Luisa Gargano

On the Size of Shares for Secret Sharing Schemes

A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of partecipants can recover the secret, but any non-qualified subset has absolutely no information on the secret. The set of all qualified subsets defines the access structure to the secret. Sharing schemes are useful in the management of cryptographic keys and in multy-party secure protocols.We analyze the relationships among the entropies of the sample spaces from which the shares and the secret are chosen. We show that there are access structures with 4 participants for which any secret sharing scheme must give to a participant a share at least 50% greater than the secret size. This is the first proof that there exist access structures for which the best achievable information rate (i.e., the ratio between the size of the secret and that of the largest share) is bounded away from 1. The bound is the best possible, as we construct a secret sharing scheme for the above access structures which meets the bound with equality.

Asynchronous deterministic rendezvous in graphs

Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|Lmin|2) when D is known and O((D + |Lmax|)3) if D is unknown, where |Lmin| and |Lmax| are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O(n|Lmin|), if the size n of the ring is known, and of cost O(n|Lmax|), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|Lmin|) if the topology of the graph and the initial positions are known to agents.

Sperner capacities

We determine the asymptotics of the largest family of qualitatively 2-independentk-partitions of ann-set, for everyk>2. We generalize a Sperner-type theorem for 2-partite sets of Körner and Simonyi to thek-partite case. Both results have the feature that the corresponding trivial information-theoretic upper bound is tight. The results follow from a more general Sperner capacity theorem for a family of graphs in the sense of our previous work on Sperner theorems on directed graphs.

Capacities: from information theory to extremal set theory

Abstract Generalizing the concept of zero-error capacity beyond its traditional links to any sort of information transmission we give an asymptotic solution to several hard problems in extremal set theory within a unified, formally information-theoretic framework. The results include the solution of far-reaching generalizations of Renyis problem on qualitatively independent partitions.

On the Information Rate of Secret Sharing Schemes

We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1/2 + e, where e is an arbitrary positive constant. We also provide several general lower bounds on information rate and average information rate of graphs. In particular, we show that any graph with n vertices admits a secret sharing scheme with information rate Ω((log n)/n).

Efficient Collective Communication in Optical Networks

This paper studies the problems of broadcasting and gossiping in optical networks. In such networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. In this paper we consider both single-hop and multihop optical networks. In single-hop networks the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high speed communication. In multihop networks a packet may have to be routed through a few intermediate nodes before reaching its final destination. In both models we give efficient broadcasting and gossiping algorithms, in terms of time and number of wavelengths. We consider both networks with arbitrary topologies and particular networks of practical interest. Several of our algorithms exhibit optimal performances.

Fault tolerant routing in the star and pancake interconnection networks

The main scope of this paper is to show that a recently proposed interconnection network injoys similar properties. We will prove that if Sn is the n-star graph then for any minimal length routing ρ diameter D(R(Sn,ρ)/F) is at most 3, as long as |F|<n. Moreover, we will prove that if ρn is the n-prancake graph then D(R(Pn,p)/F)≤4, as long as |F|<n and ρ belongs to a particular class of routings. Our results confirm earlier published topology

On the construction of statistically synchronizable codes

The problem of constructing statistically synchronizable codes over arbitrary alphabets and for any finite source is considered. It is shown how to efficiently construct a statistically synchronizable code whose average codeword length is within the least likely codeword probability from that of the Huffman code for the same source. Moreover, a method is given for constructing codes having a synchronizing codeword. The method yields synchronous codes that exhibit high synchronizing capability and low redundancy. >

Qualitative independence and Sperner problems for directed graphs

Abstract Sperners theorem about the largest family of incomparable subsets of an n -set is in fact a theorem about the largest anti-chain in the natural extension to sequences of a linear order. We replace the linear order by an arbitrary directed graph and ask for the cardinality of the largest set of incomparable sequences of length n one can form of the vertices. Two sequences are comparable if for every coordinate, all the arcs between corresponding vertices (if any) go in the same direction. Similarly, we look for the largest cardinality of sets of sequences that are incomparable in any graph from a given family. We find the asymptotic solution in some cases and give constructions in others. Our results imply new lower bounds on the cardinality of the largest family of qualitatively two-independent partitions in the sense of Renyi.

On the construction of minimal broadcast networks

Broadcast is the task of transmitting a message originated at a node in a network to all the other nodes. In this paper, we consider the problem of constructing minimal broadcast networks, that is, communication networks such that broadcast can be performed, from any node, in minimum time. The algorithms we propose allow us to improve known bounds on the minimum number of communication lines needed in minimal broadcast networks. We give some numerical evidence that our algorithms also perform well in practice.