Characterization of the Least Periods of the Generalized Self-Shrinking Sequences

In 2004, Y. Hu and G. Xiao introduced the generalized self-shrinking generator, a simple bit-stream generator considered as a specialization of the shrinking generator as well as a generalization of the self-shrinking generator. The authors conjectured that the family of generalized self-shrinking sequences took their least periods in the set {1, 2, 2}, where L is the length of the Linear Feedback Shift Register included in the generator. In this correspondence, it is proved that the least periods of such generated sequences take values exclusively in such a set. As a straight consequence of this result, other characteristics of such sequences (linear complexity or pseudorandomness) and their potential use in cryptography are also analyzed.