A Scalable Algorithm for Multiparty Interactive Communication with Private Channels

A group of n players wants to run a distributed protocol ρ over a network where communication occurs via private point-to-point channels. Unfortunately, an adversary, who knows P, is able to maliciously flip bits on the channels. Can we efficiently simulate ρ in the presence of such an adversary? We show that this is possible, even when L, the number of bits sent in ρ, 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 ρ, ρ' such that 1) ρ' fails with probability at most δ, for any δ > 0; and 2) ρ' sends Õ (L + T) bits, where the Õ notation hides a log(nL/δ) term multiplying L. Critically, our result requires that ρ be a protocol that runs correctly in an asynchronous network; in contrast, our protocol ρ' must run in a synchronous network. Additionally, we show how to improve this result when the average message size is not constant. In particular, we show how to create a protocol ρ' that sends O(L(1 + (1/α) log(nL/δ)) + T) bits, where α is the average message size. This new ρ' is adaptive in that it does not require a priori knowledge of α. We note that if α is Ω (log(nL/δ)), then this improved algorithm sends only O(L + T) bits, and is therefore within a constant factor of optimal.