Self-stabilising Byzantine clock synchronisation is almost as easy as consensus

We give fault-tolerant algorithms for establishing synchrony in distributed systems in which each of thennodes has its own clock. Our algorithms operate in a very strong fault model: we require self-stabilisation, i.e.,the initial state of the system may be arbitrary, and there can be up to f<n/3 ongoing Byzantine faults, i.e.,nodes that deviate from the protocol in an arbitrary manner. Furthermore, we assume that the local clocks ofthe nodes may progress at different speeds (clock drift) and communication has bounded delay. In this model,we study the pulse synchronisation problem, where the task is to guarantee that eventually all correct nodesgenerate well-separated local pulse events (i.e., unlabelled logical clock ticks) in a synchronised manner.Compared to prior work, we achieveexponentialimprovements in stabilisation time and the number ofcommunicated bits, and give the first sublinear-time algorithm for the problem:•In the deterministic setting, the state-of-the-art solutions stabilise in timeΘ(f)and have each nodebroadcastΘ(flogf)bits per time unit. We exponentially reduce the number of bits broadcasted pertime unit toΘ(logf)while retaining the same stabilisation time.•In the randomised setting, the state-of-the-art solutions stabilise in timeΘ(f)and have each nodebroadcastO(1)bits per time unit. We exponentially reduce the stabilisation time to polylogfwhileeach node broadcasts polylogfbits per time unit.These results are obtained by means of a recursive approach reducing the above task ofself-stabilisingpulse synchronisation in thebounded-delaymodel tonon-self-stabilisingbinary consensus in thesynchro-nousmodel. In general, our approach introduces at most logarithmic overheads in terms of stabilisation timeand broadcasted bits over the underlying consensus routine.