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Common randomness (CR), as a resource, is not commonly exploited in existing practical communication systems. In the CR generation framework, both the sender and receiver aim to generate a common random variable observable to both, ideally with low error probability. The availability of this CR allows us to implement correlated random protocols that can lead to faster and more efficient algorithms. Previous work focused on CR generation over perfect channels with limited capacity. In our work, we consider the problem of CR generation from independent and identically distributed (i.i.d.) samples of a correlated finite source with one-way communication over a Gaussian channel. We first derive the CR capacity for single-input single-output (SISO) Gaussian channels. This result is then used for the derivation of the CR capacity in the multiple-input multiple-output (MIMO) case. CR plays a key role in the identification scheme since it may allow a significant increase in the identification capacity of channels. In the identification framework, the decoder is interested in knowing \emph{whether} a specific message of special interest to him has been sent or not, rather than knowing \emph{what} the received message is. In many new applications, such as several machine-to-machine and human-to-machine systems and the tactile internet, this post-Shannon scheme is more efficient than classical transmission. In our work, we also consider a CR-assisted secure identification scheme and develop a lower bound on the corresponding secure identification capacity.