Transform domain analysis of DES

DES can be regarded as a nonlinear feedback shift register (NLFSR) with input. From this point of view, the tools for pseudo-random sequence analysis are applied to the S-boxes in DES. The properties of the S-boxes of DES under Fourier transform, Hadamard transform, extended Hadamard transform and Avalanche transform are investigated. Two important results about the S-boxes of DES are found. The rst result is that nearly two-thirds of the total 32 functions from GF(2 6) to GF(2) which are associated with the 8 S-boxes of DES have the maximal linear span 63, and the other one-third have linear span greater than or equal to 57. The second result is that for all S-boxes, the distances of the S-boxes approximated by monomial functions has the same distribution as for the S-boxes approximated by linear functions. Some new criteria for the design of permutation functions for use in block cipher algorithms are discussed.