Dear JM,

I'm not aware of the sources which say that the devil tempted Jesus on the cross. He was tempted by human beings, such as the soldiers and the criminals, I would say. Since Satan is generally alert to our weaknesses--both physical and spiritual--I will grant that the Evil One would have been around and active in Jesus' darkest human hour!

But I am more clear about the first part of your question, which appears in Matthew, Mark and Luke: Did Satan know that he was tempting the Savior in the desert?  The tempter does call Jesus the Son of God, but some people who know a lot more about Scripture than I do believe that Son of God includes all of us--and in Scripture, all of Israel. So that's not our "final answer."

I searched some references for insight into this and did not find a precise answer. I'm willing to venture this opinion: I believe that Satan was trying to figure OUT if this good man was the Savior. By the end of the three temptations (at which point Jesus was weak from fasting, but strong from prayer), he has a lot more respect for this man--and is coming to a conclusion that this man may well be the Savior. All that I have said is based on imagining that Satan thinks in a human way--and that's a big IF. Possibly he has more knowledge than I suspect....But there's a start.