On Secret Sharing Schemes, Matroids and Polymatroids

Abstract

The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. The optimization of this parameter for general access structures is an important and very difficult open problem in secret sharing. We explore in this paper the connections of this open problem with matroids and polymatroids.Matroid ports were introduced by Lehman in 1964. A forbidden minor characterization of matroid ports was given by Seymour in 1976. These results are previous to the invention of secret sharing by Shamir in 1979. Important connections between ideal secret sharing schemes and matroids were discovered by Brickell and Davenport in 1991. Their results can be restated as follows: every ideal secret sharing scheme defines a matroid, and its access structure is a port of that matroid. In spite of this, the results by Lehman and Seymour and other subsequent results on matroid ports have not been noticed until now by the researchers interested in secret sharing.Lower bounds on the optimal complexity of access structures can be found by taking into account that the joint Shannon entropies of a set of random variables define a polymatroid. We introduce a new parameter, which is denoted by κ, to represent the best lower bound that can be obtained by this method. We prove that every bound that is obtained by this technique for an access structure applies to its dual structure as well.By using the aforementioned result by Seymour we obtain two new characterizations of matroid ports. The first one refers to the existence of a certain combinatorial configuration in the access structure, while the second one involves the values of the parameter κ that is introduced in this paper. Both are related to bounds on the optimal complexity. As a consequence, we generalize the result by Brickell and Davenport by proving that, if the length of every share in a secret sharing scheme is less than 3/2 times the length of the secret, then its access structure is a matroid port. This generalizes and explains a phenomenon that was observed in several families of access structures.Finally, we present a construction of linear secret sharing schemes for the ports of the Vamos matroid and the non-Desargues matroid, which do not admit any ideal secret sharing scheme. We obtain in this way upper bounds on their optimal complexity. These new bounds are a contribution on the search of examples of access structures whose optimal complexity lies between 1 and 3/2.

Ideal Multipartite Secret Sharing Schemes

“…troids, the characterization of the ideal multipartite access structures is studied in this paper in all generality. Even though polymatroids have been used before in secret sharing [11,25], in this paper integer polymatroids are used for the first time in the characterization of ideal access structures. These combinatorial objects are proved to be a very useful tool to study multipartite matroids, which are the ones defined from ideal multipartite secret sharing schemes.…”

On the optimization of bipartite secret sharing schemes

“…In addition, the aforementioned connection between minors and secret sharing implies that σ(Γ ) ≤ σ(Γ ) and λ(Γ ) ≤ λ(Γ ). The parameters λ and κ are invariant by duality, as it was proved, respectively, in [27] and [29]. The relation between σ(Γ ) and σ(Γ * ) is an open problem.…”

“…We use the description of this method presented in [29], which is in terms of polymatroids, to obtain bounds on the information rate of bipartite access structures. We present these bounds in Section 5.…”

“…A gap in the values of the parameter κ was proved in [29]. Namely, there does not exist any access structure Γ with 1 < κ(Γ ) < 3/2.…”

“…Specifically, lower bounds on the optimal complexity can be derived from the fact that that every secret sharing scheme for a given access structure defines a polymatroid. A new parameter, κ(Γ ), was introduced in [29] to denote the best lower bound on σ(Γ ) that can be obtained by this combinatorial method. Obviously, κ(Γ ) ≤ σ(Γ ).…”