Secret sharing schemes allow a dealer to distribute a secret piece of information among several parties such that only qualified subsets of parties can reconstruct the secret. The collection of qualified subsets is called an access structure. The best known example is the k-threshold access structure, where the qualified subsets are those of size at least k. When k=2 and there are n parties, there are schemes where the size of the share each party gets is roughly log(n) bits, and this is tight even for secrets of 1 bit. In these schemes, the number of parties n must be given in advance to the dealer.

We consider the case where the set of parties is not known in advance and could potentially be infinite. Our goal is to give the t-th party arriving a small share as possible as a function of t. We present a scheme for general access structures and several schemes for variants of the k-threshold access structure in which at any point in time some bounded number of parties can recover the secret. Lastly, we discuss other classical notions such as robustness, and adapt them to the unbounded setting.

The talk is based on joint works with Moni Naor and Eylon Yogev, and with Anat Paskin-Cherniavsky.