Choices for the group in Public Key Cryptography I am only aware of two alternatives for the group used in Diffie-Hellman scheme (and similar ones) where logarithms are conjectured to be hard. Those are $\mathbb{F}_p$ and Elliptic Curves. Are there any other explored choices?
Dan Boneh's survey article on the decisional Diffie Hellman problem lists several candidate groups for which the DDH problem is hard, reproduced here:


  
  Let $p=2p_1 + 1$ where both $p$ and $p_1$ are prime. Let $Q_p$ be the subgroup of quadratic residues in $\mathbb{Z}^*_p$. It is a cyclic group of prime order. This family of groups is parameterized by $p$.
  More generally, let $p = aq+1$ where both $p$ and $q$ are prime and $q &gt; p^{1/10}$. Let $Q_{p,q}$ be the subgroup of $\mathbb{Z}_p^*$ of order $q$. This family of groups is parameterized by both $p$ and $q$.
  Let $N=pq$ where $p,q,\frac{p-1}2,\frac{q-1}2$ are prime. Let $T$ be the cyclic subgroup of order $(p-1)(q-1)$. Although $T$ does not have prime order, DDH is believed to be intractable. The group is parameterized by $N$.
  Let $p$ be a prime and $E_{a,b}/\mathbb{F}_p$ be an elliptic curve where $|E_{a,b}|$ is prime. The group is parameterized by $p,a,b$.
  Let $p$ be a prime and $J$ be a Jacobian of a hyper elliptic curve over $\mathbb{F}_p$ with a prime number of reduced divisors. The group is parameterized by $p$ and the coefficients of the defining equation.
  


In particular, #3 is not among the flavors listed in your question.