... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must use \Omega ( n2ffl2 log(n=ffl) log m) bits of storage. We also

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.

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. 1

The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD A n complexity of all strings x in some set A. The best known upper bound for this problem is 2 log(jjA n jj) + O(log(n)), due to Buhrman and Fortnow. We show that the constant factor 2 in this bound is optimal. We also give new bounds for a certain kind of random sets R ` f0; 1g n , for which we show an upper bound of log(jjR n jj) + O(log(n)). 1 Introduction Kolmogorov complexity is a notion that measures the amount of regularity in a finite string. It has turned out to be a very useful tool in theoretical computer science. A simple counting argument showing that for each length there exist random strings, i.e. strings with no regularity, has had many applications (see [LV97]). Early in the history of computational complexity resource bounded notions of Kolmogorov complexity were...

Let N ((w; r); T ) denote the minimum number of points in a (w; r)-cover-free family having T blocks. In this paper, we prove two new lower bounds on N . 1 Introduction Cover-free families were first introduced in 1964 by Kautz and Singleton [9] to investigate superimposed binary codes. These structures have been discussed in several equivalent formulations in subjects such as information theory, combinatorics and group testing by numerous researchers (see, for example, [1, 2, 4, 5, 6, 7, 8, 12]). In 1988, Mitchell and Piper [10] defined the concept of key distribution patterns, which are in fact a generalized type of cover-free family. Some papers giving constructions and bounds for these objects include [3, 4, 11, 14]. Here is the definition of a cover-free family. Definition 1.1 Let X be an n-set and let F be a set of subsets (blocks) of X. (X; F) is called a (w; r)-cover-free family (or (w; r)-CFF) provided that, for any w blocks B 1 ; \Delta \Delta \Delta ; Bw 2 F and any other ...

Abstract Motivated by the challenging task of designing "secure " vote storage mechanisms, we dealwith information storage mechanisms that operate in extremely hostile environments. In such

A cover-free family is a well-studied combinatorial structure that has many applications in computer science and cryptography. In this paper, we propose a new public key traitor tracing scheme based on cover-free families. The new traitor tracing scheme is similar to the Boneh-Franklin scheme except that in the Boneh-Franklin scheme, decryption keys are derived from Reed-Solomon codes while in our case they are derived from a cover-free family. This results in much simpler and faster tracing algorithms for single-key pirate decoders, compared to the tracing algorithms of Boneh-Franklin scheme that use Berlekamp-Welch algorithm. Our tracing algorithms never accuse innocent users and identify all traitors with overwhelming probability.

Abstract — Detection of defective members of large populations has been widely studied in the statistics community under the name “group testing”, a problem which dates back to World War II when it was suggested for syphilis screening. There, the main interest is to identify a small number of infected people among a large population using collective samples. In viral epidemics, one way to acquire collective samples is by sending agents inside the population. While in classical group testing, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in this work we assume that the decoder possesses only partial knowledge about the sampling process. This assumption is justified by observing the fact that in a viral sickness, there is a chance that an agent remains healthy despite having contact with an infected person. Therefore, the reconstruction method has to cope with two different types of uncertainty; namely, identification of the infected population and the partially unknown sampling procedure. In this work, by using a natural probabilistic model for “viral infections”, we design non-adaptive sampling procedures that allow successful identification of the infected population with overwhelming probability 1 − o(1). We propose both probabilistic and explicit design procedures that require a “small ” number of agents to single out the infected individuals. More precisely, for a contamination probability p, the number of agents required by the probabilistic and explicit designs for identification of up to k infected members is bounded by m = O(k 2 (log n)/p 2) and m = O(k 2 (log 2 n)/p 2), respectively. In both cases, a simple decoder is able to successfully identify the infected population in time O(mn). I.

For integers n, r ≥ 2 and 1 ≤ k ≤ r, a family F of subsets of [n] = {1,..., n} is called k-outof-r multiple user tracing if, given the union of any ℓ ≤ r sets from the family, one can identify at least min(k, ℓ) of them. This is a generalization of superimposed families (k = r) and of single user tracing families (k = 1). The study of such families is motivated by problems in molecular biology and communication. In this paper we study the maximum possible cardinality of such families, denoted by h(n, r, k), and show that there exist absolute constants c1, c2, c3, c4> 0 such that min ( c1 c2 r, k2 log h(n,r,k)) ≤ n ≤ min ( c3 c4 log k r, k2). In particular, for all k ≤ √ log h(n,r,k) r, n = Θ(1/r). This improves an estimate of Laczay and Ruszinkó. 1

Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n, k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2,..., n} to r bits. Monotone encodings are relevant to the study of tamperproof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multi-user tracing ((k, α)-FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant. 1