Ehhhh.... well, it's complicated. For most cryptosystems, the answer is no, because if you can solve the underlying problem efficiently you can break the security of the scheme as defined. It turns out that this isn't always a 'break' in the sense that most people understand it. For example, a 'break' might just mean the ciphertext is no longer indistinguishable from random noise, but it might be possible to prove meaningful security in a weakened model that doesn't require ciphertexts to look like random noise but, for example, requires that no bits of the plaintext are leaked with high probability. Cryptographers build schemes with very strong, conservative security guarantees for this exact reason.