# How to choose the linear approximations to build a path through a SPN in Linear cryptanalysis?

so I am reading up on Linear Cryptanalysis using Heys' Tutorial. I understood how to get "good" linear approximations of the S-boxes though I cannot seem to find a reasonable approach to combining them together. In the paper on page 12 the approximations used seem to be specifically hand-picked to work for the particular P-boxes. Anyway they pretty much appear out of nowhere and I am not sure how one would approach finding them in the first place.

This seems to be a optimization problem but I am having trouble formulating it. Please enlighten me, maybe I am on the wrong track somewhere.

I think this is what you are asking. You have to respect the permutation layer, otherwise canot apply the piling up lemma.

The approximations must be "joined together" between rounds by respecting the P-box mapping, so that they can be "added together with cancellation" to get multiround linear characteristics.

Say you apply an approximation $$a\cdot x\oplus b\cdot S(x)$$ to Sbox $$S_{1,1}$$ [round 1, 1st sbox] whose output constant $$b$$ is [1,0,0,1]. If you follow the wires at the output of this Sbox, you can see that you need to apply some linear approximation $$a'\cdot x \oplus b \cdot S(x)$$ where $$a'=[1,0,0,0]$$ to Sboxes $$S_{2,1},$$ and $$S_{2,4}$$. Then you use the fact that the key bits cancel and apply the piling up lemma to get a 2 round approximation involving Sboxes $$S_{1,1},S_{2,1},S_{2,4}$$.

Where is the optimization? You need to stick to relatively low hamming weight $$b$$ vectors to avoid turning on too many S-boxes since the piling up lemma is multiplicative and each new active S-box reduces the overall bias. You also want to find the highest individual bias characteristics.

In general, this needs to be checked exhaustively.