## 2 Exploring Dickson Basis Multiplication in GF(2\({}^{m}\))  
Consider \(\Re\) as a ring, with \(\hbar\) being a specific element contained in \(\Re\) (i.e., \(\hbar\in\Re\)). The \(k^{th}\) Dickson polynomial of the \((k+1)^{th}\) type, represented as \(\mathbb{D}_{\ell,k}(\rho,\hbar)\), serves as a significant entity in algebra and number theory. These polynomials are recognized for their fascinating characteristics and numerous applications. Their formulation enables various computations and theoretical investigations, making them essential tools in both theoretical and applied mathematics. The polynomial can be expressed using the following formula [18; 19; 20]:  
\[\mathbb{D}_{\ell,k}(\rho,\hbar)=\sum_{i=1}^{\lfloor\frac{\ell}{2}\rfloor}\frac {\ell-ki}{\ell-i}\binom{\ell-i}{i}(-\hbar)^{i}\rho^{\ell-2i} \tag{1}\]  
In this context, \(\lfloor\frac{\ell}{2}\rfloor\) refers to the floor function applied to the value of \(\frac{\ell}{2}\), effectively rounding down to the nearest integer. Meanwhile, the notation \(\binom{\ell-i}{i}\) calculates the count of possible selections of \(i\) elements from a total of \(\ell-i\) distinct items, highlighting its combinatorial significance. To simplify our analysis, we will concentrate on the \(\ell^{th}\) Dickson polynomial of the first kind, which corresponds to the case where \(k=0\). This particular polynomial plays a foundational role in the study of algebraic structures and can be expressed through a specific formula that captures its essential characteristics and relationships within the broader framework of Dickson polynomials. The polynomial can be expressed as follows:  
\[\mathbb{D}_{\ell}(\rho,\hbar)=\sum_{i=1}^{\lfloor\frac{\ell}{2}\rfloor}\frac{ \ell}{\ell-i}\binom{\ell-i}{i}(-\hbar)^{i}\rho^{\ell-2i} \tag{2}\]  
In this context, \(\mathbb{D}_{0}(\rho,\hbar)=2\) and \(\mathbb{D}_{1}(\rho,\hbar)=\rho\) serve as the initial conditions for the cases when \(\ell=0\) and \(\ell=1\), respectively. Expanding on these foundational equations, the Dickson polynomials are defined recursively by by the subsequent expressions:  
\[\mathbb{D}_{i}(\rho,\hbar)= \begin{cases}2&\text{if }i=0,\\ \rho&\text{if }i=1,\\ \rho\mathbb{D}_{i-1}(\rho,\hbar)-\hbar\mathbb{D}_{i-2}(\rho,\hbar)&\text{if }i \geq 2\end{cases}  \tag{3}\]  
In the context of the finite field GF(2), we are particularly interested in the \(\ell^{th}\) Dickson polynomial where \(\hbar=1\), denoted as \(\mathbb{D}_{\ell}(\rho,1)\). For each integer \(i\), we can represent this as \(\omega_{i}=\mathbb{D}_{i}(\rho,1)\). The Dickson basis, which is derived from an irreducible polynomial \(K\) of degree \(r\) within GF(2), can be characterized as follows:  
In the context of the finite field \(GF(2)\), we delve into the study of Dickson polynomials, specifically focusing on the polynomial of order \(\ell\) when \(\hbar=1\). This is denoted as \(\mathbb{D}_{\ell}(\rho,1)\). Considering any integer \(i\), we designate \(\omega_{i}\) as \(\mathbb{D}_{i}(\rho,1)\). When analyzing an irreducible polynomial \(K\) with degree \(r\) in \(GF(2)\), we can establish a Dickson basis, represented by the collection: \(\Omega=\{\omega_{1},\omega_{2},\ldots,\omega_{r}\}\). To illustrate the concept, we can consider \(\omega_{1}=\rho\), \(\omega_{2}=\rho^{2}\), and \(\omega_{3}=\rho^{3}+\rho\) demonstrate the construction of the basis elements. As a result, any element \(X\) in the field GF(2\({}^{r}\)) can be represented in the form: \(E=e_{1}\omega_{1}+e_{2}\omega_{2}+\cdots+e_{r}\omega_{r}\), where \(e_{i}\in GF(2)\) for \(i\) index ranging from \(1\) to \(r\). By leveraging the fundamental characteristics of GF(2\({}^{r}\)), it is possible to reformulate Equation (3) to underscore these features more effectively. This reformulation can be expressed as:  
\[\omega_{i}= \begin{cases}0&\text{if }i=0,\\ \rho&\text{if }i=1,\\ \rho\omega_{i-1}+\omega_{i-2}&\text{if }i\geq 2\end{cases}