Under the constitution and the applicable law it is the Defense Deparment.
I decide who my enemies are and those decisions are based on moral precepts and the actions of my enemies. Really quite simple actually.
1. If someone visits terrorist camps ("This, officials say, was when he first visited Afghanistan.")
2. trains in said terrorist camps ("Padilla got some instruction in bomb-making, and some cash. And al-Qaeda leaders reportedly discussed with him schemes ranging from "dirty bombs" to blowing up gas stations discussions which some captive terrorist leaders appear to have shared with U.S. agents.")
3. consorts with known terrorists ("In 2001, officials say, he made contact with Abu Zubaydah, a senior al-Qaeda commander who is in American custody and apparently co-operating with the FBI.") I then simply consider his actions and do the things that will eliminate his threat to me and mine.