Cover-free Families Research Articles

Nested cover-free families for unbounded fault-tolerant aggregate signatures

Aggregate signatures are used to create one short proof of authenticity and integrity from a set of digital signatures. However, one invalid signature in the set invalidates the entire aggregate, giving no information on which signatures are valid. Hartung et al. (2016) [6] propose a fault-tolerant aggregate signature scheme based on combinatorial group testing. Given a bound d on the number of invalid signatures among n signatures to be aggregated, this scheme uses d-cover-free families to determine which signatures are invalid. These combinatorial structures guarantee a moderate increase on the size of the aggregate signature that can reach the best possible compression ratio of O(nlog⁡n), for fixed d, coming from an information theoretical bound. The case where the total number of signatures grows dynamically (unbounded scheme) was not satisfactorily solved in their original paper, since explicit constructions had constant compression ratios. In the present paper, we propose efficient solutions for the unbounded scheme, relying on sequences of d-cover-free families that we call nested families. Some of our constructions yield high compression ratio close to the best known upper bound. We also propose the use of (d,λ)-cover-free families to support the loss of up to λ−1 parts of the aggregate.

Revocable identity-based encryption with bounded decryption key exposure resistance: Lattice-based construction and more

In general, identity-based encryption (IBE) does not support an efficient revocation procedure. In ACM CCS'08, Boldyreva et al. proposed revocable identity-based encryption (RIBE), which enables us to efficiently revoke (malicious) users in IBE. In PKC 2013, Seo and Emura introduced an additional security notion for RIBE, called decryption key exposure resistance (DKER). Roughly speaking, RIBE with DKER guarantees that the security is not compromised even if an adversary gets (a number of) short-term decryption keys. Therefore, DKER captures realistic scenarios and is an important notion.In this paper, we introduce bounded decryption key exposure resistance (B-DKER), where an adversary is allowed to get a-priori bounded number of short-term decryption keys in the security game. B-DKER is a weak version of DKER, but it seems to be sufficient for practical use. We obtain the following results:•We propose a lattice-based (anonymous) RIBE scheme with B-DKER, which is the first lattice-based construction resilient to decryption key exposure. Our lattice-based construction is secure under the learning with errors assumption. A previous lattice-based construction satisfies anonymity but is vulnerable even with a single decryption key exposure.•We propose the first pairing-based RIBE scheme that simultaneously realizes anonymity and B-DKER. Our pairing-based construction is adaptively secure under the symmetric external Diffie-Hellman assumption.Our two constructions rely on cover free families to satisfy B-DKER, whereas all the existing works rely on the key re-randomization property to achieve DKER.

Embedding cover-free families and cryptographical applications

Cover-free families are set systems used as solutions for a large variety of problems, and in particular, problems where we deal with \begin{document}$ n $\end{document} elements and want to identify \begin{document}$ d $\end{document} defective ones among them by performing only \begin{document}$ t $\end{document} tests ( \begin{document}$ t \leq n $\end{document} ). We are especially interested in cryptographic problems, and we note that some of these problems need cover-free families with an increasing size \begin{document}$ n $\end{document} . Solutions that propose the increase of \begin{document}$ n $\end{document} , such as monotone families and nested families, have been recently considered in the literature. In this paper, we propose a generalization that we call embedding families, which allows us to increase both \begin{document}$ n $\end{document} and \begin{document}$ d $\end{document} . We propose constructions of embedding families using polynomials over finite fields embedded via extension fields; we study how different parameter combinations can be used to prioritize increase of \begin{document}$ d $\end{document} or of the compression ratio as \begin{document}$ n $\end{document} grows. We also provide new constructions for monotone families with improved compression ratio. Finally, we show how to use embedded sequences of orthogonal arrays and packing arrays to build embedding families.

Union–intersection-bounded families and their applications

Cover-free families have been widely studied over recent decades due to their applications in numerous subjects. In this paper, we introduce the concept of (s,t;d)-union–intersection-bounded families, which is a generalization of t-cover-free families. We provide a general upper bound on the maximum size of an (s,t;d)-union–intersection-bounded family, and show a probabilistic lower bound for the case that the ground set is sufficiently large. They have the same order of magnitude for certain cases. We also discuss the applications of (s,t;d)-union–intersection-bounded families in broadcast encryption, and derive a better upper bound for (1,t;d)-union–intersection-bounded families (also known as superimposed distance codes).

Efficient Usage of Cover Free Families in Broadcast Encryption

Efficient Usage of Cover Free Families in Broadcast Encryption

Constraining the number of positive responses in adaptive, non-adaptive, and two-stage group testing

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of “yes” responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of “yes” responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of “yes” responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p, d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts.

On the complexity of neighbourhood learning in radio networks

Consider a synchronous radio network of n stationary nodes represented by an undirected graph with maximum degree Δ. Suppose that each node has a unique ID from {1,…,U}, where U≫n. In the neighbourhood learning task, each node must produce a list of the IDs of its neighbours in the network. We prove new lower bounds on the number of slots needed by certain classes of deterministic algorithms that solve this task. First, we show that O(U)-slot round-robin algorithms are optimal for the class of collision-free algorithms. Then, we consider algorithms where each node fixes its entire transmission schedule at the start. For such algorithms, we prove a Ω(Δ2log⁡Δlog⁡U)-slot lower bound on schedule length that holds in very general models, e.g., when nodes possess collision detectors, messages can be of arbitrary size, and nodes know the schedules being followed by all other nodes. We also prove a similar result for the SINR model of radio networks. To prove these results, we consider a generalization of cover-free families of sets. We also show a separation between the class of fixed-schedule algorithms and the class of algorithms where nodes can choose to leave out some transmissions from their schedule.

Locating modifications in signed data for partial data integrity

We consider the problem of detecting and locating modifications in signed data to ensure partial data integrity. We assume that the data is divided into n blocks (not necessarily of the same size) and that a threshold d is given for the maximum amount of modified blocks that the scheme can support. We propose efficient algorithms for signature and verification steps which provide a reasonably compact signature size, for controlled sizes of d with respect to n. For instance, for fixed d the standard signature size gets multiplied by a factor of O(log⁡n), while allowing the identification of up to d modified blocks. Our scheme is based on nonadaptive combinatorial group testing and cover-free families.

A Variation of Cover-Free Families and Its Applications.

A cover-free family is a family of subsets of a finite set in which no one is covered by the union of r others. We study a variation of cover-free family: A binary matrix is (r, w]-consecutive-disjunct if for any w cyclically consecutive columns [Formula: see text] and another r cyclically consecutive columns [Formula: see text], there exists one row intersecting [Formula: see text] but none of [Formula: see text]. In group testing, the goal is to determine a small subset of positive items D in a large population [Formula: see text] by group tests. By applying consecutive-disjunct matrices, we solve threshold group testing of consecutive positives in [Formula: see text] group tests nonadaptively, and the decoding complexity is [Formula: see text] where u is a threshold parameter in threshold group testing and it is assumed that |D|≤d and [Formula: see text]. Meanwhile, we obtain that for group testing of consecutive positives, all positives can be identified in [Formula: see text] group tests nonadaptively and the decoding complexity is [Formula: see text].

Generic Constructions and Transformations of Decryption Consistent Encryption

ABSTRACTBinding encryption provides an effective way to broadcast a secret to an authorized group of users, which guarantees decryption consistency, i.e., that all users can obtain the same message without any interaction among them. In this paper, we first give the definition of security of a binding encryption, and then construct a primitive that achieves security against chosen plaintext attacks (CPA) and decryption consistency. This primitive is derived from a CPA-secure public-key encryption (PKE) scheme. Then, we present several transformations of binding encryption, which ensure some advanced security requirements such as anonymity, strong decryption consistency, and non-malleability.(1) Negative results that: (a) A CCA (chosen-ciphertext attack) secure PKE does not imply a CCA-secure multi-receiver encryption (ME) or binding encryption; (b) an anonymous CPA-secure PKE implies an anonymous ME but does not imply any anonymous binding encryption; (c) a CCA-secure anonymous PKE does not imply a CCA-secure anonymous ME or anonymous binding encryption.(2) A generic construction for a binding encryption that: (a) Uses CPA-secure PKE and symmetric encryption to construct a CPA-secure ME; (b) uses a CPA-secure ME and a polynomially verifiable function to construct a CPA-secure binding encryption.(3) Three transformations from (anonymous) CCA-secure PKE to (anonymous) CCA-secure binding encryption: the first one is based on a strong one-time signature, the second one is derived from a trapdoor pseudo-random function, while the third one is based on (information-theoretically secure) cover-free families, and hence it does not require any additional computational assumptions.

Testers

We develop a new notion called tester of a class of multivariate polynomials M ⊆F[χ 1 ,..., χ n ] that maps the elements a ε A of an F-algebra A to a finite number (the size of the tester) of elements b 1 , . . . , b t in a smaller sub-algebra B ⊂ A where the property f ( a ) ≠ 0 is preserved for all f ε M . I.e., for all f ε M and a ε A if f ( a ) ≠ 0 then f ( b i ) ≠ 0 for some i . We use tools from elementary algebra and algebraic function fields to construct testers of almost optimal size in deterministic polynomial time in the size of the tester. We then apply testers to deterministically construct new set of objects with some combinatorial and algebraic properties that can be used to derandomize some algorithms. We show that those new constructions are almost optimal and for many of them meet the union bound of the problem. Constructions include, d-restriction problems, perfect hash, universal sets, cover-free families, separating hash functions, polynomial restriction problems, black box polynomial identity testing for polynomials and circuits over small fields and hitting sets.

Secure frameproof codes through biclique covers

Graph Theory For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 of size at most r then the intersection of these sets is nonempty. A d-biclique cover of size v of a graph G is a collection of v-complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t≥2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t,r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r,w;d) cover-free family is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.

On the Rekeying Load in Group Key Distributions Using Cover-Free Families

Key distributions based on cover-free families have been recently proposed for secure rekeying in group communication systems after multiple simultaneous user ejections. Existing literature has not quantified how difficult this rekeying operation might be. This study provides upper bounds on the number messages necessary to rekey a key distribution based on symmetric combinatorial designs after one or two simultaneous user ejections. Connections are made to results from finite geometry to show that these bounds are tight for certain key distributions. It is shown that in general determining the minimal number of messages necessary to rekey a group communication system based on a cover-free family is NP-hard.

Some new bounds for cover-free families through biclique covers

An (r,w;d) cover-free family (CFF) is a family of subsets of a finite set X such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. The minimum size of a set X for which there exists an (r,w;d)−CFF with t blocks is denoted by N((r,w;d),t).In this paper, we show that the value of N((r,w;d),t) is equal to the d-biclique covering number of the bipartite graph It(r,w) whose vertices are all w- and r-subsets of a t-element set, where a w-subset is adjacent to an r-subset if their intersection is empty. Next, we provide some new bounds for N((r,w;d),t). In particular, we show that for r≥w and r≥2N((r,w;1),t)≥cr+ww+1+r+w−1w+1+3r+w−4w−2logrlog(t−w+1), where c is approximately 12. Also, we determine the exact value of N((r,w;d),t) for t≤r+w+rw and also for some values of d. Finally, we show that N((1,1;d),4d−1)=4d−1 if and only if there exists a Hadamard matrix of order 4d.

Transport schemes for topology-transparent scheduling

Transport protocols provide reliable, end-to-end communication between a source and a destination in a network. The Transmission Control Protocol (TCP) uses backward error correction, where the destination explicitly returns feedback to the source. Forward error correction (FEC) can also be used for transport; here the source includes enough redundancy in the encoding symbols to allow the destination to decode the message. In this paper, we compare the performance of two transport schemes, TCP and LT, a scheme based on rateless FEC codes, in a wireless ad hoc network when topology-transparent scheduling is used for channel access. These schedules are derived from cover-free families, a type of combinatorial design. They provide a mechanism to guarantee collision-free communication between any two nodes provided that each of the N nodes of the network has at most a specified number D of active (transmitting) neighbours. We find that LT outperforms TCP in more strenuous network conditions.

Cover-Free Families and Topology-Transparent Scheduling for MANETs

We examine the combinatorial requirements of topology-transparent transmission schedules in a mobile ad hoc network (MANET). Specifically, if each of the N nodes has at most D active neighbors, we require the schedule to guarantee a collision-free transmission to each neighbor. This requirement is met by a cover-free family. We show that existing constructions for topology-transparent schedules correspond to an orthogonal array. Moreover, we show that Steiner systems support the largest number of nodes for a given schedule length. Both of these combinatorial objects are special cases of cover-free families. Analytically and numerically, we examine slot guarantees, expected throughput, and normalized expected throughput for systems of small strength, exploring the sensitivity of the response to D. Expected throughput provides a better performance metric than the minimum throughput results obtained earlier. The impact of a more realistic model of acknowledgments is also examined. The extension of the schedule to multiple frames returns us to the orthogonal arrays. The very density of Steiner systems that afforded an improvement over orthogonal arrays in one frame impedes the best extension to more frames.

The Lovász Local Lemma and Its Applications to some Combinatorial Arrays

The Lovasz Local Lemma is a useful tool in the “probabilistic method” that has found many applications in combinatorics. In this paper, we discuss applications of the Lovasz Local Lemma to some combinatorial set systems and arrays, including perfect hash families, separating hash families, u-free systems, splitting systems, and generalized cover-free families. We obtain improved bounds for some of these set sytems. Also, we compare some of the bounds obtained from the local lemma to those using the basic probabilistic method as well as the well-known “expurgation” method. Finally, we briefly consider a “high probability” variation of the method, wherein a desired object is obtained with high probability in a suitable space.

On a Bound of Cover-Free Families

In this paper, we use refined Stirling formulas to determine a bound of cover-free families given by Stinson et al. l31r.

Generalized cover-free families

Cover-free families have been investigated by many researchers, and several variations of these set systems have been used in diverse applications. In this paper, we introduce a generalization of cover-free families which includes as special cases all of the previously used definitions. Then we give several bounds and some efficient constructions for these generalized cover-free families.

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.