This is a thread on St Hilary of Poitiers on the Primacy of Peter. From On the Trinity, book 6.
http://www.newadvent.org/fathers/330206.htm

Hilary starts by saying that Peter was set to be the foundation stone of the church. Let’s not jump to conclusions, this still requires context.

Next Hilary defines what he means by foundation of the church. He explicitly says that the confession is the foundation. He goes on to say that the confession, that Jesus is the Son of God, is made in faith.

St Hilary goes on to say that through the faith which makes the confession, this faith has the keys of the kingdom of heaven.
Through the faith that gives the confession, not through Peter himself, the gates of Hell cannot prevail against the church.

Finally, when speaking of the apostles, the same keys of the kingdom of heaven were awarded to them for their faith, their confession, the same confession of Peter.

Judging from the wording of the first image, he was Simon before the confession, and after was set to be the foundation stone of the church, named Peter in honor of this after the confession.

But with the context that the foundation is the confession in faith, and this faith awards the keys, and that all apostles received the keys, the church began with the confession that Christ is the Rock, and Simon was named Peter in honor of his confession.

Psalm 118:22, the chief cornerstone is Christ, the next stone may be Peter, and the apostles follow, all foundation stones, like the twelve foundations which Jerusalem in heaven is built on (Rev.22).

And on this immovable stone (Christ), on these twelve foundation stones (apostles), Peter explains in his first epistles in ch2 that on these foundations we are all built up in a spiritual house, first the priesthood, upon this confession in faith that Jesus is the Son of God.

This is primacy, not supremacy. They do not mean the same thing. Matthew 18:18 is clear about giving all apostles the keys to the kingdom, where John speaks of the confession they give. The apostles are equal, Peter is first among them, and as Hebrews says, Christ is the Rock.

And to vindicate my points against the accusations against me, the quote in chapter 36 which supports me.