## 2 Exploring Dickson Basis Multiplication in GF(2\({}^{m}\))
 \tag{4}\]  
A Dickson binomial is characterized as an irreducible polynomial \(K\) that can be expressed in the following simple two-component expression: \(K=\omega_{r}+1\). This particular representation is noteworthy due to its efficiency and straightforward nature, making it highly applicable in various cryptographic scenarios. Conversely, a polynomial \(K\) is considered a Dickson trinomial when it assumes the structure \(K=\omega_{r}+\omega_{n}+1\), with the stipulation that \(1\leq n\leq\frac{r}{2}\). This configuration facilitates more intricate interactions among terms, thereby enhancing overall security measures. Dickson polynomials, encompassing both binomials and trinomials, are integral to the field of lightweight cryptography. This domain seeks to enhance efficiency while maintaining robust security, particularly in environments characterized by limited resources. The present study aims to examine the application of Dickson binomials in cryptographic frameworks. It will underscore their significant contributions and advantages, highlighting their efficacy in achieving efficient and secure cryptographic solutions. **Multiplication Technique Using the Dickson Basis:** As previously stated, let the Dickson basis be represented as \(\Omega=\{\omega_{1},\omega_{2},\cdots,\omega_{r}\}\). Additionally, we will con sider the irreducible binomial given by \(K=\omega_{r}+1\), which facilitates the generation of various elements within the finite field. The elements \(E\), \(F\), and \(H\) in the finite field GF(2\({}^{r}\)) are represented in relation to this basis as follows: \(E=e_{1}\omega_{1}+e_{2}\omega_{2}+\cdots+e_{r}\omega_{r}\), \(F=f_{1}\omega_{1}+f_{2}\omega_{2}+\cdots+f_{r}\omega_{r}\), \(H=h_{1}\omega_{1}+h_{2}\omega_{2}+\cdots+h_{r}\omega_{r}\), where the coefficients \(e_{i}\), \(f_{i}\), and \(h_{i}\) are elements of GF(2) for each index \(i\) ranging from 1 to \(r\). Moreover, the element \(H\) is defined as the result of multiplying \(E\) and \(F\), followed by taking the computation modulo \(K\). This relationship can be succinctly expressed as \(H=E\times F\bmod K\). Such an operation plays a vital role in various applications within cryptography, enabling efficient computations while ensuring security in environments with limited resources. As highlighted in the works of [18; 20], the irreducible binomial \(K=\omega_{r}+1\) plays a pivotal role in the generation of elements within a finite field GF(2\({}^{r}\)). It has been demonstrated that for all integers \(i\geq 0\), the equation \(\omega_{r+i}=\omega_{i}+\omega_{r-i}\) is satisfied. Utilizing this equation, one can derive the product \(H\) more effectively. Specifically, when calculating \(H\), the elements can be expressed in terms of their corresponding basis components, allowing for a streamlined process of multiplication. By leveraging the established relationship, the computation of \(H\) can be simplified as follows:  
\[H=EF=\underbrace{\sum_{i,j=1}^{r}e_{i}f_{i}\omega_{i+j}}_{H1}+\underbrace{\sum _{i,j=1}^{r}e_{i}f_{j}\omega_{|i-j|}}_{H1} \tag{5}\]  
Equation (5) can be reformulated in matrix notation, a representation that is essential for the accurate synthesis of the systolic multiplier. This matrix representation facilitates a more systematic approach to performing calculations, ensuring that the operations can be efficiently executed in parallel. The product \(H\) is obtained from three separate matrix-vector products, identified as \(H_{1}\), \(H_{2}\), and \(H_{3}\)[18; 20]. These products are essential components in the overall computation and can be expressed as follows:  
\[H =\underbrace{ \begin{bmatrix}e_{r}&e_{r-1}&e_{r-2}&\cdots&e_{1}\\ e_{1}&e_{r}&e_{r-1}&\cdots&e_{2}\\ e_{2}&e_{1}&e_{r}&\cdots&e_{3}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ e_{r-1}&e_{r-2}&e_{r-3}&\cdots&e_{r}\end{bmatrix} }_{H1}\times \begin{bmatrix} f_{1}\\ f_{2}\\ f_{3}\\ \vdots\\ f_{r}\end{bmatrix} }_{H1}\] \[+\underbrace{