Multiparty Interactive Communication with Private Channels

A group of n players wants to run a distributed protocol ℘ over a network where communication occurs via private point-to-point channels. Can we efficiently simulate ℘ in the presence of an adversary who knows ℘ and is able to maliciously flip bits on the channels? We show that this is possible, even when L, the number of bits sent in ℘, the average message size α in ℘, and T, the number of bits flipped by the adversary are not known in advance. In particular, we show how to create a robust version of ℘, ℘ such that 1) ℘' fails with probability at most δ, for any δ>0; and 2) ℘' sends O( L (1 + (1/α) łog (n L/δ)) + T) bits. We note that if α is Ω (log (n L/δ), then ℘ sends only O(L+T) bits, and is therefore within a constant factor of optimal. Critically, our result requires that ℘ runs correctly in an asynchronous network and our protocol ℘ must run in a synchronous network.