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Say, I want to generate a lot of $4\times4$ Sboxes with linear (or differential) branch numbre of $3$. One idea is to take all the $302$ affine classes, expand each of those classes and check if any of the Sboxes has the desired branch number.

Is there any better way/known result (such as, this particular class does not contain any Sbox with linear branch number $3$) that can be used to reduce the search space?

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The differential branch number is the sum of hamming weight between input and output difference distribution table (DDT). it is similar to linear branch number.

One way to construct S-box with branch number 3 is to use concept of differential-equivalence, DDT-equivalence and the $\gamma$-equivalence as shown in paper "Two Notions of Differential Equivalence on Sboxes".

The new paper accepted "Reconstructing an S-box from its Difference Distribution Table " provides algorithm to reconstruct S-box . therefore, you could apply the concept on $\gamma$-equivalence DDT to find S-Box with branch number as you want.

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  • $\begingroup$ Nice. So, this method actually [as far as I could see at a quick skim of the paper] finds an S-box with a given Difference Distribution Table, at some computational complexity. How would generating "a lot of S-boxes with given DDT" be done here? It seems you are familiar with this algorithm, are there ways of randomizing the procedure to get many different S-boxes? $\endgroup$ – kodlu May 9 at 23:40
  • $\begingroup$ Iam familiar with the algorithm in the first paper , second paper is a new paper. in step 2 in the (print sbox)example in "Two Notions of Differential Equivalence on Sboxes" . you have two options (3 and 7) for F[1] producing two different boxes for the same DDT. Figure 2 shows the path of two sboxes. $\endgroup$ – hardyrama May 10 at 9:52

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