  Multiple parties observing correlated data seek to recover each other's data
and attain omniscience. To that end, they communicate interactively over a
noiseless broadcast channel - each bit transmitted over this channel is
received by all the parties. We give a universal interactive communication
protocol, termed the recursive data exchange protocol (RDE), which attains
omniscience for any sequence of data observed by the parties and provide an
individual sequence guarantee of performance. As a by-product, for observations
of length $n$, we show the universal rate optimality of RDE up to an
$\mathcal{O}\left(n^{-\frac 12}\sqrt{\log n}\right)$ term in a generative
setting where the data sequence is independent and identically distributed (in
time). Furthermore, drawing on the duality between omniscience and secret key
agreement due to Csiszar and Narayan, we obtain a universal protocol for
generating a multiparty secret key of rate at most $\mathcal{O}\left(n^{-\frac
12}\sqrt{\log n}\right)$ less than the maximum rate possible. A key feature of
RDE is its recursive structure whereby when a subset $A$ of parties recover
each-other's data, the rates appear as if the parties have been executing the
protocol in an alternative model where the parties in $A$ are collocated.
