  In this paper we introduce and study the \textsc{StreamingCycles} problem, a
random order streaming version of the Boolean Hidden Hypermatching problem that
has been instrumental in streaming lower bounds over the past decade. In this
problem the edges of a graph $G$, comprising $n/\ell$ disjoint length-$\ell$
cycles on $n$ vertices, are partitioned randomly among $n$ players. Every edge
is annotated with an independent uniformly random bit, and the players' task is
to output the parity of some cycle in $G$ after one round of sequential
communication.
  Our main result is an $\ell^{\Omega(\ell)}$ lower bound on the communication
complexity of \textsc{StreamingCycles}, which is tight up to constant factors
in $\ell$. Applications of our lower bound for \textsc{StreamingCycles} include
an essentially tight lower bound for component collection in (almost) random
order graph streams, making progress towards a conjecture of Peng and Sohler
[SODA'18] and the first exponential space lower bounds for random walk
generation.
