In Pedersen Key Distribution, can the public key be persistent? I am implementing a key distribution protocol described by Torben Pedersen in A Threshold Cryptosystem without a Trusted Party (EUROCRYPT'91). In the protocol, the $n$ parties distribute a public key for the ElGamal cryptosystem, where each party ends up with "part of" the private key and can work together to decrypt a message.

In the paper, Pedersen states that the parties all agree on the two primes $p$ and $q$, and the generator $g$ of $\mathbb{G}_{q}$. $h$ is then computed based on the individiual's choice of the private key $x_{i}$ (so that $h_{i} = g^{x_{i}}$ and $h = \prod_{i}^{n} h_{i}$).

If $p$ is sufficiently large, can the public parameters $p$, $q$, and $g$ be reused? It seems to me that if in each round of encryption, each party selects a new $x_{i}$, thus resulting in a new $h$, that the values of $p$, $q$, and $g$ can be persistent.

I ask this because finding a random safe prime of 1024 bits can be slow, and finding a generator of $\mathbb{G}_{q}$ can take a very long time. If these could be agreed upon well in advance and then reused, it makes the protocol more practical.

If $(p, q, g)$ can be reused, is this true for ElGamal in general?
This question probably goes much deeper than what you actually aimed for. Simply speaking: If you use $(p,g,h)$ just a couple of times, where $x$ is not revealed in any way, then yes you can reuse it. If you reuse it very often you might run into trouble. However in a general ElGamal-like encryption system, you can use a single public key as often as you want.

The tricky part in the security proof is, when you have to deal with security under "selective openings", where you consider a larger number of private/public keys, and grant the attacker the ability to open some of them and want to know if the remaining ones are still secure. A paper about this is e.g. "Encryption Schemes Secure under Selective Opening Attack" (Bellare, Yilek, 2009) or "Possibility and impossibility results for encryption and commitment secure under selective opening" (Bellare, Hofheinz, Yilek, 2009)