Lower Bound for Primes during GnuPG key generation

[Daniel Kahn Gillmor]

On Fri 2015-05-22 11:38:36 -0400, vedaal at nym.hush.com wrote:
> https://primes.utm.edu/howmany.html   (The Prime Number Theorem, Consequence Two: The nth prime is about n log n )
>
> So, to give a trivial example,  If the interval of primes chosen is from  2^2047 to 2^2049, then this interval is only
>
> log(2) [ 2049^2 - 2047^2]  = 5678      which is a fairly small number of primes to check, for this type of attack to find the GnuPG keypair.

I think you're calculating the wrong thing.  That same link points out
that the number of primes less than x can be approximated as
pi(x) = x/(log(x)-1).

Very rough approximation below, dealing with this stuff in integer so i
don't have to worry about floating point precision:

-------------
#!/usr/bin/python
import math
def pi(x):
    return x//(int(math.log(x) - 1))

print(pi(2**2049) - pi(2**2047))
-------------

Produces:

34145667701866559944044383798802377522892758536014431538437128764517106455003913618433496010529759521130797881149503110281852350331307674834631513015472234360367041589931067679100152094894630389610217047672380307383983307748628563937362347485005455333604234204637401603112241209544524188755360669738591593193745235562705749858506233297205248008712262199741471705643342281979549220061203824401583102466100146307704833584671889641794368007460424297084011860069297821103169614694882157095281778056383498229906388753003349920901696154376284354875775139586287926960791086951258972553145862357082919346528294049800053111

That's a lot of primes to choose from! :)

> does GnuPG automatically reject twin primes ( p, p+2) ,  and Sophie-Germain primes (p, 2p+1) ?

Why should GnuPG reject these primes?  Surely, it wouldn't want to both
elements of a pair like that (i.e. for RSA you don't want q = p+2
because it's a trivial test to factor that composite), but is there a
reason to reject using a p that meets these categories with some other,
unrelated q?

          --dkg