  A theory of electron injection into diffusive shock acceleration (DSA) for
the generation of cosmic-ray electrons at collisionless shocks is presented. We
consider a recently proposed particle acceleration mechanism called stochastic
shock drift acceleration (SSDA). We find that SSDA may be understood as a
diffusive particle acceleration mechanism at an oblique shock of finite
thickness. More specifically, it is described by a solution to the
diffusion-convection equation for particles with the characteristic diffusion
length comparable to the shock thickness. On the other hand, the same equation
yields the standard DSA if the diffusion length is much longer than the
thickness. Although SSDA predicts, in general, a spectral index steeper than
DSA, it is much more efficient for low-energy electron acceleration and is
favorable for injection. The injection threshold energy corresponds to the
transition energy between the two different regimes. It is of the order of
$0.1\text{-}1$ MeV in typical interstellar and interplanetary conditions if the
dissipation scale of turbulence around the shock is determined by the ion
inertial length. The electron injection is more efficient at high $M_\textrm{A}
/ \cos \theta_{Bn}$ where $M_\textrm{A}$ and $\theta_{Bn}$ are the Alfv\'en
Mach number and the shock obliquity. The theory suggests that efficient
acceleration of electrons to ultra-relativistic energies will be more easily
realized at high-Mach-number young supernova remnant shocks, but not at weak or
moderate shocks in the heliosphere unless the upstream magnetic field is nearly
perpendicular to the shock normal.
