I am very familiar with the EFF's case and the various arguments against it, the methods used to eavesdrop on the network, and the excuses used to pretend the actions.
> NSA to collect targeted information
> show that the connection is only used by the target
This is off topic. Targeted searches are not relevant, and assuming they have a proper warrant, is perfectly legal. I'm sure some unavoidable collection happens while executing targeted searches, and the filtering necessary is is just the mechanics of wiretapping. Obviously, this is not the illegal part.
Bulk, non-targeted collection is very different. The NSA admits they do not meet the 4th Amendment's warrant requirement, while claiming to have a general warrant from the FISA court, often while waving around Sections 215 and 702 of the Patriot Act, or Executive Order 12333. IN the end, the NSA is still claiming to have a general warrant ("writ of assistance"). As we have not had a constitutional amendment that repeals the 4th Amendment, the specific warrant requirement is still the highest law of the land.
Running your "grep" filter to find the communications of a specific target will probably run across other people's communications. The important part is that - as a targeted action - it only applies to specific locations (or routers/etc) at some some specific time. If you getting a traditional search warrant to search someone's house does not allow you to come back at some future data for another search, nor does it extend to other locations.
On the other hand, if you had a warrant to search Alice's apartment, it is likely that some of her roommate Carol's stuff will be search as well. This is unfortunate, but probably unavoidable and generally legal. The fact that there is a valid reason for the search of Carol's stuff doesn't suddenly extend to allowing a search of anybody else.