively
studied in recent years. We provide a variety of constructions for CEDFs based
on graceful labellings (-valuations) of lexicographic products , where denotes a cycle of length .
SCEDFs having more than two subsets do not exist. However, we can construct
close approximations (more specifically, certain types of circular algebraic
manipulation detection (AMD) codes) using the theory of cyclotomic numbers in
finite fields
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