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 …