  We propose a novel electron acceleration mechanism, which we call stochastic
shock drift acceleration (SSDA), that extends the standard shock drift
acceleration (SDA) for low-energy electrons at a quasi-perpendicular shock to
include the effect of stochastic pitch-angle scattering. We demonstrate that
the steady-state energy spectrum of electrons accelerated within the shock
transition region becomes a power-law in the limit of strong scattering. The
spectral index is independent of the pitch-angle scattering coefficient. On the
other hand, the maximum energy attainable through the mechanism scales linearly
with the pitch-angle scattering coefficient. These results have been confirmed
by Monte Carlo simulations that include finite pitch-angle anisotropy. We find
that the theory can reasonably well explain in-situ observations of
quasi-perpendicular Earth's bow shock. Theoretical scaling law suggests that
the maximum energy increases in proportion to the square of the shock speed,
indicating that the thermal electrons may be accelerated up to mildly
relativistic energies by the SSDA at quasi-perpendicular supernova remnant
shocks. Therefore, the mechanism provides a plausible solution to the
long-standing electron injection problem.
