Is there a problem in a PK cryptosystem if the plaintexts are very small compared to the ciphertexts? I'm wondering, suppose we have a public key cryptosystem $P = (pk,sk)$ such that $$pk: \{0,1\}^n \to \{0,1\}^{n^2},$$ that is, the ciphertexts are $n$ times longer than the plaintexts (here, $n$ is supposed to be the security parameter). My question is


  Does this imply some weakness of any kind on the cryptosystem (as $n$ grows)?


It's not that I have the intuition that there may be an issue, it's just a particular property that makes me think.

BACKGROUND

We're developing a PK cryptosystem based in multivariate polynomial equations. So far, our ciphertexts were as long as our plaintexts, but now we have this situation, where the ciphertext / plaintext length ratio is not constant. This has its own consequences in the algebraic point of view, but in the cryptographic setting... does it have any implication? Thanks!
No. ​ ​ ​ ​ Also, note that by hybrid encryption, ciphertext overhead will always be

at most ​ ​ + poly(security_parameter) , ​ ​ no matter how long the message is.

For IND-CPA of any

public-key encryption scheme, or even

symmetric encryption scheme with stateless encryption

, ciphertexts for non-empty messages must be

overwhelmingly likely to be longer than plaintexts.

Your most recent comment suggests it was broken even in the following context, but otherwise you might have had a secure trapdoor permutation family whose domains are exactly the binary strings whose length equals the length chosen during key generation.