What other one-way functions are used in cryptosystems?
The Merkle–Hellman knapsack cryptosystem was based on a variation of the subset sum problem.  (It was broken by Adi Shamir a few years after it was developed.)


  Given a set of numbers $A$ and a number b, find a subset of $A$, which sums to b.


The cryptosystem relies on the fact that in this form of the subset sum problem if the set $A$ is superincreasing (each element of the set is greater than the sum of all the previous elements), the problem is solvable in polynomial time.  Also, you can transform the superincreasing set $A$ into a non-superincreasing set $B$ using a multiplier $r$ and a modulus $q$.

Solving the subset sum problem with a non-superincreasing set ($B$) is NP-complete, so you can use $B$ as a public key to encrypt messages, then use $A$, $r$, and $q$ as a private key to decrypt them.