BOOLEAN FUNCTIONS: generate a function $f: \{0, 1\}^n \rightarrow \{0, 1\}$ from a $n \times n$ S-Box

Question

I want to calculate the hamming weight of a S-Box using this formula: $\text{hw}(f) = \sum_{x=0}^{2^n-1} f(x)$. Where $f: \{0, 1\}^n \rightarrow \{0, 1\}$

My problem is that I don't know how to get the $f$-function.

I found that helpful paper: THE DESIGN OF S-BOXES. At Page 9 (18) boolean functions are described. But she only prints the truth table (Table 2.1) without describing her $f$-function. She only says $f$ is a linear function, therefore I think $f$ depends on my S-Box?

For example, if I have this $2 \times 2$ S-Box:

0    1    2    3 
1    3    0    2 

$00 \rightarrow 01\\ 01 \rightarrow 11\\ 10 \rightarrow 00\\ 11 \rightarrow 10$

What are the $f$-functions and what is the hamming weight?

Answer 1

Your n x n S-Box can be seen as a set of $ n $ boolean vectors of size $ n $. Actually you have got $ n $ functions, each one is the truth table of one particular vector (or column). In your example there are 2 columns thus 2 functions. For each column the number of 1's is 2.

Answer 2

To know how many boolean functions you got you have to look at the output of the S-Box...

A $N \times M$ S-Box gives you $M$ functions. In the example you passed, you have a function $f_1$ given by

$00 \rightarrow 0\\ 01 \rightarrow 1\\ 10 \rightarrow 0\\ 11 \rightarrow 1$

and a function $f_2$ given by

$00 \rightarrow 1\\ 01 \rightarrow 1\\ 10 \rightarrow 0\\ 11 \rightarrow 0$

So, the concatenation of $f_1(x)$ with $f_2(x)$ gives you the output of the S-box.

The Hamming Weight of $f_1$, which is simply the number of ones in the output of that function, is $hw(f_1) = 2$. And the same is valid for $f_2$, which means $hw(f_2) = 2$.