Re: D-H key exchange - how does it work?

[Hal <hfinney@shell.portal.com>]

The problem with "strong" primes, primes for which (p-1)/2 is prime, is
that they are hard to find.  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.  They
may need a dedicated prime-searching machine to do that.

(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.)

It's much easier to find a "strongish" prime, one for which (p-1)/k is
prime, where k is on the order of 100 or so.  Take your prime q in the above
and try kq+1 for k=2,4,6,....  This only takes a few minutes after you find
q.

The question is, how good are strongish primes?  What fraction of elements
of the group will have short periods, given that p-1 has a pretty small
number of prime factors?

Also, given a strong or strongish prime, are the chances that
g^x has a small period good enough that it makes sense to check for that
case?  Any event whose chances are smaller than your computer making a
mistake is generally not worth checking for.

Hal