# How can I calculate non-linearity of an s-box element wise?

First i transform the s-box into row matrix like if i have 4×4 s-box having 16 elements into 1×16 row matix then substitute the binary values in column . multiply the matrix by mobius trasformation matrix then tell me how to trasform the s-box which is in binary form into decimal form? Also how i calculate the non linearity of certain element of that s-box.

• "Also how i calculate the non linearity of certain element of that s-box.": As far as I am aware, nonlinearity is defined for functions from $$\mathbb{F}_2^n$$ to $$\mathbb{F}_2^m$$. In the context of S-boxes, we typically consider the nonlinearity of a full S-box, and not of a single element.
Having said that, let $$S : \mathbb{F}_2^n \rightarrow \mathbb{F}_2^m$$ be our S-box. Then the nonlinearity $$\mathcal{N}$$ of $$S$$ is usually defined in terms of the linearity $$\mathcal{L}$$ of $$S$$, which is usually defined in terms of the $$bias$$ $$\mathcal{E}$$ of Boolean functions. Here is the full definition, where $$S_b$$ considers Boolean component functions of $$S$$ and $$\varphi_a$$ does $$x \mapsto ax$$:
$$\mathcal{N}(S) = 2^{n-1} - \frac{1}{2}\mathcal{L}(S) = 2^{n-1} - \frac{1}{2} \max_{a,b \in \mathbb{F}_2^{m*}} \left|\mathcal{E}(S_b + \varphi_a)\right|$$.
In this answer, I elaborated on how to compute this bias $$\mathcal{E}(S_b + \varphi_a)$$ to compute the values in the linear approximation table of $$S$$, including a fully worked out example. I'll try to avoid repeating that here.
Once all values in the linear approximation table are computed, the largest absolute number (ignoring the trivial case of $$a=b=0$$) is called the linearity. Using the definition, it is then straightforward to calculate the nonlinearity.