in TLS 1.3 RSA is no longer supported as the a method for key exchange Diffie-Hellman is a more common way nowadays for exchanging session keys Diffie-Hellman is complicated but in a nutshell it uses some advanced math involving large prime numbers to derive a shared session key without ever transmitting a public key over the network
My two questions are:
Does TLS 1.3 longer support RSA? Why?
DH uses math involving large prime numbers to derive a share session key without ever transmitting a public key over the network.
RSA is no longer supported for key establishment, but is still supported for digital signatures. The reason that it is no longer supported for key establishment is a lack of forward secrecy. RSA keys are usually generated and used for a relatively long time involving multiple sessions, if at some point in the future the private key of an RSA modulus is compromised and in the possession of an adversary, then previous messages can be read.
For reasons such as this, the only key establishment methods supported in TLS 1.3 are:
ephemeral elliptic curve Diffie-Hellman (a particular form of the Diffie-Hellman key exchange with good bandwidth and computational efficiency) or ephemeral finite field Diffie-Hellman modulo five specific primes of 2048+ bits
pre-shared keys, or
a hybrid of both pre-shared keys and ephemeral (elliptic curve) Diffie-Hellman.
The ephemeral adjective means that the private keys associated with the exchange are only used once and generated afresh with every key establishment which provides forward secrecy. Pre-shared keys can be refreshed/ratcheted for forward secrecy.
This is a poor choice of words. As you correctly note, Diffie-Hellman does require the generation of a public value. The distinction between the RSA and Diffie-Hellman public values is that an RSA key is typically static (i.e. unchanging between key establishments) and the elliptic curve public value is ephemeral (i.e. generated afresh with each connection). As Diffie-Hellman can use system parameters, generation of ephemeral keys is much simpler than it would be for RSA. Conversely, it isn't really possible to use certificates without long-lived keys and so RSA is still permitted for signing. There is also forward secrecy as future compromise of a signing key does not compromise a past key establishment.