Characterizing Ideal Weighted Threshold Secret Sharing

Abstract

Abstract. Weighted threshold secret sharing was introduced by Shamir in his seminal work on secret sharing. In such settings, there is a set of users where each user is assigned a positive weight. A dealer wishes to distribute a secret among those users so that a subset of users may reconstruct the secret if and only if the sum of weights of its users exceeds a certain threshold. A secret sharing scheme is ideal if the size of the domain of shares of each user is the same as the size of the domain of possible secrets (this is the smallest possible size for the domain of shares). The family of subsets authorized to reconstruct the secret in a secret sharing scheme is called an access structure. An access structure is ideal if there exists an ideal secret sharing scheme that realizes it.It is known that some weighted threshold access structures are not ideal, while other nontrivial weighted threshold access structures do have an ideal scheme that realizes them. In this work we characterize all weighted threshold access structures that are ideal. We show that a weighted threshold access structure is ideal if and only if it is a hierarchical threshold access structure (as introduced by Simmons), or a tripartite access structure (these structures, that we introduce here, generalize the concept of bipartite access structures due to Padró and Sáez), or a composition of two ideal weighted threshold access structures that are defined on smaller sets of users. We further show that in all those cases the weighted threshold access structure may be realized by a linear ideal secret sharing scheme. The proof of our characterization relies heavily on the strong connection between ideal secret sharing schemes and matroids, as proved by Brickell and Davenport.

Weightedness and structural characterization of hierarchical simple games

“…It is interesting to compare our results with the results by Farràs and Padró (2010) (their classification is more accurate than the one in Beimel et al (2008)). They list seven types of indecomposable simple games: kout-of-n symmetric simple games; three bipartite types B 1 , B 2 , B 3 and three tripartite ones T 1 , T 2 , T 3 .…”

“…Classification of ideal access structures proved to be extremely difficult problem and the focus of attention has moved to classification of ideal access structures in subclasses of access structures like weighted threshold access structures introduced by Shamir (1979). This classification has been presented by Beimel et al (2008) and Farràs and Padró (2010) who characterized irreducible weighted ideal access structures. All other ideal weighted threshold access structures can be obtained by combining the indecomposable ones using the operation of composition defined by Beimel et al (2008).…”

“…This classification has been presented by Beimel et al (2008) and Farràs and Padró (2010) who characterized irreducible weighted ideal access structures. All other ideal weighted threshold access structures can be obtained by combining the indecomposable ones using the operation of composition defined by Beimel et al (2008). Their proof was indirect, the cornerstone of their approach was an application of the well-known connection between ideal secret sharing schemes and matroids (Brickell and Davenport, 1990).…”

“…We characterize those disjunctive and conjunctive hierarchical games which are weighted majority games. This paper was inspired by Beimel et al (2008) and Farràs and Padró (2010) characterizations of ideal weighted threshold access structures of secret sharing schemes. …”

Ideal Hierarchical Secret Sharing Schemes

“…Some of them dealt with families of multipartite access structures. Beimel, Tassa and Weinreb [2] presented a characterization of the ideal weighted threshold access structures that generalizes the partial results in [23,29]. Another important result about weighted threshold access structures have been obtained recently by Beimel and Weinreb [3].…”