  We propose a lattice-based scheme for secret key generation from Gaussian
sources in the presence of an eavesdropper, and show that it achieves the
strong secret key capacity in the case of degraded source models, as well as
the optimal secret key / public communication rate trade-off. The key
ingredients of our scheme are the use of the modulo lattice operation to
extract the channel intrinsic randomness, based on the notion of flatness
factor, together with a randomized lattice quantization technique to quantize
the continuous source. Compared to previous works, we introduce two new notions
of flatness factor based on $L^1$ distance and KL divergence, respectively,
which might be of independent interest. We prove the existence of secrecy-good
lattices under $L^1$ distance and KL divergence, whose $L^1$ and KL flatness
factors vanish for volume-to-noise ratios up to $2\pi e$. This improves upon
the volume-to-noise ratio threshold $2\pi$ of the $L^{\infty}$ flatness factor.
