Unmanned aerial vehicles increasingly exchange commands, telemetry, and mission data through interconnected ground, edge, and aerial networks. Their communication security depends on authentication, key management, implementation assurance, and constrained resources rather than on a cryptographic primitive alone. This narrative review examines lightweight authentication for UAV networks, with emphasis on the role of elliptic curve cryptography (ECC) and the trade-offs among software, FPGA, and hybrid implementations. Representative protocol studies are organized by communicating parties, trust assumptions, supporting mechanisms, and reported evidence. Standards for authenticated key establishment, authenticated encryption, and post-quantum cryptography provide a separate reference layer. The synthesis distinguishes mathematical security claims, symbolic protocol analysis, software timing, hardware realization, and operational deployment. ECC can support compact classical public-key mechanisms, but it does not independently provide message confidentiality, replay protection, capture resistance, or quantum resistance. PUF-assisted and infrastructure-assisted designs redistribute trust and implementation costs rather than eliminating them. A reporting framework identifies the information needed for meaningful comparisons, including credential provisioning, state recovery, communication overhead, and end-to-end energy. The review presents no new authentication protocol or experimental benchmark; its contribution is a critical, implementation-aware organization of the evidence and research gaps.