Black-Box Secret Sharing from Primitive Sets in Algebraic Number Fields

  • Ronald Cramer
  • Serge Fehr
  • Martijn Stam
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3621)


A black-box secret sharing scheme (BBSSS) for a given access structure works in exactly the same way over any finite Abelian group, as it only requires black-box access to group operations and to random group elements. In particular, there is no dependence on e.g. the structure of the group or its order. The expansion factor of a BBSSS is the length of a vector of shares (the number of group elements in it) divided by the number of players n.

At CRYPTO 2002 Cramer and Fehr proposed a threshold BBSSS with an asymptotically minimal expansion factor Θ(log n).

In this paper we propose a BBSSS that is based on a new paradigm, namely, primitive sets in algebraic number fields. This leads to a new BBSSS with an expansion factor that is absolutely minimal up to an additive term of at most 2, which is an improvement by a constant additive factor.

We provide good evidence that our scheme is considerably more efficient in terms of the computational resources it requires. Indeed, the number of group operations to be performed is Õ(n 2) instead of Õ(n 3) for sharing and Õ(n 1.6) instead of Õ(n 2.6) for reconstruction.

Finally, we show that our scheme, as well as that of Cramer and Fehr, has asymptotically optimal randomness efficiency.


Secret Sharing Algebraic Number Expansion Factor Secret Sharing Scheme Chinese Remainder Theorem 
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

  • Ronald Cramer
    • 1
    • 2
  • Serge Fehr
    • 1
  • Martijn Stam
    • 3
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Mathematical InstituteLeiden UniversityThe Netherlands
  3. 3.Department of Computer ScienceUniversity of BristolUnited Kingdom

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