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In this paper, we consider the problem of secret key generation with one-waycommunication through both a rate-limited public channel and a rate-limitedsecure channels where the public channel is from Alice to Bob and Eve and thesecure channel is from Alice to Bob. In this model, we do not pose anyconstraints on the sources, i.e. Bob is not degraded to or less noisy than Eve.We obtain the optimal secret key rate in this problem, both for the discretememoryless sources and vector Gaussian sources. The vector Gaussiancharacterization is derived by suitably applying the enhancement argument, andProving a new extremal inequality. The extremal inequality can be seen ascoupling of two extremal inequalities, which are related to the degradedcompound MIMO Gaussian broadcast channel, and the vector generalization ofCosta's entropy power inequality, accordingly.

Secret Key Generation from Vector Gaussian Sources with Public and Private Communications