Randomness and Discrepancy Transforms

In this paper, a new transform of ultimately periodic binary sequences, called a discrepancy transform, is introduced in terms of the Berlekamp-Massey algorithm. First, we show that the run property of the discrepancy sequences dominates the randomness of linear span profiles of the sequences. Then, using a modified version of the Berlekamp-Massey algorithm, we provide a method to construct a large family of nonlinear permutations of GF(2 n ). Thirdly, applying these permutations as filtering functions to filtering generators, we obtain that the resulting output sequences possess good randomness and have efficient implementations at both hardware and software.