Construct a random permutation from a random function? How can we construct a (pseudo) random permutation from a (pseudo) random function ?

I've seen a method from Luby-Rackoff which allows to construct a PRP of length $2N$ from a PRF of input/output length $N$. Is there a method to keep the same length $N$ ?

I'm looking for a method that has better security bounds than truncating the output of a $N$-bits output pseudorandom function to obtain an output of $N/2$-bits and using the Luby-Rackoff construction.
There are so many possible solutions here.  Without giving your requirements more carefully, it's just not possible to tell what would count as a valid solution.  Here are a bunch of schemes that offer better security than simply truncating to $N/2$ bits and applying 4 rounds of Luby-Rackoff:


For instance, one approach is to truncate the random function to $N/2$ bits, then use 6 or more rounds of the Luby-Rackoff construction.  Paterin has a paper at FSE 1998 that apparently proves that 6 rounds gives security for up to $O(2^{3N/8})$ chosen plaintext/ciphertext queries.  He also has a later paper at Crypto 2003 that apparently proves that 10 rounds is enough for security for up to $O(2^{N(1-\epsilon)/2})$ chosen plaintext/ciphertext queries.  Don't ask me to explain the proofs; I don't understand them.  I am not aware of any concrete-security versions of these results, so it's hard to know exactly how much security this gives you in practice.
Another approach is to truncate the random function to 1 bit of output (and $N-1$ input bits), then build a maximally unbalanced Feistel network that uses $4N$ rounds.  Naor and Reingold showed (Journal of Cryptology, Jan 1999) that this will give security for up to $O(2^{N/2})$ chosen plaintext/ciphertext queries.  (Technically speaking, their result assumed that you used a pairwise-independent permutation, then $2N$ rounds of unbalanced Feistel, than another pairwise-independent permutation, so if you really care about provable security, you could use that: but I suspect their results would likely apply to $4N$ rounds of unbalanced Feistel as well.)
Even better yet, if you do enough rounds of maximally unbalanced Feistel network, you can get arbitrarily close to $O(2^N)$ security.  Truncate the random function to one that takes $N-1$ input bits and has 1 output bit, and use it in an unbalanced Feistel network.  Morris, Rogaway, and Stegers proved at Crypto 2009 that, if you do $4RN$ rounds of this, you get security up to $O(2^{N(1-1/R)})$ chosen plaintext/ciphertext queries.  For instance, if you do $40N$ rounds, you get security up to something like $O(2^{0.9N})$ chosen plaintext/ciphertext queries.  That's a very impressive result.  If maximizing security is your goal, this might be your best bet.
Another approach is to truncate the random function to one on $N/2$ bits, then apply a suitable Benes network construction.  Aiello and Venkatesan proved in Eurocrypt 1996 that, with four rounds, this gives you security for up to $O(2^{N/2})$ chosen plaintext/ciphertext queries.


One keyword that may help you find still more results is "Luby-Rackoff beyond the birthday bound".