A Flying Ad-hoc Network (FANET) consists of Unmanned Aerial Vehicles (UAVs) tasked to handle the communication jobs in a multi-hop ad-hoc fashion. Unlike its predecessors, i.e. Mobile Ad-hoc Networks (MANETs) and Vehicular Ad-hoc Networks (VANETs), a FANET promises uninterrupted connectivity, especially during events that are temporary and stipulate a massive audience reach. However, usually, the participating UAVs in a FANET environment are resource-constrained and are, therefore, prone to cyber-attacks. In order to resolve the issue and to enable a secure communication between the UAVs and the Base Station (BS), we propose a Certificateless Key-Encapsulated Signcryption (CL-KESC) scheme. The scheme is based on the concept of Certificateless Public Key Cryptography (CL-PKC). Since CL-PKC is immune to key escrow problems and thus one of the major drawbacks of the Identity-based Public Key Cryptography (ID-PKC) is addressed. Unfortunately, the existing construction models of CL-KESC rely on elliptic curve-based operations, which are computationally expensive for small UAVs. To counter the issue, in this paper, we present a new construction model of CL-KESC based on Hyperelliptic Curve Cryptography (HECC). HECC is an advanced version of the elliptic curve and is characterized by smaller parameter and key size. The key size stretches to a maximum of 80-bits, as opposed to the elliptic curve that demands a 160-bits key size. The proposed scheme proved to be superior, chiefly in terms of security and performance, as demonstrated by the results obtained from the security verification and by carrying out comparative analysis with the existing counterparts.