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2023-06-30
### About generating dhparams with 4096 bits instead of 2048 bits

When you use dhparam, OpenSSL not only generates DH parameters; it also wants to assert his social status by taking care to use for the modulus a so-called "strong prime", which is useless for security but requires an awful lot more computational effort. A "strong prime" is a prime p such that (p-1)/2 is also prime. The prime generation algorithm looks like this:StackexchangeWhen going from 2048-bit to 4096-bit, the density of strong primes is divided by 4, and the primality tests will also be about 4 times slower, so if generating a 2048-bit DH modulus takes 1 hour on average, the same machine with the same software will use an average of 16 hours for a 4096-bit DH modulus. This is only averages; each individual generation may be faster or slower, depending on your luck.

- Generate a random odd integer p.
- Test whether p is prime. If not, loop.
- Test whether (p-1)/2 is prime. If not, loop.
- Random odd 4096-bit integers are probability about 1/2000 to be prime, and since both p and (p-1)/2 must be prime, this will need on average generating and testing for primality about 4 millions of odd primes. This is bound to take some time.
The reasonable solution would be to add the -dsaparam option.

openssl dhparam -dsaparam -out /etc/ssl/private/dhparam.pem 4096

This option instructs OpenSSL to produce "DSA-like" DH parameters (p is such that p-1 is a multiple of a smaller prime q, and the generator has multiplicative order q). This is considerably faster because it does not need to nest the primality tests, and thus only thousands, not millions, of candidates will be generated and tested.

As far as academics know, DSA-like parameters for DH are equally secure; there is no actual advantage to using "strong primes" (the terminology is traditional and does not actually imply some extra strength).

Similarly, you may also use a 2048-bit modulus, which is already very far into the "cannot break it zone". The 4096-bit modulus will make DH computations slower (which is not a real problem for a VPN; these occur only at the start of the connection), but won't actually improve security.

To some extent, a 4096-bit modulus may woo auditors, but auditors are unlikely to be much impressed by a Raspberry-Pi, which is way too cheap anyway.

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