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In Matsui's paper (Linear Cryptanalysis Method for DES cipher), lemma 1.

  1. $NS(a, b)$ is even
  2. if $a=1,32$,or $33$, then $NS(a,b)=32$ for all $b$

He said that the following lemma is now trivial from the definition of S-boxes.

  • But I can't prove this lemma. How can I prove this lemma?
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The following apply to all DES Sboxes.

  1. Note that $\overline{a}\cdot \overline{x}$ and $\overline{b}\cdot \overline{S(x)}$ are both linear boolean functions which thus have truth tables with even Hamming weight. The xor of the two quantities is also even, since if they have hamming distance $d$ this is 64 minus (the difference of two even hamming weights) by inclusion exclusion.

  2. The $a=1,32,33$ masking patterns are the ones where either LSB or MSB or both of input bits in the corresponding vector $\overline{a}$ are set to one while the remaining bits are set to zero. Recall that for each of the four such possibilities ($a=0$ as well) the DES Sbox output is a permutation of four bits. This also gives a balanced hamming weight, for all output masks $b$.

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