Black-Box Secret Sharing from Primitive Sets in Number Fields

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

At CRYPTO 2002 Cramer and Fehr proposed a threshold BBSSS with anasymptotically minimal expansion factor Theta(log n).

In this paper we propose a BBSSS that is based on a new paradigm,namely, primitive sets in algebraic number fields. This leadsto a new BBSSS with an expansion factor that is absolutely minimalup to an additive term of at most 2, which is an improvement by aconstant additive factor.We provide good evidence that our scheme is considerably moreefficient in terms of the computational resources it requires.Indeed, the number of group operations to be performed is$\tilde{O}(n^2)$ instead of $\tilde{O}(n^3)$ for sharing and$\tilde{O}(n^{1.6})$ instead of $\tilde{O}(n^{2.6})$ for reconstruction.

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