Order-preserving Renaming in Synchronous Message Passing Systems with   Byzantine Faults
  Renaming is a fundamental problem in distributed computing, which consists of a set of processes picking distinct names from a given namespace. The paper presents algorithms that solve order-preserving renaming in synchronous message passing systems with Byzantine processes. To the best of our knowledge, this work is the first to address order-preserving renaming in the given model. Although this problem can be solved by using consensus, it is known that renaming is "weaker" than consensus, therefore we are mainly concerned with the efficiency of performing renaming and make three contributions in this direction. We present an order-preserving renaming algorithm for $N > 3t$ with target namespace of size $N+t-1$ and logarithmic step complexity (where $N$ is the number of processes and $t$ is an upper bound on the number of faults). Similarly to the existing crash-tolerant solution, our algorithm employs the ideas from the approximate agreement problem. We show that our algorithm has constant step complexity if $N>t^2+2t$ and achieves tight namespace of size $N$. Finally, we present an algorithm that solves order-preserving renaming in just 2 communication steps, if $N > 2t^2 + t$. 
Oksana Denysyuk and Luis Rodrigues
cs.DS