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IEEE Transactions on Information Theory

Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing

Oriol Farràs, Tarik Kaced, Sebastià Martín Mollevi, Carles Padró

Published Jun 29, 2020• 16 citations
Publisher

Abstract

We present a new improvement in the linear programming technique to derive lower bounds on the information ratio of secret sharing schemes. We obtain non-Shannon-type bounds without using information inequalities explicitly. Our new technique makes it possible to determine the optimal information ratio of linear secret sharing schemes for all access structures on 5 participants and all graph-based access structures on 6 participants. In addition, new lower bounds are presented also for some small matroid ports and, in particular, the optimal information ratios of the linear secret sharing schemes for the ports of the Vamos matroid are determined.

DOI
10.1109/tit.2020.3005706

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•Sep 29, 2012•1 citation
We present a new improvement in the linear programming technique to derive lower bounds on the information ratio of secret sharing schemes. We obtain non-Shannon-type bounds without using information inequalities explicitly. Our new technique makes it possible to determine the optimal information ratio of linear secret sharing schemes for all access structures on 5 participants and all graph-based access structures on 6 participants. In addition, new lower bounds are presented also for some small matroid ports and, in particular, the optimal information ratios of the linear secret sharing schemes for the ports of the Vamos matroid are determined.
Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.
Secret sharing is a central topic in information-theoretical cryptography. In a secret sharing scheme a random secret is distributed among participants so that only the qualified subsets of parties can recover the secret. In this paper we improve previously known [13] lower bounds on the size of the largest share for several specific access structures. Our approach is based on non-Shannon-type information inequalities. We employ the linear programming (LP) technique that permits to apply new information inequalities indirectly. To reduce computational complexity of the involved linear programs we extensively use symmetry considerations.
Abstract The efficiency of a secret sharing scheme is measured via its information ratio. The optimal information ratio (OIR) of an access structure is the infimum of the information ratios of all secret sharing schemes realizing it. When the infimum is taken over all linear secret sharing schemes, it is called the optimal linear information ratio (OLIR). We review the problem of finding the OIRs and OLIRs of graph access structures on six participants. Study of such access structures was initiated by van Dijk et al. (1998) [43] . In a sequence of works, the OIRs of nine access structures, out of the 18 initially unsolved non-isomorphic ones, were determined. Very recently, the known lower bounds on the OLIRs of the other nine cases have been improved (Farras et al., 2018 [20] ), determining the OLIRs for another two cases. In this paper, for each of the remaining seven cases, we provide a new upper bound on the OIR which matches that of the recently found lower bound on the OLIR. Improved upper bounds are achieved by constructing a linear secret sharing scheme, using a suitable decomposition technique for each access structure. A new decomposition technique, called ( λ , ω ) -weighted decomposition, is also introduced which is a generalization of all known decomposition methods.
This paper is on developing some computer-assisted proof methods involving non-classical inequalities for Shannon entropy. Two areas of the applications of information inequalities are studied: Secret sharing schemes and hat guessing games. In the former a random secret value is transformed into shares distributed among several participants in such a way that only the qualified groups of participants can recover the secret value. In the latter each participant is assigned a hat colour and they try to guess theirs while seeing only some of the others'. The aim is to maximize the probability that every player guesses correctly, the optimal probability depends on the underlying sight graph. We use for both problems the method of non-Shannon-type information inequalities going back to Z. Zhang and R. W. Yeung. We employ the linear programming technique that allows to apply new information inequalities indirectly, without even writing them down explicitly. To reduce the complexity of the problems of linear programming involved in the bounds we extensively use symmetry considerations. Using these tools, we improve lower bounds on the ratio of key size to secret size for the former problem and an upper bound for one of the ten vertex graphs related to an open question by Riis for the latter problem.
In a bipartite secret sharing scheme, the set of participants is divided into two parts, and all participants in each part play an equivalent role. The ideal bipartite access structures were characterized by Padr´o and S´aez, but it is not known which is the optimal information rate of non ideal bipartite access structures. Determining the optimal information rate of general access structures is one of the major problems in secret sharing. We study this open problem restricted to the bipartite access structures, obtaining a new method to compute bounds on the optimal information rate. Namely, by using the connection between secret sharing schemes and polymatroids, we show a linear programming problem that determines, for each access structure, an upper bound on the information rate. Moreover, we show new optimal constructions for certain bipartite access structures.
A perfect secret-sharing scheme is a method to distribute a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the joint share of participants in any unqualified subset is statistically independent of the secret. The collection of all qualified subsets is called the access structure of the perfect secret-sharing scheme. In a graph-based access structure, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme realizing the access structure based on G is defined as AR = (Pv2V (G) H(v))/(|V (G)|H(s)), where s is the secret and v is the share of v, both are random variables from and H is the Shannon entropy. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing a given access structure is called the optimal average information ratio of that access structure. Most known results about the optimal average information ratio give upper bounds or lower bounds on it. In this present structures based on bipartite graphs and determine the exact values of the optimal average information ratio of some infinite classes of them.