Is there any tech that uses a signature as a private key?

I’ve been pondering how the Lightning Network will work when one of the parties in a channel tries to cheat. It seems that publishing a transaction that shouldn’t be published (a “bad act”) must involve publishing something the victim can use to unwind everything. In trying to identify what it is that the bad act provides to the victim, the “signature as a private key” seemed like a possibility.

Even if the “signature as a private key” idea isn’t used in LN, it seems like a useful concept, but I couldn’t find anything on the internet discussing it. I tried a Google search and then came here.

Recent Questions – Bitcoin Stack Exchange