On the binary sequences with high GF(2) linear complexities and low GF(p) linear complexities

Klapper [1] showed that there are binary sequences of period q − 1(q is a prime power p, p is an odd prime) with the maximal possible linear complexity q−1 when considered as sequences over GF (2), while the sequences have very low linear complexities when considered as sequences over GF (p). This suggests that the binary sequences with high GF (2) linear complexities and low GF (p) linear complexities are note secure in cryptography. In this note we give some simple constructions of the binary sequences with high GF (2) linear complexities and low GF (p) linear complexities. We also prove some lower bounds on the GF (p) linear complexities of binary sequences and a lower bound on the number of the binary sequences with high GF (2) linear complexities and low GF (p) linear complexities .