Statistical and Algebraic Properties of DES
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D. Davies and S. Murphy found that there are at most 660 different probability distributions on the output from any three adjacent S-boxes after 16 rounds of DES . In this paper it is shown that there are only 72 different distributions for S-boxes 4, 5 and 6. The distributions from S-box triplets are linearly dependent and the dependencies are described. E.g. there are only 13 linearly independent distributions for S-boxes 4, 5 and 6. A coset representation of DES S-boxes which reveals their hidden linearity is studied. That may be used in algebraic attacks. S-box 4 can be represented by significantly fewer cosets than the other S-boxes and therefore has more linearity. Open cryptanalytic problems are stated.
Fauskanger, Stian; Semaev, Igor A.. Statistical and Algebraic Properties of DES. Lecture Notes in Computer Science 2016 ;Volum 9589. s. 93-107