Efficient counting with optimal resilience

TL;DR

This paper introduces deterministic algorithms for synchronous $c$-counting in networks with Byzantine faults, achieving optimal resilience, linear stabilization time, and significantly reduced state and message complexity.

Contribution

The work presents new deterministic algorithms for Byzantine-tolerant synchronous counting with optimal resilience and minimal resource usage, improving upon prior randomized and less efficient solutions.

Findings

Deterministic algorithms with optimal resilience for Byzantine faults.

Linear stabilization time proportional to the number of faulty nodes.

Significant reduction in state complexity and message size compared to previous methods.

Abstract

Consider a complete communication network of $n$ nodes, where the nodes receive a common clock pulse. We study the synchronous $c$-counting problem: given any starting state and up to $f$ faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo $c$ in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilisation time in $f$ (asymptotically optimal), (4) use a small number of states, and consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomisation, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilisation.