Factorization of integers is hard, but finding irreducibles is expensive. Is there a ring where factorization is assumed hard but finding irreducibles is much cheaper than over $\Bbb Z$?
It could provide an alternative ring to do RSA/Rabin over.
A suggestion is the polynomial ring over $\Bbb Z_{pq}$ where $p$ and $q$ are prime. $\Bbb Z_{pq}$ has zero divisors, so I don't know what kind of creature you get. You still have to find $p$ and $q$ to start with, but still interesting.
Another suggestion is because polynomial factorization over finite fields is tractable, you could look for infinite fields, probably of nonzero characteristic.
