Generating Diffie-Hellman parameters

Starting with gnutls 3.0 the Diffie-Hellman parameter generation has been changed. That was mandated by the move from libgcrypt to nettle. Nettle didn't support Diffie-Hellman parameter generation, so I had to implement it within gnutls. For that I generate a prime of the form p=2wq+1, where w and q are also primes and q has the size in bits of the security parameter (could be 160,256 etc. bits, based on the size of p). Then I search for a generator of the q subgroup using an algorithm which typically gives a large generator --few bits less than the size of p.

This method has the advantage that a server when selecting a private key value x, instead of selecting 0 < x < p-1, it can select a smaller x within the multiplicative subgroup of order q, i.e., 0 < x < q-1. The security level of x is that of the security parameter, which in gnutls is calculated as in ECRYPT recommendations. However until now we never wrote the size of the security parameter in the Diffie-Hellman parameters file, so it was impossible for the server to guess the order of q, since the PKCS #3 file only contained the generator and the prime. However PKCS #3 has a  privateValueLength field exactly for this purpose, but it is not used by gnutls or any other implementation I'm aware of.

DHParameter ::= SEQUENCE {
  prime INTEGER, -- p
  base INTEGER, -- g
  privateValueLength INTEGER OPTIONAL 
}

By populating and using it, the performance improvement was quite impressive. The following table demonstrates that.

Prime length	Private key length	Transactions/sec with DHE-RSA
1248	1248	122.75
1248	160	189.91

So starting from 3.0.9 gnutls' generated parameters for Diffie-Hellman should perform better. However one question that I had to answer, is what is more important, keeping x small as we do, or having a small generator? Libgcrypt and openssl generate parameters in a way that the generator is kept small, e.g. g=5 or g=2. For that I generated 1248 bit parameters using openssl which happened to have g=2.

Prime length	Generator length	Private key length	Transactions/sec with DHE-RSA
1248	1246 bits	1248	122.75
1248	1246 bits	160	189.91
1248	2 bits	1248	125.94

So it seems keeping the generator small doesn't really have an impact to performance comparing to using a smaller but still secure subgroup.