Secrecy capacity region of the Gaussian multi-receiver wiretap channel

We consider the Gaussian multi-receiver wiretap channel and evaluate its secrecy capacity region. This evaluation requires the identification of underlying auxiliary random variables. For this purpose, we first visit the converse proof of the scalar Gaussian broadcast channel, and show that this proof cannot be extended to this secrecy context. The failure of this extension comes from the insufficiency of the entropy-power inequality to resolve the ambiguity regarding the auxiliary random variables. Instead, we provide two converse proofs. The first one uses the alternative representation of the mutual information as an integration of the minimum-mean-square-error (MMSE) along with the properties of the MMSE. The second one uses the relationship between the differential entropy and the Fisher information via the de Bruin identity along with the properties of the Fisher information.