I'm looking for weak groups in discrete logarithm, that $x$ can be extracted from $Y$ in polynomial time where $Y \equiv g^x \pmod{p}$ .
I thought one way is to produce a prime $p$ that $p-1$ is an smooth integer which then makes discrete logarithm problem easy using Pohlig-Hellman algorithm but i couldn't find any algorithm for generating such primes. Trivially, we can generate random numbers and then check if it's prime and if it's prime, check if $p-1$ can be factorized to a set of small prime numbers. It can be done in another way, considering a set of small prime numbers and generating random exponents for every prime in the set, multiplying all primes powered to their exponent. By doing so, we have a number $r$ that can be factorized to small prime numbers and then we can check if $r+1$ is prime.
Clearly these methods are just trial and error and may take a long time to find such prime, specially when it comes to generating 256 bit prime numbers. Is there any better algorithm ?
What are other ways to generate a 256 bit long weak modulus?