## 1 Introduction and Related Work  
The advent of the Internet of Things (IoT) has sparked a paradigm shift in sectors such as healthcare, entertainment, commercial appliances, transportation, agriculture, and housing. This revolutionary technology facilitates the effortless aggregation of data from interconnected devices, paving the way for cloud-based analysis and informed decision-making. Industries have been profoundly transformed as a result, with the IoT acting as a catalyst for innovation and connectivity in the digital era. However, the sensitive nature of IoT data necessitates robust security measures across all tiers of the network. This poses a challenge due to the resource limitations of IoT edge nodes. To address this, extensive research efforts have focused on developing tailored security methodologies for these resource-constrained nodes. Among the diverse cryptographic techniques available, Elliptic Curve Cryptography (ECC) has emerged as a highly optimized solution for IoT edge nodes. Finite-field multiplication is a key building block in the realm of finite-field arithmetic operations, which play a crucial role within ECC algorithms. Beyond its primary function of multiplication, it enables various other fundamental field operations, such as field inversion, field exponentiation, and field division. These operations are essential for the successful implementation of ECC algorithms, enabling efficient and secure cryptographic computations. By harnessing the power of finite-field multiplication, ECC algorithms leverage a comprehensive suite of field operations to deliver resilient and dependable cryptographic solutions. Consequently, there is a growing interest in adopting compact yet highly efficient cryptographic algorithms that can seamlessly operate within the resource limitations of IoT edge nodes. By embracing such algorithms, IoT networks can ensure the integrity and security of data in diverse applications across industries. The binary extension field, GF(2\({}^{k}\)), plays a pivotal role in numerous crucial applications of computer algebra, including public-key cryptography and error-correcting codes [1; 2; 3; 4; 5]. These applications heavily rely on efficient multiplication operations within GF(2\({}^{k}\)) [6; 7; 8]. Consequently, researchers have extensively explored strategies to minimize the space and delay overhead associated with this computationally expensive operation, which poses challenges in terms of area and delay complexity [9; 10]. The literature abounds with discussions and studies dedicated to devising techniques and algorithms that optimize the performance of GF(2\({}^{k}\)) multiplication [11; 12; 13]. These efforts aim to reduce resource requirements while maintaining high computational efficiency, enabling the seamless integration of Galois field computations into various applications [9; 10; 11]. By addressing the space and delay complexities associated with multiplication in GF(2\({}^{k}\)), researchers strive to unlock the full potential of public-key cryptography and other computational tasks relying on the Galois field. The binary extension field, GF(2\({}^{k}\)), offers multiple base representations, including the polynomial basis (PB), normal basis (NB), and dual basis (DB), each with its own distinct characteristics and considerations. Among these, the polynomial basis (PB) emerges as a favourable choice due to its simplicity and compatibility with typical hardware constructions, eliminating the need for base conversion. By adopting the polynomial basis, hardware designers can streamline the implementation of GF(2\({}^{k}\)) components without compromising efficiency [14]. Multiplication within the finite field GF(2\({}^{k}\)) presents inherent complexities that involve two distinct stages: polynomial multiplication and modulo reduction utilizing an irreducible polynomial.