Byzantine-Resilient Counting in Networks

TL;DR

This paper introduces two distributed algorithms for Byzantine-resilient network size estimation, one deterministic and one randomized, each with different assumptions, tolerances, and performance guarantees, applicable to specific network types.

Contribution

The paper presents the first deterministic and randomized algorithms for Byzantine counting that work under different network conditions and adversarial strengths, with proven optimality and probabilistic guarantees.

Findings

Deterministic algorithm finishes in O(log n) rounds, tolerates up to O(n^{1 - γ}) Byzantine nodes.

Randomized algorithm works in almost all d-regular graphs, tolerates up to n^{1/2 - ξ} Byzantine nodes.

Both algorithms produce a constant factor estimate of log n with high probability.

Abstract

We present two distributed algorithms for the {\em 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 {\em 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 \emph{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 {\em randomized} and most good nodes send only small-sized messages (Throughout this paper, a small-sized message is defined to be one that contains $O(\log{n})$ bits in addition to at most a constant number of node IDs.). This algorithm works in \emph{almost all} $d$-regular graphs. It tolerates up to $B(n) = n^{\frac{1}{2} - \xi}$ (note that $n$ and $B(n)$ are unknown to the algorithm) arbitrarily (adversarially) placed Byzantine nodes, where $\xi$ is any arbitrarily small (but fixed) positive constant. This algorithm takes $O(B(n)\log^2{n})$ rounds and outputs a (fixed) constant factor estimate of $\log{n}$ with probability at least $1 - o(1)$. The said estimate is known to most nodes, i.e., $\geq (1 - \beta)n$ 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.