Finding Linearly Generated Subsequences

Abstract

We develop a new algorithm to compute determinants of all possible Hankel matrices made up from a given finite length sequence over a finite field. Our algorithm fits within the dynamic programming paradigm by exploiting new recursive relations on the determinants of Hankel matrices together with new observations concerning the distribution of zero determinants among the possible matrix sizes allowed by the length of the original sequence. The algorithm can be used to isolate \emph{very} efficiently linear shift feedback registers hidden in strings with random prefix and random postfix for instance and, therefore, recovering the shortest generating vector. Our new mathematical identities can be used also in any other situations involving determinants of Hankel matrices. We also implement a parallel version of our algorithm. We compare our results empirically with the trivial algorithm which consists of computing determinants for each possible Hankel matrices made up from a given finite length sequence. Our new accelerated approach on a single processor is faster than the trivial algorithm on 160 processors for input sequences of length 16384 for instance.