Computing the linear complexity for sequences with characteristic polynomial f ^v

Abstract

We present several generalisations of the Games–Chan algorithm. For a fixed monic irreducible polynomial f we consider the sequences s that have as a characteristic polynomial a power of f. We propose an algorithm for computing the linear complexity of s given a full (not necessarily minimal) period of s. We give versions of the algorithm for fields of characteristic 2 and for arbitrary finite characteristic p, the latter generalising an algorithm of Ding et al. We also propose an algorithm which computes the linear complexity given only a finite portion of s (of length greater than or equal to the linear complexity), generalising an algorithm of Meidl. All our algorithms have linear computational complexity. The proposed algorithms can be further generalised to sequences for which it is known a priori that the irreducible factors of the minimal polynomial belong to a given small set of polynomials.