Byzantine-Resilient Counting in Networks

We present two distributed algorithms for the Byzantine counting problem, which is concerned with estimating the size of a network in the presence of a large number of Byzantine nodes. In an n-node network (n is unknown), our first algorithm, which is deterministic, finishes in O(log n) rounds and is time-optimal. This algorithm can tolerate up to O(n(1-gamma)) arbitrarily (adversarially) placed Byzantine nodes for any arbitrarily small (but fixed) positive constant gamma. It outputs a (fixed) constant factor estimate of log n that would be known to all but o(1) fraction of the good nodes. This algorithm works for any bounded degree expander network. However, this algorithms assumes that good nodes can send arbitrarily large-sized messages in a round. Our second algorithm is randomized and most good nodes send only small-sized messages.(1) This algorithm works in almost all d-regular graphs. It tolerates up to B(n) = n(1/2-xi) (note that n and B(n) are unknown to the algorithm) arbitrarily (adversarially) placed Byzantine nodes, where is any arbitrarily small (but fixed) positive constant. This algorithm takes O(B(n) log(2) n) rounds and outputs a constant factor estimate of log n with probability at least 1 - o(1). The said estimate is known to most nodes, i.e., >= (1 - beta)n nodes for any arbitrarily small (but fixed) positive constant beta. To complement our algorithms, we also present an impossibility result that shows that it is impossible to estimate the network size with any reasonable approximation with any non-trivial probability of success if the network does not have sufficient vertex expansion. Both algorithms are the first such algorithms that solve Byzantine counting in sparse, bounded degree networks under very general assumptions. Both algorithms are fully local and need no global knowledge. Our algorithms can be used for the design of efficient distributed algorithms resilient against Byzantine failures, where the knowledge of the network size - a global parameter - may not be known a priori.