KEA assumption details In order to understand the construction of a zK-SNARK, I have recently been trying to understand the KEA1 assumption in The Knowledge-of-Exponent Assumptions and 3-Round Zero-Knowledge Protocols by Mihir Bellare and Adriana Palacioy.

I do not understand the meaning of this assumption. For example, why is this "hard problem" used in cryptography? 

On page three of the paper, the authors roughly outlined the assumption, but this didn't clarify things for me. I have a strong math background, so if someone could explain the KEA1 assumption to me in a mathematical way, that would be great.
I think why this assumption is hard is that in the prime field $F_p$, you can't find directly $a$ from $g^a \bmod p$ , where $g$ is generator.
So if prime is too large it takes a lot of time to find $a$, because you need to try all the elements in $1..p$.

that is based on DLP(Discrete Logarithm Problem).