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

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.