I'm not talking about the RSA, El-gamal, nor any specific encryption scheme. Rather, my question, as related to this and this threads, is why the idea of Public-Key encryption scheme cannot be secure under CPA (which has equal power as eavesdropper in the public-key area) without the assumption that $P \neq NP$.
If $P=NP$ the task to compute a decision of "whether a secret key $S_k$ is the complement of the given public key $P_k$?" takes a poly time (where complement may be refer to inverse in the Euler group in RSA or discrete-log in El-Gamal or any other term but my question is general) is equal (in regard to the time complexity) to the task of "finding the actual complement $S_k$ from a given $P_k$ and additional public settings (as the group description etc.)" which will also take poly time.
- Does my last state seems logical/correct?
- How can I further formalize it?