On the optimization of bipartite secret sharing schemes

Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the tripartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.Peer ReviewedPostprint (author's final draft

Finding lower bounds on the complexity of secret sharing schemes by linear programming

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 schemePeer ReviewedPostprint (author's final draft

Complexity of universal access structures

An important parameter in a secret sharing scheme is the number of minimal qualified sets. Given this number, the universal access structure is the richest possible structure, namely the one in which there are one or more participants in every possible Boolean combination of the minimal qualified sets. Every access structure is a substructure of the universal structure for the same number of minimal qualified subsets, thus universal access structures have the highest complexity given the number of minimal qualified sets. We show that the complexity of the universal structure with $n$ minimal qualified sets is between $n/\log_2n$ and $n/2.7182\dots$ asymptotically

On the optimization of bipartite secret sharing schemes

Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the bipartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.Postprint (author’s final draft

The complexity of the connected graph access structure on seven participants

In this paper, we study an important problem in secret sharing that determines the exact value or bound for the complexity. First, we used induced subgraph complexity of the graph G with access structure, Gama, to obtain a lower bound on the complexity of the graph G. Secondly, by applying decomposition techniques we obtain an upper bound on the complexity of the graph G. We determine the exact values of the complexity for each of the ten graph access structures on seven participants. Also, we improve the value bound of the complexity of the six graph access structures with seven participants

Secret sharing schemes: Optimizing the information ratio

Secret sharing refers to methods used to distribute a secret value among a set of participants. This work deals with the optimization of two parameters regarding the efficiency of a secret sharing scheme: the information ratio and average information ratio. Only access structures (a special family of sets) on 5 and 6 participants will be considered. First, access structures with 5 participants will be studied, followed by the ones on 6 participants that are based on graphs. The main goal of the paper is to check existing lower bounds (and improve some of them) by using linear programs with the sage solver. Shannon information inequalities have been used to translate the polymatroid axioms into linear constraints