Comment: Bound for linear complexity of BBS sequences

Montoya Vitini et al gave a lower bound for the linear complexity of Blum Blum Shubb sequences We show this result is incorrect The lower bound for the linear complexity of Blum Blum Shubb se quences given by Montoya Vitini et al depends on Proposition of this paper This Proposition can be phrased as A binary sequence of length p q p q odd primes with certain properties has linear complexity at least p q The properties of the odd primes p q are not relevent to the justi cation of Proposition given in and the claim is not true for any odd primes p q We give a counterexample for the odd primes p and q that satisfy the criteria for primes given in We consider the linear feedback shift register LFSR represented by the polynomial X Q X Q X X X X of degree where Qi denotes the i cyclotomic polynomial over GF This LFSR generates the following periodic sequence s of period p q s has linear complexity and not at least p q as claimed by the proposition A minimal polynomial for a sequence of period p q must divide X p q X Qp X Qq X Qp q X The justi cation of Proposition given in claims that if a minimal polyno mial for a sequence of period p q does not divide Qp X Qq X Qp q X then it must be of the form A X Qp X Qq X Qp q X where A X di vides Qp X Qq X Qp q X This claim is false as for example the LFSR represented by the polynomial X Qp X Qq X generates a sequence of period p q