Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups - Cramer, Ronald; Fehr, Serge (BRICS Report Series, 2002)
A 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 Z and are designed independently of the group G from which the secret and the shares are sampled. This means that perfect completeness and perfect privacy are guaranteed 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 Z 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.