Something that is something else cannot be by something else, so the faith of Christ is not the righteousness of God. They are distinct.
The righteousness of God is the gift imparted.
The "faith of Christ" is the avenue by which the gift comes.
The recipients are "all them that believe".
What do they believe? That what they have been told about Jesus's atoning death and resurrection is the truth, that their sins are covered by His blood shed on the cross, and that they have a glorious eternal future with God. They take it as truth, and that "taking" is exercising faith. While that faith is indeed the gift of God (Ephesians 2:8), that faith is not merely in any random thing, but is specifically placed in the Person and finished work of Christ.
The correct understanding of "the faith of Christ" is faith in Christ.
The righteousness of God is the gift imparted.
The "faith of Christ" is the avenue by which the gift comes.
The recipients are "all them that believe".
What do they believe? That what they have been told about Jesus's atoning death and resurrection is the truth, that their sins are covered by His blood shed on the cross, and that they have a glorious eternal future with God. They take it as truth, and that "taking" is exercising faith. While that faith is indeed the gift of God (Ephesians 2:8), that faith is not merely in any random thing, but is specifically placed in the Person and finished work of Christ.
The correct understanding of "the faith of Christ" is faith in Christ.