  Let $G$ be the special linear group $\mathrm{SL}(2,q)$. We show that if
$(a_1,\ldots,a_t)$ and $(b_1,\ldots,b_t)$ are sampled uniformly from large
subsets $A$ and $B$ of $G^t$ then their interleaved product $a_1 b_1 a_2 b_2
\cdots a_t b_t$ is nearly uniform over $G$. This extends a result of the first
author, which corresponds to the independent case where $A$ and $B$ are product
sets. We obtain a number of other results. For example, we show that if $X$ is
a probability distribution on $G^m$ such that any two coordinates are uniform
in $G^2$, then a pointwise product of $s$ independent copies of $X$ is nearly
uniform in $G^m$, where $s$ depends on $m$ only. Extensions to other groups are
also discussed.
  We obtain closely related results in communication complexity, which is the
setting where some of these questions were first asked by Miles and Viola. For
example, suppose party $A_i$ of $k$ parties $A_1,\dots,A_k$ receives on its
forehead a $t$-tuple $(a_{i1},\dots,a_{it})$ of elements from $G$. The parties
are promised that the interleaved product $a_{11}\dots a_{k1}a_{12}\dots
a_{k2}\dots a_{1t}\dots a_{kt}$ is equal either to the identity $e$ or to some
other fixed element $g\in G$, and their goal is to determine which of the two
the product is equal to. We show that for all fixed $k$ and all sufficiently
large $t$ the communication is $\Omega(t \log |G|)$, which is tight. Even for
$k=2$ the previous best lower bound was $\Omega(t)$. As an application, we
establish the security of the leakage-resilient circuits studied by Miles and
Viola in the "only computation leaks" model.
