The Power of Shunning
Authors: Bangalore Laasya , Choudhury Ashish , Patra Arpita 
Published: May 2020

The problem of Byzantine Agreement (BA) is of interest to both the distributed computing and cryptography communities. Following well-known results from distributed computing literature, the BA problem in the
asynchronous
network setting encounters inevitable non-termination issues. The impasse is overcome via randomization that allows construction of BA protocols in two flavors of termination guarantee—with overwhelming probability and with probability one. The latter type, termed as
almost-surely terminating
BA, is the main focus of this article. An eluding problem in the domain of almost-surely terminating BA is achieving a constant expected running time. Our primary contribution in this work makes significant progress in this direction.

In a setting with
n
parties and an adversary with
unbounded
computing power controlling at most
t
parties in a Byzantine fashion, we present two almost-surely terminating BA protocols in the asynchronous setting:

○ With the
optimal resilience
of
t
<
n
/3, our first protocol runs for an expected
O
(
n
) time. The existing protocols in the same setting either run for an expected
O
(
n
2
) time (Abraham et al., PODC 2008) or require exponential computing power from the honest parties (Wang, CoRR 2015). In terms of communication complexity, our construction outperforms all the known constructions with
t
<
n
/3 that offer almost-surely terminating feature.

○ With the resilience of
t
<
n
/3 + ϵ for
any
ϵ > 0, our second protocol runs for an expected
O
(1/ϵ) time. The expected running time of our protocol turns constant when ϵ is a constant fraction. The known constructions with a constant expected running time either require ϵ to be at least 1 (Feldman-Micali, STOC 1988 and Patra-Pandu Rangan, PODC 2010), implying
t
<
n
/4, or call for exponential computing power from the parties (Wang, CoRR 2015).

We follow the traditional route of building BA via common coin protocol that in turn reduces to
Asynchronous Verifiable Secret-Sharing
(AVSS). Our constructions are built on a variant of AVSS that is termed as
shunning
. A shunning AVSS fails to offer the properties of AVSS when the corrupt parties strike, but allows the honest parties to
locally
detect and shun a set of corrupt parties for any future communication. Our shunning AVSS with
t
<
n
/3 and
t
<
n
/3 + ϵ guarantee Ω(
n
) and, respectively, Ω(ϵ
t
2
) conflicts to be revealed when failure occurs. Turning this shunning AVSS to a common coin protocol efficiently constitutes yet another contribution of this work.

As a secondary contribution, we show the power of the shunning technique and present a highly efficient cryptographically secure shunning AVSS, which is used further to design an asynchronous BA protocol with the optimal resilience of
t
<
n
/3 in the cryptographic setting. Our construct achieves an amortized expected communication complexity of
O
(
n
2
) bits for reaching agreement on a single bit while consuming a constant expected running time. This property has been achieved for the first time in the cryptographic setting and that, too, with standard cryptographic assumptions. The best-known existing construction (Cachin et al., CCS 2002), while still needing more communication complexity than ours, is proven secure only in the Random-Oracle Model (ROM).