  We show how to compress communication in selection protocols, where the goal
is to agree on a sequence of random bits using only a broadcast channel. More
specifically, we present a generic method for converting any selection
protocol, into another selection protocol where each message is ``short'' while
preserving the same number of rounds, the same output distribution, and the
same resilience to error. Assuming that the output of the protocol lies in some
universe of size $M$, in our resulting protocol each message consists of only
$\mathsf{polylog}(M,n,d)$ many bits, where $n$ is the number of parties and $d$
is the number of rounds. Our transformation works in the presence of either
static or adaptive Byzantine faults.
  As a corollary, we conclude that for any $\mathsf{poly}(n)$-round collective
coin-flipping protocol, leader election protocol, or general selection
protocols, messages of length $\mathsf{polylog}(n)$ suffice (in the presence of
either static or adaptive Byzantine faults).
