  Byzantine Agreement (BA) is one of the most fundamental problems in
distributed computing, and its communication complexity is an important
efficiency metric. It is well known that quadratic communication is necessary
for BA in the worst case due to a lower bound by Dolev and Reischuk. This lower
bound has been shown to be tight for the unauthenticated setting with $f < n/3$
by Berman et al. but a considerable gap remains for the authenticated setting
with $n/3 \le f < n/2$.
  This paper provides two results towards closing this gap. Both protocols have
a quadratic communication complexity and have different trade-offs in
resilience and assumptions. The first protocol achieves the optimal resilience
of $f < n/2$ but requires a trusted setup for threshold signature. The second
protocol achieves near optimal resilience $f \le (1/2 - \varepsilon)n$ in the
standard PKI model.
