depends on the marginal distribution of the channel and not on the joint distribution we can minimize over all joint distributions with fixed marginal distributions.

Gaussian Case
=============

![Secret-key agreement codebook for the dirty paper channel. The transmit sequence $\rvx^n$ is selected so that $\rvu^n = \rvx^n+\rvs^n$ is a sequence in the codebook $\cC$. The smaller spheres above denote the noise uncertainty at the legitimate receiver. Their centres are the codewords in $\cC$. The larger sphere denotes the noise uncertainty at the eavesdropper. Our binning of smaller spheres guarantees that the noise uncertainty sphere of the eavesdropper has all possible messages, resulting in (asymptotically) perfect equivocation.[]{data-label="fig:dpc"}](pp2x)

We develop the lower and upper bounds on secret-key agreement capacity for the Gaussian channel model.

Proof of Prop. \[prop:lbGaussNoDisc\]
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Recall that $\rvs \sim \cN(0,Q)$. Choose $\rvx \sim \cN(0,P)$ to be a Gaussian random variable independent of $\rvs$ and let $E[\rvx\rvs] = \rho \sqrt{PQ}$. Select $\rvu = \rvx + \alpha \rvs$ and the lower bound follows by evaluating $$\begin{aligned}
R &= I(\rvu;\rvyr) - I(\rvu;\rvye) \\
&= h(\rvu|\rvye) - h(\rvu|\rvyr)\end{aligned}$$ Further evaluating each of the terms above with $\rvu = \rvx + \al \rvs$, note that $$\begin{aligned}
&h(\rvu|\rvye)= h(\rvx  + \al \rvs |\rvx + 