  Reconfigurable intelligent surface (RIS) is a disruptive technology to
enhance the performance of physical-layer key generation (PKG) thanks to its
ability to smartly customize the radio environments. Existing RIS-assisted PKG
methods are mainly based on the idealistic assumption of an independent and
identically distributed (i.i.d.) channel model at both the base station (BS)
and the RIS. However, the i.i.d. model is inaccurate for a typical RIS in an
isotropic scattering environment and neglecting the existence of channel
spatial correlation would possibly degrade the PKG performance. In this paper,
we establish a general spatially correlated channel model and propose a new
channel probing framework based on the transmit and the reflective beamforming.
We derive a closed-form key generation rate (KGR) expression and formulate an
optimization problem, which is solved by using the low-complexity Block
Successive Upper-bound Minimization (BSUM) with Mirror-Prox method. Simulation
results show that compared to the existing methods based on the i.i.d. fading
model, our proposed method achieves about $5$ dB transmit power gain when the
spacing between two neighboring RIS elements is a quarter of the wavelength.
Also, the KGR increases significantly with the number of RIS elements while
that increases marginally with the number of BS antennas.
