There is a Research paper on image encryption based on finite algebraic structures. They gave following method to generate lookup table ( also known as S-box).
S-box which is used in the proposed approach is based on the action of general linear group $GL(2,\mathbb F_{2^8})$ on finite field $\mathbb F_{2^8}$ of order 256.
w:$GL(2,\mathbb F_{2^8})$× ${\mathbb {F} _{2^8}} \to {\mathbb {F} _{2^8}}$
w(M,y)= $F_M(y)$
where $F_M(y)=\frac{\alpha(y)+\beta}{\gamma(y)+\delta} $ and $\alpha ,\ \beta ,\ \gamma\ and\ \delta$ are the elements of $\mathbb F_{2^8}$. $F_M$ is a bijective mapping from $\mathbb F_{2^8}$ to $\mathbb F_{2^8}$, and the resultant values of $F_M$ are then converted into a 16x16 lookup table, which is the required S-box.
I have not studied group theory, so after doing some research on internet I was able to understand some of it. But I still don't understand how will I get the final lookup table. I am having difficulty understanding what is $\alpha ,\ \beta ,\ \gamma\ and\ \delta$ here.
Any help would be appreciated, even if you know some source from where I can get started.