  The security of finite-length keys is essential for the implementation of
device-independent quantum key distribution (DIQKD). Presently, there are
several finite-size DIQKD security proofs, but they are mostly focused on
standard DIQKD protocols and do not directly apply to the recent improved DIQKD
protocols based on noisy preprocessing, random key measurements, and modified
CHSH inequalities. Here, we provide a general finite-size security proof that
can simultaneously encompass these approaches, using tighter finite-size bounds
than previous analyses. In doing so, we develop a method to compute tight lower
bounds on the asymptotic keyrate for any such DIQKD protocol with binary inputs
and outputs. With this, we show that positive asymptotic keyrates are
achievable up to depolarizing noise values of $9.33\%$, exceeding all
previously known noise thresholds. We also develop a modification to
random-key-measurement protocols, using a pre-shared seed followed by a "seed
recovery" step, which yields substantially higher net key generation rates by
essentially removing the sifting factor. Some of our results may also improve
the keyrates of device-independent randomness expansion.
