Are there any public-key encryption schemes based on DLog? There are public-key encryption schemes based on many different mathematical hardness assumptions, like the hardness of Decisional Diffie-Hellman problem, the hardness of the Factoring problem, the hardness of the RSA problem, etc.
But my question is, are there any public-key encryption schemes based on the hardness of the Discrete Log problem?  Does the hardness of the Discrete Log problem even imply the existence of secure public-key encryption?
It is a major open research question whether such a scheme exists, and how to construct one (see, for example, Open Problem 9.10). Of course, we do have schemes like (hashed) ElGamal, which are based on the conjectured hardness of the (computational or decisional) Diffie-Hellman problem. But it is unknown whether either of these problems is equivalent to the discrete logarithm problem (except, of course, in groups where discrete log is easy).