Growth of civil air traffic worldwide poses a great challenge for the supporting Communication, Navigation and Surveillance (CNS) infrastructure. Analogue systems have to be replaced by digital means to optimize spectrum efficiency and automation is becoming much more important to be able to handle the amount of participants in the air traffic system. As safety and security are strongly intertwined in aviation, cybersecurity is one key enabler for digitalization in civil aviation. As such we investigate mutual authentication and key agreement methods for the digital aeronautical ground-based communications system LDACS. Thereby, we compare the suitability of three different Diffie-Hellman key exchange flavors - Ephemeral Diffie-Hellman (DHKE), Elliptic Curve Diffie-Hellman (ECDH), Supersingular Isogeny Diffie–Hellman (SIDH) -, used in the STS protocol, for digital aeronautical communication in terms of latency and security data overhead. We conclude, the STS protocol based on a central public-key infrastructure (PKI) based trust solution with SIDH for post-quantum security to be best suited for long term security. However, due to the smaller key sizes, ECDH is the more resource efficient candidate and may play a role in low resource authentication scenarios for LDACS.