Is there any hash function that relies on integer exponentiation instead of XOR? I'm trying to implement a hash function in an esoteric language that has no 32-bit or 64-bit operations. Keccak, for example, is too slow in that language. I'd need a design that minimizes the number of 1-bit operations. Alternatively, I could also use a hash function that relies on integer exponentiation, which is not expensive on this language. Is there any hash function like that?
You can use the discrete log-based hash given by $H(x,y) = g^x h^y$, where $g$ and $h$ are two randomly chosen group elements. If your group has known prime order $p$, then this function is collision-resistant if and only if discrete logarithm is hard in your group. In particular, if you find a collision, $g^x h^y = g^{x'} h^{y'}$, then you get $g^{x-x'} = h^{y'-y}$. This means that $x - x' = \log_g h \cdot (y' - y) \mod p$. Since $p$ is prime and $y \neq y'$, we can invert $y'-y$ modulo $p$ to compute the logarithm.

This is assuming that you want a collision-resistant hash function. If you want something stronger, then this might not be suitable for you. E.g., this obviously has the property that $H(x+1,y) = g \cdot H(x,y)$, which could be really bad in some applications.