Unmanned aerial vehicles (UAVs) are becoming more and more popular in a wide range of industries. Traditional security mechanism does not suffice to meet necessary criteria and constraints for UAV-based applications. This is owning to the fact that UAV-based applications have restricted computational capabilities, real-time operations, and data exchange across wireless networks. This paper discusses the methodology of designing and developing security protocols for drone applications using the Elliptic Curve Cryptography (ECC). ECC is a preferred in resource-constrained application as it provides a robust security with a smaller key size as compared to traditional methods. The paper presents a design of custom key exchange protocol. The proposed protocol has been formally verified using AVISPA and Scyther. The proposed protocol has also been implemented on a testbed of UAV network comprising of a drone and Ground Control Station (GCS).