What is the discrete log used for?

Question

Perusing Wikipedia, I stumbled on the discrete logarithm. I looks interesting that we'd be able have a function that could solve $b^k=g$ for integers $b,k,$ and $g$. However, Wikipedia says "No efficient classical algorithm for computing general discrete logarithms $\log_bg$ is known." I'm just wondering, if we can't calculate it (except by trial and error with presumably small numbers, is this even a useful concept? What can we do with the discrete logarithm?

Answer 1

We could break Diffie-Hellman protocol for data exchange with no shared secret.

Public Information:

• Prime number $p$
• Generator $g\in{Z}^*p$

Protocol:

Advantage of Alice and Bob over Eve:

• Alice and Bob can easily compute $k=g^{ab}$
• Eve intercepts $g^a$ and $g^b$, but cannot easily compute $g^{ab}$

Assumptions: