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D-H key exchange - how does it work?

   It takes hours and hours of searching to find
   a 1024 bit strong prime on a workstation.  Granted, you don't need to change
   very often perhaps, but some people would like to change every day.  

If they really want to change that often, they can buy a dedicated
machine.  There's no good cryptographic reason to change that often,
if the modulus is large enough.  In addition, changing the modulus can
have unpleasant effects on traffic analysis, if not done properly.

   (The best way I know to find strong primes is to find a prime q and then
   check 2q+1 for primality.  Finding 1024 bit primes takes a long time, and
   the chances that 2q+1 is prime is very low.)

Well, there are faster ways.  One can combine the sieve for q with a
sieve for p.  The biggest problem is that there are just a lot fewer
primes with the above property.

   The question is, how good are strongish primes?  

Just fine.  The complexity of taking discrete logs is dependent on the
largest prime factor of the modulus.

   What fraction of elements
   of the group will have short periods, given that p-1 has a pretty small
   number of prime factors?

If q is the largest prime factor, then about p/q will have short
periods, namely, those divisible by q.  When p=2q+1, there is one
element of order 1 (namely 1), one element of order 2 (namely -1, aka
2q), and every other element has order 2q or q.  For primes of the
form p=kq+1, there are about k with short periods.