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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.