Are computers deployed with a list of precomputed prime numbers

Question

With my understanding of public key encryption, a server generates a public key from the product of two large prime numbers, p and q, to produce n.

n is then sent to the client that wishes to securely connect to the server.

The client picks a random number x and computes x^3 modulo n and passes this back to the server.

The server then calculates x from using p and q and then x is used as the private key to encrypt all traffic and both client and server know this without it ever being exposed.

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Question

For each client that wants to connect to the server, the server needs to create a public key for each connection. On a busy server this would require the generation of many p's and q's.

Whilst computers are fast, if both p and q are large numbers within the range of 2^1024 determining whether these numbers are prime would take a lot of effort, and repeated effort if lots of connections are made to the server (thinking Google, Facebook etc.).

Also if the the random picking of p or q turn out to be non-prime, then the server will need to pick another random number and determine whether that number is prime. This looks like there might be quite a lot of wasted computational time.

So rather than the server having to compute two random primes, for each connection, can/does the server pick from a list of pre-computed primes that are installed on the server ? If so, can we trust these prime numbers are actually prime, and, picked randomly ?

Answer 1

You misunderstand PKI, the server does not create a new public key for each client and connection.

Furthermore WRT creating RSA primes see the comments to the question by Rob Napier:

"generating large primes is not nearly as expensive as you likely think."
and his link generating large prime numbers for RSA.