Tran van Trung

Secure Frameproof Codes, Key Distribution Patterns, Group Testing Algorithms and Related Structures

Abstract Frameproof codes were introduced by Boneh and Shaw as a method of “digital fingerprinting” which prevents a coalition of a specified size c from framing a user not in the coalition. Stinson and Wei then gave a combinatorial formulation of the problem in terms of certain types of extremal set systems. In this paper, we study frameproof codes that provide a certain (weak) form of traceability. We extend our combinatorial formulation to address this stronger requirement, and show that the problem is solved by using ( i , j )-separating systems, as defined by Friedman, Graham and Ullman. Using constructions based on perfect hash families, we give the first efficient explicit constructions for these objects for general values of i and j . We also review nonconstructive existence results that are based on probabilistic arguments. Then we look at two other, related concepts, namely key distribution patterns and nonadaptive group testing algorithms. We again approach these problems from the point of view of extremal set systems, and we describe a natural common setting in which these two problems are complementary special cases. This approach also demonstrates a close relationship between these two problems and frameproof codes. Explicit constructions are given, and some nonconstructive existence results are reviewed. In the case of key distribution patterns, our explicit constructions are the most efficient ones known.

Some New Results on Key Distribution Patterns and BroadcastEncryption

This paper concerns methods by which a trusted authority can distribute keys and/or broadcast a message over a network, so that each member of a privileged subset of users can compute a specified key or decrypt the broadcast message. Moreover, this is done in such a way that no coalition is able to recover any information on a key or broadcast message they are not supposed to know. The problems are studied using the tools of information theory, so the security provided is unconditional (i.e., not based on any computational assumption).In a recent paper st95a, Stinson described a method of constructing key predistribution schemes by combining Mitchell-Piper key distribution patterns with resilient functions; and also presented a construction method for broadcast encryption schemes that combines Fiat-Naor key predistribution schemes with ideal secret sharing schemes. In this paper, we further pursue these two themes, providing several nice applications of these techniques by using combinatorial structures such as orthogonal arrays, perpendicular arrays, Steiner systems and universal hash families.

On t -Covering Arrays

This paper concerns construction methods for t-covering arrays. Firstly, a construction method using perfect hash families is discussed by combining with recursion techniques and error-correcting codes. In particular, by using algebraic-geometric codes for this method we obtain infinite families of t-covering arrays which are proved to be better than currently known probabilistic bounds for covering arrays. Secondly, inspired from a result of Roux [16] and also from a recent result of Chateauneuf and Kreher [6] for 3-covering arrays, we present several explicit constructions for t-covering arrays, which can be viewed as generalizations of their results for t-covering arrays.

The existence of symmetric block designs with parameters (41, 16, 6) and (66, 26, 10)

Abstract In [1, 2] it is indicated that the existence of block designs with parameters (41, 16, 6) and (66, 26, 10) is still unknown. The purpose of this paper is to prove the existence of block designs with these parameters. The idea for the construction of these block designs is to assume that they admit certain automorphism group.

On a Class of Traceability Codes

Traceability codes are designed to be used in schemes that protect copyrighted digital data against piracy. The main aim of this paper is to give an answer to a Staddon–Stinson–Weis problem of the existence of traceability codes with q< w2 and b>q. We provide a large class of these codes constructed by using a new general construction method for q-ary codes.

Bounds for separating hash families

This paper aims to present new upper bounds on the size of separating hash families. These bounds improve previously known bounds for separating hash families.

Directed t -Packings and Directed t -SteinerSystems

The aim of this paper is an investigation of directed t-packings and in particular of directed t-Steiner systems. A new upper bound on the number of points k for directed t-Steiner systems T(t,k,k) is obtained. We disprove a conjecture of Levenshtein on T(t,k,k) for t ≥ 3 by showing that a T(4,6,6) exists. Furthermore, it is proved that the symmetric group S6 can be partitioned into 30 disjoint T(4,6,6)s. Extensive computer search shows that the tight upper bound on K for t =4,5 is 6 and for t=6 is 7. The non-existence of further small directed t-Steiner systems is established, and large directed t-packings for t,4,5,6 are constructed.

A new biplane of order 9 with a small automorphism group

A biplane of order n is a set of u = 1 + (n + l)(n + 2)/2 points and u lines so that each line contains exactly k = n + 2 points and any two distinct lines intersect in 2 points. The largest known biplane was constructed 14 years ago by Aschbacher [I] with k = 13 (order 11). In fact, his biplane is not self-dual and so there are exactly two biplanes with k = 13 which are known. Biplanes of order n < 8 have been classified and there are exactly 10 biplanes with those orders [2]. Up to now, there have been only four biplanes of order 9 known [3]. Since there is no biplane of order 10, it

Improved bounds for separating hash families

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