DHE key exchange. Is it really secure? Context
I recently came across a video on the net showing the process of key exchange in Diffie Hellmann.
Alice and Bob want to communicate. Eve is the (wo)man in the middle.
This video is the basis of my question
http://youtu.be/YEBfamv-_do?t=6m22s
Question
Alice sends the modulo and the prefix to bob. Eve gets that.
What prevents Eve to have a giant table of all the calculations possible given a prefix and a modulo and the public value and intersect thos possibilities with the public value from bob ?
Yes the table would be very big but you can then iterate over the intersecting possibilities and try the numbers on the message up to the point to get something right.
i may miss something, but it is not that hard, even with very big numbers.
So what did I miss ?
Thanks
A lot of modern cryptography is based on some mathematical assumptions and aims to achieve what is called Computational Security. That means that the adversary (Eve) could get some information about the plaintext with a negligible probability and the adversary is modeled as someone with bounded computational power, storage and bounded time. 
So all the (encryption) schemes are dimensioned in sort to achieve such level of security: that means that all mathematical elements are chosen to be large enough to make the best known attack feasible in a reasonable amount of time.

Even if this is a weaker notion of security, this assumption on the adversary is easy to accept in practice.

Other approach exists: Information theoretic Security or Perfect Security

So coming back to your solution based on a big table, if the scheme is correctly dimensioned the table you are supposed to use to attack the scheme would be so large that it cannot be stored in a (very very) large disk or to be (too) long to be generated.