In 1975, Wyner published two very different papers that are unexpectedly connected. One introduced the wiretap channel, showing that information-theoretic secrecy is possible without a secret key by taking advantage of channel noise. This is the foundation for much of physical-layer security. The other paper introduced a notion of common information relevant to generating random variables at different terminals. In that work he introduced a soft-covering tool for proving achievability. Coincidently, soft covering has now become the tool of choice for proving strong secrecy in wiretap channels, although Wyner didn't appear to make any connection between the two results.
We present a sharpening of the soft-covering tool by showing that the soft-covering phenomenon happens with doubly-exponential certainty with respect to a randomly generated codebook. Through the union bound, this enables security proofs in settings where many security constraints must be satisfied simultaneously. The "type II" wiretap channel is a great example of this, where the eavesdropper can actively influence his observations but security must hold in all cases. We demonstrate the effectiveness of this tool by deriving the secrecy capacity of wiretap channels of type II with a noisy main channel---previously an open problem. Additionally, this stronger soft covering allows information-theoretic security proofs to be easily upgraded to semantic security, which is the gold standard in cryptography.