According to the lecture slide by Shafi Goldwasser a prime is a strong prime if:
$$p = 2q + 1$$ for some prime q.
For me it, seems a bit arbitrary that is the definition of a strong prime in cryptography. Is there a computational complexity reason of why this is the definition of a strong prime in cryptography? What are the advantages of this type of prime over other different types of prime numbers? Does anyone has a intuition or a rigorous mathematical argument to defend why this is a good definition of a "hard prime" in the context of cryptography? In terms of security, how is this a good definition? I am not even sure what the word "hard" is adding to the definition.
I guess I am trying to look for a more theoretical argument to support whatever this definition is trying to achieve. Arguments on the lines on "it does well in practice" will not suffice to answer my question. I am looking for a more mathematical reason/intuition for why this is a good definition.
This is not the only definition of "strong prime", according to Wikipedia. Anyway, the motivation seems to be that Pollard's p-1 algorithm can find a number's prime factors $p$ such that $p-1$ has only small prime factors. If $p-1 = 2q$ where $q$ is prime, this won't happen.