We investigate the secret key agreement from correlated vector Gaussian
sources in which the legitimate parties can use the public communication with
limited rate. For the class of protocols with the one-way public communication,
we show that the optimal trade-off between the rate of key generation and the
rate of the public communication is characterized as an optimization problem of
a Gaussian random variable. The characterization is derived by using the
enhancement technique introduced by Weingarten et.al. for MIMO Gaussian
broadcast channel.

We consider the distributed source coding system of $L$ correlated Gaussian
sources $Y_i,i=1,2,\cdots,L$ which are noisy observations of correlated
Gaussian remote sources $X_k, k=1,2,\cdots,K$. We assume that $Y^{L}={}^{\rm
t}(Y_1,Y_2,$ $\cdots, Y_L)$ is an observation of the source vector $X^K={}^{\rm
t}(X_1,X_2,\cdots, X_K)$, having the form $Y^L=AX^K+N^L$, where $A$ is a
$L\times K$ matrix and $N^L={}^{\rm t}(N_1,N_2,\cdots,N_L)$ is a vector of $L$
independent Gaussian random variables also independent of $X^K$.