A small problem in security protocol

In many protocols under academic discussion (like NSPK protocol,
Big-mouth-frog protocol, etc) there is no an apparent field in some
messages that shows which step in which protocol this message is and
who is the sender of this message, is it a problem?

I got this problem for if there is no such a tag, it will bring a
problem that when one agent receive a message in the protocol, how does
he know use which key in his own private key and other agents' public
(maybe decades or even hundreds) to decrypt the message like message 1
in NSPK protocol in an environment where many different protocols and
multi instances of same protocol are executed at the same time. It is
impossible to try each key for most asymmetric key cryptograph
algorithms are highly resource-cost.

Considering the case in Woo Lam mutual authentication protocol of
missing the sender's identification:
1. P -> Q : P, N1
2. Q -> P : Q, N2
3. P -> Q : {P, Q, N1, N2}Kps
4. Q -> S : {P, Q, N1, N2}Kps, {P, Q, N1, N2}Kqs

When Q get message 3 being encrypted by Kps which is a shared key
between P and S and Q didn't know it, consider multi instances of this
protocol are carry out at the same time, how does Q know this message 3
is from P and compose the second half part of message 4 using P's ID in
{P, Q, N1, N2}Kqs? And further considering in a multi protocol
environment, Q may even doesn't know message 3 is a message in Woo Lam
mutual authentication protocol.

Maybe this problem of showing the sender of message can be resolved
easily by adding the sender ID in the message, but will it bring new
security problem to the protocol? If the protocol analysis won't
consider the situation of multi protocol environment, is it right and
won't there be an attack scheme of cross-protocol attacking?

.

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