# A secure public-key cryptosystem

Is there any public-key cryptosystem that is (possibly) secure against $$NP\cap coNP$$ adversary?

Is there any public-key cryptosystem that is (possibly) secure against $$NP\cap coNP$$ adversary?
Hash based signatures come to mind; essentially, to break them, you need to find a (second) preimage of the underlying hash function, and that problem is not known to be within $$NP\cap coNP$$. Of course, the current instantiations are based on fixed-sized hash functions (and hence are not within $$NP$$); however generalizing the concept to variable length hash functions should be fairly straight-forward.