Hard discrete logarithm problem I want to give an introductory talk on Diffie-Helman key exchange. Along the way, I will mention that there exist groups, like $(\mathbb Z/p\mathbb Z)^\times$ for which it is believed that the discrete logarithm problem (DLP) is hard. Trying to anticipate a question from the audience, suppose someone asks:


  Are there groups for which the DLP is surely known to be hard?


What is a good way to answer such a question, and in the case of no asnwer, how do I convince the audience that the Diffie-Helman key exchange is still worth the hustle? 
I understand this is not good question to ask here, so I bring my honest apologies.
I think this is a very good question, and an interesting one. Not sure if my answer will go in the direction you meant.

Generally when first discussing the DLP for groups, you'd want to find an attack which uses only the most basic properties of a group. That is, you think of having a black box into which you can feed any two group elements, and it will compute the group operation on them. You see the result, but have no idea how it was computed. 

Now suppose we have such a black box group, and assume that it is cyclic of prime order $r$. Then this paper proves that you need on average at least $O(\sqrt{r})$ calls to the black box to solve the DLP. For those groups we are sure that the DLP is hard, for some definition of "sure" and "hard".

This may be somewhat counterintuitive. Note that for a prime $p$, $\mathbb{Z}/p\mathbb{Z}$ with addition is a prime order group for which the DLP is trivial. Also for every black box group $G$ such that $|G|=p$ there exist an isomorphism $\phi:G\cong \mathbb{Z}/p\mathbb{Z}$. The problem is that we cannot reduce the problem from the black box group $G$ to $\mathbb{Z}/p\mathbb{Z}$, since we cannot compute the isomorphism $\phi$ efficiently.