Updating the parameters of a threshold scheme by minimal broadcast

Threshold schemes allow secret data to be protected among a set of participants in such a way that only a prespecified threshold of participants can reconstruct the secret from private information (shares) distributed to them on a system setup using secure channels. We consider the general problem of designing unconditionally secure threshold schemes whose defining parameters (the threshold and the number of participants) can later be changed by using only public channel broadcast messages. In this paper, we are interested in the efficiency of such threshold schemes, and seek to minimize storage costs (size of shares) as well as optimize performance in low-bandwidth environments by minimizing the size of necessary broadcast messages. We prove a number of lower bounds on the smallest size of broadcast message necessary to make general changes to the parameters of a threshold scheme in which each participant already holds shares of minimal size. We establish the tightness of these bounds by demonstrating optimal schemes.