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

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...

• Type: Conference
• Year: 2005
• Where: CRYPTO
• Authors: Ronald Cramer, Serge Fehr, Martijn Stam
• Keywords: CRYPTO 2005, Expansion Factor, Group Elements, Group Operations