Were The Petrine Epistles Authored By Peter?

Traditionally, Peter was thought to be martyred in AD 64 or 65 by crucifixion, and there seems to be some evidence to support this, include a letter by Clement of Rome written later in the first century, and no reason to suppose otherwise. Much of the argument for Petrine authorship revolves around the dating of the letters. A date later than AD 65 clearly indicates a letter was not authored by Peter.

The issue of motive is an interesting one. Why would an anonymous author want to pass of his letter as that of Peter? The most likely answer is that the author was a sincere Christian, who felt his letter was important, and perhaps was what Peter would have said, and gave it Peter's name to lend it authority within the church.

It is worth noting that we do have a Gospel of Peter, which, like the letters, explicitly claims to be the work of the apostle. Christianity nevertheless rejects the Gospel of Peter, so the church itself acknowledges that some texts that claim Petrine authorship were not actually written by the apostle.
https://en.wikipedia.org/wiki/Gospel_of_Peter

Similarly, the Apocalypse of Peter is no longer considered canon, although it originally was, despite claiming Petrine authorship.
https://en.wikipedia.org/wiki/Apocalypse_of_Peter

1 Peter

There are various reasons for supposing Peter was not the author of 1 Peter. To start with, the theology is Paul's not Peter's.

Secondly, there is no mention of Jesus on a personal level. It does, however, mention the "sufferings of Christ and the subsequent glories" in a general way (see also 1 Peter 2:21-24 in particular). It reads as someone who knows Jesus suffered, and is aware of the theology, but not as someone who was there at the time. Even 1 Peter 5:1 ("a witness of the sufferings of Christ, as well as a partaker in the glory that is going to be revealed"), where the author claims to be a witness, there is nothing personal; there is no sense of the author drawing on his own experience.

The style of writing and the philosophy exposed is considered by many to be too advanced for a Galilean fisherman. I will acknowledge he could have used a secretary, and his philosophy could have developed over decades in the church. More significant is that he uses the Septuagint as a source for Old Testament quotes, which certainly is bizarre for a Hebrew-speaking Jew.

In 1 Peter 5:1, the author refers to himself as an elder, a position that appeared later in the church, further indicating a later authorship.

In 1 Peter 1:1 the author mentions "the provinces of Pontus, Galatia, Cappadocia, Asia and Bithynia," Various sources on the internet (eg here indicate that this sequence of states was established by Emperor Vespasian in AD 72. I have found only limited support for that claim outside articles dating the epistle, so count it as suspect:
http://archive.org/stream/cambridgeancient015566mbp/cambridgeancient015566mbp_djvu.txt

There is no mention of Mosaic law; this was a big issue in the early church, as Acts and the Pauline letters make clear, and that suggests the letter is later, after the issue had been resolved.

The letter finishes "By Silvanus, a faithful brother as I regard him, I have written briefly to you", and some suggest this indicates Peter dictated the letter to Silvanus, which accounts for the good Greek. However, the Greek indicates that Silvanus was the courier, not the scribe.

2 Peter

While scholarship is divided for 1 Peter; for 2 Peter the situation is quite different!

Very, very few people today would argue for Petrine authorship of this book. The Word Biblical Commentary, which is a conservative commentary series, argues against Petrine authorship. Of all of the books of the Bible, this is the one that is most difficult to defend in regard to authorship.
https://www.biblicaltraining.org/library/general-epistles-2-peter-authorship/new-testament-acts-revelation/robert-stein

From an article arguing for Petrine authorship:

J. N. D. Kelly in his commentary on 2 Peter confesses that “scarcely anyone nowadays doubts that 2 Peter is pseudonymous.”1 Indeed, from the very start this epistle has had a difficult journey. It was received into the New Testament canon with hesitation, considered second-class Scripture by Luther, reluctantly accepted by Calvin, rejected by Erasmus, and now is repudiated as pseudonymous by modern scholarship.

http://www.biblicalstudies.org.uk/pdf/2peter_kruger.pdf

This is not just a modern view. Here is Origen:

Peter, on whom the Church of Christ is built, against which the gates of Hades shall not prevail, has left one acknowledged epistle, and, it may be, a second also; for it is doubted.

Why so much doubt? for one thing, 2 Peter 3:15-16 refer to Paul's letters as scripture; the early church would not have regarded them as such, indicating this was written relatively late.

There is some evidence 2 Peter is based on Jude, again giving a later date. This article makes the case that they are similar because both were authored by Jude (so in 2 Peter 3:1, this is the second letter after Jude, not 1 Peter):
http://www.desiringgod.org/articles/who-wrote-2-peter

Differences in style and vocabulary indicate 1 Peter and 2 Peter have difference authors.

While 2 Peter was eventually accepted as canon, it was considered "antilegomena" by the early church, i.e., there was much debate as to its status even then.