Distribution of safe primes generated using different techniques Is there any difference in the distribution of safe primes generated by creating prime $q$ and testing $2q+1$ for primality, compared to generating a larger prime $p$ and testing $(p-1)/2$ instead? The former is what is used in practice for efficiency. For the purposes of this question, I am assuming primality is determined by creating an odd integer, subjecting it to a sufficient number of Miller-Rabin tests, and incrementing it by two if it is composite before testing it again.
$\{p \in \mathbb Z \mid \text{$p$ is prime and $(p - 1)/2$ is prime}\} = \{2q + 1 \in \mathbb Z \mid \text{$q$ is prime and $2q + 1$ is prime}\}$

With your intervals suitably adjusted, the algorithm considers the same candidates with the same probability whether you check $p$ or $q$ for primality first.