## 2 Background and Related Work  
Digital communication demands for security and efficiency combined with scalability needs drive the ongoing development of cryptographic methods. Elliptic curve cryptography (ECC) enjoys widespread acceptance because it provides robust security with smaller keys compared to traditional public-key systems like RSA and DSA, with various basic implementations [7; 8]. During the last twenty years, ECC technology has been implemented across multiple protocols, including secure sockets layer (SSL), mobile authentication systems, digital signature processes, and key exchange mechanisms, according to official studies and reports that are made [9; 10]. Since the development of Diffie-Hellman (DH) key exchange protocol, secure communications have benefited from substantial advancements in key agreement protocols. The original Diffie-Hellman protocol formed the basis for public-key cryptography but has been replaced in high-performance settings by elliptic curve cryptography implementations over multiplicative groups of integers modulo a prime. The variants use elliptic curves' mathematical properties to deliver the same security level with smaller keys and reduced computational requirements. The GN-authenticated key agreement (GN-AK) protocol uses ECC to enable entities to mutually authenticate and create a shared secret. The GN-AK protocol includes a two-step or three-step procedure where both participants create temporary scalars and compute ECC points before sharing partial data and authenticating messages through digital signature verification. The security mechanism of the protocol depends on the computational difficulty of solving the Elliptic Curve Discrete Logarithm Problem, which makes it impossible for eavesdroppers to deduce session keys. The GN-AK protocol maintains security based on standard assumptions but faces performance difficulties in certain implementation restrictions because it uses classical scalar multiplication techniques, which are not suited for limited environments like smart cards and IoT devices. The most computationally intensive operation in ECC involves performing scalar multiplication to calculate \(kP\) from a point \(P\) and scalar \(k\). The classical scalar multiplication approaches double-and-add and non-adjacent form (NAF) exhibit performance limitations in constrained environments. Mitigating the challenges led researchers to investigate various alternative representations and algorithms to minimize elliptic curve operation requirements. Recent studies on IoT security in medical settings have focused on RFID authentication techniques using elliptic curve cryptography because they provide efficient computational solutions for devices with limited resources. The referenced investigation [11] examines multiple ECC authentication systems for RFID applications and finds three solutions that balance security with performance needs suitable for medical IoT infrastructure deployment. These schemes demonstrate different efficiency levels, which are determined by the specific infrastructure conditions along with the interoperability between systems and the particular clinical environment where they operate. Comparable approaches have also been proposed in other studies in the literature [12; 13; 14; 15; 16], in research contexts targeting similar problems and functional requirements, highlighting the general tendency to respond to common challenges through conceptually similar solutions. In the field of remote user authentication, recent research has focused on the use of elliptic curve cryptography in combination with smart devices, such as smart cards, to obtain authenticated key exchange (AKE) schemes with a high level of security and efficiency. A relevant study in this direction [17] proposes an improved authentication scheme, which reinterprets the concept of \"strong forward security\" and provides a formal demonstration of its security, while also highlighting reduced time, storage, and communication costs compared to existing solutions.