  In this paper we consider the problem of extracting secret key from an
eavesdropped source $p_{XYZ}$ at a rate given by the conditional mutual
information. We investigate this question under three different scenarios: (i)
Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness
with the eavesdropper Eve ($Z$), (ii) Alice and Bob are allowed one-way public
communication, and (iii) Alice and Bob are allowed two-way public
communication. Distributions having a key rate of the conditional mutual
information are precisely those in which a "helping" Eve offers Alice and Bob
no greater advantage for obtaining secret key than a fully adversarial one. For
each of the above scenarios, strong necessary conditions are derived on the
structure of distributions attaining a secret key rate of $I(X:Y|Z)$. In
obtaining our results, we completely solve the problem of secret key
distillation under scenario (i) and identify $H(S|Z)$ to be the optimal key
rate using shared randomness, where $S$ is the G\'acs-K\"orner Common
Information. We thus provide an operational interpretation of the conditional
G\'acs-K\"orner Common Information. Additionally, we introduce simple example
distributions in which the rate $I(X:Y|Z)$ is achievable if and only if two-way
communication is allowed.
