Hiding the identity of a party within the Kerberos authentication scheme

In the Kerberos authentication protocol, as described here:

would it be safer to replace step (1) with:

$$A \rightarrow T : A, E_{K_A{_{T}}}(B, N_A)$$

so that a passive adversary does not know the identity of party B with which A wishes to communicate?

Or even add an extra layer of security by encrypting the whole message with a globally distributed symmetric key:

$$K_G$$ thus message (1) becoming: $$A \rightarrow T : E_{K_G}(A, B, N_A)$$

where KG is a global key shared with all potential clients, and the trusted third party T decrypts any incoming request with this key; it assumes that all inbound ciphertexts are initially encrypted once with the respective key.

• Good question. I have no answer, but improving privacy is always a good thing. Therefore, thumbs up!
– user27950
Oct 17 '15 at 22:57