  Broadcast protocols enable a set of $n$ parties to agree on the input of a
designated sender, even facing attacks by malicious parties. In the
honest-majority setting, randomization and cryptography were harnessed to
achieve low-communication broadcast with sub-quadratic total communication and
balanced sub-linear cost per party. However, comparatively little is known in
the dishonest-majority setting. Here, the most communication-efficient
constructions are based on Dolev and Strong (SICOMP '83), and sub-quadratic
broadcast has not been achieved. On the other hand, the only nontrivial
$\omega(n)$ communication lower bounds are restricted to deterministic
protocols, or against strong adaptive adversaries that can perform "after the
fact" removal of messages.
  We provide new communication lower bounds in this space, which hold against
arbitrary cryptography and setup assumptions, as well as a simple protocol
showing near tightness of our first bound.
  1) We demonstrate a tradeoff between resiliency and communication for
protocols secure against $n-o(n)$ static corruptions. For example,
$\Omega(n\cdot {\sf polylog}(n))$ messages are needed when the number of honest
parties is $n/{\sf polylog}(n)$; $\Omega(n\sqrt{n})$ messages are needed for
$O(\sqrt{n})$ honest parties; and $\Omega(n^2)$ messages are needed for $O(1)$
honest parties.
  Complementarily, we demonstrate broadcast with $O(n\cdot{\sf polylog}(n))$
total communication facing any constant fraction of static corruptions.
  2) Our second bound considers $n/2 + k$ corruptions and a weakly adaptive
adversary that cannot remove messages "after the fact." We show that any
broadcast protocol within this setting can be attacked to force an arbitrary
party to send messages to $k$ other parties. This rules out, for example,
broadcast facing 51% corruptions in which all non-sender parties have sublinear
communication locality.
