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Characterizing Ideal Weighted Threshold Secret Sharing

  • Amos Beimel
  • Tamir Tassa
  • Enav Weinreb
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3378)

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.

Keywords

Secret Sharing Access Structure Secret Sharing Scheme Secure Multiparty Computation Strong User 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Amos Beimel
    • 1
  • Tamir Tassa
    • 1
    • 2
  • Enav Weinreb
    • 1
  1. 1.Dept. of Computer ScienceBen-Gurion UniversityBeer ShevaIsrael
  2. 2.Division of Computer ScienceThe Open UniversityRa’ananaIsrael

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