How does an AMRAAM communicate securely with the aircraft that launched it?
Do you have a need to know?

Fortunately, there is a simpler answer to your question: the original AMRAAM does not communicate with the host aircraft. Only the host aircraft communicates data to the missile. This data is transmitted in the form of fixed-format messages, repeated multiple times with redundancy. 

Security would fall outside the scope here, except in most general terms. In general terms, when secured, unidirectional datalinks use fixed key symmetric encryption. A missile's downlink isn't necessarily secured, as it's only transmitting the enemy's position for guidance correction.

Newer and more complex missiles that do communicate bidirectionally typically use Link 16. That is a secure encrypted protocol. It's well-documented and you can find a fairly good overview of the protocol on the net.