Paper 2011/218

Cryptographic Analysis of All 4 x 4 - Bit S-Boxes

Markku-Juhani O. Saarinen


We present cryptanalytic results of an exhaustive search of all $16!$ bijective 4-bit S-Boxes. Previously affine equivalence classes have been exhaustively analyzed in 2007 work by Leander and Poschmann. We extend on this work by giving further properties of the optimal S-Box linear equivalence classes. In our main analysis we consider two S-Boxes to be cryptanalytically equivalent if they are isomorphic up to the permutation of input and output bits and a XOR of a constant in the input and output. We have enumerated all such equivalence classes with respect to their differential and linear properties. These equivalence classes are equivalent not only in their differential and linear bounds but also have equivalent algebraic properties, branch number and circuit complexity. We describe a ``golden'' set of S-boxes that have ideal cryptographic properties. We also present a comparison table of S-Boxes from a dozen published cryptographic algorithms.

Available format(s)
Secret-key cryptography
Publication info
Published elsewhere. Accepted to Selected Areas in Cryptography (SAC) 2011, August 11-12, 2011 Toronto, Ontario, Canada.
S-BoxDifferential cryptanalysisLinear cryptanalysisExhaustive permutation search.
Contact author(s)
mjos @ iki fi
2011-07-12: last of 9 revisions
2011-05-07: received
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      author = {Markku-Juhani O.  Saarinen},
      title = {Cryptographic Analysis of All 4 x 4 - Bit S-Boxes},
      howpublished = {Cryptology ePrint Archive, Paper 2011/218},
      year = {2011},
      note = {\url{}},
      url = {}
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