Perfect Hash Families Research Articles

Bounds And Constructions For Threshold Shared Generation Of Authenticators

Shared generation of authenticator systems (or SGA-systems) generalise Simmons' traditional authentication codes when authenticators for source states are produced by collaboration of a group of senders, rather than a single sender. In this paper we study threshold SGA-systems. We derive information-theoretic and combinatorial lower bounds on the probability of success and the size of the key space, and give two key efficient constructions for SGA-systems based on den Boer A-codes and error-correcting codes. We also give a recursive construction method using perfect hash families to construct SGA-systems for large groups.

Broadcast authentication for group communication

Traditional point-to-point message authentication systems have been extensively studied in the literature. In this paper, we consider authentication systems for group communication. The basic primitive is a multireceiver authentication system with dynamic senders (DMRA-code). In a DMRA-code any member of the group can broadcast an authenticated message to the rest of the group such that every other member of the group can individually verify the authenticity of the message. We give a flexible ‘synthesis’ construction for DMRA-codes by combining an A-code and a key distribution pattern. tDMRA-codes extend this model when there are up to t senders. We give two constructions, one algebraic and one by ‘synthesis’ of an A-code and a perfect hash family. We show universality of ‘synthesis’ constructions for unconditional and computational models of security which means that our results have wider range applications. Finally, to demonstrate the usefulness of DMRA model, we modify a secure dynamic conference-key distribution system to construct a secure dynamic conference system that provides secrecy and authenticity of communication among conferencees. The system is key-efficient as its key requirement remains nearly the same as the original conference key distribution system and so authentication is effectively obtained without any extra cost. We discuss possible extensions to this work.

Some baby-step giant-step algorithms for the low hamming weight discrete logarithm problem

In this paper, we present several baby-step giant-step algorithms for the low hamming weight discrete logarithm problem. In this version of the discrete log problem, we are required to find a discrete logarithm in a finite group of order approximately 2m, given that the unknown logarithm has a specified number of l's, say t, in its binary representation. Heiman and Odlyzko presented the first algorithms for this problem. Unpublished improvements by Coppersmith include a deterministic algorithm with complexity O (m (t/2m/2)), and a Las Vegas algorithm, with complexity O (√t (m/2 t/2)).We perform an average-case analysis of Coppersmith's deterministic algorithm. The average-case complexity achieves only a constant factor speed-up over the worst-case. Therefore, we present a generalized version of Coppersmith's algorithm, utilizing a combinatorial set system that we call a splitting system. Using probabilistic methods, we prove a new existence result for these systems that yields a (nonuniform) deterministic algorithm with complexity O(t3/2 (log m) (t/2m/2)))). We also present some explicit constructions for splitting systems that make use of perfect hash families.

Combinatorial properties of frameproof and traceability codes

In order to protect copyrighted material, codes may be embedded in the content or codes may be associated with the keys used to recover the content. Codes can offer protection by providing some form of traceability (TA) for pirated data. Several researchers have studied different notions of TA and related concepts in previous years. Strong versions of TA allow at least one member of a coalition that constructs a pirate decoder to be traced. Weaker versions of this concept ensure that no coalition can frame a disjoint user or group of users. All these concepts can be formulated as codes having certain combinatorial properties. We study the relationships between the various notions, and we discuss equivalent formulations using structures such as perfect hash families. We use methods from combinatorics and coding theory to provide bounds (necessary conditions) and constructions (sufficient conditions) for the objects of interest.

Explicit Constructions of Perfect Hash Families from Algebraic Curves over Finite Fields

Let A be a set of order n and B be a set of order m. An (n, m, w)-perfect hash family is a set H of functions from A to B such that for any X⊆A with |X|=w, there exists an element h∈H such that h is one-to-one when restricted to X. Perfect hash families have many applications to computer science, such as database management, circuit complexity theory and cryptography. In this paper, we provide explicit constructions of perfect hash families based on algebraic curves over finite fields. In particular, using the Garcia–Stichtenoth curves, we obtain infinite classes of (n, m, w)-perfect hash families with |H|=O(logn) for fixed m and w, which are among the most efficient explicit constructions for perfect hash families known in the literature. We also exhibit examples to show the efficiency of the new constructions and their applications to the constructions of cover-free families.

Perfect Hash Families: Probabilistic Methods and Explicit Constructions

An (n, q, t)-perfect hash family of size s consists of a set V of order n, a set F of order q, and a sequence φ1, φ2, …, φs of functions from V to F with the following property. For all t-subsets X⊆V, there exists i∈{1, 2, …, s} such that φi is injective when restricted to X. An (n, q, t)-perfect hash family of minimal size is known as optimal. The paper presents a probabilistic existence result for perfect hash families which improves on the well known result of Mehlhorn for many parameter sets. The probabilistic methods are strong enough to establish the size of an optimal perfect hash family in many cases. The paper also gives several explicit constructions of classes of perfect hash families.

Secure frameproof codes, key distribution patterns, group testing algorithms and related structures

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.

New constructions for perfect hash families and related structures using combinatorial designs and codes

In this paper, we consider explicit constructions of perfect hash families using combinatorial methods. We provide several direct constructions from combinatorial structures related to orthogonal arrays. We also simplify and generalize a recursive construction due to Atici, Magliversas, Stinson and Wei [3]. Using similar methods, we also obtain efficient constructions for separating hash families which result in improved existence results for structures such as separating systems, key distribution patterns, group testing algorithms, cover-free families and secure frameproof codes. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:189–200, 2000

Optimal Linear Perfect Hash Families

LetVbe a set of ordernand letFbe a set of orderq. A setS⊆{φ:V→F} of functions fromVtoFis an (n,q,t)-perfect hash familyif for allX⊆Vwith |X|=t, there existsφ∈Swhich is injective when restricted toX. Perfect hash families arise in compiler design, in circuit complexity theory and in cryptography. LetSbe an (n,q,t)-perfect hash family. The paper provides lower bounds on |S|, which better previously known lower bounds for many parameter sets. The paper exhibits new classes of perfect hash families which show that these lower bounds are realistic.

Combinatorial Properties and Constructions of Traceability Schemes and Frameproof Codes

In this paper, we investigate combinatorial properties and constructions of two recent topics of cryptographic interest, namely frameproof codes for digital fingerprinting and traceability schemes for broadcast encryption. We first give combinatorial descriptions of these two objects in terms of set systems and also discuss the Hamming distance of frameproof codes when viewed as error-correcting codes. From these descriptions, it is seen that existence of a c-traceability scheme implies the existence of a c-frameproof code. We then give several constructions of frameproof codes and traceability schemes by using combinatorial structures such as t-designs, packing designs, error-correcting codes, and perfect hash families. We also investigate embeddings of frameproof codes and traceability schemes, which allow a given scheme to be expanded at a later date to accommodate more users. Finally, we look briefly at bounds which establish necessary conditions for existence of these structures.

Some recursive constructions for perfect hash families

An (n, m, w)-perfect hash family is a set of functions F such that for each f ϵ F, and for any X ⊆ {1,…,n} such that |X| = w, there exists at least one f ϵ F such that f|x is one-to-one. Perfect hash families have been extensively studied by computer scientists for over 15 years, mainly due to their applications in database management. In particular, much attention has been given to finding efficient algorithms to construct perfect hash families. In this article, we study perfect hash families from a combinatorial viewpoint, and describe some new recursive constructions for these objects. © 1996 John Wiley & Sons, Inc.

Some recursive constructions for perfect hash families

An (n, m, w)-perfect hash family is a set of functions F such that for each f ϵ F, and for any X ⊆ {1,…,n} such that |X| = w, there exists at least one f ϵ F such that f|x is one-to-one. Perfect hash families have been extensively studied by computer scientists for over 15 years, mainly due to their applications in database management. In particular, much attention has been given to finding efficient algorithms to construct perfect hash families. In this article, we study perfect hash families from a combinatorial viewpoint, and describe some new recursive constructions for these objects. © 1996 John Wiley & Sons, Inc.