We develop a paradigm for studying multi-player deterministic communication,
based on a novel combinatorial concept that we call a {\em strong fooling
set}. Our paradigm leads to optimal lower bounds on the per-player
communication required for solving multi-player $\textsc{equality}$
problems in a private-message setting. This in turn gives a very strong---$O(1)$
versus $\Omega(n)$---separation between private-message and one-way blackboard
communication complexities.

Applying our communication complexity results, we show that for deterministic
data streaming algorithms, even loose estimations of some basic statistics of an
input stream require large amounts of space. For instance, approximating the
frequency moment $F_k$ within a factor $\alpha$ requires
$\Omega(n/\alpha^{1/(1-k)})$ space for $k 1$. In particular, approximation
within any {\em constant} factor $\alpha$, however large, requires {\em linear}
space, with the trivial exception of $k = 1$. This is in sharp contrast to the
situation for randomized streaming algorithms, which can approximate $F_k$ to
within $(1\pm\varepsilon)$ factors using $\widetilde{O}(1)$ space for $k \le 2$
and $o(n)$ space for all finite $k$ and all constant $\varepsilon > 0$. Previous
linear-space lower bounds for deterministic estimation were limited to small
factors $\alpha$, such as $\alpha < 2$ for approximating $F_0$ or $F_2$.

We also provide certain space/approximation tradeoffs in a deterministic setting
for the problems of estimating the empirical entropy of a stream as well as the
size of the maximum matching and the edge connectivity of a streamed graph.