Whose righteousness should we be wearing?
Isaiah 64:6
But we are all like an unclean thing, and all our righteousnesses are like filthy rags; we all fade as a leaf, and our iniquities, like the wind, have taken us away.
Righteousness may deliver from death, but whose righteousness are we talking about? The above scripture makes it clear that our individual level of righteousness is on par with "filthy rags"—and when you're looking to protect yourself from death, filthy rags make for a lousy breastplate.
Jeremiah 23:6
In His days Judah will be saved, and Israel will dwell safely; now this is His name by which He will be called: THE LORD OUR RIGHTEOUSNESS.
Also consider scriptures such as Job 36:3; Psalm 5:8; 23:3; 24:5; and 71:16, which show that true righteousness comes from God. It is God's righteousness, and not our own, which must serve as our breastplate and defense against Satan.