Proactive Secret Sharing with Constant Communication

This paper presents the first protocols for Proactive Secret Sharing (PSS) that only require constant (in the number of parties, n) communication per party per epoch. By harnessing the power of expander graphs, we are able to obtain strong guarantees about the security of the system. We present the following PSS protocols: - A PSS protocol that provides privacy (but no robustness) against an adversary controlling O(n) parties per epoch. - A PSS protocol that provides robustness (but no privacy) against an adversary controlling O(n) parties per epoch. - A PSS protocol that provides privacy against an adversary controlling O(n(a)) parties per epoch and provides robustness against an adversary controlling O(n(1-a)) parties per epoch, for any constant 0 <= a <= 1. Instantiating this with a = 1/2 gives a PSS protocol that is proactively secure (private and robust) against an adversary controlling O(root n) parties per epoch. Additionally, we discuss how secure channels, whose existence is usually assumed by PSS protocols, are challenging to create in the mobile adversary setting, and we present a method to instantiate them from a weaker assumption.