What a lovely hat

Paper 2002/036

Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups

Ronald Cramer and Serge Fehr

Abstract

A {\em black-box} secret sharing scheme for the threshold access structure $T_{t,n}$ is one which works over any finite Abelian group $G$. Briefly, such a scheme differs from an ordinary linear secret sharing scheme (over, say, a given finite field) in that distribution matrix and reconstruction vectors are defined over the integers and are designed {\em independently} of the group $G$ from which the secret and the shares are sampled. This means that perfect completeness and perfect privacy are guaranteed {\em regardless} of which group $G$ is chosen. We define the black-box secret sharing problem as the problem of devising, for an arbitrary given $T_{t,n}$, a scheme with minimal expansion factor, i.e., where the length of the full vector of shares divided by the number of players $n$ is minimal. Such schemes are relevant for instance in the context of distributed cryptosystems based on groups with secret or hard to compute group order. A recent example is secure general multi-party computation over black-box rings. In 1994 Desmedt and Frankel have proposed an elegant approach to the black-box secret sharing problem based in part on polynomial interpolation over cyclotomic number fields. For arbitrary given $T_{t,n}$ with $0<t<n-1$, the expansion factor of their scheme is $O(n)$. This is the best previous general approach to the problem. Using low degree integral extensions of the integers over which there exists a pair of sufficiently large Vandermonde matrices with co-prime determinants, we construct, for arbitrary given $T_{t,n}$ with $0<t<n-1$ , a black-box secret sharing scheme with expansion factor $O(\log n)$, which we show is minimal.