On the security of the gollmann cascades

The purpose of this paper is to evaluate the security of the Gollmann m-sequence cascades of k stages. We give some theoretical results, which can be utilized to construct the transition matrix T _n of the conditional probabilities between the input and output strings of a stage. And then, we describe an attack algorithm for guessing the initial state of the first LFSR with desired reliability, using the transition matrix S _n = T _n ^k−1 of the conditional probabilities between the input string of the second stage and the output of the final stage of the given k-stage cascade. We finally evaluate the security of the cascades against this attack. Menicocci recently conjectured that there do not exist the complete analysis of the Gollmann cascades of more than 4 stages and it is infeasible to attack the 10-stage cascades with LFSRs of degree 100. Our experimental results show that the 9-stage cascades with LFSRs of degree 100 are completely breakable and the 10-stage cascades may be insecure.