Not yet reviewed by Pith; the record is open.

Consensus is one of the most thoroughly studied problems in distributed computing, yet there are still complexity gaps that have not been bridged for decades. In particular, in the classical message-passing setting with processes' crashes, since the seminal works of Bar-Joseph and Ben-Or [1998] \cite{Bar-JosephB98} and Aspnes and Waarts [1996, 1998] \cite{AspnesW-SICOMP-96,Aspnes-JACM-98} in the previous century, there is still a fundamental unresolved question about communication complexity of fast randomized Consensus against a (strong) adaptive adversary crashing processes arbitrarily online. The best known upper bound on the number of communication bits is $\Theta(\frac{n^{3/2}}{\sqrt{\log{n}}})$ per process, while the best lower bound is $\Omega(1)$. This is in contrast to randomized Consensus against a (weak) oblivious adversary, for which time-almost-optimal algorithms guarantee amortized $O(1)$ communication bits per process \cite{GK-SODA-10}. We design an algorithm against adaptive adversary that reduces the communication gap by nearly linear factor to $O(\sqrt{n}\cdot\text{polylog } n)$ bits per process, while keeping almost-optimal (up to factor $O(\log^3 n)$) time complexity $O(\sqrt{n}\cdot\log^{5/2} n)$. More surprisingly, we show this complexity indeed can be lowered further, but at the expense of increasing time complexity, i.e., there is a {\em trade-off} between communication complexity and time complexity. More specifically, our main Consensus algorithm allows to reduce communication complexity per process to any value from $\text{polylog } n$ to $O(\sqrt{n}\cdot\text{polylog } n)$, as long as Time $\times$ Communication $= O(n\cdot \text{polylog } n)$. Similarly, reducing time complexity requires more random bits per process, i.e., Time $\times$ Randomness $=O(n\cdot \text{polylog } n)$.