## 1 Introduction
### Literature Review  
The selection of base representation for components in GF(2\({}^{m}\)) plays a crucial role in determining the multiplication efficiency within finite fields. Various representations, such \begin{table}
\begin{tabular}{c c c} \hline \hline
**Aspect** & **Transitional Designs** & **Our Proposal** \\ \hline Resource Efficiency & High computational demand, unsuitable for IoT devices & Optimized hybrid systolic array with reduced power and space requirements \\ \hline Complexity & Quadratic complexity leading to integration challenges & Linear complexity for easier implementation in compact devices \\ \hline Miniaturization & Limited capability for miniaturization in existing designs & Compact design tailored for resource-constrained environments \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of current traditional designs to the proposed solution. as polynomial basis (PB), normal basis (NB), dual basis (DB), and redundant basis (RB), each offer unique advantages tailored to different applications [18; 19; 20]. Notably, the normal basis (NB) stands out as particularly advantageous for cryptographic applications. One of the key benefits of normal bases is their proficiency in executing squaring operations efficiently. This efficiency is achieved by utilizing simple cyclic shifts for squaring, which require fewer computational demands and processing time. This characteristic is especially important in security protocols, where squaring occurs frequently, underscoring the normal basis's suitability for rapid processing. In addition, normal bases ensure a uniform representation of components. This uniformity results in predictable patterns during arithmetic operations, facilitating easier hardware optimization. Such predictability is essential for crafting high-performance digital circuits, as it helps maintain stable operation and resource management. Conversely, other representations, like polynomial basis (PB), can lead to inconsistencies in task complexity, making the design and refinement of hardware solutions more challenging [21; 22; 23; 24; 25]. Although normal bases present numerous benefits, they also come with certain limitations, particularly in the multiplication of components. The multiplication procedure using a normal basis tends to be more complex than with other representations, which can hinder performance, especially in scenarios that require frequent multiplication operations. To address these challenges, advanced versions of normal bases, including optimal normal basis (ONB) and Gaussian normal basis (GNB), have been introduced. These modifications aim to improve multiplication efficiency while preserving the beneficial characteristics that make normal bases advantageous. Considering the drawbacks of normal bases, the Dickson basis emerges as a promising substitute for developing effective finite field multipliers in various applications [18]. The space complexity related to Dickson-based multipliers typically aligns well with that of optimal normal basis (ONB) multipliers, rendering this option especially appealing for resource-limited settings, such as embedded systems and IoT devices. Scholars, including [PERSON] and [PERSON] [18], have explored lightweight Dickson polynomials, including binomials and trinomials, to enhance the performance of these multipliers further. Additionally, [PERSON] et al. [19] have made significant strides by creating high-throughput bidirectional systolic array multipliers that leverage the Dickson basis, showcasing its relevance and effectiveness in cryptographic applications. However, the bidirectional architecture frequently observed in existing designs, such as those by [PERSON] [19], poses notable challenges when it comes to incorporating effective error-detection strategies. These strategies are essential for safeguarding systems against side-channel attacks that may exploit weaknesses in elliptic curve cryptography.