## 1 Introduction  
Elliptic curve cryptography stands out as a leading public-key cryptographic method in today's era. ECC stands out from other cryptographic systems because it achieves similar security levels through much smaller key sizes. The system achieves faster computation speeds while reducing memory needs and bandwidth consumption. Many applications, such as mobile devices and smart cards, as well as Internet of Things (IoT) environments, widely use ECC because performance and power efficiency are crucial there.  
The Elliptic Curve Discrete Logarithm Problem (ECDLP) serves as the fundamental basis for ECC security. The problem of determining the scalar \(k\) becomes computationally impossible when only provided with a point \(P\) and its multiple \(kP\) on an elliptic curve. Numerous ECC-based cryptographic protocols for digital signatures, encryption schemes, and key exchange mechanisms rely on this hardness assumption.  
Key agreement protocols serve as essential components for creating secure communication channels in these applications. The GN-authenticated key agreement (GN-AK) [1] protocol serves as a prime example of this scheme because it allows two parties to mutually authenticate and establish a shared session key. This has had applications in several functional systems, and its principles are indirectly relevant in auxiliary implementation models--for instance, secure communication or authentication layers embedded within broader infrastructures such as those used in healthcare or medical monitoring environments [2, 3, 4, 5]. The GN-AK protocol depends on regular scalar multiplication methods that provide security but require substantial computational resources. The computational demands of standard scalar multiplication create a bottleneck in environments with limited resources.  
Various enhancements to the scalar multiplication process have been introduced by researchers to solve this problem. The use of double-base chains (DBCs) as scalar representations through sums of powers of two and three enables researchers to decrease the necessary elliptic curve operations. In [6] was proposed a new method that uses powers of \(\frac{1}{2}\), in place of powers of 2 which allows for point halving to be utilized as a quicker substitution for point doubling. The experimental data confirmed that scalar multiplication computational costs were substantially lowered when using the new double-base chain with \(\frac{1}{2}\) and 3 as bases.  
In this work is proposed an improved GN-AK protocol that combines the new double-base scalar multiplication method alongside point halving. Our analysis covers the mathematical basis of the method in detail while we modify the original GN-AK protocol for the new scalar representation and present an extensive evaluation of both security and performance. The new protocol maintains the cryptographic security features of the original GN-AK protocol but delivers significantly better computational performance.  
The remainder of this paper is organized as follows: Section 2 discusses related work and existing improvements in scalar multiplication and key agreement. Section 3 reviews the mathematical foundations of elliptic curve scalar multiplication. Section 4 details the double-base chain with point halving. Section 5 presents the original GN-AK protocol. Section 6 introduces our modified protocol. Sections 7 and 8 present security and performance evaluations, respectively. Section 9 contains Discussion and Future Work, and Section 10 contains Conclusions.