As easy as ABC: Optimal (A)ccountable (B)yzantine (C)onsensus is easy!

It is known that the agreement property of the Byzantine consensus problem among
$n$
processes can be violated in a non-synchronous system if the number of faulty processes exceeds
$t_{0}$
= ┌
$n$
/3┐ − 1 [10], [19]. In this paper, we investigate the accountable Byzantine consensus problem in non-synchronous systems: the problem of solving Byzantine consensus whenever possible (e.g., when the number of faulty processes does not exceed
$t_{0}$
) and allowing correct processes to obtain proof of culpability of (at least)
$t_{0}+ 1$
faulty processes whenever correct processes disagree. We present four complementary contributions: 1) We introduce ABC: a simple yet efficient transformation of any Byzantine consensus protocol to an accountable one. ABC introduces an overhead of only two all-to-all communication rounds and
$O(n^{2})$
additional bits in executions with up to
$t_{0}$
faults (i.e., in the common case). 2) We define the accountability complexity, a complex-ity metric representing the number of accountability-specific messages that correct processes must send. Fur-thermore, we prove a tight lower bound. In particular, we show that any accountable Byzantine consensus protocol incurs cubic accountability complexity. Moreover, we illustrate that the bound is tight by applying the ABC transformation to any Byzantine consensus protocol. 3) We demonstrate that, when applied to an optimal Byzan-tine consensus protocol, ABC constructs an accountable Byzantine consensus protocol that is (1) optimal with respect to the communication complexity in solving consensus whenever consensus is solvable, and (2) op-timal with respect to the accountability complexity in obtaining accountability whenever disagreement occurs. 4) We generalize ABC to other distributed computing prob-lems besides the classic consensus problem. We charac-terize a class of agreement tasks, including reliable and consistent broadcast [5], that ABC renders accountable.