The concept of an Internet of Drones (IoD) is becoming increasingly important in various domains, including surveillance and logistics. Effective communication between the interconnected systems is the essence of the Internet of Drones, however, due to the resource constraints of drones and the dynamic nature of the operating environment, security of communication within IoD networks is indeed the top priority. Considering these challenges on the part of IoD communication, a novel Hyperelliptic Curve Cryptography (HECC)-based authentication protocol is proposed in this paper to secure the data exchange between two drones and to ensure efficient communication. The proposed HECC protocol is compared with three existing protocols, Elliptic curve cryptography-based protocol with public secure authentication scheme (ECCCPSAS), Non-linear wireless unmanned aerial systems authentication scheme (NLWUAS), and the Multi-Agent System (MAS). The comparison is made among average performance metrics such as communication overhead, packet delivery ratio, throughput, and end-to-end delay. The experimental outcome assures that the HECC protocol outperforms the other protocols in terms of all metrics. HECC provides Minimum communication overhead, Maximum packet delivery ratio, Maximum throughput, and Minimum end-to-end delay. The HECC-based authentication protocol enhances communication security in IoD networks by reducing computational overload as well as improving packet delivery, and is beneficial for resource-constrained environments like drones. This efficiency is particularly valuable in practical scenarios like autonomous fleets of drones for observation or delivery, as it reduces delays and energy use. It is suggested by the results of our study that, for drone communication in IoD networks, HECC is a much more secure and scalable solution as compared to all other alternatives. Ongoing research will focus on the practical deployment of the protocol and support for future quantum-resistant algorithms.