Secret sharing with public reconstruction

All known constructions of information theoretic t-out-of-n secret-sharing schemes require secure, private communication channels among the parties for the reconstruction of the secret. We investigate the cost of performing the reconstruction over public communication channels. A naive implementation of this task distributes 2n-2 one times pads to each party. This results in shares whose size is 2n-1 times the secret size. We present three implementations of such schemes that are substantially more efficient. A scheme enabling multiple reconstructions of the secret by different subsets of parties, with factor O(n/t) increase in the shares' size. A one-time scheme, enabling a single reconstruction of the secret, with O(log(n/t)) increase in the shares' size. A one-time scheme, enabling a single reconstruction by a set of size exactly t, with factor O(1) increase in the shares' size. We prove that the first implementation is optimal (up to constant factors) by showing a tight Ω(n/t) lower bound for the increase in the shares' size