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what is the branch number of the binary identity matrix?

For example, $ I $ is 4x4 binary identity matrix,

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

Here, the branch number is $min_x ( wt(x)+wt(Ix))=2 $. Is that correct?

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Yes. This a degenerate matrix that provides no mixing and has minimal branch number.

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  • $\begingroup$ that means we can find a matrix which has a minimal branch number (2) and provides mixing. $\endgroup$ – S.Mu Jul 31 '18 at 13:22
  • $\begingroup$ Identity provides no mixing. each sbox influences a unique sbox in the next round. some sbox must be influenced by at least two sboxes from previous round for mixing. $\endgroup$ – kodlu Jul 31 '18 at 22:12
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Yes, its branch number is 2, which is the minimal possible branch number.

Note that every permutation matrix (that is, one element per row or column is one, all others are zero) has branch number 2.

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