What other one-way functions are used in cryptosystems? For RSA and El Gamal (and most other public key cryptosystems), one of the key ideas is that factoring and finding discrete logarithms are hard. There are other systems that rely on certain properties of lattices.

What are the other one-way-ish functions that have cryptosystems designed around them?
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.