A320
Air Traffic Control Fully Certified
Communication Breakdown
ATC Issue All Types; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Issued New Clearance
Human Factors
An A320 was in position on Runway 4L; a B757 was in position on Runway 31L KE Intersection. I issued the B757 takeoff clearance; the A320 took the B757 clearance and did read it back with his call sign. The B757 immediately asked for a confirmation of the take off clearance.  I then restated the B757's clearance to take off. He then stated that he someone else took his take off clearance. At this point I noticed the A320 had begun his departure roll on Runway 4L. He may have gone a hundred feet down the runway when he stopped and asked if he was cleared for take off. I stated that I did not clear him for take off but he was at this time now cleared for take off. There was no loss of separation or possibility of it. The A320 and the B757 both departed with no further issues. Pilots and controllers must be more aware of hear back read back errors. I did not realize that the A320 read back the B757 clearance. I heard what I thought was the B757 acknowledging the clearance. The call signs were similar but not similar enough that I pointed them out to both pilots. The A320 heard what he was waiting to hear even though the clearance was not for him.
Aircraft cleared position and hold 4L at JFK. Second aircraft cleared position and hold 31L; KE Intersection. ATC issued takeoff clearance; both aircraft responded; at which point we delayed takeoff until clarification was given by ATC; normal departure without incident.
JFK Controller described a confused take off event when an Air Carrier initiated a take off without being cleared; confused by another another Air Carrier's clearance departing an intersecting runway.