Inverse public-key encryption
What you are looking for is so called white-box cryptography. That said, you should definitely check if you can't get away with digital signatures. The reason is that white box cryptography is a special case of program obfuscation that is quite hard to achieve. Existing, practical schemes do not rely on hard problems but rather on not being broken so far. So you do not get rigorous security guarantees but only the statement that nobody that tried did succeed (which says nothing about how many tried and who tried...).