  Sub-GeV dark matter candidates are of increasing interest, because
long-favored candidates such as GeV-scale WIMPs have not been detected. For
low-mass dark matter, model-independent constraints are weak or nonexistent. We
show that for such candidates, because the number density is high, cosmic ray
propagation can be affected by elastic scattering with dark matter. We call
this type of search `reverse direct detection,' because dark matter is the
target and Standard Model particles are the beam. Using a simple propagation
model for galactic cosmic rays, we calculate how dark matter affects cosmic ray
spectra at Earth, and set new limits on the dark matter-proton and dark
matter-electron cross sections. For protons, our limit is competitive with
cosmological constraints, but is independent. For electrons, our limit covers
masses not yet probed, and improves on cosmological constraints by one to two
orders of magnitude. We comment on how future work can significantly improve
the sensitivity of cosmic-ray probes of dark matter interactions.
