A $d$-broadcast primitive is a communication primitive that allows a
sender to send a value from a domain of size $d$ to a set of parties.
A broadcast protocol emulates the $d$-broadcast primitive using only
point-to-point channels, even if some of the parties cheat, in
the sense that all correct recipients agree on the same value $v$
(consistency), and if the sender is correct, then $v$ is the value
sent by the sender (validity). A celebrated result by Pease, Shostak
and Lamport states that such a broadcast protocol exists if and only if $t <
n/3$, where $n$ denotes the total number of parties and $t$ denotes
the upper bound on the number of cheaters.
This paper is concerned with broadcast protocols for any number of
cheaters ($t 3$ no broadcast amplification
is possible, i.e., $\phi_n(d)=d$ for any $d$.
However, if other parties than the sender can also broadcast some
short messages, then broadcast amplification is possible for
\emph{any}~$n$. Let $\phi^*_n(d)$ denote the minimal $d'$ such that
$d$-broadcast can be constructed from primitives $d'_1$-broadcast,
\ldots, $d'_k$-broadcast, where $d'=\prod_i d'_i$ (i.e., $\log
d'=\sum_i \log d'_i$). Note that $\phi^*_n(d)\leq\phi_n(d)$.
We show that broadcasting $8n\log n$ bits in
total suffices, independently of $d$, and that at least $n-2$ parties,
including the sender, must broadcast at least one bit. Hence
$\min(\log d,n-2) \leq \log \phi^*_n(d) \leq 8n\log n$.
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