Ronald Cramer (Leiden University)
Black-box secret sharing from primitive sets in algebraic number fields
A {\em 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 groupelements 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 $\Theta(\log n)$.We present a BBSSS that is based on a new paradigm, namely, {\em 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. The construction uses techniques from algebraic number theory as well as algebraic geometry.
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 $\tilde{O}(n^2)$ 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.
This presentation is based on a paper to appear at CRYPTO 2005, by Ronald Cramer, Serge Fehr and Martijn Stam, with contributions by Hendrik Lenstra, Jr.