Generating and distributing a secret without trusting the source
You'd use a KBKDF on the clients, using the earlier secret as Input Keying Material (IKM) using the random as salt or Info input. KDF's generally have an output size parameter as well.
Currently HKDF is considered a good KDF, and if you use the full function it supports a salt.
Basically you would be implementing a so-called ratchet, where each key depends on the previous one, but doesn't provide any information about the previous keying material.