As you know, the RSA system is based on the fact that factoring a number $n$ cannot be done in polynomial time ($P(\ln(n))$, not $P(n)$). The factoring problem is known to be in $NP$, but we don't know if it is $NP-complete$. My question is:
Is there a cryptosystem based on a problem known to be $NP-complete$?
Is there a cryptosystem based on a problem known to be $PSPACE-complete$ (or any other kind of completeness for a complexity class containing $NP$)?
I know some alternative cryptosystems exist, but I couldn't find any theoretical complexity proof about them. It looks to me that it would be safer to find such systems, just in case $P=NP$ (I personally think $P\neq NP$, but as my grandma says, "better safe than sorry")
NP Complete need not mean safer, since you want strength not only in the worst case (for the attacker) but in all cases. Some cryptosystems based on NP complete problems (most notoriously based on the knapsack problem) have been broken.
Having said that, cryposystems based on the Discrete Logarithm problem in prime order groups, as well as those based on the additive group of elliptic curves are believed to be "stronger than RSA", in the sense that the best known algorithms for attacking them are exponential in the size of the input, while the best known factoring algorithms, relevant to breaking RSA, are superpolynomial but subexponential, e.g., the Number Field Sieve.
You can possibly browse the crypto.stackexchange group for more information on these topics.