In distributed computing, the renaming problem requires $n$ nodes with unique identities from a large namespace $[N]$ to acquire new, distinct identities from a smaller target namespace $[M]$. A solution is strong if $M=n$, and is order-preserving if the relative order of identities is maintained. In the synchronous message-passing model, although many fault-tolerant renaming algorithms achieve logarithmic time complexity, they universally incur a high message complexity of $Ω(n^2)$. Recent work breaks the quadratic barrier, but demands linear runtime and relies on shared randomness. This paper addresses the challenge of designing renaming algorithms that are simultaneously time-efficient, message-efficient, and Byzantine fault-tolerant, assuming only message authentication. We present two randomized algorithms for strong and order-preserving renaming that tolerate up to $(1/3-δ)n$ Byzantine failures for any constant $δ>0$. Our first algorithm, which assumes shared randomness, terminates in $O(\text{poly-log}(n))$ rounds with $\tilde{O}(n)$ total communication cost. This matches known lower bounds within poly-logarithmic factor. Our second algorithm eliminates the shared randomness assumption and achieves $O(\text{poly-log}(n))$ runtime with $\tilde{O}(n+\min\{nf,T\})$ total communication cost, where $f$ is the actual number of faulty nodes and $T$ is the amount of messages faulty nodes sent. This gives the first Byzantine renaming algorithm that achieves both poly-logarithmic runtime and subquadratic communication cost for a wide range of parameter regimes, without shared randomness. A key technical enabler is a novel and scalable committee election primitive that could be easily integrated into other algorithms to solve various distributed computing problems with low cost and strong fault-tolerance.

翻译：在分布式计算中，重命名问题要求具有来自大命名空间 $[N]$ 的唯一标识的 $n$ 个节点，从较小的目标命名空间 $[M]$ 获取新的、不同的标识。如果 $M=n$，则解是强解；如果保持标识的相对顺序，则是保序解。在同步消息传递模型中，尽管许多容错重命名算法实现了对数时间复杂性，但它们普遍会招致 $\Omega(n^2)$ 的高消息复杂性。最近的工作突破了二次屏障，但需要线性运行时间并依赖共享随机性。本文解决了设计同时具有时间效率、消息效率和拜占庭容错的重命名算法的挑战，仅假设消息认证。我们提出了两种用于强重命名和保序重命名的随机算法，它们能容忍最多 $(1/3-δ)n$ 个拜占庭故障，其中 $δ>0$ 为任意常数。我们的第一个算法假设共享随机性，在 $O(\text{多对数}(n))$ 轮内终止，总通信开销为 $\tilde{O}(n)$。这在对数多项式因子内与已知下界匹配。我们的第二个算法消除了共享随机性假设，实现了 $O(\text{多对数}(n))$ 的运行时间，总通信开销为 $\tilde{O}(n+\min\{nf,T\})$，其中 $f$ 是实际故障节点的数量，$T$ 是故障节点发送的消息量。这给出了首个在广泛参数范围内无需共享随机性即可同时实现多对数运行时间和次二次通信开销的拜占庭重命名算法。一个关键技术推动因素是新颖且可扩展的委员会选举原语，该原语可轻松集成到其他算法中，以低成本和强容错性解决各种分布式计算问题。